Multi-wave mixing processes: from ultrafast polarization beats to electromagnetically induced transparency

Yanpeng Zhang Min Xiao

Multi-Wave Mixing Processes From Ultrafast Polarization Beats to Electromagnetically Induced Transparency

Yanpeng Zhang Min Xiao

Multi-Wave Mixing Processes

From Ultrafast Polarization Beats to Electromagnetically Induced Transparency

With 134 figures

AUTHORS: Prof. Yanpeng Zhang Key Laboratory for Physical Electronics and Devices of the Ministry of Education Xi’an Jiaotong University, Xi’an 710049, China E-mail: [email protected]

Prof. Min Xiao Department of Physics University of Arkansas, Fayetteville Arkansas 72701, USA E-mail: [email protected]

ISBN 978-7-04-025795-3 Higher Education Press, Beijing ISBN 978-3-540-89527-5 Springer Berlin Heidelberg New York e ISBN 978-3-540-89528-2 Springer Berlin Heidelberg New York

Library of Congress Control Number: 2008939220 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. c 2009 Higher Education Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg  Co-published by Higher Education Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg Springer is a part of Springer Science+Business Media springer.com The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper

Preface

Nonlinear optics covers very broad research directions and has been a very active area of research for about fifty years since the invention of the first laser at the beginning of 1960s. There are several excellent text books devoted to various aspects of nonlinear optics including Nonlinear Optics by R. W. Boyd, Nonlinear Optics by Y. R. Shen, Quantum Electronics by A. Yariv, Principles of Nonlinear Optical Spectroscopy by S. Mukamel, and Nonlinear Fiber Optics by G. P. Agrawal. Multi-wave mixings in gases, liquids, and solid materials are important parts of the nonlinear optical process. Typically, lower-order nonlinear optical processes always dominate since they are more efficient than higher-order ones. So normally only two-wave, three-wave, and four-wave mixing (FWM) processes, depending on symmetries of nonlinear materials, are studied, and their basic principles are covered in detail in those textbooks. FWM comes from the third-order nonlinearity, which is one of the most popular nonlinear phenomena, and can be easily observed in materials with the inversion central symmetry (in which the third-order nonlinearity is the lowest nonlinear one). In certain specially designed material systems, or in certain phase conjugation configurations, the FWM efficiency can be very high, reaching 100% or even with gain. With newly developed short-pulse high-power lasers, and new materials designed and optimized for certain nonlinearities, higher-order wave-mixing processes, such as the sixwave mixing (SWM, corresponding to the fifth-order nonlinearity) and even the eight-wave mixing (EWM, corresponding to the seventh-order nonlinearity), have been experimentally investigated in recent years. Since typically higher-order nonlinear coefficients are much smaller than lower-order ones, in order to observe higher-order wave-mixing signals, one needs to eliminate (or at least greatly suppress) lower-order signals by various techniques, or to perform a heterodyne detection between the weak signal from a higherorder nonlinear process and a much stronger lower-order signal (as a local oscillator). Of course, very high-order harmonic generations have been used recently to generate UV and even x-ray wavelengths by employing very short and high intensity laser pulses. However, high-order nonlinearities described in this book do not include this region of extreme nonlinear processes. The authors have worked, both theoretically and experimentally, on nonlinear optics for many years, especially on high-order nonlinear wave-mixing processes in the past few years. They have worked on the electromagneti-

vi

Preface

cally induced transparency (EIT)- or atomic coherence-enhanced multi-wave mixing processes in multi-level atomic systems, which have many advantages over traditional multi-wave mixing processes in the nonlinear media. By specially selecting atomic energy levels and phase-matching conditions, they have shown co-existing FWM and SWM signals in several open- and close-cycled four-level atomic systems by making use of the EIT concept of two-photon Doppler-free configurations in Doppler-broadened atomic media. Using uniquely designed spatial laser beam patterns and configurations, generated FWM and SWM signals can fall into a same EIT window with low absorption, and propagate in the same direction with the same frequency. SWM processes can be greatly enhanced by playing with atomic coherences between different energy levels, and relative strengths between FWM and SWM signals can be completely controlled. SWM signals can even be enhanced to have same amplitudes as FWM signals in the same system at the same time. Spatial and temporal interferences, as well as interference in the frequency domain, between FWM and SWM signals have been experimentally demonstrated in four-level atomic systems. An efficient energy exchange during propagation between generated FWM and SWM signals (and with the probe beam) was also observed and studied in detail. In this monograph, the authors will describe treatments of multi-wave mixing processes using the perturbation approach and show how, by manipulating phase-matching and EIT conditions using various laser beams, co-existing FWM and SWM (or even EWM) processes can be achieved, and how to control their interplays. It has been shown that one can control the relative strength of generated FWM and SWM signals (by using the amplitudes and the frequency detuning of pump beams) and the relative phase between them (by adjusting the time delay of one of the pump beam involved only in the FWM process). This monograph will mainly focus on the relevant work recently done in the authors’ group. Other than EIT-enhanced co-existing high-order nonlinear wave-mixing processes in multi-level atomic systems, several other topics will also be discussed, including femtosecond and attosecond polarization beats between two FWM, or two SWM processes, and their heterodyne detections in multi-level atomic systems, as well as Raman- or Rayleigh-enhanced polarization beats in the liquid systems. Some potential applications of the fast polarization beats and EIT-enhanced co-existing high-order nonlinear wave-mixing processes are also discussed in the book. The authors believe that although several good textbooks on the general topics of nonlinear optics exist, the current book treats a special topic of co-existing multi-wave mixing processes in multi-level systems and will have high values to serve a special group of readers. Especially, the topic of EIT- or atomic coherence-enhanced multi-wave nonlinear optical processes and their interplays have not been touched by any existing books. This monograph serves as a reference book intended for advanced undergraduates, graduate students, and researchers working in the related field of nonlinear optics,

Preface

vii

nonlinear optical spectroscopy, and quantum optics. We take this opportunity to thank many researchers and collaborators who have worked on the research projects as described in this book. We specially thank Leijian Shen and Ling Li for their great helps in compiling this book. Yanpeng Zhang Min Xiao November 2008

Contents

1

Introduction· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

1

1.1 Nonlinear Susceptibility · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

1

1.2 Four-wave Mixing · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

5

1.3 Generalized Resonant MWM in Multi-level Atomic Systems · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

7

1.4 Enhanced Nonlinearity via Electromagnetically Induced

2

Transparency · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

12

References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

15

Femtosecond Polarization Beats · · · · · · · · · · · · · · · · · · · · · ·

18

2.1 Effects of Field-correlation on Polarization Beats · · · · · · · · · ·

18

2.1.1 PBFS in a Doppler-broadened System · · · · · · · · · · · · ·

23

2.1.2 Photon-echo· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

26

2.1.3 Experiment and Result· · · · · · · · · · · · · · · · · · · · · · · ·

27

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields · · · ·

30

2.2.1 Photon-echo· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

38

2.2.2 Experiment and Result· · · · · · · · · · · · · · · · · · · · · · · ·

41

2.3 Higher-order Correlations of Markovian Stochastic Fields on

3

Polarization Beats · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

44

2.3.1 HOCPB in a Doppler-broadened System · · · · · · · · · · ·

50

2.3.2 Photon-echo· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

51

2.3.3 Experiment and Result· · · · · · · · · · · · · · · · · · · · · · · ·

57

References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

60

Attosecond Polarization Beats · · · · · · · · · · · · · · · · · · · · · · · ·

63

3.1 Polarization Beats in Markovian Stochastic Fields · · · · · · · · ·

63

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Contents

3.2 Perturbation Theory · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

66

3.3 Second-order Stochastic Correlation of SFPB· · · · · · · · · · · · ·

73

3.4 Fourth-order Stochastic Correlation of SFPB · · · · · · · · · · · · ·

88

3.5 Discussion and Conclusion · · · · · · · · · · · · · · · · · · · · · · · · · ·

96

References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 103 4

Heterodyne/Homodyne Detection of MWM · · · · · · · · · · · · 106 4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order Polarization Beats · · · · · · · · · · · · · · · · · · · · · · · 106 4.1.1 Liouville Pathways · · · · · · · · · · · · · · · · · · · · · · · · · · · 108 4.1.2 Color-locking Stochastic Correlations · · · · · · · · · · · · · 113 4.1.3 Purely Homogeneously-broadened Medium · · · · · · · · · 116 4.1.4 Extremely Doppler-broadened Limit · · · · · · · · · · · · · · 122 4.1.5 Discussion and Conclusion · · · · · · · · · · · · · · · · · · · · · 130 4.2 Color-locking Phase Control of Fifth-order Nonlinear Response · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 131 4.3 Seventh-order Nonlinear Response· · · · · · · · · · · · · · · · · · · · · 136 References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 141

5

Raman- and Rayleigh-enhanced Polarization Beats · · · · · · 144 5.1 Raman-enhanced Polarization Beats · · · · · · · · · · · · · · · · · · · 145 5.1.1 Chaotic Field · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 149 5.1.2 Raman Echo · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 154 5.1.3 Phase-diffusion Field · · · · · · · · · · · · · · · · · · · · · · · · · 156 5.1.4 Gaussian-amplitude Field · · · · · · · · · · · · · · · · · · · · · · 159 5.1.5 Experiment and Result· · · · · · · · · · · · · · · · · · · · · · · · 166 5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 170 5.2.1 Stochastic Correlation Effects of RFWM · · · · · · · · · · · 175 5.2.2 Homodyne Detection of Sum-frequency RASPB · · · · · · 188 5.2.3 Heterodyne Detection of the Sum-frequency RASPB · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 197 5.2.4 Discussion and Conclusion · · · · · · · · · · · · · · · · · · · · · 202 References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 205

Contents

6

xi

Coexistence of MWM Processes via EIT Windows · · · · · · · 207 6.1 Opening FWM and SWM Channels · · · · · · · · · · · · · · · · · · · 207 6.2 Enhancement of SWM by Atomic Coherence · · · · · · · · · · · · · 216 6.3 Observation of Interference between FWM and SWM · · · · · · 223 6.4 Controlling FWM and SWM Processes · · · · · · · · · · · · · · · · · 227 References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 233

7

Interactions of MWM Processes · · · · · · · · · · · · · · · · · · · · · · 235 7.1 Competition between Two FWM Channels · · · · · · · · · · · · · · 235 7.2 Efficient Energy Transfer between FWM and SWM Processes · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 242 7.3 Spatial and Temporal Interferences between Coexisting FWM and SWM Signals· · · · · · · · · · · · · · · · · · · · · · · · · · · · 251 References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 258

8

Multi-dressed MWM Processes · · · · · · · · · · · · · · · · · · · · · · · 261 8.1 Matched Ultraslow Pulse Propagations in Highly-Efficient FWM · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 261 8.1.1 Time-dependent, Adiabatic Treatment for Matched Probe and NDFWM Signal Pulses · · · · · · · · · · · · · · · 267 8.1.2 Steady-state Analysis · · · · · · · · · · · · · · · · · · · · · · · · · 273 8.1.3 Discussion and Outlook · · · · · · · · · · · · · · · · · · · · · · · 281 8.2 Generalized Dressed and Doubly-dressed MWM Processes · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 283 8.2.1 Generalized Dressed-(2n–2)WM and Doublydressed-(2n–4)WM Processes · · · · · · · · · · · · · · · · · · · 284 8.2.2 Interplays Among Coexisting FWM, SWM, and EWM Processes· · · · · · · · · · · · · · · · · · · · · · · · · · · · · 287 8.3 Interacting MWM Processes in a Five-level System with Doubly-dressing Fields · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 292 8.3.1 Three Doubly-Dressing Schemes · · · · · · · · · · · · · · · · · 294 8.3.2 Aulter-Townes Splitting, Suppression, and Enhancement· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 299

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Contents

8.3.3 Competition between Two Coexisting Dressed MWM · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 307 8.3.4 Conclusion and Outlook · · · · · · · · · · · · · · · · · · · · · · · 313 References · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 315 Index · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 317

1 Introduction

The main subject of this book centers around mainly two topics. The first topic (Chapters 2 – 5) covers the ultrafast polarization beat due to the interaction between multi-colored laser beams and multi-level media. Both difference-frequency femtosecond and sum-frequency attosecond polarization beats can be observed in multi-level media depending on the specially arranged relative time delay in multi-colored laser beams. Effects of different stochastic noise models for the lasers on the polarization beat signal are carefully studied. Polarization beats between MWM processes are among the most important ways to study transient property of the medium. The second topic (Chapters 6 – 8) relates to the co-existence and interplay between efficient multi-wave mixing (MWM) processes enhanced by atomic coherence in multi-level atomic systems. The co-existing higher-order nonlinear optical process can be experimentally controlled and becomes comparable or even greater in amplitude than the lower-order wave-mixing process by means of atomic coherence and the multi-photon interference. Furthermore, the spatial-temporal interference and efficient energy exchange during propagation are shown to exist between the generated four-wave mixing (FWM) and six-wave mixing (SWM) signals. The multi-dressed wave-mixing process is also investigated in a multi-level atomic system to enhance or suppress the MWM process. Only the MWM process with multi-colored laser beams and related effects in multi-level media will be covered in this book. Experimental results will be presented and compared with theoretical calculations throughout the book. Also, emphasis will be given only to works done by the authors’ group in the past few years. Before starting the main topics of the book, some basic physical concepts and mathematical techniques, which are useful and needed in the later chapters, will be briefly presented in this introduction chapter.

1.1 Nonlinear Susceptibility Nonlinear optics is the study of optical phenomena that occur in a material system as a consequence of nonlinear response to the input light. Typically, only the coherent laser light is sufficiently intense to provide the nonlinear

2

1 Introduction

changes to the material’s optical properties. In fact, the beginning of the field of nonlinear optics has often been taken to be at the discovery of the second-harmonic generation (SHG) by Fraken, et al. in 1961 [1], just shortly after the demonstration of the first working laser by Maiman in 1960 [1]. Nonlinear optical phenomena are “nonlinear” in the sense that they occur when the response of a material system to an applied optical field depends on a nonlinear manner to (or high-order power of) the strength of the input optical field. For example, SHG occurs in many optical crystals as a result of the applied strong optical field with the generated second-harmonic signal intensity (at twice the frequency of the applied light) proportional to the square of the applied light intensity. For inversion symmetric materials, such as atoms, the second-order nonlinearity usually does not exist, so the lowest nonlinear effect is the third-order one. Today, lasers with very high intensities and very short pulse durations are readily available, for which concepts and approximations of the traditional nonlinear optics can no longer apply. In this regime of extreme “nonlinear optics”, a large variety of novel and unusual effects arise, such as frequency doubling in inversion symmetric materials or high-harmonic generations in gases, which can lead to attosecond electromagnetic pulses or pulse trains [2]. Other examples of “nonlinear optics” cover diverse areas such as solid-state physics, liquids, atomic and molecular physics, relativistic free electrons in a vacuum, and even the vacuum itself [3, 4]. In this book, we only deal with nonlinear multi-wave mixing processes at relatively lower – orders in multilevel media, in which traditional principles and approximations of nonlinear optics (as described in the textbooks [3, 4]) are still hold. In order to describe more precisely what we mean by the optical nonlinearity, let us consider how the dipole moment per unit volume, or polarization P , of a material system depends on the strength E of the applied optical field. The induced polarization depends nonlinearly on the electric field strength of the applied field in a manner that can be described by the relation P = PL + PN L [3, 4]. Here, PL = P (1) = ε0 χ(1) · E . P = P (2) + P (3) + · · · = ε (χ(2) : EE + χ(3) ..EEE + · · · ) 0

NL

When we only consider the atomic system (which is isotropic and has inversion symmetry), we can write the total polarization as P = ε0 χE in general, where the total effective optical susceptibility can be described by a generalized expression of ∞  χ(2j+1) |E|2j χ= j=0

The lowest term χ(1) (j = 0) is independent of the field strength and is known as the linear susceptibility. The next two terms in the summation, χ(3) and χ(5) , are known as third- and fifth-order nonlinear optical susceptibilities, respectively.

1.1 Nonlinear Susceptibility

3

The index of refractive of many optical materials depends on the intensity of the light due to nonlinear responses, which can be described by the relation n = n0 +

∞ 

n ¯ 2j |E|2j

j=1

The nonlinear indices n ¯ 2j are influenced by the intensity of the light [5]. An alternative way of defining the intensity-dependent refractive index is by the equation ∞  n2j I j n = n0 + j=1

where I denotes the intensity of the applied field, given by ε0 c 2 |E| I= 2 2 n ¯ 2j [3]. Hence, n ¯ 2j and n2j are related by n2j = ε0 c The linear and nonlinear refractive indices are directly related to the linear and nonlinear susceptibilities. It is generally true that n2 = 1 + χ, and by introducing ∞  2j n ¯ 2j |E| n= j=0

on the left-hand side and χ=

∞ 

2j

χ(2j+1) |E|

j=0

on the right-hand side of this equation, it gives ⎛ ⎞2 ∞ ∞   ⎝ n ¯ 2j |E|2j ⎠ = 1 + χ(2j+1) |E|2j j=0

j=0 2j

Correct to terms of up to the order of |E| , this general expression gives the following relations, i.e.,  n20 = 1 + Reχ(1) ⇒ n0 = 1 + Reχ(1) Reχ(3) 2 Reχ(3) ⇒ n2 = 2n0 ε0 c 2n0 (5) 2 ¯2 ¯ 22 Reχ − n 2 Reχ(5) − n 2n0 n ¯4 + n ¯ 22 = Reχ(5) ⇒ n ¯4 = ⇒ n4 = 2n0 ε0 c 2n0 (7) n2 n ¯4 Reχ − 2¯ 2¯ n2 n ¯ 4 + 2n0 n ¯ 6 = Reχ(7) ⇒ n ¯6 = ⇒ n6 2n0 n2 n ¯4 2 Reχ(7) − 2¯ = ε0 c 2n0 2n0 n ¯ 2 = Reχ(3) ⇒ n ¯2 =

4

1 Introduction

Finally the expression n=

∞ 

2j

n ¯ 2j |E|

j=0

becomes 2

4

n (E)  n0 +[Re(χ(3) )/2n0 ] |E| +{Re(χ(5) )−[Re(χ(3) )/2n0 ]2 } |E| /2n0 +· · · where n0 represents the usual, weak-field (linear) refractive index and the new nonlinear optical constants n2j (sometimes called the second-, fourth-, sixth-. . . order indices of refraction) give the rate at which the refractive index increases with increasing optical intensity. For a typical nonlinear medium, ¯ 4 have orders of magnitude of 10−7 m2 /V2 and 10−13 m4 /V4 , respecn ¯ 2 and n tively [5]. The change of optical properties due to the second-order refractive index n2 (or third-order nonlinear susceptibility) is typically called the optical Kerr effect, by analogy to the traditional Kerr electro-optic effect, in which the refractive index of a material changes by an amount that is proportional to n, the square of the strength of an applied static electric field (i.e., n ≈ n0 + Δ¯ 2 4 Δ¯ n≈n ¯ 2 |E| + n ¯ 4 |E| , n ¯ 4 is typically neglected in low power range) [3]. Higher-order nonlinear susceptibilities are typically much smaller than the lower-order one by several orders of magnitude. In general, order of magnitude comparison is given by χ(n) /χ(n−1) ≈ 10−7 So, typically the lowest-order non-zero nonlinear term always dominates in studying nonlinear optical properties in a medium, and higher-order nonlinear terms are simply neglected. However, in recent years higher-order nonlinearities have shown to make a significant contribution to nonlinear optical properties, even when the lower-order nonlinear term is not zero, especially at higher optical intensity [3, 4]. For example, in the N-type four-level system [5], as it can be appreciated from Re(χ), the real part of the total suscep2 tibility of the medium grows linearly with |E| at low powers (due to the 2 effect of only the positive term n ¯ 2 |E| , which is the lowest nonlinear term.), 4 but it decreases at high powers (due to the negative term n ¯ 4 |E| ), while the losses are comparatively small in this range [5]. In such a case, there is a balance for the diffraction plus self-focusing at low field amplitudes and self-defocusing at larger amplitudes. This type of competition can be found in media with the so called cubic-quintic-type nonlinearity, which can be very important in the propagation property of high intensity optical pulse, and optical soliton formation. As we will show in Chapters 6 – 8, different-order nonlinear optical processes can co-exist even with low power cw laser beams, enhanced by atomic coherence in the multi-level atomic systems [6 – 14].

1.2 Four-wave Mixing

5

1.2 Four-wave Mixing The four-wave mixing (FWM) refers to the nonlinear optical process with four interacting electromagnetic waves (i.e., with three applied fields to generate the fourth field). In the weak interaction limit, FWM is a thirdorder nonlinear optical process and is governed by the third-order nonlinear susceptibility [3]. Unlike the second-order process, the third-order process is allowed in all media, with or without inversion symmetry. Therefore, in many optical media (such as the ones with the inversion symmetry), such third-order nonlinear wave-mixing processes are the lowest-order ones, which are the dominant nonlinear interactions. FWM processes are well studied in many material systems and their general properties can be found in wellwritten textbooks [3, 4], so we will not review the general topics of FWM processes here. Instead, in the following, we will only discuss some special cases of the FWM process, which are relevant mainly to certain parts of this book (Chapters 2 – 5). Let us consider a special case of FWM processes. The third-order nonlinear polarization governing the process has, in general, three components with different wave vectors. E1 (ω1 , k1 ), E2 (ω2 , k2 ) and E3 (ω3 , k3 ) denote the three input laser fields. Here, ωi and ki are the frequency and the propagation wave vector of the ith beam, respectively. We can choose to have a small angle θ between input pump laser beam k2 and beam k1 . The probe laser beam (beam k3 ) propagates along a direction that is almost opposite to that of beam k1 (see Fig. 1.1). Because of the strong resonant interactions, the third-order nonlinear susceptibility χ(3) for this FWM process can be very large in certain media (such as multi-level atomic systems). As a result, this third-order nonlinear optical process is often observable with relatively weak continuous-wave (CW) laser beams. The output of the generated FWM signals can be easily understood from the following physical picture. Two of the three input waves interfere and form either a static grating or a moving grating (depending on the frequency difference between laser beams); the third input wave is then scattered off by this grating to yield the output signal wave. In most cases, contributions from the static gratings should dominate. With three input waves, three different gratings can be formed. The grating formed by the input k1 and k2 waves scatters the k3 wave to yield output signals at k3 ± (k1 − k2 ). The one formed by the k3 and k2 waves scatters the k1 wave to yield output signals at k1 ± (k3 − k2 ). The one formed by the k1 and k3 waves scatters the k2 wave to yield output signals at k2 ± (k1 − k3 ). These processes are illustrated in Fig. 1.1. Altogether, three output signal waves with different wave vectors, ks1 = −k1 + k2 + k3 , ks2 = k1 − k2 + k3 , and ks3 = k1 + k2 − k3 , can be expected. As is common for optical wave-mixing processes, phase-matching condition (Δk = 0) is of prime importance here, since it greatly enhances the signal output under the phase-matched condition. By carefully considering three output situations, as shown in Fig. 1.1, one can easily see that only the

6

1 Introduction

generated output at ks2 = k1 − k2 + k3 is always phases matched among the three possible output waves, which is the usual case for efficient phase conjugation [4].

Fig. 1.1. Schematic diagram of phase-conjugate FWM.

More specifically, if beams 1 and 2 have the same frequency (i.e., ω1 = ω2 ) and a small angle θ is set between them, the coherence length of the generated FWM signal at ks2 is given by lcf = 2c/[n(ω1 /ω3 )|ω1 − ω3 |θ2 ] which is much larger than that of the other two outputs at ks1 and ks3 . However, the nonlinear interactions between laser beam 1 and beam 2 with an absorbing medium can give rise to the molecular-reorientation and the thermal non-resonant static gratings (i.e., QM and QT in liquid media) [4], respectively. In this case, the FWM signal at ks2 = k1 − k2 + k3 is the results of diffractions of beam 3 with a frequency ω3 by these two gratings. On the other hand, the interference pattern generated by beam 2 and beam 3 can move with a phase velocity |ω3 − ω2 |. Now, if the frequency difference Δa = ω3 − ω2 ≈ 0, two resonant moving gratings QRM and QRT with a large angle are formed by the interference between beam 2 and beam 3. Beam 1 is then diffracted by them to enhance the FWM signals. This is the Rayleigh-enhanced FWM with the wave vector ks2 = k1 − k2 + k3 at the direction of beam 4 (which will be discussed in Chapter 5). However, if the frequency difference Δb = ω3 − ω2 − Ω R ≈ 0 (where ΩR is the Raman resonant frequency of the medium), one large resonant moving grating, QR , is formed by the interference between beam 2 and beam 3, which will excite the Raman-active vibrational mode of the Kerr medium and enhance the FWM signal (i.e., Raman-enhanced FWM, which will be discussed in Chapter 5.). FWM interactions are the basic nonlinear optical processes to be discussed in this book. We will present near resonant third-order FWM interactions in two-level and three-level systems with degenerate and non-degenerate configurations. In later chapters, enhancements of FWM processes due to electromagnetically induced transparency (EIT) or atomic coherence in multi-level atomic systems [15 – 18], as well as competitions between different FWM channels [11], will be discussed. Higher-order wave-mixing processes, such as six-wave mixing (SWM, fifth-order nonlinearity) and eight-wave mixing (EWM, seventh-order nonlinearity), can also be generated and enhanced in multi-level atomic systems via atomic coherence.

1.3 Generalized Resonant MWM in Multi-level Atomic Systems

7

1.3 Generalized Resonant MWM in Multi-level Atomic Systems Multi-level atomic systems can generally be divided into two categories, i.e., close-cycled (such as Λ-type, ladder-type, N-type, double -Λ, and multi-level folded-type) systems and open-cycled (such as V-type, Y-type, inverted Ytype, and K-type) systems. Atomic coherences can be induced in both of such multi-level atomic systems. Enhanced FWM processes have been demonstrated in many of these multi-level atomic systems via atomic coherence, especially in Λ-type [16], double -Λ type [15, 17], ladder-type [6 – 12], and N-type [18] atomic systems. Typically, in the close-cycled multi-level atomic systems, lower-order nonlinear wave-mixing processes (such as FWM and SWM) can be turned off by special arrangement in laser beams, so the higher-order one (either SWM or EWM) can be observed [15 – 20]. Recently, new experiments have been designed and implemented, which can generate co-existing FWM and SWM processes in either open-cycled [7 – 12] or close-cycled [6] multi-level atomic systems by manipulating laser beam configurations and induced atomic coherence among the energy levels (which will be discussed in Chapters 6 – 8). Simultaneous opening of EIT windows and induced atomic coherence among relevant energy levels are essential in generating co -existing FWM and SWM signals with comparable intensities [7]. Such co -existing wave -mixing processes of different orders (such as FWM and SWM processes), and interplays between them, open the door for new interesting research directions in nonlinear optics. As the details of these systems will be discussed in later chapters, a general example will be given here in the following to illustrate the techniques used to treat such multi-level atomic systems. Let us consider a close-cycled (n+1)-level cascade system, as shown in Fig. 1.2. The transition from state |i − 1 to state |i is driven by two laser fields Ei (ωi , ki ) and Ei (ωi , ki ), with Rabi frequencies Gi and Gi , respectively. The Rabi frequencies are defined as Gi = εi μij / Gi = εi μij / where μij is the transition dipole moment between level i and level j(or i−1). Fields En and En (with the same frequency) propagate along beam 2 and beam 3, respectively, with a small angle θ between them [see Fig. 1.2(a)]. Fields E2 , E3 to En−1 propagate along the direction of beam 2, while a weak probe field E1 (beam 1) propagates along the opposite direction of beam 2. The simultaneous interactions of the multi-level atoms with fields E1 , E2 to En will induce atomic coherence between states |0 and |n through the resonant n-photon transitions. This n-photon coherence is then probed by the field En and, as a result, a 2n-wave-mixing (2n-WM) signal of frequency ω1 in beam 4 is generated almost exactly opposite to the direction of beam

8

1 Introduction

3, satisfying the phase-matching condition k2n = k1 + kn − kn .

Fig. 1.2. (a) Schematic diagram for phase-conjugate 2n-WM process; (b) Energylevel diagram for 2n-WM in a close-cycled (n+1)-level cascade system.

Using the Master equation for the evolution of this system, we can write 1 ˆ ∂ ρˆ(t) ˆ = [H ˆ(t)] − Γˆ ρ 0 + H1 (t), ρ ∂t i ˆ 1 = −E μ where H ˆ is the dipole interaction [4]. Then, we can expand the density operator ρˆ(t) using the series expansion technique [3, 4] and write ρˆ(t) = ρˆ(0) (t) + ρˆ(1) (t) + ρˆ(2) + · · · + ρˆ(r) (t) + · · ·

(1.1)

By introducing this expansion into the initial Heisemberg equation, the density-matrix equation takes the form i

∂ (0) (ˆ ρ (t) + ρˆ(1) (t) + ρˆ(2) + · · · + ρˆ(r) (t) + · · · ) ∂t ˆ0 + H ˆ 1 , ρˆ(0) (t) + ρˆ(1) (t) + ρˆ(2) + · · · + ρˆ(r) (t) + · · · ] − = [H iΓ(ˆ ρ(0) (t) + · · · + ρˆ(r) (t) + · · · )

(1.2)

Thus ⎧ ∂ ⎪ ˆ 0 , ρˆ(0) (t)] − iΓ[ˆ ⎪ i ρˆ(0) (t) = [H ρ(0) (t)] ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂ (1) ˆ 0 , ρˆ(1) (t)] + [H ˆ 1 , ρˆ(0) (t)] − iΓˆ i ρˆ (t) = [H ρ(1) (t) ∂t ⎪ ⎪ ⎪ ············ ⎪ ⎪ ⎪ ⎪ ∂ (r) ⎩ ˆ 0 , ρˆ(r) (t)] + [H ˆ 1 , ρˆ(r−1) (t)] − iΓˆ ρ(r) (t) i ρˆ (t) = [H ∂t

(1.3)

Then, we can obtain the series ρ(0) , · · ·, ρ(r) by solving the above equations step by step (from lower to higher orders). According to the density-matrix dynamic equations, one can write the above dynamic equations in the matrix

1.3 Generalized Resonant MWM in Multi-level Atomic Systems

form with the matrices given ⎡ 0 μ1 0 · · · ⎢ ⎢ μ1 0 μ2 · · · ⎢ ⎢ ⎢ 0 μ2 0 · · · μ ˆ=⎢ .. .. ⎢ .. ⎢ . . . ⎢ ⎢ 0 0 0 ··· ⎣ 0 0 0 ··· ⎡ (r) (r) ρ00 ρ01 · · · ⎢ (r) (r) ⎢ρ ⎢ 10 ρ11 · · · (r) ρˆ = ⎢ . .. ⎢ . . ⎣ . (r)

(r)

Γˆ ρ

(r)

9

by 0

0

⎤ ⎡

⎥ 0 ⎥ ⎥ ⎥ 0 0 ⎥ .. .. ⎥ ⎥ . . ⎥ ⎥ 0 μn ⎥ ⎦ μn 0 ⎤ (r) ρ0,n ⎥ (r) ρ1,n ⎥ ⎥ .. ⎥ ⎥ . ⎦ (r) · · · ρn,n 0

ρn,0 ρn,1 ⎡ (r) (r) Γ0 ρ00 Γ10 ρ01 · · · ⎢ (r) ⎢ Γ ρ(r) Γ1 ρ11 ··· ⎢ 10 10 =⎢ .. .. ⎢ . . ⎣ (r) (r) Γn,0 ρn,0 Γn,1 ρn,1 · · ·

⎢ ⎢ ˆ0 = ⎢ H ⎢ ⎢ ⎣

···

0

E1 · · · .. .

0 .. .

E0

0

0 .. . 0

(r)

Γn,0 ρ0,n

0

···

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

En

⎤

⎥ (r) Γn,1 ρ1,n ⎥ ⎥ ⎥ .. ⎥ . ⎦ (r) Γn, ρn,n

For the diagonal element ρii , Γi represents the longitudinal relaxation rate (e.g., Γ0 , Γ1 , Γ2 , Γ3 , · · · , Γn ); However, for the off-diagonal element ρij , then Γij is the transverse relaxation rate (e.g., Γ10 , Γ20 , Γ30 , Γ40 , Γ21 , Γ31 , · · · , Γn,n−1 ). μi is the transition dipole moment. Then, the dynamic equation can be written as ˆ0 ˆ 0 ρˆ(r) − ρˆ(r) H ˆ 0 , ρˆ(r) ] = H [H ⎡ (r) 0 ρ01 (E0 − E1 ) ⎢ ⎢ ρ(r) 0 10 (E1 − E0 ) ⎢ =⎢ . .. .. ⎢ . ⎣ (r)

··· ···

(r)

ρn−1,0 (En−1 − E0 ) ρn,1 (En − E1 ) · · ·

(r)

ρ0,n (E0 − En )

⎥ (r) ρ1,n (E1 − En ) ⎥ ⎥ ⎥ .. ⎥ . ⎦ 0

ˆ 1 , ρˆ(r−1) (t)]= − E[ˆ μ1 , ρˆ(r−1) ] = −E[ˆ μρˆ(r−1) − ρˆ(r−1) μ ˆ] [H ⎡ (r−1) (r−1) (r−1) (−ρ01

+ ρ10

)μ1

⎢ (−ρ(r−1) + ρ(r−1) )μ + ρ(r−1) μ 1 2 ⎢ 11 00 20 = −E ⎢ . ⎣ . . (r−1)

−ρn,1

(r−1)

μ1 + ρn−1,0 μn

··· ···

···

(r−1)

−ρ0,n−2 μn−1 + ρ1,n−1 μ1

(r−1)

(r−1)

(r−1)

⎤ ⎥

−ρ1,n−2 μn−1 + ρ0,n−1 μ1 + ρ2,n−1 μ2 ⎥ ⎥ . ⎦ . . (r−1) (r−1) (−ρn,n−1 + ρn−2,n )μn

According to equation i

⎤

∂ (r) ˆ 0 , ρˆ(r) (t)] + [H ˆ 1 , ρˆ(r−1) (t)] − iΓij ρˆ(r) (t) ρˆ (t) = [H ij ij ij ∂t ij

10

1 Introduction

one can write ⎧ ∂ρ10 1 ⎪ ⎪ = {ρ10 (E1 − E0 ) − E[−μ1 ρ11 + μ1 ρ00 + μ2 ρ20 ]} − Γ10 ρ10 ⎪ ⎪ ∂t i ⎪ ⎪ ⎪ ⎪ ⎪ 1 ∂ρ20 ⎪ ⎪ = [ρ20 (E2 − E0 ) − E(−μ1 ρ21 + μ2 ρ10 + μ3 ρ30 )] − Γ20 ρ20 ⎪ ⎪ ∂t i ⎪ ⎪ ⎪ ⎨ ············ (1.4) ⎪ 1 ∂ρ n−1,0 ⎪ ⎪ = [ρn−1,0 (En−1 − E0 ) − E(−μ1 ρn−1,1 + ⎪ ⎪ i ⎪ ∂t ⎪ ⎪ ⎪ ⎪ μ ⎪ n−1 ρn−2,0 + μn ρn,0 )] − Γn−1,0 ρn−1,0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ρn,0 = 1 [ρn,0 (En − E0 ) − E(−μ1 ρn,1 + μn ρn−1,0 )] − Γn,0 ρn,0 ∂t i As an example of this general description, we now consider a five-level folded atomic system, as shown in Fig. 1.3.

Fig. 1.3. Energy-level diagram of a close-cycled (folded) five-level atomic system.

Based on the above derivation, we can write the density-matrix equations as

⎧ ∂ρ10 ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ∂ρ ⎪ ⎪ ⎨ 20 ∂t ∂ρ ⎪ 30 ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎩ ∂ρ40 ∂t

1 {ρ10 (E1 − E0 ) − E[μ1 ρ00 + μ2 ρ20 − μ1 ρ11 ]} − Γ10 ρ10 i 1 = [ρ20 (E2 − E0 ) − E(μ2 ρ10 + μ3 ρ30 − μ1 ρ21 )] − Γ20 ρ20 i 1 = [ρ30 (E3 − E0 ) − E(μ3 ρ20 + μ4 ρ40 − μ1 ρ31 )] − Γ30 ρ30 i 1 = [ρ40 (E4 − E0 ) − E(μ4 ρ30 − μ1 ρ41 )] − Γ40 ρ40 i =

(1.5)

In the bare-state picture, these equations of motion for the atomic polarizations and populations (atomic responses) are considered up to different orders of Liouville pathways that provide a diagrammatic representation to designate the time evolution of the density matrix of the system [13]. Thus, we can employ perturbation theory to calculate the density-matrix elements. In this five-level system, the perturbation chains (i.e., Liouville pathways with perturbation theory) are written as following [14]: (0) ω

(1) ω

(2) −ω

(3)

1 2 2 ρ10 −−→ ρ20 −−−→ ρ10 I : ρ00 −−→

1.3 Generalized Resonant MWM in Multi-level Atomic Systems (0) ω

(1) ω

(2) −ω

(3) ω

(4) −ω

(5)

(0) ω

(1) ω

(2) −ω

(3) ω

(4) −ω

(5) ω

11

1 2 3 3 2 II : ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ30 −−→ ρ20 −−−→ ρ10

(6) −ω

(7)

1 2 3 4 4 3 2 III : ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ30 −−→ ρ40 −−−→ ρ30 −−→ ρ20 −−−→ ρ10

Here, in order to proceed further and to simplify the mathematics, we will (0) neglect the ground-state depletion (ρ00 ≈ 1), which is a good approximation for the case with a weak probe beam, and not consider the propagation characteristics of the pulsed pump, probe and generated FWM fields. Also, we only retain near-resonant dipole interaction terms in the derivation of the complex susceptibility, known as the rotating-wave approximation (RWA). Because of the selectivity imposed by the RWA, each pulse interaction contributes in a unique way to the phase-matching directions of the nonlinear signal. The perturbation chains (I) – (III) correspond to these FWM, SWM and EWM processes, respectively. Solving Eq. (1.5) in steady state, we obtain (3)

−iG1 G2 (G2 )∗ eikF ·r (Γ10 + iΔ1 )2 [Γ20 + i(Δ1 + Δ2 )]

(1.6)

(5)

iG1 G2 (G2 )∗ G3 (G3 )∗ eikS ·r (Γ10 + iΔ1 )2 [Γ20 + i(Δ1 + Δ2 )]2 [Γ30 + i(Δ1 + Δ2 − Δ3 )]

(1.7)

ρ10 = ρ10 =

ρ10 = −iG1 G2 (G2 )∗ G3 (G3 )∗ G4 (G4 )∗ eikE ·r /(Γ10 + iΔ1 )2 [Γ20 + i(Δ1 + (7)

Δ2 )]2 [Γ30 + i(Δ1 + Δ2 − Δ3 )]2 [Γ40 + i(Δ1 + Δ2 − Δ3 + Δ4 )] (1.8) The phase-matching conditions for these nonlinear optical processes are kF = k1 + k2 − k2

kS = k1 + k2 − k2 + k3 − k3 kE = k1 + k2 − k2 + k3 − k3 + k4 − k4 (3)

(5)

(7)

The response functions (ρ10 , ρ10 , and ρ10 ) of perturbation chains (I) – (III) can be deduced from using double-sided Feynman diagrams (DSFD), as shown in Fig. 1.4 [13, 14]. The time evolution of density-matrix elements of optically driven atoms or molecules can be represented schematically by either the Liouville space coupling representation [chains (I) – (III)] or the DSFD (see Fig. 1.4). Each diagram represents a distinct Liouville space pathway. Figure 1.4 shows the diagrammatic representations corresponding to the third, fifth, and seventh orders of the resonant dipole interactions applied to the atomic system with five electronic states, respectively. In the Liouville space coupling representation [chains (I) – (III)], the state of the system is designated by a position in Liouville space, with indices corresponding to the ket-bra “axis”. Up and down transitions on the ket are excited by positive and negative frequency fields, whereas negative and positive frequency fields induce up and down transitions on the bra. Here, the negative frequency fields are the conjugate of the corresponding positive frequency fields. The DSFD (as

12

1 Introduction

shown in Fig. 1.4) can be described as follows: the vertical left and right lines of the diagram represent the time evolutions (bottom to top) of the ket and bra, respectively. These applied electric fields are indicated with arrows oriented toward the left if propagating with a negative wave vector and toward the right for a positive wave vector. Each interaction with the electric field produces a transition between the two electronic states of either the bra or the ket. The ability to track the evolutions of the bra and ket simultaneously makes the density-matrix representation a most appropriate tool for describing many dynamical phenomena in nonlinear optical processes.

Fig. 1.4. Double-sided Feynman diagrams. (a), (b), and (c) represent FWM, SWM, and EWM processes, respectively.

1.4 Enhanced Nonlinearity via Electromagnetically Induced Transparency When coherent laser fields interact with multi-level atomic systems, atomic coherence between different energy levels can be induced [14]. This laserinduced atomic coherence is essential in creating novel effects related to EIT [21 – 25]. EIT-related phenomena in three-level atomic systems have been extensively studied since 1990 [26]. By employing two-photon Doppler-free configurations, EIT has been observed with weak CW diode lasers in threelevel cascade-type (with two laser counter-propagating through) and Λ-type (with two laser co-propagating through) atomic vapor cells [27, 28]. Sharp dispersion properties due to the EIT [29] and enhancing efficiencies of nonlinear optical processes (such as non-degenerate FWM) by using EIT effect have all been experimentally demonstrated [16, 17]. There are three basic configurations for three-level atomic systems, i.e., ladder-type, V-type, and Λ-type systems, as shown in Fig. 1.5. A strong coupling (pump) laser beam (with frequency ω2 and Rabi frequency G2 ) interacts with the atomic transition between the state |2 and state |3, and

1.4 Enhanced Nonlinearity via Electromagnetically Induced Transparency

13

a weak probe laser (frequency ω1 and Rabi frequency G1 ) interacts with the transition between states |1 and |2. The strong coupling beam dramatically modifies the transmission properties of the probe beam. For a given coupling beam strength, a transparency window is created for the probe beam near the two-photon resonance condition of Δ1 + Δ2 = 0 (with a dip in χ ), as shown in Fig. 1.6 for a Λ-type three-level atomic system, where Δ1 and Δ2 are the frequency detuning of the probe and coupling beams, respectively. A sharp normal dispersion slope is created at the center of the EIT window, which can be used to slow down the group velocity of probe light pulses [25, 29]. Both of the absorption reduction and dispersion enhancement are important in increasing the efficiencies of nonlinear optical processes. The slowing down of the optical pulses inside the medium increases the effective interaction length, and the opened EIT window will allowed the generated signal beam to propagate through the medium with greatly reduced absorption.

Fig. 1.5. (a) Sketche of three-level ladder-type atomic system; (b) sketche of threelevel Λ-type atomic system; (c) sketche of three-level V-type atomic system. Δ1 and G1 (Δ2 and G2 ) are the frequency detuning and the Rabi frequency of the probe (the coupling) field, respectively.

Fig. 1.6. Measured three-level linear responses (χ and χ ) versus frequency detuning of probe beam with Δ2 = 0.

Actually, the third-order nonlinear optical coefficient of the three-level atomic system has been shown to be greatly enhanced comparing to its twolevel counterpart [30]. By placing three-level [as shown in Fig. 1.5(c)] Λ-type 87 Rb atoms inside an optical ring cavity and making use of the asymmetric transmission profiles of an optical cavity caused by the intracavity nonlinear medium, the Kerr nonlinear index of refraction (n2 ) has been measured, as shown in Fig. 1.7 [30]. The significant modification and enhancement of third-

14

1 Introduction

order nonlinearity are shown for the three-level EIT system near two-photon resonance, which have very sensitive dependence on the frequency detuning of the probe and coupling laser beams. Such reduction in the linear absorption and greatly enhancement in Kerr nonlinearity are caused by the light-induced atomic coherence and quantum interference between transition probabilities, which allow one to study nonlinear optical properties at low light levels.

Fig. 1.7. Measured Kerr-nonlinear coefficient n2 versus frequency detuning of probe beam with Δ2 = 0 and G2 = 2π × 72 MHz. Solid squares are with coupling beam and open circles are without coupling beam.

For atoms inside a vapor cell, the Doppler effect can have dramatic effects in the atomic coherence and EIT-related effects. Actually, Dopplerbroadening at room temperature can wipe out most of the coherence effects, except with extremely large coupling laser power (which needs to use pulsed lasers) [21]. However, with the uses of two-photon Doppler-free configurations, first-order Doppler effect can be eliminated, such EIT and related effects can be observed with cw diode lasers near atomic resonance [27, 28]. To achieve the two-photon Doppler-free configuration, the coupling and the probe lasers have to counter-propagate through the atomic vapor cell for the three-level ladder-type system [27] and co-propagate through the atomic vapor cell for Λ-type (or V-type) system [28]. Such two-photon Doppler-free configurations are also very important in studying the novel nonlinear steadystate (such as optical bistability, multistability) and dynamical (instability, from period doubling to chaos, and stochastic resonance) effects in three-level atomic systems [24]. Such controllable linear and nonlinear optical properties exist not only in three-level atomic systems, as shown above, but also in more complicated four- and five-level atomic systems, where the four- and five-level systems can be considered as various composite three-level systems. For example, a four-level close-cycled N-type system can be considered to be consist of a three-level Λ-type and a three-level V-type sub-systems, in which the Kerr nonlinearity has also be shown to be greatly increased [31]. Similarly, a fourlevel open-cycled inverted-Y-type system is made of two coupled ladder-type sub-systems, sharing the same probe beam [32]. In Chapters 6 – 8, we will show that in such composite four- and five-level atomic systems, not only

References

15

FWM processes can be greatly enhanced, but also efficient higher-order nonlinear wave-mixing processes, such as SWM and even EWM, can be generated, which can co-exist with the lower-order FWM processes [6 – 9]. By adjusting the strengths and the frequency detuning of the coupling laser beam, SWM signals can be enhanced to be in the same order as the FWM signal intensity, and some time even stronger in signal intensity. Interferences between FWM and SWM signals in frequency, temporal, and spatial domains were observed and studied [8, 10]. Energy exchanges between the probe beam, and the generated FWM and SWM beams are demonstrated [12]. The efficient coupling between these high-order nonlinear wave-mixing processes makes them exchange energy in propagation before reaching their respective equilibrium values at long propagation distance (or high optical density) through the medium. Light-induced atomic coherence [33] and quantum interferences between different energy levels are underlying mechanisms for such enhanced higherorder nonlinear optical wave-mixing processes. Also, specially-designed spatial configurations for laser beams (in square-box patterns) have been used to make use of two-photon Doppler-free configurations in selectively enhancing and suppressing certain wave-mixing channels. These simultaneous opening of dual EIT windows in a four-level system for the generated FWM and SWM signals allows two different order nonlinear signals to propagate through the atomic medium without significant absorptions [9]. There have been some theoretical interests in generating efficient and coexisting high-order nonlinear optical processes in multi-level atomic systems for 3-qubit quantum computation and liquid light condensate, and stable 2-dimensional soliton formation. Also, such co-existing third- and fifth-order nonlinearities are important in the propagation of high intensity laser pulses through the so-called cubic-quintic type media. In Chapters 6 – 8, we will describe theoretical and experimental studies of such enhanced and co-existing higher-order nonlinear multi-wave mixing processes in different multi-level atomic systems and in different laser beam configurations.

References [1] [2] [3] [4] [5] [6]

Franken P A, Hill A E, Peters C W, Weinreich G. Generation of optical harmonics. Phys. Rev. Lett., 1961, 7: 118 – 119. Antoine P, L’Huillier A, Lewenstein M. Attosecond pulse trains using high – order harmonics. Phys. Rev. Lett., 1996, 77: 1234 – 1237. Boyd R W. Nonlinear Optics. New York: Academic Press, 1992. Shen Y R. The Principles of Nonlinear Optics. New York: Wiley, 1984. Michinel H, Paz-Alonso M J, Perez-Garcia V M. Turning light into a liquid via atomic coherence. Phys. Rev. Lett., 2006, 96: 023903. Zhang Y P, Xiao M. Controlling four-wave and six-wave mixing processes in multilevel atomic systems. Appl. Phys. Lett., 2007, 91: 221108.

16

[7] [8] [9]

[10] [11]

[12]

[13]

[14] [15]

[16]

[17]

[18] [19] [20]

[21] [22] [23]

[24] [25] [26] [27]

1 Introduction

Zhang Y P, Xiao M. Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system. Appl. Phys. Lett., 2007, 90: 111104. Zhang Y P, Brown A W, Xiao M. Observation of interference between fourwave mixing and six-wave mixing. Opt. Lett., 2007, 32: 1120 – 1122. Zhang Y P, Brown A W, Xiao M. Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows. Phys. Rev. Lett., 2007, 99: 123603. Anderson B, Zhang Y P, Khadka U, Xiao M. Spatial interference between four- and six-wave mixing signals. Opt. Lett., 2008, 33: 2029 – 2031. Zhang Y P, Anderson B, Brown A W, Xiao M. Competition between two four-wave mixing channels via atomic coherence. Appl. Phys. Lett., 2007, 91: 061113. Zhang Y P, Anderson B, Xiao M. Efficient energy transfer between four-wavemixing and six-wave-mixing processes via atomic coherence. Phys. Rev. A, 2008, 77: 061801. Zhang Y P, Gan C L, Xiao M. Title: Modified two-photon absorption and dispersion of ultrafast third-order polarization beats via twin noisy driving fields. Phys. Rev. A, 2006, 73: 053801. Zhang Y P, Xiao M. Generalized dressed and doubly-dressed multiwave mixing. Opt. Exp., 2007, 15: 7182 – 7189. Hemmer P R, Katz D P, Donoghue J, Cronin-Golomb M, Shahriar M S, Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium. Opt. Lett., 1995, 20: 982. Li Y, Xiao M. Enhancement of non-degenerate four-wave mixing using electromagnetically induced transparency in rubidium atoms. Opt. Lett., 1996, 21: 1064. Lu B, Burkett W H, Xiao M. Nondegenerate four-wave mixing in a doubleLambda system under the influence of coherent population trapping. Opt. Lett., 1998, 23: 804. Kang H, Hernandez G, Zhu Y F. Superluminal and slow light propagation in cold atoms. Phys. Rev. A, 2004, 70: 061804. Kang H, Hernandez G, Zhu Y F. Slow-light six-wave mixing at low light intensities. Phys. Rev. Lett., 2004, 93: 073601. Zuo Z C, Sun J, Liu X, Jiang Q, Fu S G, Wu L A, Fu P M. Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phaseconjugate geometry. Phys. Rev. Lett., 2006, 97: 193904. Harris S E. Electromagnetically induced transparency. Phys. Today, 1997, 50: 36 – 42. Marangos J P. Electromagnetically induced transparency. J. of Mod. Optics, 1998, 45: 471 – 503. Fleischhauer M, Imamoglu A, Marangos J P. Electromagnetically induced transparency: optics in coherent media. Rev. of Mod. Phys., 2005, 77: 633 – 673. Joshi A, Xiao M. Controlling nonlinear optical processes in multi-level atomic systems. Progress in Optics, 2006, 49: 97 – 175. Boyd R W, Gauthier D J. “Slow” and “fast” light. Progress in Optics, 2002, 43: 497 – 530. Harris S E, Field J E, Imamoˇ glu A. Nonlinear optical processes using electromagnetically induced transparency. Phys. Rev. Lett., 1990, 64: 1107 – 1110. Gea-Banacloche J, Li Y, Jin S, Xiao M. Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment. Phys. Rev. A, 1995, 51: 576 – 584.

References

[28] [29]

[30] [31] [32]

[33]

17

Li Y Q, Xiao M. Electromagnetically induced transparency in a three-level L-type system in rubidium atoms. Phys. Rev. A, 1995, 51: 2703. Xiao M, Li Y Q, Jin S, Gea-Banacloche J. Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms. Phys. Rev. Lett., 1995, 74: 666 – 669. Wang H, Goorskey D, Xiao M. Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system. Phys. Rev. Lett., 2001, 87: 073601. Kang H S, Zhu Y F. Observation of large Kerr nonlinearity at low light intensities. Phys. Rev. Lett., 2003, 91: 093601. Joshi A, Xiao M. Electromagnetically induced transparency and its dispersion properties in a four-level inverted-Y atomic system. Physics Letter A, 2003, 317: 370 – 377. Xiao M. Novel linear and nonlinear optical properties of electromagnetically induced transparency systems. IEEE J. Quantum Electron., 2003, 9: 86 – 92.

2 Femtosecond Polarization Beats

When two or more transition pathways exist in multi-level systems excited by multiple laser beams, the generated wave-mixing signals, if arranged appropriately in phase-matching conditions and spatial configurations, can have the same frequency and propagate in the same direction. Therefore, the total detected signal, proportional to the total polarization, will have interference terms determined by the relative time delay between different transition pathways. The beating signal in the total polarization (which we refer to as polarization beats) can have a very fast time scale giving by the energy difference between different resonant transition frequencies. In this chapter, we describe how the different order of coherence functions of laser fields can affect the detected polarization beat. Different stochastic models for the laser fields under different conditions are discussed. Experimental results in multilevel atomic systems are presented to illustrate the concept of polarization beats in femtosecond time scale.

2.1 Effects of Field-correlation on Polarization Beats Polarization beats, which originate from the interference between the macroscopic polarizations, have attracted a lot of attention in recent years. It is closely related to the quantum beat spectroscopy, which appears in the conventional time-resolved fluorescence and in the time-resolved nonlinear laser spectroscopy. Quantum beat spectroscopy has been applied to the quasi-twolevel [1,2] and cascade three-level systems [3]. In the quasi-two-level case, the excited and the ground states consist of sub-level structures. Quantum beat manifests itself as an oscillation of the signal with frequency corresponding to the energy-level splittings. For example, Debeer, et al. performed the first ultrafast modulation spectroscopy (UMS) experiment in a sodium vapor [1]. The beating signal exhibits a 1.9 ps modulation corresponding to the sodium D-line splitting when the time delay between two doublefrequency pump beams increases. Fu, et al. [2] then analyzed the UMS with the phase-conjugation geometry in a Doppler-broadened system. They found that a Doppler-free precision in the measurement of the energy-level splitting could be achieved. Based on the interference between the one-photon

2.1 Effects of Field-correlation on Polarization Beats

19

and two-photon processes, UMS technique has also been applied to a cascade three-level system [3]. UMS in the cascade three-level case shows a beating between the resonant frequencies of this system. If the energy separation between the ground state and the intermediate state is previously known, then the energy separation between the intermediate state and the excited state can be deduced from the beating signal. Based on the interference between two two-photon processes, we show how to extend the UMS technique to a four-level system. In this section, we describe the effect of field-correlation on polarization beat spectroscopy in a four-level system (PBFS). We present a second-order coherence function theory to elucidate the basic physics of the effects of fieldcorrelation on polarization beats in the four-level system. The asymmetric behavior of the polarization beats [4 – 6] will be discussed, which can be attributed to the shift of the zero time delay due to the dispersion of the optical components in the delay-line. If PBFS is employed for the energy-level difference measurement, the advantages are that the energy-level difference between states can be widely separated and a Doppler-free precision in the measurement can be achieved. Investigations of the relationship between PBFS and other Doppler-free techniques have also been carried out in both frequency and time domains. It is found that PBFS is closely related to the two-photon absorption spectroscopy with a resonant intermediate state [3] and the sum-frequency trilevel photon-echo [7] when the pump laser beams are narrow band and broadband, respectively. However, it possesses the main advantages of these techniques in the frequency domain and in the time domain. In the current case, PBFS is a polarization beat phenomenon originating from the interference between two two-photon processes. Let us consider a four-level system (see Fig. 2.1) with a ground state |0, an intermediate state |1, and two excited states |2 and |3. This four-level system is typically called a “Y-type” system. States between |0 and |1, and between |1 and |2 (|3) are coupled by dipole transitions with resonant frequencies Ω1 and Ω2 (Ω3 ), respectively, while transitions between |2 and |3, and between |0 and |2 (|3) are dipole forbidden. We consider, in this four-level Y-type system, a double-frequency time-delayed four-wave mixing (FWM) experiment in which beam 2 and beam 3 consist of two frequency components ω2 and ω3 , while beam 1 has frequency ω1 (see Fig. 2.1). We assume that ω1 ≈ Ω1 and ω2 ≈ Ω2 (ω3 ≈ Ω3 ), therefore ω1 and ω2 (ω3 ) will drive the transitions from |0 to |1 and from |1 to |2 (|3), respectively. In this double-frequency time-delayed FWM process, beam 1 with frequency ω1 together with ω2 and ω3 frequency components of beam 2 induce coherences between states |0 and |2, as well as between |0 and |3 by two separate two-photon transitions, which are then probed by the ω2 and ω3 frequency components of beam 3. These are two-photon FWM processes with a common resonant intermediate state |1 and the frequency of the generated signal equals to ω1 . We are interested in the dependence of the beating signal intensity on the relative time delay

20

2 Femtosecond Polarization Beats

between beam 2 and beam 3.

Fig. 2.1. Four-level Y-type system used in PBFS.

The complex electric fields of beam 2 (Ep2 ) and beam 3 (Ep3 ) can be written as Ep2 = ε2 u2 (t) exp[i(k2 · r − ω2 t)] + ε3 u3 (t) exp[i(k3 · r − ω3 t)] Ep3 = ε2 u2 (t − τ ) exp[i(k2 · r − ω2 t + ω2 τ )] + ε3 u3 (t − τ + δτ ) exp[i(k3 · r − ω3 t + ω3 τ − ω3 δτ )] with two field components in each beam. Beam 3 is delayed by a time, as is shown in Fig. 2.2. Here, εi , ki (εi ,ki ) are the constant field amplitude and the wave vector of the ωi component in beam 2 (beam 3), respectively. ui (t) is a dimensionless statistical factor that contains phase and amplitude fluctuations. We assume that the ω2 (ω3 ) components of the Ep2 and Ep3 fields come from a single laser source and τ is the time delay of beam 3 with respect to beam 2. δτ denotes the difference in the zero time delay. Beam 1 is assumed to be a quasi-monochromatic light, so the complex electric field of beam 1 can be written as Ep1 = ε1 exp[i(k1 · r − ω1 t)].

Fig. 2.2. Schematic diagram of geometry of laser beams for PBFS.

We employ the perturbation theory to calculate the density-matrix elements of the interaction system. Following the perturbation chains: (I) (0) ω1 (1) ω2 (2) −ω2 (3) (0) ω1 (1) ω3 (2) −ω3 (3) ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ10 and (II) ρ00 −−→ ρ10 −−→ ρ30 −−−→ ρ10 , (3)

we can obtain the third-order off –diagonal density-matrix element ρ10 , which has the wave-vector of either k2 − k2 + k1 or k3 − k3 + k1 . The total density-matrix element is the sum of these two possible processes with (3) (3) ρ10 = ρ(I) + ρ(II) . ρ(I) and ρ(II) , corresponding to ρ10 of the perturbation chains (I) and (II), respectively, are   −iμ1 μ22 (I)  ∗  ρ10 = ε ε (ε ) exp i[(k + k − k ) · r − ω t − ω τ ] × 1 2 2 1 2 1 2 2 3  ∞  ∞  ∞ dt3 dt2 dt1 × 0

0

0

2.1 Effects of Field-correlation on Polarization Beats

21

exp{−iv · [k1 (t1 + t2 + t3 ) + k2 (t2 + t3 ) − k2 t3 ]} × exp[−(Γ10 + iΔ1 )t3 ] × exp[−(Γ20 + iΔ1 + iΔ2 )t2 ] × exp[−(Γ10 + iΔ1 )t1 ]u2 (t − t2 − t3 )u∗2 (t − t3 − τ ) (II)

ρ10 =

(2.1)

  −iμ1 μ23  ∗  ε ε (ε ) exp i[(k + k − k ) · r − ω t − ω τ + ω δτ ] × 1 3 1 3 1 3 3 3 3 3     ∞

∞

dt3 0

∞

dt2 0

dt1 ×

0

exp{−iv · [k1 (t1 + t2 + t3 ) + k3 (t2 + t3 ) − k2 t3 ]} × exp[−(Γ10 + iΔ1 )t3 ] × exp[−(Γ30 + iΔ1 + iΔ3 )t2 ] × exp[−(Γ10 + iΔ1 )t1 ]u3 (t − t2 − t3 )u∗3 (t − t3 − τ + δτ )

(2.2)

Here, v is the atomic velocity; μ1 ,μ2 , μ3 are dipole-moment matrix-elements between states |0 and |1, |1 and |2, |1 and |3, respectively; Γ10 , Γ20 , Γ30 are transverse relaxation rates of the transitions from |0 and |1, |0 to |2, |and 0 to |3, respectively; Δ1 = Ω1 − ω1 , Δ2 = Ω2 − ω2 , Δ3 = Ω3 − ω3 are frequency detunings between different transitions. The nonlinear polarization P (3) , which is responsible for the phase- conjugate FWM signal, is given by averaging over the velocity distribution function  +∞ (3) w(v). Thus, P (3) = N μ1 −∞ dvw(v)ρ10 (v). N is the atomic density. For √ a Doppler-broadened atomic system, we have w (v) = exp[−(v/υ)2 ]/( πu),  here, υ = 2kB T /m with m being the mass of an atom, kB Boltzmann’s constant and T the absolute temperature. The FWM signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic proces |P (3) |2 , which involves fourth-order coherence function of ui (t) in the phase–conjugation geometry. While the FWM signal intensity in Debeer’s self-diffraction geometry is related to the sixth-order coherence function of the incident fields [1]. We assume that beam 2 (beam 3) is a multimode thermal source (i.e., laser field with a large number of uncorrelated modes), thus it corresponds to the chaotic field which undergoes both amplitude and phase fluctuations. As a result, ui (t) has Gaussian statistics with its fourth-order coherence function satisfying [8] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 ) + i = 2, 3 ui (t1 )u∗i (t4 )ui (t1 )u∗i (t3 ) In this case, we are only interested in the τ -dependent part of the signal. The FWM signal intensity can be well approximated by the absolute square of the stochastic average of the polarization |P (3) |2 , which can be broken down as second-order correlation functions of ui (t). Here, a second-order correlation function theory is developed to study the effects of laser coherence. This theory is valid when we are only interested in the τ -dependent parts of

22

2 Femtosecond Polarization Beats

the beat signal. By further assuming the fields in beam 2 (beam 3) to have Lorentzian line shape, we have [1] ui (t1 ) u∗i (t2 ) = exp(−αi |t1 − t2 |)

i = 2, 3

1 δωi with δωi being the linewidth of the laser with frequency ωi . 2 Then, the stochastic average of the polarization is given by P (3)  = (I) P + P (II) , where

here, αi =

 P (I) = S1 (r) exp [−i (ω1 t + ω2 τ )]



+∞

dvw (v) −∞



∞

0



∞

dt3

∞

dt2 0

dt1 ×

0

exp [−iθI (v)] exp [− (Γ10 + iΔ1 ) t3 ] exp [− (Γ20 + iΔ2 + iΔ1 ) t2 ] × exp [− (Γ10 + iΔ1 ) t1 ] exp (−α2 |t2 − τ |)

(2.3)

and P (II) = S2 (r) exp [−i (ω1 t + ω3 τ − ω3 δτ )] ×  ∞  +∞  ∞  ∞ dvw (v) dt3 dt2 dt1 exp [−iθII (v)] × −∞

0

0

0

exp [− (Γ10 + iΔ1 ) t3 ] exp [− (Γ30 + iΔ3 + iΔ1 ) t2 ] × exp [− (Γ10 + iΔ1 ) t1 ] exp (−α3 |t2 − τ + δτ |)

(2.4)

with S1 (r) = −iN μ21 μ22 ε1 ε2 (ε2 )∗ exp[i(k1 + k2 − k2 ) · r]/3 S2 (r) = −iN μ21 μ23 ε1 ε3 (ε3 )∗ exp[i(k1 + k3 − k3 ) · r]/3 θI (v) = v · [k1 (t1 + t2 + t3 ) + k2 (t2 + t3 ) − k2 t3 ] θII (v) = v · [k1 (t1 + t2 + t3 ) + k3 (t2 + t3 ) − k3 t3 ]

Let us now consider the case of beam 2 and beam 3 to be narrow-band so that α2 << Γ20 and α3 << Γ30 . For simplicity, we neglect the Doppler effect first. After performing the tedious integration, the beat signal intensity becomes I (τ ) ∝ |B1 |2 exp (−2α2 |τ |) + |ηB2 |2 exp (−2α3 |τ − δτ |) + exp (−α3 |τ − δτ |) exp (−α2 |τ |) × {ηB1∗ B2 exp [−i (ω3 − ω2 ) τ + i ω3 δτ ] + η ∗ B1 B2∗ exp [i (ω3 − ω2 ) τ − i ω3 δτ ]} 2

B1 = 1/{(Γ10 + iΔ1 ) [Γ20 + i (Δ2 + Δ1 )]} 2

B2 = 1/{(Γ10 + iΔ1 ) [Γ30 + i (Δ3 + Δ1 )]}  ∗ ∗ η = S2 (r) S1 (r) ≈ μ23 ε3 (ε3 ) [μ22 ε2 (ε2 ) ]

(2.5)

2.1 Effects of Field-correlation on Polarization Beats

23

Equation (2.5) indicates that the beat signal is modulated by a frequency ω3 −ω2 as τ is varied. In this case, ω2 and ω3 are tuned to resonant frequencies of the transitions from |1 to |2, and from |1 to |3, respectively, and then the modulation frequency equals to Ω3 −Ω2 . In other words, one can obtain a beating between the resonant frequencies of a four-level system. A Dopplerfree precision can be achieved in the measurement of Ω3 − Ω2 [2]. The temporal behavior of the beat signal is asymmetric with the maximum of the signal shifted from τ = 0, which can be attributed to the shift of the zero time delay due to the dispersion of the optical components in the delay-line.

2.1.1 PBFS in a Doppler-broadened System The beat signal can also be calculated from a different viewpoint. Under the large Doppler-broadening limit (i.e., k1 υ → ∞), we have √  +∞ 2 π δ (t1 + t2 + t3 − ξ1 t2 ) dvw (v) exp [−iθI (v)] ≈ (2.6) k1 u −∞ √  +∞ 2 π δ (t1 + t2 + t3 − ξ2 t2 ) dvw (v) exp [−iθII (v)] ≈ (2.7) k1 u −∞ where, ξ1 = k2 /k1 , ξ2 = k3 /k1 . We assume that ξ1 > 1,ξ2 > 1, δτ > 0. When Eqs. (2.6) and (2.7) are substituted into Eqs. (2.3) and (2.4), we obtain: (i) τ > δτ  √  2 π (3) (I) (II) P  = P + P = S1 (r) exp [−i (ω1 t + ω2 τ )] (ξ1 − 1) × k1 u  exp (−α2 |τ |) a a 2 + exp [− (Γ20 − Γ10 + iΔ2 ) |τ |] × a (Γ20 − Γ10 − α2 + iΔa2 )  −τ (Γa20 − Γ10 − α2 + iΔa2 ) − 1 + 2 (Γa20 − Γ10 − α2 + iΔa2 )  τ (Γa20 − Γ10 + α2 + iΔa2 ) + 1 + 2 (Γa20 − Γ10 + α2 + iΔa2 ) √ 2 π S2 (r) exp [−i (ω1 t + ω3 τ − ω3 δτ )] (ξ2 − 1) × k1 u  exp (−α3 |τ − δτ |) + exp [− (Γa30 − Γ10 + iΔa3 ) |τ − δτ |] × a (Γ30 − Γ10 − α3 + iΔa3 )2  − (τ − δτ ) (Γa30 − Γ10 − α3 + iΔa3 ) − 1 + 2 (Γa30 − Γ10 − α3 + iΔa3 )

24

2 Femtosecond Polarization Beats

(τ − δτ ) (Γa30 − Γ10 + α3 + iΔa3 ) + 1

 (2.8)

2

(Γa30 − Γ10 + α3 + iΔa3 )

Here, Γa20 = Γ20 +ξ1 Γ10 , Δa2 = Δ2 +ξ1 Δ1 , Γa30 = Γ30 +ξ2 Γ10 , Δa3 = Δ3 +ξ2 Δ1 . (ii) 0 < τ < δτ  √  2 π P (3)  = P (I) + P (II) = S1 (r) × k1 u  exp (−α2 |τ |) exp [−i (ω1 t + ω2 τ )] (ξ1 − 1) 2 + a (Γ20 − Γ10 − α2 + iΔa2 )  −τ (Γa20 − Γ10 − α2 + iΔa2 ) − 1 a a exp [− (Γ20 − Γ10 + iΔ2 ) |τ |] + (Γa20 − Γ10 − α2 + iΔa2 )2  τ (Γa20 − Γ10 + α2 + iΔa2 ) + 1 + 2 (Γa20 − Γ10 + α2 + iΔa2 ) √ 2 π S2 (r) exp [−i (ω1 t + ω3 τ − ω3 δτ )] (ξ2 − 1) × k1 u 1 (2.9) exp (−α3 |τ − δτ |) a 2 (Γ30 − Γ10 − α3 + iΔa3 ) (iii) τ < 0 P (3)  = P (I) + P (II)  √  2 π = S1 (r) exp [−i (ω1 t + ω2 τ )] (ξ1 − 1) × k1 u 1 exp (−α2 |τ |) a 2 + (Γ20 − Γ10 − α2 + iΔa2 ) √ 2 π S2 (r) exp [−i (ω1 t + ω3 τ − ω3 δτ )] (ξ2 − 1) × k1 u 1 exp (−α3 |τ − δτ |) a 2 (Γ30 − Γ10 − α3 + iΔa3 )

(2.10)

Let us first consider the case with beam 2 and beam 3 being narrow band, so that α2 << Γ20 and α3 << Γ30 . The beat signal intensity is then given by (ξ1 − 1)2 exp (−2α2 |τ |) |η|2 (ξ2 − 1)2 exp (−2α3 |τ − δτ |) I (τ ) ∝  + 2 +  2 2 2 2 2 (Γa20 − Γ10 ) + (Δa2 ) (Γa30 − Γ10 ) + (Δa3 ) exp (−α2 |τ |) × exp (−α3 |τ − δτ |) {q exp [−i (ω3 − ω2 ) τ + iω3 δτ ] + q ∗ exp [i (ω3 − ω2 ) τ − iω3 δτ ]}

(2.11)

where, q = η (ξ1 − 1) (ξ2 − 1) /{[(Γa20 − Γ10 ) − iΔa2 ]2 [(Γa30 − Γ10 ) + iΔa3 ]2 }. Equation (2.11) is consistent with Eq. (2.5) above.

2.1 Effects of Field-correlation on Polarization Beats

25

Now, let us consider the case for beam 2 and beam 3 to be broadband, so that α2 >> Γ20 and α3 >> Γ30 . Under this condition, the beat signal rises to its maximum value quickly, and then decays with a time constant mainly determined by transverse relaxation times of the system. Although the beat signal modulation is quite complicated in general, one can see, from −1 Eqs. (2.8) and (2.9), that at the tail of the signal (i.e., τ >> α−1 2 , τ >> α3 ), the beat signal intensity can be written as: (i) τ > δτ  2 α2 (ξ1 − 1) τ I (τ ) ∝ exp [−2 (Γa20 − Γ10 ) |τ |] + 2 2 a α2 + (Δ2 )  2 2 α3 (ξ2 − 1) (τ − δτ ) |η| exp [−2 (Γa30 − Γ10 ) |τ − δτ |] + 2 α23 + (Δa3 )    α3 (ξ2 − 1) τ α2 (ξ1 − 1) τ τ (τ − δτ ) × 2 2 α22 + (Δa2 ) α23 + (Δa3 ) exp [− (Γa20 − Γ10 ) |τ |] exp [− (Γa30 − Γ10 ) |τ − δτ |] ×  η exp [−i (Ω3 − Ω2 ) τ − i (ξ2 − ξ1 ) Δ1 τ + i (Ω3 + ξ2 Δ1 ) δτ ] +  η ∗ exp [i (Ω3 − Ω2 ) τ + i (ξ2 − ξ1 ) Δ1 τ − i (Ω3 + ξ2 Δ1 ) δτ ] (2.12) Equation (2.12) indicates that the modulation frequency of the beat signal equals to Ω3 − Ω2 when Δ1 = 0 and δτ = 0. The overall accuracy of using PBFS with broadband lights to measure the energy-level splitting between two excited states (which are dipole forbidden from the ground state) is limited by homogeneous linewidths of these excited states [3]. The temporal behavior of the beat signal is asymmetric with the maximum of the signal shifted from τ = 0, as discussed before. (ii) 0 < τ < δτ 2  α2 (ξ1 − 1) τ exp [−2 (Γa20 − Γ10 ) |τ |] (2.13) I (τ ) ∝ 2 α22 + (Δa2 ) Equation (2.13) shows that the beat signal here becomes similar to the FWM signal intensity in which beam 2 and beam 3 only consist of ω2 frequency components. (iii) τ < 0 I (τ ) ∝

2

2

2

(ξ1 − 1) exp (−2α2 |τ |) |η| (ξ2 − 1) exp (−2α3 |τ − δτ |) + +  2  2 2 2 α22 + (Δa2 ) α23 + (Δa3 )  exp (−α2 |τ |) × exp (−α3 |τ − δτ |) q  exp [−i (ω3 − ω2 ) τ + i ω3 δτ ] + ∗

(q  ) exp [i (ω3 − ω2 ) τ − i ω3 δτ ]

(2.14)

26

2 Femtosecond Polarization Beats

where, q  = η (ξ1 − 1) (ξ2 − 1) /[(α2 − iΔa2 )2 (α3 − iΔa3 )2 ]. Equation (2.14) is consistent with Eq. (2.5). Therefore, the requirement for the existence of a τ -dependent beat signal for τ < 0 is for the phasecorrelated field component (pulses with different frequencies) in beam 2 and beam 3 to temporally overlap. Since beams 2 and 3 are mutually coherent, the temporal behavior of the beat signal should coincide with the case when beam 2 and beam 3 are nearly monochromatic.

2.1.2 Photon-echo It is interesting to understand the underlying physics in PBFS with incoherent lights. More attentions have been paid to the studies of various ultrafast phenomena by using incoherent light sources recently [9,10]. For the phase-matching conditions k2 − k2 + k1 and k3 − k3 + k1 , two sumfrequency trilevel photon echoes exist for the perturbation chains (I) and (II) [7]. Under the large Doppler-broadening limit (i.e., k1 υ → ∞) and assuming beam 2 (and beam 3) to have Gaussian line shape, we then have:  ! "2  α i ui (t1 )u∗i (t2 ) = exp − √ (t1 − t2 ) 2 ln 2  = exp −[βi (t1 − t2 )]2 i = 2, 3  P

(I)

= S1 (r) exp [−i (ω1 t + ω2 τ )]



+∞

dvw (v) −∞



∞

dt3 0



∞

dt2 0

∞

dt1 ×

0

exp [−iθI (v)] exp [− (Γ10 + iΔ1 ) t3 ] exp [− (Γ20 + iΔ2 + iΔ1 ) t2 ] × exp [− (Γ10 + iΔ1 ) t1 ] exp[−β22 (t2 − τ )2 ]

(2.15)

P (II) = S2 (r) exp [−i (ω1 t + ω3 τ − ω3 δτ )] ×  +∞  ∞  ∞  ∞ dvw (v) dt3 dt2 dt1 × −∞

0

0

0

exp [−iθII (v)] exp [− (Γ10 + iΔ1 ) t3 ] exp [− (Γ30 + iΔ3 + iΔ1 ) t2 ] × (2.16) exp [− (Γ10 + iΔ1 ) t1 ] exp[−β32 (t2 − τ + δτ )2 ] Let us now consider the case for beam 2 and beam 3 to be broadband, so that α2 >> Γ20 and α3 >> Γ30 . Then, one can approximate √ (2.17) exp[−β22 (t2 − τ )] ≈ πδ (t2 − τ ) /β2  √ 2 exp[−β3 (t2 − τ + δτ )] ≈ πδ (t2 − τ + δτ ) β3 (2.18) When Eqs. (2.6), (2.7), (2.17), (2.18) are substituted into Eqs. (2.15) and (2.16), we obtain:

2.1 Effects of Field-correlation on Polarization Beats

27

(i) τ > δτ  $2 2 2# I (τ ) ∝ [(ξ1 − 1) τ /β2 ] exp (−2Γa20 |τ |) + |η| (ξ2 − 1) (τ − δτ ) β3 × #  $ exp (−2Γa30 |τ − δτ |) + (ξ1 − 1) (ξ2 − 1) β2 β3 τ (τ − δτ ) ×  exp(−Γa20 |τ |) exp (−Γa30 |τ − δτ |) η exp[−i (Ω3 − Ω2 ) τ − i (ξ2 − ξ1 ) Δ1 τ + i (Ω3 + ξ2 Δ1 ) δτ ] + η ∗ exp [i (Ω3 − Ω2 ) τ + i (ξ2 − ξ1 ) Δ1 τ − i (Ω3 + ξ2 Δ1 ) δτ ]

 (2.19)

which is consistent with Eq. (2.12). (ii) 0 < τ < δτ , 2δ (t2 − τ + δτ ) /α3 = 0. Under these conditions, photonecho doesn’t  exist for the perturbation chain (II). Then, one can write I(τ ) ∝ [(ξ1 − 1)τ β2 ]2 exp(−2Γa20 τ  ), which is consistent with Eq. (2.13). (iii) τ < 0, 2δ (t2 − τ ) α2 = 0. Under these conditions, photon-echo doesn’t exist for either perturbation chain (I) or (II). This case is consistent with Eq. (2.5).

2.1.3 Experiment and Result The experiment of PBFS was performed in sodium vapor. In this atomic system, the ground state 3S1/2 , the intermediate state 3P3/2 , and two excited states 6S1/2 and 5D3/2,5/2 from a four-level Y-type system. Three dye lasers (DL1, DL2, and DL3), pumped by the second harmonic of a Quanta-Ray YAG laser, were used to generate the needed frequencies of ω1 , ω2 , and ω3 . DL1, DL2, and DL3 have linewidths of about 0.1 nm and pulse widths of about 5 ns. DL1 was tuned to 589.0 nm (the wavelength of the 3S1/2 − 3P3/2 transition); DL2 was tuned to 515.4 nm (the wavelength of the 3P3/2 − 6S1/2 transition); and DL3 was tuned to 498.3 nm (the wavelength of the 3P3/2 − 5D3/2,5/2 transition). A beam splitter was used to combine the ω2 and ω3 components derived from DL2 and DL3, respectively, for beam 2 and beam 3, which intersect with a small angle in the oven containing the Na vapor. The time delay τ between beam 2 and beam 3 was varied by an optical delay line. Beam 1, propagating along the direction opposite to that of beam 2, was derived from DL1. All the incident beams were linearly polarized in the same direction. The beat signal has the same polarization as the incident beams and propagates along the direction almost opposite to that of beam 3. The beat signal was detected by a photodiode. The beat signal intensity was measured as a function of the time delay between beam 2 and beam 3, as shown in Fig. 2.3. The result shows that as τ varies, the beat signal intensity modulates in a sinusoidal form with a period of 50 fs. The modulation frequency can be obtained more directly by making a Fourier transformation of the PBFS data. Figure 2.4 presents the Fourier spectrum of the data in which τ is varied for a range of 15 ps, from which the

28

2 Femtosecond Polarization Beats

modulation frequency of 126 (ps)−1 can be determined, which corresponds to the beating between the resonant frequencies of the transitions from 3P3/2 to 6S1/2 and from 3P3/2 to 5D3/2,5/2 . Again, the temporal shape of the beat signal is quite asymmetric with the maximum of the signal shifted from τ = 0. This asymmetry is due to the difference in the zero time delay between beam 2 and beam 3 for the ω2 and ω3 frequency components. To confirm this, the τ -dependence of the FWM signal was measured when beam 2 and beam 3 consist of only one frequency component.

Fig. 2.3. Beat signal intensity versus relative time delay. Adopted from Ref. [11].

Fig. 2.4. Fourier spectrum of the experimental data in which τ is varied for a range of 15 ps. Adopted from Ref. [11].

Figures 2.5 and 2.6 present the result when the frequencies of beam 2 and beam 3 are ω2 and ω3 , respectively. The difference in the zero-time delay is obvious in these figures. This effect is due to the large difference between the wavelengths of DL2 and DL3, so the dispersions of the optical components become important. This can be understood as follows: let’s consider the case that the optical paths of beam 2 and beam 3 are equal for the ω2 component. Owing to the difference between the zero time delays for the ω2 and ω3 frequency components, the optical paths of beam 2 and beam 3 will now be different by cδτ for the ω3 component. As a result, there is an extra phase factor ω3 δτ for the ω3 frequency component. The difference between the zero time delays for the ω2 and ω3 frequency components corresponds to the

2.1 Effects of Field-correlation on Polarization Beats

29

propagation of the beams in the glass (mainly the prism in the optical delay line) [4, 5].

Fig. 2.5. FWM signal intensity versus relative time delay when the pump beams consist of only ω2 component.

PBFS can be considered as a technique, which possesses the main features of the laser spectroscopy in the frequency domain and in the time domain. First, PBFS is closely related to the Doppler-free two-photon absorption spectroscopy in tuning ω2 and ω3 to the resonant frequency when narrow-band lights are used. However, unlike the techniques in the frequency domain, here we are interested in the temporal behavior of the signal, and the frequencies of the lasers do not need to be calibrated. In this sense PBFS is similar to the spectroscopy in the time domain. PBFS is related intrinsically to the sumfrequency trilevel photon-echo situation when broadband lights are used. In this case, when pulse laser beams 2 and 3 are separated temporally, then before the application of beam 3 the polarization exhibits free evolution. As a result, the modulation frequency is directly related to the energy level of the system regardless of whether the beams have a narrow-band or broadband linewidth. The advantage of the PBFS over other time-domain techniques is that the temporal resolution is not limited by the laser pulse widths.

Fig. 2.6. FWM signal intensity versus relative time delay when the pump beams consist of only ω3 component.

In this section, a second-order coherence function theory has been developed to study the effects of laser coherence on polarization beats in a four-

30

2 Femtosecond Polarization Beats

level Y-type atomic system. It was found that the temporal behavior of the beat signal depends on stochastic properties of lasers and transverse relaxation rates of the atomic energy-level system. We have considered the cases that pump beams have either narrow band or broadband linewidths, and found that for both cases a Doppler-free precision in determining the energylevel splitting between two upper excited states (which are dipole forbidden from the ground state) can be achieved. We also discussed the asymmetric behavior of polarization beat signals. It is worth mentioning that the asymmetric behavior of the polarization beat signal in a four-level system does not affect the overall accuracy of using the PBFS to determine the energy-level splitting. Furthermore, PBFS can tolerate small perturbations of the optical path due to mechanical vibrations and distortions of the optical components,  as long as these are small compared with c |ω3 − ω2 |.

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields The atomic responses to stochastic optical fields are now largely well understood. Methods exist to calculate the second-order moments of the atomic density-matrix elements for a wide variety of field statistics, including phasediffusing fields, phase-diffusing fields with colored noise, chaotic fields, real Gaussian fields, and phase-jump fields [12 – 17]. When the laser field is sufficiently intense so that many photon interactions occur, the laser spectral bandwidth or spectral shape, obtained from the second-order correlation function, is inadequate to characterize the field. Rather than using higher-order correlation functions explicitly, as in most discussions of finitebandwidth effects, we employ soluble models for fluctuating light fields. The chaotic field model and the Brownian-motion phase-diffusion model are considered in parallel with a detailed discussion on a V-type three-level atom system. A unified theory of treating field statistics is developed which involves fourth-order coherence functions to study the influence of partial-coherence properties of pump beams on polarization beats. In this section, we describe how the fourth-order field-correlation functions can have effect on polarization beats in a V-type three-level system (PBVTS). First, we assume that the laser sources are of chaotic fields. A chaotic field, which is used to describe a multimode laser source, is characterized by the fluctuations in both the amplitude and the phase of the field [2]. Another commonly used stochastic model is the phase-diffusion model, which has been used to describe an amplitude-stabilized laser source [8]. For this model it is assumed that the amplitude of the laser field is a constant, while its phase fluctuates as a random process. Based on these two types of models, we can study the influences of various quantities, such as laser linewidth, transverse relaxation rate, and longitudinal relaxation rate, on polarization

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields

31

beats. One of the relevant problems is the stationary FWM with incoherent light sources, which was proposed by Morita, et al. [9] to achieve an ultrafast temporal resolution of relaxation processes. Since they assumed that the laser linewidth is much longer than the transverse relaxation rate, that theory cannot be used to study the effect of the light bandwidth on the Bragg reflection signal. Asaka, et al. [18] considered the finite linewidth effect. However, the constant background contribution has been ignored in their analysis. The fourth-order coherence function theory presented in this section includes both the finite light bandwidth effect and the constant background contribution. The different roles of the phase fluctuation and amplitude fluctuation have been pointed out in the time domain. PBVTS originates from the interference between two one-photon processes. Let us consider a V-type three-level system (Fig. 2.7) with a ground state |0, and two excited states |1 and |2. States between |0 and |1, and between |0 and |2 are coupled by dipole transitions with resonant frequencies Ω1 and Ω2 , respectively, while transition between |1 and |2 is dipole forbidden. The schematic diagram of the beam geometry for PBVTS is shown in Fig. 2.8. In this V-type three-level system, we consider a double-frequency time-delayed FWM experiment in which beam 1 and beam 2 each consist of two frequency components ω1 and ω2 . We assume that ω1 ≈ Ω1 and ω2 ≈ Ω2 , therefore ω1 and ω2 will drive the transitions from |0 to |1 and from |0 to |2, respectively. There are two processes involved in this double-frequency time-delayed FWM scheme. First, the ω1 frequency component of beam 1 and beam 2 induce a population grating between states |0 and |1, which is probed by beam 3 with the same frequency ω1 . This is a one-photon resonant degenerate FWM (DFWM) and the generated signal (beam 4) has the same frequency ω1 . Second, the ω2 frequency component of beam 1 and beam 2 induces a population grating between states |0 and |2, which is probed by beam 3. This is a one-photon resonant nondegenerate FWM (NDFWM) and the frequency of the generated signal equals to ω1 again.

Fig. 2.7. V-type three-level configuration used in studying PBVTS.

Fig. 2.8. Schematic diagram of geometry for laser beams used in PBVTS.

The complex electric fields of beam 1 (Ep1 ) and beam 2 (Ep2 ) can be

32

2 Femtosecond Polarization Beats

written as Ep1 = A1 (r, t) exp(−iω1 t) + A2 (r, t) exp(−iω2 t) = ε1 u1 (t) exp[i(k1 · r − ω1 t)] + ε2 u2 (t) exp[i(k2 · r − ω2 t)] Ep2 = A1 (r, t) exp(−iω1 t) + A2 (r, t) exp(−iω2 t) = ε1 u1 (t − τ ) exp[i(k1 · r − ω1 t + ω1 τ )] + ε2 u2 (t − τ ) exp[i(k2 · r − ω2 t + ω2 τ )]

Here, εi , ki (εi ,ki ) are the constant field amplitude and the wave vector of the ωi component in beam 1 (beam 2), respectively. ui (t) is a dimensionless statistical factor that contains phase and amplitude fluctuations. We assume that ω1 (ω2 ) components of Ep1 and Ep2 come from a single laser source and τ is the time delay of beam 2 with respect to beam 1. Beam 3 is assumed to be a quasi-monochromatic light and the complex electric field of it can be written as Ep3 = A3 (r, t) exp(−iω1 t) = ε3 exp[i(k3 · r − ω1 t)]. ω1 , ε3 , and k3 are the frequency, the field amplitude, and the wave vector of the field in beam 3, respectively. The monochromatic field in beam 3 is infinitely self-correlated, but is completely uncorrelated with the fields in the twin beam 1 and beam 2. By employing the perturbation theory the density-matrix elements can be calculated. Using the following perturbation chains: (0) A

 ∗ (1) (A )

(2) A

(3)

1 3 ρ10 −−−1− → ρ00 −−→ ρ10 (I) ρ00 −−→  ∗ (0) (A )

A

(2) A

1 3 (II) ρ00 −−−1− → (ρ10 )∗ −−→ ρ00 −−→ ρ10

(0) A

(1)

 ∗ (1) (A )

(2) A

(3)

(3)

1 3 (III) ρ00 −−→ ρ10 −−−1− → ρ11 −−→ ρ10  ∗ (0) (A )

A

(2) A

1 3 (IV) ρ00 −−−1− → (ρ10 )∗ −−→ ρ11 −−→ ρ10

(0) A

(1)

 ∗ (1) (A )

(2) A

(3)

(3)

2 3 (V) ρ00 −−→ ρ20 −−−2− → ρ00 −−→ ρ10  ∗ (0) (A )

A

(2) A

2 3 (VI) ρ00 −−−2− → (ρ20 )∗ −−→ ρ00 −−→ ρ10

(1)

(3)

(3)

we can obtain the third-order off–diagonal density-matrix element ρ10 via different pathways, which has wave vector of either k1 −k1 +k3 or k2 −k2 +k3 . The nonlinear polarization P (3) (responsible for the phase-conjugate FWM signal) is given by averaging over the velocity distribution function w(v). Thus  +∞ (3) dvw(v)ρ10 (v). P (3) = N μ1 −∞

v is the atomic velocity and N is the atomic  √ density. For a Doppler-broadened atomic system, w (v) = exp[−(v/u)2 ] ( πu).

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields

33

The FWM signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic process |P (3) |2 , which involves second- and fourth-order coherence functions of ui (t) in the phaseconjugation geometry. The FWM signal intensity in the self-diffraction geometry is related to the sixth-order coherence function of the incident fields [1]. We first assume that beam 1 (beam 2) contains multimode thermal laser field with a large number of uncorrelated modes, thus it corresponds to the chaotic field which undergoes both amplitude and phase fluctuations. As a result, ui (t) obeys circular complex Gaussian statistics and its fourth-order coherence function satisfies [8] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 ) + i = 1, 2 ui (t1 )u∗i (t4 )ui (t2 )u∗i (t3 )

(2.20)

Thus, the four-point time correlation function is broken down into the sum of two terms, each consisting of a product of two two-point time correlators. Under the stationary condition, each of the factored four-point correlators has one term having a product of two τ -dependent two -point correlators and one term with no τ -dependence. Furthermore, by assuming that fields in beam 1 and beam 2 have the Lorentzian line shape, we can write [2] ui (t1 ) u∗i (t2 ) = exp (−αi |t1 − t2 |)

i = 1, 2

(2.21)

Where αi = δωi /2 with δωi being the linewidth of the laser with frequency ωi . The total polarization is then given by P (3) = P (I) + P (II) + P (III) + P (IV ) + P (V ) + P (VI) where



P (I) = S1 (r) exp [−i (ω3 t + ω1 τ )]

P (II)

dvw (v)



∞

dt3

−∞



∞

∞

dt2

0

0

dt1 ×

0

exp [−iθI (v)] × H1 (t1 ) H2 (t2 ) H3 (t3 ) u1 (t − t1 − t2 − t3 ) × (2.22) u∗1 (t − t2 − t3 − τ )  +∞  ∞  ∞  ∞ = S1 (r) exp [−i (ω3 t + ω1 τ )] dvw (v) dt3 dt2 dt1 × H1∗

P (III)



+∞

−∞

0

0

0

exp [−iθII (v)] × (t1 ) H2 (t2 ) H3 (t3 ) u1 (t − t2 − t3 ) × u∗1 (t − t1 − t2 − t3 − τ )  +∞  ∞  ∞  = S1 (r) exp [−iω3 t − iω1 τ ] dvw (v) dt3 dt2 −∞

0

0

(2.23) ∞

dt1 ×

0

exp [−iθI (v)] × H1 (t1 ) H4 (t2 ) H3 (t3 ) u1 (t − t1 − t2 − t3 ) × u∗1 (t − t2 − t3 − τ )

(2.24)

34

2 Femtosecond Polarization Beats

 P (IV ) = S1 (r) exp [−iω3 t − iω1 τ ]

P (V )



+∞

−∞



∞

dvw (v)



∞

dt3 0

0

exp [−iθII (v)] × H1∗ (t1 ) H4 (t2 ) H3 (t3 ) u1 (t − t2 − t3 ) × u∗1 (t − t1 − t2 − t3 − τ )  +∞  ∞  ∞  = S2 (r) exp [−iω3 t − iω2 τ ] dvw (v) dt3 dt2 −∞

∞

dt2

0

0

dt1 ×

0

(2.25) ∞

dt1 ×

0

exp [−iθIII (v)] × H5 (t1 ) H2 (t2 ) H3 (t3 ) u2 (t − t1 − t2 − t3 ) × u∗2 (t − t2 − t3 − τ )

(2.26) 

P (VI) = S2 (r) exp [−iω3 t − iω2 τ ] exp [−iθIV (v)] ×



+∞

dvw (v)

−∞ [H5 (t1 )]∗ H2 (t2 ) H3



∞

0



∞

dt3

∞

dt2 0

dt1 ×

0

(t3 ) u∗2 (t − t1 − t2 − t3 − τ ) ×

u2 (t − t2 − t3 ).

(2.27)

In these expressions, the factors are defined by S1 (r) = −iN μ41 ε1 (ε1 )∗ ε3 exp[i(k1 − k1 + k3 ) · r]/4

S2 (r) = −iN μ21 μ22 ε2 (ε2 )∗ ε3 exp[i(k2 − k3 + k3 ) · r]/2 θI (v) = v · [k1 (t1 + t2 + t3 ) − k1 (t2 + t3 ) + k3 t3 ] θII (v) = v · [−k1 (t1 + t2 + t3 ) + k1 (t2 + t3 ) + k3 t3 ] θIII (v) = v · [k2 (t1 + t2 + t3 ) − k2 (t2 + t3 ) + kt3 ]

θIV (v) = v · [−k2 (t1 + t2 + t3 ) + k2 (t2 + t3 ) + k3 t3 ] H1 (t) = exp [− (Γ10 + iΔ1 ) t] H2 (t) = exp (−Γ0 t) H3 (t) = exp [− (Γ10 + iΔ3 ) t] H4 (t) = exp (−Γ1 t) H5 (t) = exp [− (Γ20 + iΔ2 ) t]

μ1 (μ2 ) is the dipole-moment matrix element between states |0 and |1 (|0 and |2); Γ0 (Γ1 ) is the population relaxation rate of state |0 (|1); Γ10 (Γ20 ) is the transverse relaxation rate of the transition from |0 to |1 (|0 to |2); the frequency detunings are defined by Δ1 = Ω1 − ω1 , Δ2 = Ω2 − ω2 , Δ3 = Ω3 − ω3 . Let us first consider the case when fields in beam 1 and beam 2 are narrow band, so that α1 << Γ10 , α2 << Γ20 . For simplicity, here we neglect the Doppler effect. After performing the tedious integration, the beat signal intensity is then given by 2

2

I (τ ) ∝ |P (3) |2  ∝ A1 + |ηA2 | + |A3 | exp (−2α1 |τ |) + 2

|ηA4 | exp (−2α2 |τ |) + exp [−(α1 + α2 ) |τ |] × {ηA∗3 A4 exp [−i (ω2 − ω1 ) τ ] + η ∗ A3 A∗4 exp [i (ω2 − ω1 ) τ ]}

(2.28)

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields

35

 A3 = Γ10 (Γ0 + Γ1 ) [Γ0 Γ1 (Γ10 + iΔ3 )(Γ210 + Δ21 )]  A4 = Γ20 [Γ0 (Γ10 + iΔ3 )(Γ220 + Δ22 )]  Γ210 [(Γ0 + Γ1 )2 − 16α21 ] − A1 = 2 2 2 2 (Γ10 + Δ1 ) (Γ10 + Δ23 )(Γ20 − 4α21 )(Γ21 − 4α21 ) 2α1 (2Γ0 + Γ1 ) − (Γ0 + Γ10 − iΔ3 )(Γ10 + iΔ3 )(Γ20 − 4α21 )Γ0 (Γ0 + Γ1 )  2α1 (Γ0 + 2Γ1 ) (Γ1 + Γ10 − iΔ3 )(Γ10 + iΔ3 )(Γ21 − 4α21 )Γ1 (Γ0 + Γ1 ) ! Γ220 1 A2 = 2 − (Γ20 + Δ22 )2 (Γ210 + Δ23 )(Γ20 − 4α22 ) " 2α2 (Γ0 + Γ10 − iΔ3 )(Γ10 + iΔ3 )(Γ20 − 4α22 )Γ0   η = S2 (r) S1 (r) = μ22 ε2 (ε2 )∗ [μ21 ε1 (ε1 )∗ ] exp{ir[−(k1 − k1 ) + (k2 − k2 )]} which is for spatial dependence of the fields. Equation (2.28) consists of three parts. The first part (first and second terms), depending on fourth-order coherence functions (which originate from the amplitude fluctuation of the chaotic field), is independent of the relative time-delay between beams 1 and 2. The second part (third and fourth terms), depending on the fourth-order coherence functions (which originate from the phase fluctuation of the chaotic field), indicates an exponential decay of the beat signal as |τ | increases. The third part (fifth and sixth terms), depending on the second-order coherence functions (which are determined by the laser line shape), gives rise to the modulation of the beat signal. Equation (2.28) shows that the beat signal oscillates not only temporally but also spatially with a period 2π/|(k1 − k1 ) − (k2 − k2 )| along the direction of (k1 − k1 ) − (k2 − k2 ), which is almost perpendicular to the propagation direction of the beat signal. The polarization-beat model has assumed that both pump beams are plane waves. Therefore, DFWM and NDFWM signals, which propagate along the directions of ks1 = k1 − k1 + k3 and ks2 = k2 − k2 + k3 , respectively, are plane waves also. Since DFWM and NDFWM signals propagate along slightly different directions, the interference between them leads to a spatial oscillation. However, the spatial dependence in η can be neglected in the typical experiment [19], i.e., η = μ22 ε2 (ε2 )∗ [μ21 ε1 (ε1 )∗ ]. Equation (2.28) only indicates that the beat signal modulates with a frequency ω2 − ω1 as τ is varied. In this case, ω1 and ω2 are tuned to the resonant frequencies of the transitions from |0 to |1 and from |0 to |2, respectively, so the modulation frequency in the polarization beat signal equals to Ω2 − Ω1 . This means that one can obtain a beating signal between the resonant frequencies of the two transitions in a V-type three-level system. A Doppler-free precision can be achieved in the

36

2 Femtosecond Polarization Beats

measurement of Ω2 − Ω1 [3,17]. Let us now consider the case when beams 1 and 2 are broadband, so that α1 , α2 >> Γ10 , Γ20 >> Γ0 , Γ1 . Under these conditions, the beat signal rises to its maximum value quickly and then decays with a time constant mainly determined by the transverse relaxation rate of the system. Although the expression of the beat signal modulation is complicated in general, at the tail −1 of the signal (i.e., τ >> α−1 1 , τ >> α2 ) the result can be simplified to 2

2

I (τ ) ∝ |P (3) |2  = B1 + |η| B2 + |B3 | exp (−2Γ10 |τ |) + 2

|ηB4 | exp(−2Γ20 |τ |) + exp[−(Γ20 + Γ10 ) |τ |] × {ηB3∗ B4 exp[−i(Ω2 − Ω1 )τ ] + η ∗ B3 B4∗ exp[i(Ω2 − Ω1 )τ ]}

(2.29)

where B1 = B2 =

α21 (α21 + Δ21 + Δ23 )(Γ20 + 6Γ0 Γ1 + Γ21 ) 2Γ0 Γ1 Γ10 (Γ0 + Γ1 )(Γ210 + Δ23 )(α21 + Δ21 )[α21 + (Δ3 − Δ1 )2 ][α21 + (Δ3 + Δ1 )2 ] 2Γ0 Γ20 (Γ210

+

Δ23 )(α22

α22 (α22 + Δ22 + Δ23 ) + Δ22 )[α22 + (Δ3 − Δ2 )2 ][α22 + (Δ3 + Δ2 )2 ]

α1 (Γ0 + Γ1 ) B3 = Γ0 Γ1 (Γ10 + iΔ3 )(α21 + Δ21 ) α2 B4 = Γ0 (Γ10 + iΔ3 )(α22 + Δ22 )

Equation (2.29) also consists of three parts. The first part (first and second terms), depending on fourth-order coherence functions (which originate from the amplitude fluctuation of the chaotic field), is independent of the relative time-delay τ . The second part (third and fourth terms), depending on the fourth-order coherence functions (which originate from the phase fluctuation of the chaotic field), shows an exponential decay of the beat signal as |τ | increases. The third part (fifth and sixth terms), depending on the second-order coherence functions (which are determined by the laser line shape), produces the modulation of the beat signal. Equation (2.29) shows that the modulation frequency of the beat signal is equal to Ω2 − Ω1 . The overall accuracy of using this PBVTS with broadband lights to measure the energy-level splitting between two excited states is limited by the homogeneous linewidths of the system [3,17]. The beat signal can also be calculated from a different viewpoint. Under the large Doppler-broadening limit (i.e., k3 u → ∞), we have √  +∞ 2 π δ (t3 − ξ1 t1 ) dvw (v) exp [−iθI (v)] ≈ (2.30) k3 u −∞ √  +∞ 2 π δ (t3 + ξ1 t1 ) dvw (v) exp [−iθII (v)] ≈ (2.31) k3 u −∞ √  +∞ 2 π δ (t3 − ξ2 t1 ) dvw (v) exp [−iθIII (v)] ≈ (2.32) k3 u −∞

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields



+∞

−∞

dvw (v) exp [−iθIV

√ 2 π δ (t3 + ξ2 t1 ) (v)] ≈ k3 u

37

(2.33)

  where ξ1 = k1 k3 , ξ2 = k2 k3 . When Eqs. (2.30)∼(2.33) are substituted into Eqs. (2.22)∼(2.27), we find P (II) = P (IV ) = P (VI) = 0. Then I(τ ) ∝ |P (3) |2  >= |P (I) + P (III) + P (V ) |2 

(2.34)

First, we consider the case when the fields in beams 1 and 2 are narrow band, so that α1 << Γ10 ,α2 << Γ20 . After performing tedious integrations, the beat signal intensity is calculated to be 2

2

I(τ ) ∝ |P (3) |2  = C1 + C2 |η| + |C3 | exp(−2α1 |τ |) + |ηC4 |2 exp(−2α2 |τ |) + exp[−(α1 + α2 ) |τ |] × {ηC3∗ C4 exp [−i (ω2 − ω1 ) τ ] + η ∗ C3 C4∗ exp [i (ω2 − ω1 ) τ ]}

(2.35)

where ! 1 1 1 + + (Γa10 )2 + (Δa1 )2 Γ0 (Γ0 + 2α1 ) Γ1 (Γ1 + 2α1 ) " 2(Γ0 Γ1 − 4α21 ) 4α1 (Γ20 + Γ21 − 8α21 ) − (Γ20 − 4α21 )(Γ21 − 4α21 ) (Γ0 + Γ1 )(Γ20 − 4α21 )(Γ21 − 4α21 ) 1 C2 = Γ0 (Γ0 + 2α2 )[(Γa20 )2 + (Δa2 )2 ] Γ 0 + Γ1 C3 = Γ0 Γ1 (Γa10 + iΔa1 ) 1 C4 = Γ0 (Γa20 + iΔa2 ) C1 =

Here, Γa10 = Γ10 +ξ1 Γ10 , Δa1 = Δ1 +ξ1 Δ3 , Γa20 = Γ20 +ξ2 Γ10 , Δa2 = Δ2 +ξ2 Δ3 . This result is consistent with Eq. (2.28). We then consider the case when the fields in beams 1 and 2 are broadband, so that α1 , α2 >> Γ10 , Γ20 >> Γ0 , Γ1 . (i) τ > 0 In this case, the beat signal rises to its maximum quickly and then decays with a time constant mainly determined by the transverse relaxation rates of the system. Again, the complicated expression of the beat signal can be −1 simplified, at the tail of the signal (i.e., τ >> α−1 1 ,τ >> α2 ), to be the form I (τ ) ∝ |P (3) |2  = D1 + |η|2 D2 + |D3 |2 exp(−2Γa10 |τ |) + 2

|ηD4 | exp(−2Γa20 |τ |) + exp[−(Γa20 + Γa10 ) |τ |]ηD3 D4 × {exp[−i(Ω2 − Ω1 )τ − i(ξ2 − ξ1 )Δ3 τ ] × exp[i(Ω2 − Ω1 )τ + i(ξ2 − ξ1 )Δ3 τ ]}

(2.36)

38

2 Femtosecond Polarization Beats

where α21 (Γ20 + 6Γ0 Γ1 + Γ21 ) 4Γ0 Γ1 Γa10 (Γ0 + Γ1 )[α21 + (Δa1 )2 ]2 α22 D2 = 4Γ0 Γa20 [α22 + (Δa2 )2 ]2 α1 (Γ0 + Γ1 ) D3 = Γ0 Γ1 [α21 + (Δa1 )2 ] α2 D4 = Γ0 [α22 + (Δa2 )2 ]

D1 =

Again, under the large Doppler-broadening limit, Equation (2.36) indicates that, although P (II) = P (IV ) = P (VI) = 0 for the total polarization, there are three parts contribute to the total signal. The first and second terms are independent of the relative time-delay τ , and they come from the amplitude fluctuation of the chaotic field of fourth-order coherence functions. The third and fourth terms which depend on fourth-order coherence functions due to the phase fluctuation of the chaotic field have exponential decay of the beat signal as |τ | increases. Last, the fifth and sixth terms is the third part which comes from the second-order coherence function determined by the laser line shape. Based on this part, when Δ3 = 0 one can modulate the beat signal with the frequency Ω2 − Ω1 in femtosecond time scale by controlling the value of the time-delay τ . (ii) τ < 0 Under this condition, the simplified beat signal is given by 2

2

I(τ ) ∝ |P (3) |2  = 4D1 + 4D2 |η| + |E1 | exp(−2α1 |τ |) + 2

|ηE2 | exp(−2α2 |τ |) + exp[−(α1 + α2 ) |τ |] × {ηE1∗ E2 exp [−i (ω2 − ω1 ) τ ] + η ∗ E1 E2∗ exp [i (ω2 − ω1 ) τ ]}

(2.37)

where, E1 = (Γ0 + Γ1 )/[Γ0 Γ1 (α1 + iΔa1 )], E2 = 1/[Γ0 (α2 + iΔa2 )]. This result is also consistent with Eq. (2.28). Therefore, the requirement for the existence of a τ -dependent beat signal for τ < 0 is for the phasecorrelated subpulses in beams 1 and 2 to overlap temporally. Since beam 1 and beam 2 are mutually coherent, the temporal behavior of the beat signal should coincide with the case when beams 1 and 2 are both nearly monochromatic.

2.2.1 Photon-echo A medium has been made to emit spontaneously a short, intense burst of radiation, which we will call a photon echo, after being excited by two short, intense light pulses. It will be interesting to understand the underlying

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields

39

physics in PBVTS with incoherent lights. Recently, lots of attentions have been paid to the studies of various ultrafast phenomena by using incoherent light sources [9]. For the phase-matching condition k1 − k1 + k3 , the threepulse stimulated photon-echoes exist for the perturbation chains (I) and (III), For the phase-matching condition k2 − k2 + k3 , the three-pulse stimulated photon-echo exists for the perturbation chain (V) [7]. Under the extreme Doppler-broadening limit (i.e., k3 u → ∞) and assuming beam 1 (beam 2) to have Gaussian line shape, then one can have [11]  ! "2  αi ∗ ui (t1 )ui (t2 ) = exp − √ (t1 − t2 ) 2 ln 2  = exp −[βi (t1 − t2 )]2 i = 1, 2 Let us consider the case when beams 1 and 2 are broadband fields, so that α1 , α2 >> Γ10 , Γ20 >> Γ0 , Γ1 . Then  √ i = 1, 2 (2.38) ui (t1 )u∗i (t2 ) = exp[−βi2 (t1 − t2 )2 ] ≈ πδ (t1 − t2 ) βi When Eqs (2.20) and (2.38) are substituted into Eq. (2.34), the following results can be obtained: (i) τ > 0 2

Γ20 + Γ21 + 6Γ0 Γ1 |η| + 2 a + 4(Γ0 + Γ1 )Γ0 Γ1 Γa10 α21 4α2 Γ20 Γ0   2 2 1 Γ 0 + Γ1 a exp (−2Γ10 |τ |) + exp (−2Γa20 |τ |) + α1 Γ0 Γ1 α2 Γ0

I (τ ) ∝ |P (3) |2  =

Γ 0 + Γ1 exp [− (Γa20 + Γa10 ) |τ |] × α2 α1 Γ20 Γ1 {η exp [−i (Ω2 − Ω1 ) τ − i (ξ2 − ξ1 ) Δ3 τ ] + η ∗ exp [i (Ω2 − Ω1 ) τ + i (ξ2 − ξ1 ) Δ3 τ ]}

(2.39)

which is consistent with Eq. (2.36). (ii) τ < 0 I (τ ) ∝ |P (3) |2  =

2

Γ20 + Γ21 + 6Γ0 Γ1 |η| + 2 a 4(Γ0 + Γ1 )Γ0 Γ1 Γa10 α21 4α2 Γ20 Γ0

Photon-echo does not exist for the perturbation chains (I), (III), and (V). The requirement for the existence of a τ -dependent beat signal for τ < 0 is for the phase-correlated sub-pulses in beams 1 and 2 to overlap temporally. Therefore, this case is consistent with Eq. (2.28) as discussed earlier. We have assumed that the laser sources are chaotic fields in the above calculations. A chaotic field, which is used to describe a multimode laser source, is characterized by the fluctuations of both the amplitude and the

40

2 Femtosecond Polarization Beats

phase of the field. Another commonly used stochastic model is the phasediffusion model, which has been used to describe an amplitude-stabilized laser source. This model assumes that the amplitude of the laser field is a constant, while its phase fluctuates as a completely random process. If the lasers have Lorentzian line shape, the fourth-order coherence function can be written as [8] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = exp[−αi (|t1 − t3 | + |t1 − t4 | + |t2 − t3 | + |t2 − t4 |)] × i = 1, 2 exp[αi (|t1 − t2 | + |t3 − t4 |)]

(2.40)

We now consider the case with fields in beams 1 and 2 to be broadband, so that α1 , α2 >> Γ10 , Γ20 >> Γ0 , Γ1 . Then, the second-order coherence function is given by ui (t1 ) u∗i (t2 ) = exp (−αi |t1 − t2 |) ≈

2δ(t1 − t2 ) αi

i = 1, 2

(2.41)

When Eqs. (2.40) and (2.41) are substituted into Eq. (2.34), one can have the following results: (i) τ > 0  I(τ ) ∝ |P (3) |2  = 

1 α2 Γ0

2

Γ 0 + Γ1 α1 Γ0 Γ1

2 exp (−2Γa10 |τ |) +

exp (−2Γa20 |τ |) +

Γ 0 + Γ1 exp [− (Γa20 + Γa10 ) |τ |] × α2 α1 Γ20 Γ1

 η exp[−i (Ω2 − Ω1 ) τ − i (ξ2 − ξ1 ) Δ3 τ ] + η ∗ exp [i (Ω2 − Ω1 ) τ + i (ξ2 − ξ1 ) Δ3 τ ]

(2.42)

(ii) τ < 0, I(τ ) ∝ |P (3) |2  = 0 So, Photon-echoes do not exist for the perturbation chains (I), (III), and (V) for τ < 0. Equation (2.42) consists of two parts. The first part (first and second terms), depending on fourth-order coherence functions due to the phase fluctuation of the phase-diffusion field, has an exponential decay of the beat signal as |τ | increases. The second part (third and fourth terms), depending on second-order coherence functions determined by the laser line shape, gives rise to the modulation of the beat signal. This case is consistent with the results of the second-order coherence function theory [3,17], where the constant background contribution has been ignored in the analysis. Therefore, the fourth-order coherence function theory of chaotic field is of vital importance in PBVTS.

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields

41

2.2.2 Experiment and Result The experiment of PBVTS has been performed in a sodium vapor, in which the ground state 3S1/2 and two excited states 3P1/2 and 3P3/2 form a V-type three-level system. Two dye lasers (DL1 and DL2), pumped by the second harmonic of a Quanta-Ray YAG laser, were used to generate frequencies at ω1 and ω2 . DL1 and DL2 have linewidths of about 0.01 nm and pulse widths of 5 ns. DL1 was tuned to the wavelength of 589.6 nm for the 3S1/2 − 3P1/2 transition. DL2 was tuned to the wavelength of 589 nm for the 3S1/2 − 3P3/2 transition. A beam splitter was used to combine the ω1 and ω2 components (derived from DL1 and DL2, respectively) for beam 1 and beam 2, which intersect in the oven containing the Na vapor. The relative time delay τ between beam 1 and beam 2 can be varied. Beam 3, which is derived from DL1, propagates along the direction opposite to that of beam 1. All the incident beams are linearly polarized in the same direction. The beat signal has the same polarization as the incident beams and propagates along a direction almost opposite to that of beam 2. The generated beat signal was detected by a photodiode. First, a DFWM experiment was performed with beam 1 and beam 2 both consisting only the ω1 frequency component. From the DFWM spectrum we tune ω1 to the resonant frequency Ω1 , whose center wavelength is 589.6 nm. Next, we performed a NDFWM experiment in which beam 1 and beam 2 both consist of only the ω2 frequency component, and we measured the NDFWM spectrum by scanning ω2 (see Fig. 2.9). From the NDFWM spectrum we tune ω2 to the resonant frequency Ω2 , whose center wavelength is 589 nm. Then, we performed the PBVTS experiment by measuring the beat signal intensity as a function of the relative time delay when beam 1 and beam 2 consist of both frequencies ω1 and ω2 at the same time. Figures 2.10 (a) and (b) present the results of the polarization beat experiment, in which τ has been varied for ranges of 400 ps and 30 ps, respectively. The solid curve in Fig. 2.10 (b) is the theoretical calculation given by Eq. (2.38) with α1 = α2 = 2.7×1010s−1 , ω2 −ω1 = 3.26×1012s−1 , A1 = 0.2, η = A3 = A4 = 1 and A2 = 0.5. At zero relative time delay, thees fields of twin beams 1 and 2 perfectly overlap at the atomic medium, resulting in maximal interferometric contrast. As |τ | is increased, the interferometric contrast diminishes on the time scale that reflects material memory, which is usually much longer than the correlation time of the light [6]. The beat signal intensity modulates in a sinusoidal form with period of 1.93 ps. The modulation frequency can be obtained more directly by making a Fourier transform of the PBVTS data. Figure 2.11 shows the Fourier spectrum of the data in which τ is varied for a range of 400 ps with α1 = α2 = 2.7 × 1010 s−1 , A1 = 0.2, A2 = 0.5, η = A3 = A4 = 1 and ω2 − ω1 = 3.26 × 1012 s−1 . Then, the modulation frequency of 3.262 × 1012 s−1 can be obtained, corresponding to the beating frequency between the resonant frequencies of the transitions from 3S1/2 to 3P1/2 and from 3S1/2 to 3P3/2 .

42

2 Femtosecond Polarization Beats

Fig. 2.9. Spectrum of FWM when beam 1 and beam 2 consist of only ω1 or ω2 in which the center wavelengths are 589 nm and 589.6 nm, respectively. Adopted from Ref. [20].

Fig. 2.10. (a) Experimentally measured beat signal intensity versus time delay τ for a range of 400 ps. (b) The beat signal intensity versus time delay τ for a range of 30 ps. The filled squares are the experimental data, and the solid curve is the theoretical calculation. Adopted from Ref.[20].

Now, we discuss the major difference between the PBVTS and the UMS [1] with the self-diffraction geometry from a physical viewpoint. The frequencies and wave vectors of the UMS signal are ωs1 = 2ω1 − ω1 , ωs2 = 2ω2 − ω2 , and ks1 = 2k1 − k1 , ks2 = 2k2 − k2 , respectively, which indicate that one photon is absorbed from each of two mutually correlated pump beams. On the other hand, the frequencies and wave vectors of the PBVTS signal are ωs1 = ω1 − ω1 + ω1 , ωs2 = ω2 − ω2 + ω1 , and ks1 = k1 − k1 + k3 , ks2 = k2 − k2 + k3 , respectively, therefore photons are absorbed from and emitted to the mutually correlated beam 1 and beam 2, respectively. This difference between the PBVTS and the UMS has a profound influence on the field-correlation effect. Note that roles of beam 1 and beam 2 are interchangeable in the UMS, this interchangeable feature also makes the second-order coherence function theory failure in the UMS. Because of u(t1 )u(t2 ) = 0, the absolute square of the stochastic average of the polarization v can not be used to describe the temporal behavior of the UMS. Therefore, the fourthorder coherence function theory is of vital importance in the UMS. We have presented the coherence function theory and experimental results for the atomic response in polarization beats with the phase-conjugation ge-

2.2 Correlation Effects of Chaotic and Phase-diffusion Fields

43

Fig. 2.11. The filled squares are the experimental data of the Fourier spectrum in which τ is varied for a range of 400 ps. The solid curve is the theoretical calculation. Adopted from Ref. [20].

ometry using chaotic fields. The τ -dependent PBVTS signal is accompanied by a constant background. As laser linewidth increases the τ -independent background of the PBVTS signal increases also, which make the study of the temporal behavior of the PBVTS difficult. Let IS be the maximum intensity of the τ -dependent signal and IB the intensity of the constant background. We define η  = IB IS as the ratio between IB and IS . In the limit of η = A3 = A4 = 1, α2 << Γ20 and Γ10 , Γ20 >> Γ0 , Γ1 , IB =

Γ0 − 2α2 (Γ0 + Γ1 )2 − 16α21 − + Γ0 (Γ20 − 4α22 )Γ220 (Γ20 − 4α21 )Γ210 (Γ21 − 4α21 )

2α1 (2Γ0 + Γ1 ) 2α1 (Γ0 + 2Γ1 ) − Γ0 (Γ20 − 4α21 )Γ210 (Γ0 + Γ1 ) Γ1 (Γ21 − 4α21 )Γ210 (Γ0 + Γ1 ) ! "2 Γ20 (Γ0 + Γ1 ) + Γ1 Γ10 IS = Γ0 Γ1 Γ20 for Δ1 = Δ2 = Δ3 = 0 and η = 1. In the limit of α1 , α2 >> Γ10 , Γ20 >> Γ0 , Γ1 , 1 Γ20 + 6Γ0 Γ1 + Γ21 + 2Γ0 Γ20 α22 2Γ0 Γ1 (Γ0 + Γ1 )Γ10 α21 ! "2 α2 (Γ0 + Γ1 ) + Γ1 α1 IS = Γ0 Γ1 α1 α2

IB =

for Δ1 = Δ2 = Δ3 = 0 and η = 1. In order to observe the τ dependence of the PBVTS signal, the signal-tonoise ratio of the experiment should be larger than η  . In the above experiment using chaotic fields with α1 << Γ10 ,α2 << Γ20 , we have η  ≈ 2.4, corresponding to α1 = α2 = 2.7 × 1010 s−1 , while the signal-to-noise ratio is about 5. Thus, we have adopted chaotic and phase-diffusion models to study the effects of laser’s fourth-order coherence properties on the PBVTS. The dif-

44

2 Femtosecond Polarization Beats

ferent roles of the phase fluctuation and amplitude fluctuation have been pointed out in the time domain. fourth-order coherence functions of chaotic fields are of vital importance in the PBVTS. Finally, differences between the PBVTS and the DeBeer’s UMS have also been discussed from the underlying physics.point of view.

2.3 Higher-order Correlations of Markovian Stochastic Fields on Polarization Beats The statistical property of the noisy light field is of particular importance for nonlinear optical processes since these are often sensitive to the higher-order correlation in the field. Thees effects of such higher-order correlations have been studied in several nonlinear optical processes characterized by either Markovian or non-Markovian fluctuations [13, 14, 21 – 23]. The Markovian field is described statistically in terms of the marginal and the conditional probability densities [24, 25]. The atomic response to non-Markovian fields is much less well understood [13]. This is primarily due to the fact that the complete hierarchy of conditional probabilities must be known in order to fully describe a non-Markovian process. Some non-Markovian processes can be made to be Markovian by extension to higher-order correlations. The atomic response to Markovian stochastic optical fields is now basically well understood [14, 21 – 23]. When the laser field is sufficiently intense that multi-photon interactions occur, the laser spectral bandwidth or spectral shape, obtained from the second-order correlation function, is inadequate to characterize the field. Rather than to use higher-order correlation functions explicitly, three different Markovian fields are considered here: (a) the chaotic field; (b) the phase-diffusion field; and (c) the Gaussian-amplitude field. The chaotic field undergoes both amplitude and phase fluctuations and corresponds to a multimode laser field with a large number of uncorrelated modes, or light emitted from a single-mode laser operated below threshold. Since a chaotic field does not possess any intensity stabilization mechanism, the field can take on any values in a two-dimensional region of the complex plane centered around the origin [23]. The phase-diffusion field undergoes only phase fluctuations and corresponds to an intensity-stabilized single-mode laser field. The phase of the laser field, however, has no natural stabilization mechanism [23]. The Gaussian-amplitude field has only amplitude fluctuations. Although pure amplitude fluctuations can not be produced by a nonadiabatic process, we consider this Gaussian-amplitude field for two reasons. First, because it allows us to isolate those effects due solely to the amplitude fluctuations. Second, because it is an example of a field which undergoes stronger amplitude (intensity) fluctuations than a chaotic field. By comparing the results for the chaotic and the Gaussian-amplitude fields we can determine the effect of increasing amplitude fluctuations [24, 25].

2.3 Higher-order Correlations of Markovian Stochastic Fields on Polarization · · ·

45

Here, we discuss the role of noises in incident fields on the wave-mixing signal – particularly in the time domain. This important topic has been already treated extensively in the literature including the introduction of a new diagrammatic technique (called factorized time correlator diagrams) [26 – 32]. That technique has been used to treat higher-order noise correlators when circular Gaussian statistics can apply. There should be two classes of such two-component beams. In one case the field components are derived from separate lasers, so their mixed (cross) correlators should vanish. In the second case two field components are derived from a single laser source with mixing dyes whose spectral output is doubly peaked. We will only consider the first case here with two-color beams each derived from a separate laser source. The uses of such multi-color noisy lights in FWM have already been investigated both theoretically and experimentally [27, 28], in which multicolored beams (a single laser source for the multi-peaked “tailored” light) are from the same laser source in the self-diffraction geometry. In those previous works, the case of cascade three-level system with phase-conjugation geometry using three types of noisy models has not been treated, and the beam 3 was not noisy in those works [27, 28]. Using a cascade three-level atomic system, we consider effects of higherorder correlations on polarization beat signals using the chaotic field, the Brownian-motion phase-diffusion field, and the Gaussian-amplitude field, respectively. We develop a unified theory, which involves sixth-order coherence function, to study the influences of partial-coherence properties of pump beams on polarization beats. Our treatment of the effects of higher-order correlation on polarization beats (HOCPB) includes both the finite light bandwidth influence and the constant background contribution. HOCPB is a polarization phenomenon [33, 34] originated from the interference between one-photon and two-photon processes. Let’s consider the cascade three-level system (see Fig. 2.12) with a ground state |0, an intermediate state |1, and an excited state |2. Transitions between |0 and |1, and between |1 and |2 are dipole allowed with resonant frequencies Ω1 and Ω2 , respectively, while transition between |0 and |2 is dipolar forbidden. We consider, in this cascade three-level system, a double-frequency time-delayed FWM experiment in which beams A and B consist of two frequency components ω1 and ω2 , while beam 3 has frequency ω3 (see Fig. 2.13). We assume that ω1 ≈ Ω1 (ω3 ≈ Ω1 ) and ω2 ≈ Ω2 , therefore ω1 (ω3 ) and ω2 will drive transitions from |0 to |1 and from |1 to |2, respectively. We further assume that beam 3 is split from the laser source for the frequency ω1 component in beam A and beam B. There are two processes involved in such doublefrequency time-delayed FWM. First, the ω1 frequency component of beams A and B induce a population grating between states |0 and |1, which is probed by beam 3 of frequency ω3 . This is an one-photon resonant DFWM and the generated signal (beam 4) has frequency ω3 . Second, beam 3 and the ω2 frequency component of beam A induce a two-photon coherence between states |0 and |2, which is then probed by the ω2 frequency component of

46

2 Femtosecond Polarization Beats

beam B. This is a two-photon NDFWM with a resonant intermediate state and the frequency of the generated signal is equal to ω3 again.

Fig. 2.12. Cascade three-level configuration used for HOCPB.

The complex electric fields of beam A (Ep1 ) and beam B (Ep2 ) can be written as Ep1 = A1 (r, t) exp(−iω1 t) + A2 (r, t) exp(−iω2 t) = ε1 u1 (t) exp[i(k1 · r − ω1 t)] + ε2 u2 (t) exp[i(k2 · r − ω2 t)] Ep2 = A1 (r, t) exp(−iω1 t) + A2 (r, t) exp(−iω2 t) = ε1 u1 (t − τ ) exp[i(k1 · r − ω1 t + ω1 τ )] + ε2 u2 (t − τ ) exp[i(k2 · r − ω2 t + ω2 τ )]

where εi ,ki (εi ,ki ) are the constant field amplitude and the wave vector of the ωi component in beam A (beam B), respectively. ui (t) is a dimensionless statistical factor that contains phase and amplitude fluctuations. τ is the time delay of beam B with respect to beam A. The complex electric field of beam 3 can be written as Ep3 = A3 (r, t) exp(−iω3 t) = ε3 u3 (t) exp[i(k3 · r − ω3 t)], with ω3 , ε3 , and k3 being the frequency, the field amplitude, and the wave vector of beam 3, respectively. Since ω1 and ω3 come from the same laser source, we have u1 (t) = u3 (t).

Fig. 2.13. Schematic diagram of the laser beams for HOCPB.

We employ perturbation theory to calculate the density-matrix elements. For above described one-photon and two-photon FWM processes, we can write following perturbation chains: (0) A

 ∗ (1) (A )

(2) A

(3)

1 3 ρ10 −−−1− → ρ00 −−→ ρ10 (I) ρ00 −−→  ∗ (0) (A )

A

(2) A

1 3 (II) ρ00 −−−1− → (ρ10 )∗ −−→ ρ00 −−→ ρ10

(0) A

(1)

 ∗ (1) (A )

(2) A

(3)

(3)

1 3 (III) ρ00 −−→ ρ10 −−−1− → ρ11 −−→ ρ10  ∗ (0) (A )

A

(2) A

1 3 (IV) ρ00 −−−1− → (ρ10 )∗ −−→ ρ11 −−→ ρ10

(1)

(3)

2.3 Higher-order Correlations of Markovian Stochastic Fields on Polarization · · · (0) A

(1) A

 ∗ (2) (A )

47

(3)

3 2 (V) ρ00 −−→ ρ10 −−→ ρ20 −−−2− → ρ10

Chains (I) – (IV) correspond to the one-photon resonant DFWM, while chain (V) corresponds to the two-photon resonant NDFWM. Now, let us consider other possible perturbation chains, where gratings induced by beam 3 and ω1 or ω2 frequency components of beam B are responsible for the generation of the FWM signals. These gratings have much smaller fringe spaces, which equal approximately to one half of the wavelengths of incident lights. For a Doppler-broadened system, gratings will be washed out by the atomic motion. Therefore, it is appropriate to neglect the FWM signals from these perturbation chains. In addition, some perturbation chains involve the coherence between the excited states |1 and |2. For a system with the relaxation time of ρ00 much longer than that of ρ21 (or ρ12 ) such FWM signal can be reduced further. We have also neglected the contributions from the perturbation chains which give rise to signals with frequencies ω4 = ±(ω2 − ω1 ) + ω3 , since it can be separated from the FWM signal with frequency ω3 by a monochromator or a narrow-band filter. Furthermore, the more strict requirement on the phase-matching condition and the involvement of ρ21 (or ρ12 ) also make such process unimportant. In addition, for the static grating (the FWM signal has the same frequency as the probe beam) the coherence length is usually longer than the thickness of a typical sample. For the moving grating (the signal from moving grating has frequencies of ω3 ± (ω2 − ω1 )) the coherence length is usually much smaller than the thickness of a typical sample. In this case the contributions to the FWM signals from these moving gratings can be neglected. From the appropriate perturbation chain, one can obtain the third-order (3) off-diagonal density-matrix element ρ10 which has the wave vector k1 −k1 +k3 or k2 − k2 + k3 . The nonlinear polarization P (3) responsible for the phaseconjugate FWM signal is given by averaging over the velocity distribution function W (v). Thus,  +∞ (3) dvw(v)ρ10 (v) P (3) = N μ1 −∞

with v being the atomic velocity and N the atomic density. For a Dopplerbroadened atomic system, we have √ w (v) = exp[−(v/u)2 ]/( πu) The total polarization is given by P (3) = P (I) + P (II) + P (III ) + P (IV ) + P (V ) Here, P (I) , P (II) , P (III) , P (IV ) , and P (V ) , corresponding to the polarizations of the perturbation chains (I), (II), (III), (IV), and (V), respectively, are  +∞  ∞  ∞  ∞ P (I) = S1 (r) exp[−i(ω3 t + ω1 τ )] dvw(v) dt3 dt2 dt1 × −∞

0

0

0

48

2 Femtosecond Polarization Beats

exp[−iθI (v)]H1 (t1 ) H2 (t2 ) H3 (t3 ) u1 (t − t1 − t2 − t3 ) ×

P (II)

u∗1 (t − t2 − t3 − τ )u3 (t − t3 )  +∞  = S1 (r) exp[−i(ω3 t + ω1 τ )] dvw(v) −∞

P (III )

∞



∞

dt3

0

0

0

0

dt1 ×

0

(2.44) ∞

dt1 ×

0

exp[−iθI (v)]H1 (t1 ) H4 (t2 ) H3 (t3 ) u1 (t − t1 − t2 − t3 ) × (2.45) u∗1 (t − t2 − t3 − τ )u3 (t − t3 )  +∞  ∞  ∞  ∞ = S1 (r) exp[−iω3 t − iω1 τ ] dvw(v) dt3 dt2 dt1 × −∞

0

0

0

exp[−iθII (v)]H1∗ (t1 ) H4 (t2 ) H3 (t3 ) u1 (t − t2 − t3 ) ×

P (V )

∞

dt2

exp[−iθII (v)]H1∗ (t1 ) H2 (t2 ) H3 (t3 ) u1 (t − t2 − t3 ) × u∗1 (t − t1 − t2 − t3 − τ )u3 (t − t3 )  +∞  ∞  ∞  = S1 (r) exp[−iω3 t − iω1 τ ] dvw(v) dt3 dt2 −∞

P (IV )

(2.43) 

u∗1 (t − t1 − t2 − t3 − τ )u3 (t − t3 )  +∞  = S2 (r) exp[−iω3 t − iω2 τ ] dvw(v) −∞

0

(2.46) 

∞



∞

dt3

∞

dt2 0

dt1 ×

0

exp[−iθIII (v)]H3 (t1 ) H5 (t2 ) H3 (t3 ) u2 (t − t2 − t3 ) × u∗2 (t − t3 − τ )u3 (t − t1 − t2 − t3 )

(2.47)

where S1 (r) = −iN μ41 ε1 (ε1 )∗ ε3 exp[i(k1 − k3 + k3 ) · r]/4 S2 (r) = −iN μ21 μ22 ε2 (ε2 )∗ ε3 exp[i(k2 − k2 + k3 ) · r]/4 θI (v) = v · [k1 (t1 + t2 + t3 ) − k1 (t2 + t3 ) + k3 t3 ] θII (v) = v · [−k1 (t1 + t2 + t3 ) + k1 (t2 + t3 ) + k3 t3 ] θIII (v) = v · [k3 (t1 + t2 + t3 ) + k2 (t1 + t2 ) − k2 t3 ] H1 (t) = exp [− (Γ10 + iΔ1 ) t] , H2 (t) = exp (−Γ0 t) H3 (t) = exp[− (Γ10 + iΔ3 ) t], H4 (t) = exp (−Γ1 t) H5 (t) = exp[− (Γ20 + iΔ2 + iΔ3 ) t] μ1 (μ2 ) is the dipole-moment matrix element between states |0 and |1 (|1 and |2); Γ0 (Γ1 ) is the population relaxation rate of state |0 (|1); Γ10 (Γ20 ) is the transverse relaxation rate of the transition from |0 to |1 (|0 to |2). The frequency detunings are Δ1 = Ω1 − ω1 , Δ2 = Ω2 − ω2 , Δ3 = Ω1 − ω3 . The total FWM signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic process, given as |P (3) |2 , which involves second-, fourth- and sixth-order coherence functions

2.3 Higher-order Correlations of Markovian Stochastic Fields on Polarization · · ·

49

of ui (t) in the phase–conjugation geometry. However, the FWM signal intensity in the self-diffraction geometry is related only to sixth-order coherence functions of the incident fields. For the macroscopic system where phase matching takes place, this signal must be drawn from the P (3) , developed on one “atom”, multiplied by the (P (3) )∗ that is developed on another “atom” located elsewhere in space, with summation over all such pairs [26, 30]. We first assume that the laser sources are multimode thermal-like sources. Then, ui (t) has Gaussian statistics with its sixth- and fourth-order coherence functions satisfying [24, 25] ui (t1 )ui (t2 )ui (t3 )u∗i (t4 )u∗i (t5 )u∗i (t6 ) = ui (t1 )u∗i (t4 )ui (t2 )ui (t3 )u∗i (t5 )u∗i (t6 ) + ui (t1 )u∗i (t5 )ui (t2 )ui (t3 )u∗i (t4 )u∗i (t6 ) + ui (t1 )u∗i (t6 )ui (t2 )ui (t3 )u∗i (t4 )u∗i (t5 )

(2.48)

ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 ) + ui (t1 )u∗i (t4 )ui (t2 )u∗i (t3 )

(2.49)

By further assuming the laser sources in beams A, B and 3 to have Lorentzian line shape, the second-order coherence functions can be written in the simple form [2] ui (t1 ) u∗i (t2 ) = exp (−αi |t1 − t2 |)

(2.50)

where αi = δωi /2 with δωi being the linewidth of the laser with frequency ωi . The form of the second-order coherence function, which is determined by the laser line shape, as expressed in Eq. (2.50), is the general feature of the three different stochastic models [24, 25]. We now consider the case with laser sources being narrow band, so that α1 , α2 << Γ10 , Γ20 and Γ0 , Γ1 << Γ10 , Γ20 . For simplicity, we first neglect the Doppler effect. After performing the tedious integration and some algebra, the total beat signal intensity becomes I (τ, r) ∝ |P (3) |2  ∝ B1 + |η|2 B2 + B3 exp(−2α1 |τ |) + |η|2 B2 exp(−2α2 |τ |) + exp[−(α1 + α2 )|τ |] × B2 B3 {η exp[iΔk · r + i(ω2 − ω1 )τ ] + (2.51) η ∗ exp[−iΔk · r − i(ω2 − ω1 )τ ]}  where Δk = (k1 − k1 ) − (k2 − k2 ) and η = μ22 ε∗2 ε2 (μ21 ε∗1 ε1 ). B1 , B2 , and B3 mainly depend on laser linewidths and relaxation rates of the transition and are constants. Equation (2.51) consists of five terms. The first and third terms depend on the sixth-order coherence function of u1 (t) for DFWM, while the second and fourth terms depend on the fourth-order coherence function on u2 (t) and second-order coherence functions on u1 (t) for NDFWM. The first and

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second terms, originating from amplitude fluctuations of the chaotic fields, are independent of the relative time-delay between two beams A and B. The third and fourth terms include an exponential decay of the beat signal as |τ | increases. The fifth term, depending on fourth-order coherence function of u1 (t) and second-order coherence functions of u2 (t), generates the modulation of the beat signal. Equation (2.51) indicates that the beat   signal oscillates not only temporally, but also spatially with a period 2π Δk, here Δk ≈ 2π|λ1 − λ2 |θ λ2 λ1 . θ is the angle between beam A and beam B. The beat signal modulates temporally with a frequency ω2 − ω1 as τ is varied. When ω1 and ω2 are tuned to the resonant frequencies of the transitions from |0 to |1 and from |1 to |2, respectively, the modulation frequency equals to Ω2 − Ω1 . Then, we can obtain the beating between resonant frequencies of a cascade three-level system.

2.3.1 HOCPB in a Doppler-broadened System The beat signal can also be calculated from a different way. Under the large Doppler-broadening limit (i.e., k3 u → ∞), the integrals for the velocity distribution can be evaluated as √  +∞ 2 π δ (t3 − ξ1 t1 ) dvw (v) exp [−iθI (v)] ≈ (2.52) k3 u −∞ √  +∞ 2 π δ (t3 + ξ1 t1 ) dvw (v) exp [−iθII (v)] ≈ (2.53) k3 u −∞ √  +∞ 2 π dvw (v) exp [−iθIII (v)] ≈ δ (t1 + t2 + t3 − ξ2 t2 ) (2.54) k3 u −∞  Here, we have assumed ξ2 > 1, ξ1 = k1 k3 , ξ2 = k2 /k3 . When Eqs. (2.52)– (2.54) are substituted into Eqs. (2.43)–(2.47), we calculate the total polarization beat signal in term of (2.55) I (τ, r) ∝ |P (3) |2  = |P (I) + P (III ) + P (V ) |2  We first consider the case for the laser sources to be narrow band, so that α1 , α2 << Γ10 , Γ20 and Γ0 , Γ1 << Γ10 , Γ20 . After the tedious calculation, the beat signal intensity is given by 2

I(τ, r) ∝ |P (3) |2  = B4 + B5 |η| + B6 exp(−2α1 |τ |) + 2

|η| B7 exp(−2α2 |τ |) + exp[−(α1 + α2 ) |τ |] × B6 B7 {η exp[iΔk · r + i (ω2 − ω1 ) τ ] + η ∗ exp[−iΔk · r − i (ω2 − ω1 ) τ ]}

(2.56)

where B4 , B5 , B6 and B7 are constant mainly depending on the laser linewidths and relaxation rates of the transitions. This result is consistent with Eq. (2.51).

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51

When the laser sources are broadband (α1 , α2 >> Γ10 , Γ20 >> Γ0 , Γ1 ), the beat signal will rise to its maximum quickly and then decays with a time constant mainly determined by the transverse relaxation rate of the system. −1 In general, at the tail of the signal (i.e. τ >> α−1 1 , τ >> α2 ), we have (i) τ > 0 2

I (τ, r) ∝ |P (3) |2  = B8 + |η| B9 + B10 exp (−2Γa10 |τ |) + |η|2 B11 exp [−2 (Γa20 − Γ10 ) |τ |] + B10 B11 exp [− (Γa20 + Γa10 − Γ10 ) |τ |] ×  η exp[iΔk · r + i(Ω2 − Ω1 )τ − i(ξ2 − ξ1 )Δ3 τ ] + η ∗ exp [−iΔk · r − i(Ω2 − Ω1 )τ + i(ξ2 − ξ1 )Δ3 τ ]

(2.57)

where B8 , B9 , B10 and B11 are constants, which mainly depend on the laser linewidths and relaxation rates of the transitions. Also, Γa10 = Γ10 + ξ1 Γ10 , Γa20 = Γ20 + ξ2 Γ10 . Equation (2.57) also consists of five terms. The first and third terms depend on the sixth-order coherence function of u1 (t) for DFWM, while the second and fourth terms are dependent on the fourth-order in u2 (t) and second-order in u1 (t) for NDFWM. The first and second terms, originating from the amplitude fluctuation of the chaotic field, are independent of the relative time-delay between the two beams A and B. The third and fourth terms show an exponential decay of the beat signal as |τ | increases. The fifth term, depending on the fourth-order in u1 (t) and second-order in u2 (t), gives rise to the modulation of the beat signal. The modulation frequency of the beat signal equals to Ω2 − Ω1 when Δ3 = 0. The overall accuracy of determining the energy-level difference of the excited states by using HOCPB with broadband lights is limited by the homogeneous linewidths. (ii) τ < 0 I(τ, r) ∝ |P (3) |2  = B8 + B9 |η|2 + B12 exp(−2α1 |τ |) + |η|2 B13 exp(−2α2 |τ |) + exp[−(α1 + α2 )|τ |] × B12 B13 {η exp[iΔk · r + i(ω2 − ω1 )τ ] + η ∗ exp[−iΔk · r − i(ω2 − ω1 )τ ]}

(2.58)

where B12 and B13 are constants mainly depending on the laser linewidths and relaxation rates of the transitions. This result is consistent with Eq. (2.51). To see a τ -dependent beat signal for τ < 0, the phase-correlated pulses in beam A and beam B need to overlap temporally.

2.3.2 Photon-echo It is interesting to understand underlying physics in HOCPB with broadband nontransform limited quasi-cw (noisy) lights. For the phase-matching

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condition k1 − k1 + k3 the three-pulse stimulated photon-echo exists for the perturbation chains (I) and (III). For the phase-matching condition k2 −k2 +k3 the sum-frequency tri-level photon-echo exists for the perturbation chain (V). The chaotic field has a complex Gaussian stochastic property. Under the extreme Doppler-broadening limit (i.e., k3 u → ∞) and by assuming the laser source to have Gaussian line shape, then one can write [11]  ! "2   αi ∗ ui (t1 )ui (t2 ) = exp − √ = exp − [βi (t1 − t2 )]2 (t1 − t2 ) 2 ln 2 We now consider the case when laser sources are broadband, so that α1 , α2 >> Γ10 , Γ20 >> Γ0 , Γ1 , then √ $ # 2 π ∗ 2 ui (t1 )ui (t2 ) = exp −βi (t1 − t2 ) ≈ δ (t1 − t2 ) (2.59) βi By substituting Eqs. (2.48), (2.49) and (2.59) into Eq. (2.54), we can calculate the beat signal intensity as (i) τ > 0 2

2

I (τ, r) ∝ |P (3) |2  = A1 + |η| A2 + |A3 | exp(−2Γa10 |τ |) + 2

|A4 | exp[−2(Γa20 − Γ10 ) |τ |] + A3 A4 exp[−α1 |ξ1 − ξ2 | |τ |] × exp[−(Γa20 + Γa10 − Γ10 ) |τ |]{η exp[iΔk · r + i(Ω2 − Ω1 )τ + i(ξ2 − ξ1 )Δ3 τ ] + η ∗ exp[−iΔk · r − i(Ω2 − Ω1 )τ − i(ξ2 − ξ1 )Δ3 τ ]} (2.60)  2 a 2 a where,  2 Aa1 = (Γ0 2+ Γ1 ) Γ0 Γ1 α1 Γ10 ξ1+ 2/[α1 Γ10 ξ1 (Γ0 + Γ1 )], A2 = (ξ2 − 1) [α2 (Γ20 − Γ10 ) ], A3 = 2(Γ0 + Γ1 ) (α1 Γ0 Γ1 ), and A4 = 2(ξ2 − 1)|τ | α2 . This result has a similar form as Eq. (2.57). (ii) τ < 0 2

I (τ, r) ∝ |P (3) |2  = A1 + |η| A2 + 4{exp(−2Γ0 |τ |) + exp(−2Γ1 |τ |) + 2 exp[−(Γ0 + Γ1 ) |τ |]}/[Γ10 (ξ1 + 1)2 α31 ] Photon-echo doesn’t exist for perturbation chains (I), (III), and (V) for τ < 0. The requirement for the existence of a τ -dependent beat signal for τ < 0 is the same as the earlier cases, where phase-correlated subpulses in beams A and B need to overlap temporally. Also, the temporal behavior of the beat signal should be the same as when the beams A and B are nearly monochromatic, since beam A and beam B are mutually coherent, which is consistent with Eq. (2.51). By assuming laser sources to be phase-diffusion fields and to have Lorentzian line shape, the sixth- and fourth-order coherence functions can be written as [24, 25] ui (t1 )ui (t2 )ui (t3 )u∗i (t4 )u∗i (t5 )u∗i (t6 ) = exp[−αi (|t1 − t4 | + |t1 − t5 | + |t1 − t6 | + |t2 − t4 | + |t2 − t5 | +

2.3 Higher-order Correlations of Markovian Stochastic Fields on Polarization · · ·

53

|t2 − t6 | + |t3 − t4 | + |t3 − t5 | + |t3 − t6 |)] × exp[αi (|t1 − t2 | + |t1 − t3 | + |t2 − t3 | + |t4 − t5 | + |t4 − t6 | + |t5 − t6 |)]

(2.61)

ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = exp[−αi (|t1 − t3 | + |t1 − t4 | + |t2 − t3 | + |t2 − t4 |)] × exp[αi (|t1 − t2 | + |t3 − t4 |)]

(2.62)

Under the broadband condition for laser beams (so that α1 , α2 >> Γ10 , Γ20 >> Γ0 , Γ1 ), the second-order coherence function reduce to ui (t1 ) u∗i (t2 ) = exp (−αi |t1 − t2 |) ≈ 2δ(t1 − t2 )/αi

i = 1, 2

(2.63)

Substituting Eqs. (2.61)–(2.63) into Eq. (2.54), the polarization beat intensity can be obtained, for different cases, to be: (i) τ > 0 2

2

I(τ, r) ∝ |P (3) |2  = |A5 | exp(−2Γa10 |τ |) + |ηA6 | exp[−2(Γa20 − Γ10 ) |τ |] + A5 A6 × exp[−α1 |ξ1 − ξ2 | |τ |] × exp[−(Γa20 + Γa10 − Γ10 ) |τ |] × {η exp[iΔk · r + i (Ω2 − Ω1 ) τ + i (ξ2 − ξ1 ) Δ3 τ ] + η ∗ exp[−iΔk · r − i (Ω2 − Ω1 ) τ − i (ξ2 − ξ1 ) Δ3 τ ]}

(2.64)

  where A5 = (Γ0 + Γ1 ) (α1 Γ0 Γ1 ) and A6 = τ (ξ2 − 1) α2 . (ii) τ < 0 4 {exp (−2Γ0 |τ |) + exp (−2Γ1 |τ |) + Γa10 α31 ξ1 2 exp [− (Γ0 + Γ1 ) |τ |]}

I (τ, r) ∝ |P (3) |2  =

Photon-echo doesn’t exist for perturbation chains (I), (III), and (V) for τ < 0 case. Equation (2.64) consists of three terms. The first term depends on the sixth-order coherence function of u1 (t) for DFWM, while the second term depends on the fourth-order in u2 (t) and second-order in u1 (t) in coherence functions for NDFWM. The first and second terms include exponential decays of the beat signal as |τ | increases. The third term depends on the fourth-order in u1 (t) and second-order in u2 (t) in the coherence functions, which shows the modulation of the beat signal. This case is consistent with the result of the second-order coherence function theory, where the constant background contribution has been ignored. Therefore, we can conclude that the sixthorder coherence function theory of chaotic field is of vital importance in HOCPB. The Gaussian-amplitude field has a constant phase but its real amplitude undergoes fluctuations with Gaussian statistics. If lasers have Lorentzian line

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2 Femtosecond Polarization Beats

shape, the sixth- and fourth-order coherence functions can be reduced to [24, 25] ui (t1 )ui (t2 )ui (t3 )ui (t4 )ui (t5 )ui (t6 ) = ui (t1 )ui (t4 )ui (t2 )ui (t3 )ui (t5 )ui (t6 ) + ui (t1 )ui (t5 )ui (t2 )ui (t3 )ui (t4 )ui (t6 ) + ui (t1 )ui (t6 )ui (t2 )ui (t3 )ui (t4 )ui (t5 ) + ui (t1 )ui (t2 )ui (t3 )ui (t4 )ui (t5 )ui (t6 ) + ui (t1 )ui (t3 )ui (t2 )ui (t4 )ui (t5 )ui (t6 )

(2.65)

ui (t1 )ui (t2 )ui (t3 )ui (t4 ) = ui (t1 )ui (t3 )ui (t2 )ui (t4 ) + ui (t1 )ui (t4 )ui (t2 )ui (t3 ) + ui (t1 )ui (t2 )ui (t3 )ui (t4 )

(2.66)

Again, when Eqs. (2.63), (2.64) and (2.66) are substituted into Eq. (2.54), the polarization beat signal intensity takes the following form: (i) τ > 0 2

I (τ, r) ∝ |P (3) |2  = A7 + |η| A8 + exp(−2Γa10 |τ |){A9 + A10 (exp[−(2ξ1 − 1)Γ0 |τ |] − exp[−(2ξ1 + 1)Γ0 |τ |]) + A11 (exp[−(2ξ1 − 1)Γ1 |τ |] − exp[−(2ξ1 + 1)Γ1 |τ |])} + A12 exp[−2(Γa20 − Γ10 ) |τ |] + A13 exp[−α1 |ξ1 − ξ2 | |τ |] × exp[−(Γa20 + Γa10 − Γ10 ) |τ |] × {η exp[iΔk · r + i(Ω2 − Ω1 )τ + i(ξ2 − ξ1 )Δ3 τ ] + η ∗ exp[−iΔk · r − i(Ω2 − Ω1 )τ − i(ξ2 − ξ1 )Δ3 τ ]}

(2.67)

where  A7 = 4(Γ20 + Γ21 + 6Γ0 Γ1 ) [α21 Γa10 Γ0 Γ1 (Γ0 + Γ1 )]  A8 = 4(ξ2 − 1) (α2 Γa20 )2  A9 = 4(ξ1 + 1)(Γ0 + Γ1 )2 [α21 (2ξ1 + 1)Γ20 Γ21 ]  A10 = (3Γ0 + Γ1 ) [2Γ20 (Γ0 + Γ1 )] A12 = (ξ2 − 1)2 |τ |2 + (ξ2 + 1)/4(Δ2 + ξ2 Δ3 )2  A13 = 4(ξ2 − 1)|τ |(Γ0 + Γ1 ) (α1 α2 Γ0 Γ1 ) (ii) τ < 0 I (τ, r) ∝ |P (3) |2  = A7 + |η|2 A8 . It is clear that photon-echo doesn’t exist for the perturbation chains (I), (III), and (V). Equation (2.67) consists of five terms. The first and third terms depend on the sixth-order coherence functions in u1 (t) for DFWM, while the second and fourth terms are dependent on the fourth-order in u2 (t) and second-order

2.3 Higher-order Correlations of Markovian Stochastic Fields on Polarization · · ·

55

in u1 (t) in the coherence functions for NDFWM. The first and second terms, which originate from the amplitude fluctuations of the Gaussian-amplitude fields, are independent of the relative time-delay between the two beams A and B. The third and fourth terms have exponential decays of the beat signal as |τ | increases. The fifth term, depending on the fourth-order in u1 (t) and second-order in u2 (t), gives rise to the modulation of the beat signal. Equation (2.67) with the parameters Ω2 − Ω1 = 140 (ps)−1 , Δk = 12.25 (mm)−1 , η = ξi = 1, Ai = 0.6, Γa10 = 13.5 (ps)−1 , Γa20 − Γ10 = 14.5 (ps)−1 , Γ0 = 2.7 (ps)−1 , Γ1 = 2.9 (ps)−1 also shows that beat signal oscillates not only temporally with a period of 2π/|Ω2 − Ω1 | = 44.9 fs, but also spatially with a period of 2π/Δk = 0.51 mm along the direction Δk, which is almost perpendicular to the propagation direction of the beat signal (Fig. 2.14). A three-dimensional plot of the beat signal intensity I(τ, r) versus time delay τ and transverse distance r has a large constant background caused by the intensity fluctuations of the Gaussian-amplitude fields. At zero relative time delay (τ = 0), the twin beams (originated from the same laser source) can have a perfect overlap at the sample with their corresponding noise patterns, which gives a maximum interferometric contrast. As |τ | is increased, the interferometric contrast diminishes on the time scale that reflects the material memory, which is usually much longer than the correlation time of the light [31].

Fig. 2.14. (a) A three-dimensional plot of the beat signal intensity I(τ, r) versus time delay τ and the transverse distance r for the Gaussian-amplitude fields. (b) A two-dimensional representation of the beat signal intensity I(τ, r). Adopted from Ref. [35].

It is important to note that these three types of stochastic fields can have the same spectral density and thus the same second-order coherence functions. The fundamental differences in the statistical properties of these fields are manifested only in the higher-order coherence functions. The term “higher-order” refers to all orders higher than the second. According to Gaussian statistics a chaotic field can be fully described by the second-order coherence functions. However, the phase-diffusion field and the Gaussian-

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amplitude field require more higher-order coherence functions [22,23]. In this section, different stochastic models of laser fields only affect the fourth- and sixth-order coherence functions. Figure 2.15 presents the beat signal intensity versus relative time delay with parameters Ω2 − Ω1 = 140 (ps)−1 , Δk = 0, −1 η = ξi = 1, Ai = 0.6, (a) Γa10 = 2.7(ps) , Γa20 − Γ10 = 2.9 (ps)−1 , Γ0 = −1 −1 a 1.35 (ps) , Γ1 = 1.45 (ps) , (b) Γ10 = 13.5 (ps)−1 , Γa20 −Γ10 = 14.5 (ps)−1 , Γ0 = 2.7 (ps)−1 , Γ1 = 2.9 (ps)−1 . Three curves represent the chaotic field (solid line), phase-diffusion field (dashed line), and Gaussian-amplitude field (dotted line), respectively. The polarization beat signal is shown to be particularly sensitive to statistical properties of Markovian stochastic light fields with arbitrary bandwidth. This is quite different from the fourth-order partial-coherence effect in the formation of integrated-intensity gratings with pulsed light sources [35], which have been shown to be insensitive to specific radiation models. The constant background in the beat signal intensity for a Gaussian-amplitude field or a chaotic field is much larger than that of the signal for a phase-diffusion field in Fig. 2.15. The physical explanation for such phenomenon is that the Gaussian-amplitude field undergoes stronger intensity fluctuations than a chaotic field. Also, the intensity (amplitude) fluctuations of the Gaussian-amplitude field or the chaotic field are always much larger and contribute more to the detected signal intensity than pure phase fluctuations of the phase-diffusion field.

Fig. 2.15. The beat signal intensity versus relative time delay. The three curves represent the chaotic field (solid line), phase-diffusion field (dashed line), and Gaussian-amplitude field (dotted line), respectively.

The main purpose of the above discussion is to show an important fact that the amplitude fluctuation plays a critical role in the temporal behavior of the HOCPB signal. Furthermore, we have tried to point out different roles of the phase fluctuation and amplitude fluctuation in the time domain. This is quite different from the time-delayed FWM with incoherent lights in a two-level system, in which case phase fluctuations of light fields are crucial. The HOCPB is more analogous to the Raman-enhanced polarization beats, in which the amplitude fluctuations of the light fields are also more important. Because of ui (t) = 0 and u∗i (t) = 0 in this case, the stochastic

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57

average of the absolute square of the polarization |P (3) |2 , which involves second-order coherence functions of ui (t), can not be used to fully describe the temporal behavior of the HOCPB. The second-order coherence function theory is valid when we are only interested in the τ -dependent part of the beat signal. Therefore, the sixth-order coherence function theory is needed to study the HOCPB. The application of the theory with higher-oder coherence functions to the HOCPB experiment has yielded a better fit to the data than an expression involving only the second-order coherence functions.

2.3.3 Experiment and Result We present experimental results for the material response in the cascade three-level polarization beats with the phase-conjugation geometry using chaotic fields. Beam A and beam B are identical in make-up, as shown in Fig. 2.13, each of which is composed of two noisy fields with frequencies centered at two different colors, ω1 and ω2 , and they carry their own statistical factors u1 (t) and u2 (t), respectively. Beam A and beam B differ only in their respective wave vectors, polarization vectors, and relative time delay. At present, it is difficult to realize the polarization beat experiments with either the phase-diffusion field or the Gaussian-amplitude field. Therefore, direct experimental comparisons between the effects of using different types of fluctuating fields can not be made at this time. The experiment of the HOCPB was performed in a sodium vapor, where the ground state 3S1/2 , the intermediate state 3P3/2 , and the excited state 5S1/2 form a cascade three-level system, as shown in Fig. 2.12. Two dye lasers (DL1 and DL2), with linewidths of 0.01 nm and pulse widths of 10 ns, are used to generate two-color frequencies at ω1 and ω2 , respectively. The relevant transverse relaxation rates for the lower and upper excited states in sodium vapor are 0.175 (ps)−1 and 0.085 (ps)−1 [36], respectively. This system is under the narrow band limit. DL1 is tuned to the wavelength of 589 nm for the 3S1/2 −3P3/2 transition and DL2 is tuned to the wavelength of 616 nm for the 3P3/2 − 5S1/2 transition. A beam splitter is used to combine the ω1 and ω2 components generated from DL1 and DL2, respectively, for beam A and beam B, which intersect in the oven containing the Na vapor with a small angle. The relative time-delay τ between beam A and beam B can be varied. Beam 3 is split from DL1 with frequency ω1 and propagates along the direction opposite to that of beam A. All the incident beams are linearly polarized in the same direction. The generated beat signal propagates along a direction almost opposite to that of beam B, which is determined by the phase-matching condition, and is detected by a photodiode. First, a DFWM experiment was performed with beam A and beam B containing only the ω1 frequency component. Figure 2.16 shows the spectrum of the normal DFWM signal, with the center wavelength at the atomic resonant

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wavelength of 589 nm. The center dip in the DFWM spectrum reflects the resonant absorption behavior of the chaotic field.

Fig. 2.16. Spectrum of DFWM when beams A and B consist of only the ω1 component (with center wavelength of 589 nm). Adopted from Ref. [35].

The generated DFWM signal intensity versus relative time delay is given in Fig. 2.17 with parameters α1 =2.7×1010 s−1 , B1 =0.1 and B3 =1.

Fig. 2.17. DFWM signal intensity versus relative time delay when beams A and B consist of only ω1 frequency component. The squares are the experimental data and the solid line is the theoretical curve. Adopted from Ref. [35].

Then, a NDFWM experiment with beams A and B containing only the ω2 frequency component was performed. The NDFWM spectrum was measured by scanning ω2 (Fig. 2.18), which shows a resonant profile due to two-photon transition. From the NDFWM spectrum one can tune ω2 to the resonant frequency Ω2 , whose center wavelength is 616 nm. The NDFWM signal intensity versus relative time delay is presented in Fig. 2.19 with parameters α2 = 2.9 × 1010 s−1 , B2 = 0.2, and η = 1. After carrying out the experiments for only one frequency component in these beams, the HOCPB experiment was performed by measuring the beat signal intensity as a function of the relative time delay when beam A and beam B both contain two frequency components (ω1 and ω2 ). Figure 2.20 presents the experimentally measured result of the beat signal intensity versus relative time delay between beam A and beam B. The signal is clearly

2.3 Higher-order Correlations of Markovian Stochastic Fields on Polarization · · ·

59

Fig. 2.18. Spectrum of NDFWM signal when beams A and B have only ω2 frequency component with a center wavelength of 616 nm. Adopted from Ref. [35].

modulated sinusoidally with a period of 45 fs. The modulation frequency can be seen more directly by making a Fourier transform of the HOCPB data, as shown in Fig. 2.21 with parameters α1 = 2.7 × 1010 s−1 , α2 = 2.9 × 1010 s−1 , ω2 − ω1 = 1.4 × 1014 s−1 , Δk = 0, and η = Bi = 1, in which τ is varied for a range of 7.3 ps. The modulation frequency is 1.4 × 1014 s−1 , corresponding to the beating between the resonant frequencies of the transitions from 3S1/2 to 3P3/2 and from 3P3/2 to 5S1/2 .

Fig. 2.19. The NDFWM signal intensity versus relative time delay when beams A and B contain only ω2 frequency component. The squares are the experimental data and the solid line is the theoretical fit. Adopted from Ref. [35].

Fig. 2.20. Experimental results of beat signal intensity versus relative time delay. (a) Time delay τ is varied for a range of 7.3 ps; (b) Time delay τ is varied for a range of 0.5 ps. Adopted from Ref. [35].

60

2 Femtosecond Polarization Beats

Fig. 2.21. The squares are the experimental data for the Fourier spectrum, in which τ is varied for a range of 7.3 ps. The solid line is the theoretical curve given. Adoptedc from Ref. [35].

Now, we discuss the difference between the HOCPB and the UMS with the self-diffraction geometry. Thees frequencies and wave vectors of the UMS signals are ωs1 = 2ω1 − ω1 , ωs2 = 2ω2 − ω2 , and ks1 = 2k1 − k1 , ks2 = 2k2 − k2 , respectively, which indicate that one photon is absorbed from each of the two mutually correlated pump beams. However, frequencies and wave vectors of HOCPB signals are ωs1 = ω1 − ω1 + ω3 , ωs2 = ω2 − ω2 + ω3 , and ks1 = k1 − k1 + k3 , ks2 = k2 − k2 + k3 , respectively, in which case photons are absorbed from and emitted into the mutually correlated beams A and B, respectively. This difference between the HOCPB and the UMS has profound influence on the field-correlation effects. We note that the roles of beam A and beam B are interchangeable in the UMS [11], which makes the secondorder coherence function theory fail in the UMS. Since u(t1 )u(t2 ) = 0, the absolute square of the stochastic average of the polarization |P (3) |2 can not be used to fully describe the temporal behavior of the UMS. So, the sixth-order coherence function theory, presented above, should be of vital importance in the UMS. In summary, we have adopted the chaotic, the phase-diffusion, and the Gaussian-amplitude field models to study the effects of the sixth-order coherence functions on the polarization beats in a cascade three-level system. Different stochastic models of the laser fields only affect the sixth- and fourth-order coherence functions. We found that the constant background of the beat signal intensity originates from amplitude fluctuations of Markovian stochastic fields. The effects due to fluctuations from the Gaussianamplitude fields are larger than from the chaotic fields, which again exhibit much larger effects than that from phase-diffusion fields with pure phase fluctuations caused by spontaneous emission.

References [1]

Debeer D, Van Wagenen L G, Beach R, et al. Ultrafast modulation spectroscopy. Phys. Rev. Lett. 1986,56: 1128 – 1131.

References

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Fu P M, Yu Z H, Mi X, et al. Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry. Phys. Rev. A, 1994, 50: 698 – 708. Fu P M, Mi X, Yu Z H, et al. Ultrafast modulation spectroscopy in a cascade three-level system. Phys. Rev. A, 1995, 52: 4867 – 4870. Beach R, Debeer D, Hartmann S R. Time-delayed 4-wave mixing using intense incoherent-light. Phys. Rev. A, 1985, 32: 3467 – 3474. Mi X, Yu Z H, Jiang Q, et al. Time-delayed laser-induced double gratings with broadband lights. Opt. Common., 1995, 116: 443 – 448. Kirkwood J C, Ulness D J, Albrecht A C. Electronically nonresonant coherent Raman scattering using incoherent light: Two Brownian oscillator approaches. J. Chem. Phys., 1998, 108: 9425 – 9435. Mossberg T W, Kachru R, Hartmann S R, et al. Echoes in gaseous media: a generalized theory of rephasing phenomena. Phys. Rev. A, 1979, 20: 1976 – 1996. Picinbono B, Boileau E. Higher-order coherence functions of optical field and phase fluctuations. J. Opt. Soc. Am., 1968, 58: 784. Morita N, Yajima T. Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light. Phys. Rev. A, 1984, 30: 2525 – 2536. Mitsunaga M, Brewer R G. Generalized perturbation-theory of coherent optical-emission. Phys. Rev. A, 1985, 32: 1605 – 1603. Zhang Y P, Sun L Q, Tang T T, et al. Effects of field correlation on polarization beats. Phys. Rev. A, 2000, 61: 053819. Do B, Cha J W, Elliott D S, et al. Phase-conjugate four-wave mixing with partially coherent laser fields. Phys. Rev. A, 1999, 60: 508 – 517. Anderson M H, Vemuri G, Cooper J, et al. Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field. Phys. Rev. A, 1993, 47: 3202 – 3209. Chen C, Elliott D S, Hamilton M W. Two-photon absorption from the real Gaussian field. Phys. Rev. Lett., 1992, 68: 3531 – 3534. Agarwal G S. Nonlinear spectroscopy with cross-correlated chaotic fields. Phys. Rev. A, 1988, 37: 4741 – 4746. Agarwal G S, Kunasz C V. 4-wave mixing in stochastic fields -fluctuationinduced resonances. Phys. Rev. A, 1983, 27: 996 – 1012. Mi X, Yu Z H, Jiang Q, et al. Four-level ultrafast modulation spectroscopy. Opt. Common., 1998, 152: 361 – 364. Asaka S, Nakatsuka H, Fujiwara M, et al. Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass. Phys. Rev. A, 1984, 29: 2286 – 2289. Mi X, Yu Z H, Jiang Q, et al. Time-delayed laser-induced double gratings. J. Opt. Soc. Am. B, 1993, 10: 725 – 732. Zhang Y P, Lu K Q, Li C S, et al. Correlation effects of chaotic and phasediffusion fields on polarization beats in a V-type three-level system. J. Mod. Opt., 2001, 48: 549 – 564. Ryan R E, Bergeman T H. Hanle effect in nonmonochromatic laser-light. Phys. Rev. A, 1991, 43: 6142 – 6155. Walser R, Ritsch H, Zoller P, et al. Laser-niose-induced population fluctuations in 2-level systems - complex and real gaussian driving fields. Phys. Rev. A, 1992, 45: 468 – 476. Ryan R E, Westling L A, Blumel R, et al. 2-Photon spectroscopy: a technique for characterizing diode-laser noise. Phys. Rev. A, 1995, 52: 3157 – 3169.

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2 Femtosecond Polarization Beats

Georges A T. Resonance fluorescence in markovian stochastic fields. Phys. Rev. A, 1980, 21: 2034 – 2049. Bratfalean R, Ewart P. Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields. Phys. Rev. A, 1997, 56, 2267 – 2279. Ulness D J, Albrecht A C. Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density: analytic solution and a diagrammatic approach. Phys. Rev. A, 1996, 53: 1081 – 1095. Ulness D J, Albrecht A C. Theory of time resolved coherent Raman scattering with spectrally tailored noisy light. J. Raman Spectrosc., 1997, 28: 571 – 578. Demott D C, Ulness D J, Albrecht A C. Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light. Phys. Rev. A, 1997, 55: 761 – 771. Ulness D J, Kirkwood J C, Albrecht A C. Competitive events in fifth order time resolved coherent Raman scattering: Direct versus sequential processes. J. Chem. Phys., 1998, 108: 3897 – 3902. Kirkwood J C, Albrecht A C, Ulness D J, et al. Coherent Raman scattering with incoherent light for a multiply resonant mixture: a factorized time correlator diagram analysis. Phys. Rev. A, 1998, 58: 4910 – 4925. Kirkwood J C, Albrecht A C, Ulness D J. Fifthorder nonlinear Raman processes in molecular liquids using quasi-cw noisy light. J. Chem. Phys., 1999, 111: 253 – 271. Kirkwood J C, Albrecht A C. Down-conversion of electronic frequencies and their dephasing dynamics: interferometric four-wave-mixing spectroscopy with broadband light. Phys. Rev. A, 2000, 61: 033802. Ma H, De Araujo C B. Interference between 3rd-order and 5th-order polarizations in semiconductor-doped glasses. Phys. Rev. Lett., 1993, 71: 3649 – 3652. Ma H, Acioli L H, Gomes A S L, et al. Method to determine the phase dispersion of the 3rd-order susceptibility. Opt. Lett., 1991, 16: 630 – 632. Zhang Y P, De Araujo C B, Eyler E E. Higher-order correlation on polarization beats in Markovian stochastic fields. Phys. Rev. A, 2001, 63: 043802. Golub J E, Mossberg T W. Studies of picosecond collisional dephasing in atomic sodium vapor using broad-bandwidth transient 4-wave-mixing. J. Opt. Soc. Am. B, 1986, 3: 554 – 559.

3 Attosecond Polarization Beats

In Chapter 2, we have described femotosecond polarization beats in threeand four-level systems with two pump laser beams having two frequency components in each beam. The femtosecond beating comes from the frequency difference between two atomic transitions, therefore named difference frequency polarization beats (DFPB), when the two pump beams have a relative time delay. However, for a different situation with a time delay between two frequency components in each pump beam, the polarization beat signal will appear with a sum frequency of two atomic transitions, which is called sum-frequency polarization beats (SFPB). Such SFPB can result in beating signals with attosecond time scale; therefore, such SFPB technique has sometimes been called attosecond polarization beats (ASPB). Such SFPB technique can be used effectively for certain ultrafast laser spectroscopy measurements. In this chapter, we will describe how such SFPB appear in multilevel systems and how the signal intensity of the SFPB changes for different Markovian stochastic fields. Based on the phase-conjugate polarization interference between two-pathway excitations, the second-order or fourthorder Markovian stochastic correlations of the SFPB in attosecond time scale have been studied in a three-level V-type system. Field correlations have weakly influence on the SFPB signal when the lasers have narrow bandwidths. In contrast, when lasers have broadband linewidths, the SFPB signal shows the resonant-nonresonant cross correlation, and sensitivities of the SFPB signal to three Markovian stochastic models increase as the time delay is increased. A Doppler-free precision in the measurement of the sum frequency between optical transitions can be achieved with an arbitrary bandwidth. As an attosecond ultrafast modulation process, this SFPB technique can be extended in principle to any sum-frequency between various energy levels.

3.1 Polarization Beats in Markovian Stochastic Fields Markovian stochastic processes are ubiquitous in all branches of science. Unlike the non-Markovian fluctuations [1], which arise from the atom’s memory of its past, the atomic responses to Markovian stochastic optical fields are now well understood [2-8]. When the laser field is sufficiently intense to have

64

3 Attosecond Polarization Beats

multi-photon interactions, the laser spectral bandwidth or spectral shape, described by the second-order correlation function, will be inadequate to characterize the field. Rather than using higher-order correlation functions explicitly, three different Markovian fields are considered: (a) the chaotic field, (b) the phase-diffusion field, and (c) the Gaussian-amplitude field. The chaotic field undergoes both amplitude and phase fluctuations, which corresponds to a multimode laser field with a large number of uncorrelated modes, or a single-mode laser emitting light below the threshold. The phase-diffusion field undergoes only phase fluctuations, which corresponds to an intensitystabilized single-mode laser field. The phase of the laser field, however, has no natural stabilizing mechanism. The Gaussian-amplitude field has only amplitude fluctuations. Although pure amplitude fluctuations cannot be produced by a nonadiabatic process, we consider the Gaussian-amplitude field to isolate the effects due solely to amplitude fluctuations and to use it as an example for a field with stronger amplitude (intensity) fluctuations than a chaotic field. By comparing the results for the chaotic and the Gaussian-amplitude fields the effect of increasing amplitude fluctuations can be determined. When the signal decays are on the order of the correlation times of the noisy lights, the Markovian stochastic models become more important because dynamical information about the material must be extracted from underneath the correlation decay. A laser-based sampling system, consisting of a few-femtosecond visible light pulse and a synchronized sub-femtosecond soft X-ray pulse, allows researchers to trace the dynamical properties directly in the time domain with attosecond resolution [9]. The nearly transform-limited nanosecond optical pulses (< 100 MHz bandwidth), generated by the pulsed (10 Hz) amplification of a tunable, single-frequency, cw ring dye laser beam, are an ideal tool for high-resolution spectroscopic studies [10, 11]. For over two decades, ultrashort time resolution of material dynamics has always been accomplished by the broadband, non-transform-limited noisy light. The time resolution is determined by the ultrafast correlation time of the noisy light and not by its temporal envelope, which is typically a few nanoseconds [12 – 25]. Such a “noisy” light source is usually derived from a dye laser modified to permit oscillation over almost the entire bandwidth of the broadband light source. The typical bandwidth of the noisy light is about 100 cm−1 , and has a correlation time of 100 fs (HWHM) [1]. In fact, the multimode broadband light and the transform-limited femtosecond laser sources are different in two fundamental ways. First, the broadband source, though pulsed, is effectively nearly continuous (or quasi-cw) and is also more energetic than the femtosecond source, which has true ultrashort pulses in time. That is to say, the noisy light typically has higher average energy, but the light source with ultrashort pulses has greater instantaneous energy. The second fundamental distinction between the two kinds of light sources relates to the concept of “cross-color” coherence. Even when both sources have the identically broad spectrum, a noisy field possesses random relative phases among the available colors.

3.1 Polarization Beats in Markovian Stochastic Fields

65

There is no cross-color coherence in a noisy field and the field correlators are “color locked”. On the other hand, the light source with femtosecond pulses consists of fields characterized by well-defined relative phases among the colors. The fundamental difference is that the transform-limited femtosecond pulse laser is phase coherent, while the broadband noisy light source has random phase. Noisy light can be used to probe atomic and molecular dynamics, and it offers a unique alternative to the more conventional frequency domain (narrow bandwidth) spectroscopy and ultrashort (femtosecond pulses) time domain spectroscopy. Color-locked noisy light is an intermediate situation between cw and short-pulse methods [25]. Polarization beats, which originate from the interference between macroscopic polarizations that are excited simultaneously by twin fields, are well known [26 – 40] and have been discussed in Chapter 2. It is closely related to the quantum beat spectroscopy based on superposition-state interference. DeBeer, et al. [26, 27] performed the first difference-frequency and sumfrequency ultrafast modulation spectroscopy (UMS) experiments in those experiments twin-color lasers all operated in four or five longitudinal modes and the beating signals exhibit 1.9 ps and 980 as modulations, respectively. The Markovian field correlation effects in the difference-frequency polarization beats (DFPB) of femtosecond scale were described in detail in Chapter 2. In this chapter, we shall systematically study attosecond sum-frequency polarization beats (SFPB) in twin Markovian stochastic fields using a threelevel V-type system. To be different from DFPB in Chapter 2, for SFPB a time delay is introduced in both composite beams and as a result one can control the time delay to obtain polarization beats with the frequency which is the sum of frequencies of twin fields. On the other hand, owing to the different phase-conjugation geometry, if self-correlation signal of one frequency component shows the interplay between atomic and light responses only at positive time delay for DFPB, in contrast, such interplay is observed at negative time delay for SFPB. The subtle phase control of the light beams in SFPB is also considered. The difference between SFPB and the first polarization beat experiment of Rothenberg and Grischkowsky [29] is that in SFPB the signal is modulated not in real time but rather as a function of the delay between two frequency components in pump beams. SFPB is also related to the coherent control that has been used to control the ionization rate of an atom [41], the dissociation rate of a molecule [10, 42], and the direction of current generated in a semiconductor [43]. This method uses two coincident laser fields to induce transitions in an atom, a molecule, or a solid. One achieves control by varying the relative phase of the two fields such that the induced transition amplitudes interfere either constructively or destructively. The common point of SFPB and coherent control is that both methods involve two fields to induce two-pathway excitation. The physical processes are manipulated by variation of the phase of the fields. The difference between them is that the coherent control describes a quantum interference between transition probability amplitudes, whereas SFPB originates from the

66

3 Attosecond Polarization Beats

interference between macroscopic polarizations, which is classic in nature. Damped SFPB and DFPB involve purely material oscillators, purely radiation oscillators, and hybrid radiation-matter oscillators [24]. Their proper interpretations can provide new insight into the nonlinear wave-mixing processes and the associated material ultrafast relaxation dynamics. In this chapter, SFPB in a Doppler-broadened V-type three-level system will be theoretically described. As we mentioned above, SFPB is based on the interference between two one-photon FWM processes simultaneously induced by time-delayed correlated fluctuating twin fields. The phase control of light beams in SFPB can be extremely sensitive. If incident fields are assumed to be weak and have finite bandwidths, in the extreme Doppler-broadening limit a closed (analytic) form for the second-order or fourth-order Markovian stochastic correlations of SFPB can be obtained. The different roles of the amplitude fluctuations and the phase fluctuations in SFPB can be understood physically by a time-domain picture. In a broadband case (i.e. noisy light), the resonantly excited levels are spectrally embraced by the broadband field, and the corresponding one-photon resonant FWM signals exhibit hybrid radiation-matter terahertz detuning damping oscillation. The unbalanced, controllable dispersion compensation between two arms of the Michelson interferometer results in different autocorrelation maximums of two one-photon FWM processes, and causes the maximum of the SFPB signal to shift from its zero time delay position.

3.2 Perturbation Theory Let’s consider a V-type three-level system, as shown in Fig. 3.1(a), with a ground state |0, and two excited states |1 and |2. States between |0 and |1 and between |0 and |2 are coupled by dipole transitions with resonant frequencies Ω1 and Ω2 , respectively, while transition from |1 to |2 is dipole forbidden. The schematic diagram of the laser beam geometry for the SFPB experiment is shown in Fig. 3.1(b). We consider, in this three-level V-type system, a double-frequency time-delayed FWM experiment, in which beam 1 and beam 2 consist of two frequency components ω1 and ω2 in each beam. We further assume that ω1 ≈ Ω1 and ω2 ≈ Ω2 , therefore ω1 and ω2 drive the transitions from |0 to |1 and from |0 to |2, respectively. There are two processes involved in this double-frequency time-delayed FWM configuration. First, the ω1 frequency components of beam 1 and beam 2 induce a population grating between states |0 and |1, which is probed by beam 3 with the same frequency ω1 . This is an one-photon resonant degenerate FWM (DFWM) process and the generated signal (beam 4) has frequency ω1 . Second, the ω2 frequency components of beam 1 and beam 2 induce a population grating between states |0 and |2, which is then probed by beam 3. This is a one-photon resonant nondegenerate FWM (NDFWM) process

3.2 Perturbation Theory

67

and the frequency of the generated signal equals to ω1 also.

Fig. 3.1. (a) The three-level V-type system of the sodium atoms used in SFPB; (b) phase-conjugation geometry of laser beams for SFPB.

In a typical experiment, the two-color light sources enter a dispersioncompensated Michelson interferometer to generate identical twin composite beams. The two-component stochastic fields of beam 1 (Ep1 (r, t)) and beam 2 (Ep2 (r, t)) can be written as Ep1 = A1 (r, t) exp(−iω1 t) + A2 (r, t) exp(−iω2 t) = ε1 u1 (t) exp[i(k1 · r − ω1 t)] + Ep2

(3.1)

ε2 u2 (t − τ − δτ ) exp[i(k2 · r − ω2 t + ω2 τ + ω2 δτ )] = A1 (r, t) exp(−iω1 t) + A2 (r, t) exp(−iω2 t) = ε1 u1 (t − τ ) exp[i(k1 · r − ω1 t + ω1 τ )] + ε2 u2 (t) exp[i(k2 · r − ω2 t)]

(3.2)

where εi , ki (εi ,ki ) are the constant field amplitude and the wave vector of the ωi component in beam 1 (beam 2), respectively. ui (t) is a dimensionless statistical factor that contains phase and amplitude fluctuations, which is taken to be a complex ergodic stochastic function of t. ui (t) obeys complex circular Gaussian statistics for a chaotic field. τ is the variable relative time delay between the prompt (unprime) and delayed (prime) field components in each beam. To produce such a laser configuration with relative time delay in each beam, the frequency components of the ω1 and ω2 lights are split and recombined. Two double-frequency fields are generated in such a way that the ω1 frequency component is delayed by τ in beam 2 and the ω2 component is delayed by the same amount in the beam 1 [Fig. 3.1 (b)], which is quite different from the case for DFPB as discussed in Chapter 2. δτ denotes the difference between two autocorrelation processes at the zero time delay (δτ < 0), which comes from the unbalance dispersion effects between the two arms of the Michelson interferometer. Furthermore, beam 3 with a single frequency component is assumed to be a quasi-monochromatic light and the total complex electric field of it can be written as Ep3 = A3 (r, t) exp(−iω3 t) = ε3 u3 (t) exp[i(k3 · r − ω3 t)]

(3.3)

68

3 Attosecond Polarization Beats

where ω3 , ε3 and k3 are the frequency, the field amplitude, and the wave vector of beam 3, respectively. u3 (t) ≈ 1 for a quasi-monochromatic light. We employ perturbation theory to calculate density-matrix elements. One can write the following perturbation chains for the above described SFPB system: (0) A

 ∗ (1) (A )

(2) A

(3)

1 3 ρ10 −−−1− → ρ00 −−→ ρ10 (I) ρ00 −−→  ∗ (0) (A )

1 3 (II) ρ00 −−−1− → (ρ10 )∗ −−→ ρ00 −−→ ρ10

(0) A

(1)

A

 ∗ (1) (A )

(2) A

(2) A

(3)

(3)

1 3 (III) ρ00 −−→ ρ10 −−−1− → ρ11 −−→ ρ10  ∗ (0) (A )

1 3 (IV) ρ00 −−−1− → (ρ10 )∗ −−→ ρ11 −−→ ρ10  (0) A

(1)

A

∗ (1) (A2 )

(2) A

(2) A

(3)

(3)

3 2 (V) ρ00 −−→ ρ20 −−−− → ρ00 −−→ ρ10 ∗ (0) (A2 )

A

3 2 (VI) ρ00 −−−− → (ρ20 )∗ −−→ ρ00 −−→ ρ10

(1)

(2) A

(3)

We have attributed the FWM signals to the gratings induced by the ω1 (or ω2 ) frequency components of beam 1 and beam 2. Now, we consider other possible density operator pathways: (0) A

 ∗ (1) (A )

(2) A

(3)

3 1 (VII) ρ00 −−→ ρ10 −−−1− → ρ00 −−→ ρ10  ∗ (0) (A )

A

(2) A

3 1 (VIII) ρ00 −−−1− → (ρ10 )∗ −−→ ρ00 −−→ ρ10

(0) A

(1)

 ∗ (1) (A )

(2) A

(3)

(3)

3 1 (IX) ρ00 −−→ ρ10 −−−1− → ρ11 −−→ ρ10  ∗ (0) (A )

A

(2) A

3 1 (X) ρ00 −−−1− → (ρ10 )∗ −−→ ρ11 −−→ ρ10

(0) A

(1)

 (2) A

∗ (1) (A2 )

(3)

(3)

3 2 (XI) ρ00 −−→ ρ10 −−−− → ρ12 −−→ ρ10 ∗ (0) (A2 )

A

 (2) A

3 2 (XII) ρ00 −−−− → (ρ20 )∗ −−→ ρ12 −−→ ρ10

(1)

(3)

The grating induced by beam 3 and the ω1 (or ω2 ) frequency component of beams 2 is responsible for the generation of the FWM signal. These gratings have much smaller fringe spaces which equal approximately to one half of θ≈180◦

the wave-lengths of the incident lights {λi /[2 sin(θ/2)] −−−−−→ λi /2}. For a Doppler-broadened system, the grating will be washed out by the atomic motion. Therefore, it is safe to neglect FWM signals from the density operator pathways from (VII) to (XII). In addition, density operator pathways (XI) and (XII) involve the coherence between the excited states |1 and |2. For a system with the relaxation time of ρ00 (or ρ11 ) much longer than the relaxation time of ρ12 the FWM signals can be reduced further. We have

3.2 Perturbation Theory

69

also neglected the contributions from the density operator pathways:  (0) A

 ∗ (1) (A )

(2) A

(3)

3 2 ρ20 − −−1− → ρ21 −−→ ρ20 (XIII) ρ00 −−→  ∗ (0) (A )

A

(2) A

3 2 (XIV) ρ00 −−−1− → (ρ10 )∗ −−→ ρ21 −−→ ρ20

(1)

(3)

These two processes give rise to a signal with frequency ω4 = ω3 + (ω2 − ω1 ); therefore, it can be separated from the FWM signals with frequency ω3 by a monochromator or a narrow-band filter. Furthermore, the stricter requirement on the phase matching and the involvement of ρ12 also make this process less important, and therefore negligible. As a time-domain technique, although SFPB is similar to the quantumbeat technique, the advantage of the SFPB over the conventional quantumbeat technique is that the temporal resolution is not limited by the laser pulse width. With laser pulses of nanosecond scale time duration, femtosecond or picosecond time scale modulations were observed in the DFPB experiments [30 – 40]. Now we consider a time-delayed FWM process in a three-level V-type system [see Fig. 3.1 (a)], where beam 1 and beam 2 only consist of one frequency component either ω1 or ω2 with a broad bandwidth, so the two transitions can be excited simultaneously (i.e., α1 , α2 > |Ω2 − Ω1 |). In this case, besides perturbation chains given in expressions (I) – (XIV) above, there are other additional chains. In this section, chains (I) – (IV) and (V) – (VI) correspond to one-photon DFWM from |0 to |1 and one-photon NDFWM from |0 to |2, respectively. We obtain the total third-order off-diagonal density-matrix element (3) ρ10 which has wave vector k1 − k1 + k3 or k2 − k2 + k3 , and ρ(3) = ρ(I) + ρ(II) + ρ(III) + ρ(IV ) + ρ(V ) + ρ(VI) . Here ρ(I) , ρ(II) , ρ(III) , ρ(IV ) , ρ(V ) (3) and ρ(VI) correspond to the individual ρ10 of the perturbation chains (I), (II), (III), (IV), (V), and (VI), respectively, and can be caculated by  ρ

(I)

ρ(II)

ρ(III)

=

iμ1 

3

 exp(−iω3 t)



∞

dt3 0



∞

dt2 0

∞

dt1 ×

0

H3 (t3 )H2 (t2 )H1 (t1 )A1 (r, t − t1 − t2 − t3 ) × [A1 (r, t − t2 − t3 )]∗ A3 (r, t − t3 )  3  ∞  ∞  ∞ iμ1 = exp(−iω3 t) dt3 dt2 dt1 ×  0 0 0 H3 (t3 )H2 (t2 )[H1 (t1 )]∗ A1 (r, t − t2 − t3 ) × [A1 (r, t − t1 − t2 − t3 )]∗ A3 (r, t − t3 )  3  ∞  ∞  ∞ iμ1 = exp(−iω3 t) dt3 dt2 dt1 ×  0 0 0 H3 (t3 )H5 (t2 )H1 (t1 )A1 (r, t − t1 − t2 − t3 )×

(3.4)

(3.5)

70

3 Attosecond Polarization Beats

ρ(IV )

[A1 (r, t − t2 − t3 )]∗ A3 (r, t − t3 )  3  ∞  ∞  ∞ iμ1 = exp(−iω3 t) dt3 dt2 dt1 ×  0 0 0

(3.6)

H3 (t3 )H5 (t2 )[H1 (t1 )]∗ A1 (r, t − t2 − t3 ) ×

ρ(V )

[A1 (r, t − t1 − t2 − t3 )]∗ A3 (r, t − t3 )   2  ∞  ∞  ∞ iμ1 iμ2 = exp(−iω3 t) dt3 dt2 dt1 ×   0 0 0

(3.7)

H3 (t3 )H2 (t2 )H4 (t1 )A2 (r, t − t1 − t2 − t3 ) ×

ρ(VI)

[A2 (r, t − t2 − t3 )]∗ A3 (r, t − t3 )   2  ∞  ∞  ∞ iμ1 iμ2 = exp(−iω3 t) dt3 dt2 dt1 ×   0 0 0 H3 (t3 ) H2 (t2 ) [H4 (t1 )]∗ A2 (r, t − t2 − t3 ) × [A2 (r, t − t1 − t2 − t3 )]∗ A3 (r, t − t3 )

(3.8)

(3.9)

In these above expressions, H1 (t) = exp[− (Γ10 + iΔ1 ) t], H2 (t) = exp (−Γ0 t), H3 (t) = exp[− (Γ10 + iΔ3 ) t], H4 (t) = exp[− (Γ20 + iΔ2 ) t], and H5 (t) = exp (−Γ1 t). μ1 (μ2 ) is the dipole-moment matrix element between |0 and |1 (|0 and |2). Frequency detunings are Δ1 = Ω1 −ω1 , Δ2 = Ω2 −ω2 , Δ3 = Ω1 − ω3 . Γ0 (Γ1 ) is the population relaxation rate of the state |0 (|1). Γ10 (Γ20 ) is the transverse relaxation rate of the transition from |0 to |1 (|0 to |2), which contains material’s dephasing dynamics. In the presence of collisions, two types of relaxation processes can be observed: (1) T1 processes, which involve the losses of amplitude in the population terms ρii and (2) T2 processes, which involve the losses in the coherence term ρij . Inhomogeneous broadening, T2∗ , does not depend on collisions but can also lead to a loss of coherence. The intramolecular dephasing time might be defined as a T3 process. In a gaseous system, the position r(t ) is related to r(t) for a particular atom with constant velocity v by r(t ) = r(t) + (t − t)v. Therefore, we have ρ(I) (v, t) =

−iμ31 ε1 (ε1 )∗ ε3 exp{i[(k1 − k1 + k3 ) · r − ω3 t − ω1 τ ]} × 3  ∞  ∞  ∞ dt3 dt2 dt1 exp[−iθI (v)]H3 (t3 ) H2 (t2 ) H1 (t1 ) × 0

0

0

u1 (t − t1 − t2 − t3 )[u1 (t − t2 − t3 − τ )]∗ ρ(II) (v, t) =

−iμ31 ε1 (ε1 )∗ ε3 exp{i[(k1 − k1 + k3 ) · r − ω3 t − ω1 τ ]} × 3  ∞  ∞  ∞ dt3 dt2 dt1 exp[−iθII (v)]H3 (t3 ) H2 (t2 ) × 0

0

0

(3.10)

3.2 Perturbation Theory

[H1 (t1 )]∗ u1 (t − t2 − t3 )[u1 (t − t1 − t2 − t3 − τ )]∗ ρ(III) (v, t) =

0

0

u1 (t − t1 − t2 − t3 )[u1 (t − t2 − t3 − τ )]∗

0

0

0

(3.13)

−iμ1 μ22  ε2 (ε2 )∗ ε3 exp{i[(k2 − k2 + k3 ) · r − ω3 t + ω2 τ + 3  ∞  ∞  ∞ ω2 δτ ]} dt3 dt2 dt1 exp[−iθIII (v)]H3 (t3 ) H2 (t2 ) × 0

0

0

H4 (t1 ) u2 (t − t1 − t2 − t3 − τ − δτ )[u2 (t − t2 − t3 )]∗ ρ(VI) (v, t) =

(3.12)

−iμ31 ε1 (ε1 )∗ ε3 exp{i[(k1 − k1 + k3 ) · r − ω3 t − ω1 τ ]} × 3  ∞  ∞  ∞ dt3 dt2 dt1 exp[−iθII (v)]H3 (t3 ) H5 (t2 ) × [H1 (t1 )]∗ u1 (t − t2 − t3 )[u1 (t − t1 − t2 − t3 − τ )]∗

ρ(V ) (v, t) =

(3.11)

−iμ31 ε1 (ε1 )∗ ε3 exp{i[(k1 − k1 + k3 ) · r − ω3 t − ω1 τ ]} × 3  ∞  ∞  ∞ dt3 dt2 dt1 exp[−iθI (v)]H3 (t3 ) H5 (t2 ) H1 (t1 ) × 0

ρ(IV ) (v, t) =

71

(3.14)

−iμ1 μ22  ε2 (ε2 )∗ ε3 exp{i[(k2 − k2 + k3 ) · r − ω3 t + ω2 τ + 3  ∞  ∞  ∞ ω2 δτ ]} dt3 dt2 dt1 exp[−iθIV (v)]H3 (t3 ) × 0

0

0

H2 (t2 ) [H4 (t1 )]∗ u2 (t − t2 − t3 − τ − δτ ) × [u2 (t − t1 − t2 − t3 )]∗

(3.15)

where θI (v) = v · [k1 (t1 + t2 + t3 ) − k1 (t2 + t3 ) + k3 t3 ]

θII (v) = v · [−k1 (t1 + t2 + t3 ) + k1 (t2 + t3 ) + k3 t3 ] θIII (v) = v · [k2 (t1 + t2 + t3 ) − k2 (t2 + t3 ) + k3 t3 ]

θIV (v) = v · [−k2 (t1 + t2 + t3 ) + k2 (t2 + t3 ) + k3 t3 ] The nonlinear polarization P (3), responsible for the phase-conjugate FWM signal, is given by averaging over the velocity distribution function w(v).  +∞ (3) Thus, P (3) = N μ1 −∞ dvw(v)ρ10 (v), where v is the atomic velocity and N is the atomic density. For a Doppler-broadened atomic system, we have √ w (v) = exp[−(v/u)2 ]/( πu)  Here, u = 2kB T /m with m being the mass of an atom, kB is Boltzmann constant and T is the absolute temperature. In general, the total thirdorder polarization P (3) is composition of four DFWM and two NDFWM

72

3 Attosecond Polarization Beats

polarizations: P (3) = P (I) + P (II) + P (III) + P (IV ) + P (V ) + P (VI) The individual component can be written as  +∞  (I) P = S1 (r) exp[−i(ω3 t + ω1 τ )] dvw(v) −∞

P(II)

exp[−iθI (v)]H3 (t3 )H2 (t2 )H1 (t1 ) × u1 (t − t1 − t2 − t3 )[u1 (t − t2 − t3 − τ )]∗  +∞  = S1 (r) exp[−i(ω3 t + ω1 τ )] dvw(v)

−∞

P(IV )

−∞



∞

0

dt1 ×

0

(3.16) 

∞



∞

dt3

∞

dt2 0

dt1 ×

0

(3.17) 

∞



∞

dt3

∞

dt2

0

exp[−iθI (v)]H3 (t3 )H5 (t2 )H1 (t1 ) × u1 (t − t1 − t2 − t3 )[u1 (t − t2 − t3 − τ )]∗  +∞  = S1 (r) exp[−i(ω3 t + ω1 τ )] dvw(v)

∞

dt2

0

exp[−iθII (v)]H3 (t3 )H2 (t2 )[H1 (t1 )]∗ × u1 (t − t2 − t3 )[u1 (t − t1 − t2 − t3 − τ )]∗  +∞  = S1 (r) exp[−i(ω3 t + ω1 τ )] dvw(v)



∞

dt3

0

−∞

P(III )



∞

0

dt1 ×

0

(3.18) 

∞

∞

dt3

0

dt2 ×

0

dt1 exp[−iθII (v)]H3 (t3 )H5 (t2 )[H1 (t1 )]∗ ×

0

P(V )

u1 (t − t2 − t3 )[u1 (t − t1 − t2 − t3 − τ )]∗  +∞  = S2 (r) exp[−i(ω3 t − ω2 τ − ω2 δτ ] dvw(v) −∞



∞

(3.19) 

∞

∞

dt3

0

dt2 ×

0

dt1 exp[−iθIII (v)]H3 (t3 )H2 (t2 )H4 (t1 ) ×

0

P(VI)

u2 (t − t1 − t2 − t3 − τ − δτ )[u2 (t − t2 − t3 )]∗  +∞  = S2 (r) exp[−i(ω3 t − ω2 τ − ω2 δτ ] dvw(v) −∞



∞

(3.20) 

∞

∞

dt3

0

dt2 ×

0

dt1 exp[−iθIV (v)]H3 (t3 )H2 (t2 )[H4 (t1 )]∗ ×

0

u2 (t − t2 − t3 − τ − δτ )[u2 (t − t1 − t2 − t3 )]∗ where S1 (r) = −iN μ41 ε1 (ε1 )∗ ε3 exp[i(k1 − k1 + k3 ) · r]/4

S2 (r) = −iN μ21 μ22 ε2 (ε2 )∗ ε1 exp[i(k2 − k2 + k3 ) · r]/3

(3.21)

3.3 Second-order Stochastic Correlation of SFPB

73

In general, the SFPB (at the intensity level) can be viewed as the sum of five contributions: (i) three (resonant-resonant, nonresonant-nonresonant or resonant-nonresonant) types of τ -independent auto-correlation terms; (ii) the purely resonant τ -dependent auto-correlation terms; (iii) the purely nonresonant τ -dependent auto-correlation terms; (iv) the resonant-nonresonant τ -dependent auto-correlation terms; and (v) three (resonant-resonant, nonresonant-nonresonant or resonant-nonresonant) types of τ -dependent cross-correlation terms.

3.3 Second-order Stochastic Correlation of SFPB For the macroscopic system where phase matching takes place the signal must be drawn from the P (3) (having t time variable) developed on one “atom” multiplied by the (P (3) )∗ (having s time variable) developed on another “atom” which must be located elsewhere in space (with summation over all such pairs) [16 – 23, 33 – 40]. For the homodyne detected SFPB, the signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic process |P (3) |2  (having5 both t and s time variables), which involves second- and fourth-order coherence functions of ui (t) in phase–conjugation geometry. The ultrafast modulation spectroscopy in the self-diffraction geometry is related to the sixth-order coherence functions of the incident fields [26, 27]. In the case that we are only interested in the τ -dependent part of the SFPB signal, the SFPB signal intensity can be well approximated by the absolute square of the non-trivial stochastic average of the polarization |P (3) |2 , which involves second-order coherence functions of ui (t) [30 – 40]. Assuming that the noisy light beams have Lorentzian lineshape, we have ui (t1 ) u∗i (t2 ) = exp (−αi |t1 − t2 |)

i = 1, 2

(3.22)

1 δωi (with δωi being the linewidth of the ωi frequency com2 ponent) is the autocorrelation decay of the noisy light. The form of the second-order coherence function, which is determined by the laser line shape as expressed in Eq. (3.22), is a general feature of three different Markovian stochastic models with chaotic field, phase-diffusion filed, and Gaussianamplitude filed [7,8], repectively. Using Eq. (3.22) the stochastic average of the total polarization then given by

Here, αi =

P (3)  = P (I)  + P (II)  + P (III)  + P (IV )  + P (V )  + P (VI)  where



P(I)  = S1 (r) exp[−i(ω3 t + ω1 τ )]



+∞

dvw(v) −∞



∞

dt3 0



∞

∞

dt2 0

exp[−iθI (v)]H3 (t3 )H2 (t2 )H1 (t1 ) exp(−α1 |t1 − τ |)

dt1 ×

0

(3.23)

74

3 Attosecond Polarization Beats

 P(II)  = S1 (r) exp[−i(ω3 t + ω1 τ )]



+∞ −∞



∞

dvw(v)



∞

dt3 0

0

0

0

0

0



∞

∞

dt1 × (3.25)

∞

dt1 ×

0

exp[−iθII (v)]H3 (t3 )H5 (t2 )[H1 (t1 )]∗ exp(−α1 |t1 + τ |)  +∞  ∞  dvw(v) dt3 P(V )  = S2 (r) exp[−i(ω3 t − ω2 τ − ω2 δτ ] −∞

(3.24)

0

exp[−iθI (v)]H3 (t3 )H5 (t2 )H1 (t1 ) exp(−α1 |t1 − τ |)  +∞  ∞  ∞  P(IV )  = S1 (r) exp[−i(ω3 t + ω1 τ )] dvw(v) dt3 dt2 −∞

dt1 ×

0

exp[−iθII (v)]H3 (t3 )H2 (t2 )[H1 (t1 )]∗ exp(−α1 |t1 + τ |)  +∞  ∞  ∞  P(III)  = S1 (r) exp[−i(ω3 t + ω1 τ )] dvw(v) dt3 dt2 −∞

∞

dt2

0

(3.26) ∞

dt2 ×

0

dt1 exp[−iθIII (v)]H3 (t3 )H2 (t2 )H4 (t1 ) ×

0

exp(−α2 |t1 + τ + δτ |)

(3.27) 

P(VI)  = S2 (r) exp[−i(ω3 t − ω2 τ − ω2 δτ ] 

∞



+∞

∞

dvw(v) −∞

 dt3

0

∞

dt2 ×

0

dt1 exp[−iθIV (v)]H3 (t3 )H2 (t2 )[H4 (t1 )]∗ ×

0

exp(−α2 |t1 − τ − δτ |)

(3.28)

We discuss the SFPB in a Doppler-broadened system. After calculating the tedious integral in Eqs. (3.23)–(3.28) over t1 , t2 and t3 we obtain for: (i) τ > −δτ  +∞ L1 (v) + L3 (v) × P (3)  = S1 (r) exp(−iω3 t) dvw (v) Γ10 − α1 + i(Δ1 + k1 · v) −∞ % [2Γ10 + i(k1 − k1 ) · v] exp(−α1 |τ | − iω1 τ ) − Γ10 + α1 − i(Δ1 + k1 · v) & 2α1 exp[−Γ10 |τ | − i(Ω1 + k1 · v)τ ] + S2 (r) × Γ10 + α1 + i(Δ1 + k1 · v)  +∞ dvw (v) L2 (v) × exp(−iω3 t) %

−∞

[2Γ20 + i(k2 − k2 ) · v] exp[−α2 |τ + δτ | + iω2 (τ + δτ )] − [Γ20 + α2 + i(Δ2 + k2 · v)][Γ20 − α2 − i(Δ2 + k2 · v)] & 2α2 exp[−Γ20 |τ + δτ | + i(Ω2 + k2 · v)(τ + δτ )] (3.29) [Γ20 − i(Δ2 + k2 · v)]2 − α22

3.3 Second-order Stochastic Correlation of SFPB

75

(ii) −δτ > τ > 0  P

(3)

 = S1 (r) exp(−iω3 t) %

+∞

dvw (v) −∞

L1 (v) + L3 (v) × Γ10 − α1 + i(Δ1 + k1 · v)

k1 )

[2Γ10 + i(k1 − · v] exp(−α1 |τ | − iω1 τ ) − Γ10 + α1 − i(Δ1 + k1 · v) & 2α1 exp[−Γ10 |τ | − i(Ω1 + k1 · v)τ ] + S2 (r) × Γ10 + α1 + i(Δ1 + k1 · v)  +∞ dvw (v) L2 (v) × exp(−iω3 t) %

−∞

[2Γ20 + i(k2 − k2 ) · v] exp[−α2 |τ + δτ | + iω2 (τ + δτ )] − [Γ20 − α2 + i(Δ2 + k2 · v)][Γ20 + α2 − i(Δ2 + k2 · v)] & 2α2 exp[−Γ20 |τ + δτ | + i(Ω2 + k2 · v)(τ + δτ )] (3.30) [Γ20 + i(Δ2 + k2 · v)]2 − α22

(iii) τ < 0  P

(3)

 = S1 (r) exp(−iω3 t) %

+∞

dvw (v) −∞

L1 (v) + L3 (v) × Γ10 − α1 − i(Δ1 + k1 · v)

k1 )

[2Γ10 + i(k1 − · v] exp(−α1 |τ | − iω1 τ ) − Γ10 + α1 + i(Δ1 + k1 · v) & 2α1 exp[−Γ10 |τ | − i(Ω1 + k1 · v)τ ] + S2 (r) × Γ10 + α1 − i(Δ1 + k1 · v)  +∞ dvw (v) L2 (v) × exp(−iω3 t) %

−∞

[2Γ20 + i(k2 − k2 ) · v] exp[−α2 |τ + δτ | + iω2 (τ + δτ )] − [Γ20 − α2 + i(Δ2 + k2 · v)][Γ20 + α2 − i(Δ2 + k2 · v)] & 2α2 exp[−Γ20 |τ + δτ | + i(Ω2 + k2 · v)(τ + δτ )] (3.31) [Γ20 + i(Δ2 + k2 · v)]2 − α22

Here 1 [Γ0 + i(k1 − k1 ) · v]{Γ10 + i[Δ3 + (k1 − k1 + k3 ) · v]} 1 L2 (v) = [Γ0 + i(k2 − k2 ) · v]{Γ20 + i[Δ3 + (k2 − k2 + k3 ) · v]} 1 L3 (v) =  [Γ1 + i(k1 − k1 ) · v]{Γ10 + i[Δ3 + (k1 − k1 + k3 ) · v]} L1 (v) =

(3.32) (3.33) (3.34)

In general temporal behaviors of the broadband one-photon resonant DFWM and NDFWM signals are complicated and asymmetric about τ = 0.

76

3 Attosecond Polarization Beats

However, the situation becomes much simpler when laser linewidths are much narrower than homogeneous linewidths of transitions (i.e., α1 << Γ10 and α2 << Γ20 ). Under the approximations of Γ10 |τ | >> 1 and Γ20 |τ | >> 1, One can write, for τ > −δτ , −δτ > τ > 0 and τ < 0, P (3)  = S1 (r) exp(−iω3 t){B1 exp(−α1 |τ | − iω1 τ ) + ηB2 exp[−α2 |τ + δτ | − iΔk · r + iω2 (τ + δτ )]}

(3.35)

Here, η = μ22 ε2 (ε2 )∗ /[μ21 ε1 (ε1 )∗ ] is the ratio of the Rabi frequencies for NDFWM and DFWM processes; Δk = (k1 − k1 ) − (k2 − k2 ) is the spatial modulation factor. The constant Bi mainly depends on laser linewidths and relaxation rates of transitions. B1 and B2 are τ -independent factors given by  +∞ [2Γ10 + i(k1 − k1 ) · v][L1 (v) + L3 (v)] (3.36) B1 = dvw (v) [Γ10 + i(Δ1 + k1 · v)][Γ10 − i(Δ1 + k1 · v)] −∞  +∞ [2Γ20 + i(k2 − k2 ) · v]L2 (v) B2 = (3.37) dvw (v) [Γ20 + i(Δ2 + k2 · v)][Γ20 − i(Δ2 + k2 · v)] −∞ In a typical experiment involving one-photon DFWM and NDFWM processes [see Figs. 3.1 and 3.2], the intersection angle between these twincomposite beam 1 and beam 2 is small, and beam 3 basically propagates along the direction opposite to beam 1. Therefore, it is appropriate to make the following approximations: k3 ≈ −k3 z, k1 ≈ k1 z, k1 ≈ k1 z, k2 ≈ k2 z, and k2 ≈ k2 z. According to Eqs. (3.32), (3.34)and(3.36) the conditions that an atom can interact simultaneously with three incident beams are Δ1 + k1 · v ≈ 0, Δ1 + k1 · v ≈ 0 and Δ1 + (k1 − k1 + k3 ) · v = Δ1 − k1 · v ≈ 0. Since k1 ≈ k1 , only atoms with velocities centered on k1 · v = 0 are effective in generating the conjugate signal. Therefore, as in the case of saturated absorption spectroscopy, the Doppler-free DFWM spectrum has a peak located at Δ1 = 0. We then fix the frequency of the beam 3 and perform the NDFWM experiment with beam 1 and beam 2 containing only the ω2 frequency component. Since only atoms with velocities satisfying k1 · v ≈ 0 interact with beam 3, from Eqs. (3.33) and (2.37) the condition for beam 1 and beam 2 to interact with the same group of atoms is Δ2 = 0. Again, the NDFWM spectrum is Doppler-free since only atoms in specific velocity group contribute to the NDFWM signal. The above point is confirmed in the extreme Doppler-broadening limit. As we will discuss later [the first or second term of Eq. (3.45)], the Doppler-free linewidth (FWHM) equals approximately to 2Γa10 and 2Γa20 for DFWM and NDFWM processes, respectively. If the Doppler effect is neglected (in the limit of no inhomogeneous broadening), we have B1 ≈ 2Γ10 (Γ0 + Γ1 )/[(Γ10 + iΔ3 )(Γ210 + Δ21 )Γ0 Γ1 ] B2 ≈ 2Γ20 /[(Γ20 + iΔ3 )(Γ220 + Δ22 )Γ0 ]

3.3 Second-order Stochastic Correlation of SFPB

77

Fig. 3.2. Double-sided Feynman diagrams representing the Liouville pathways for P (I) , P (II) , P (III) , P (IV ) , P (V ) , and P (VI) , respectively. (a) One-photon DFWM of SFPB or DFPB; (b) one-photon NDFWM of SFPB, and DFPB.

According to Eq. (3.35), the SFPB signal intensity with the τ -dependence is (the τ -independent term is absent in the second-order stochastic correlation) 2

2

I (τ, r) ∝ |P (3) |2 ∝ |B1 | exp (−2α1 |τ |) + |ηB2 | exp (−2α2 |τ + δτ |) + exp(−α1 |τ | − α2 |τ + δτ |){B1∗ ηB2 × exp[−i(Δk · r − (ω2 + ω1 ) τ − ω2 δτ )] + B1 η ∗ B2∗ exp[i(Δk · r − (ω2 + ω1 ) τ − ω2 δτ )]}

(3.38)

With a balanced dispersion (i.e., δτ = 0), the SFPB signal depends only on laser characteristics and exhibits a damping oscillation with the sum frequency ω2 + ω1 and damping rate α1 + α2 for both τ > 0 and τ < 0. The theoretical limit at which the modulation frequency can be measured is determined by the laser linewidth [i.e., π(α1 + α2 )]. For narrow-band laser sources the modulation frequency can be measured with great accuracy. Therefore, the precision of using SFPB to measure Ω2 + Ω1 is determined by how well ω1 and ω2 can be tuned to Ω1 and Ω2 . One-photon resonant DFWM can provide a Doppler-free spectrum with a peak located at Δ3 = 0 [44]. When ω3 is set to the center of the Doppler profile, as discussed above, ω2 can also be tuned to Ω2 with a Doppler-free accuracy.

78

3 Attosecond Polarization Beats

However, as the laser bandwidths are comparable with the homogeneous linewidth of the transition in Eqs. (3.29)–(3.31), the one-photon resonant term [The factor exp(−Γ10 |τ | − iΩ1 τ ) or exp[−Γ20 |τ + δτ | + iΩ2 (τ + δτ )] reflects the free evolution of the one-photon coherence resonance for DFWM or NDFWM and becomes important. When the laser sources are broadband so that α1 >> Γ10 and α2 >> Γ20 , the SFPB beat signal rises to its maximum quickly and then decays with a time constant mainly determined by the transverse relaxation time of the atomic system. From Eqs. (3.29)–(3.31) and under the condition α1 |τ | >> 1 and α2 |τ | >> 1, the modulated SFPB signal can be simplified as I (τ, r) ∝ |P (3) |2 ∝ |B3 |2 exp(−2Γ10 |τ |) + |ηB4 |2 exp(−2Γ20 |τ + δτ |) + exp(−Γ10 |τ | − Γ20 |τ + δτ |){B3∗ × ηB4 exp[−i(Δk · r − (Ω2 + Ω1 )τ − Ω2 δτ )] + B3 η ∗ B4∗ exp[i(Δk · r − (Ω2 + Ω1 )τ − Ω2 δτ )]}

(3.39)

For simplicity, we neglect the Doppler effect (pure homogeneous broadening) in B3 and B4 , and write B3 ≈ 2α1 (Γ0 + Γ1 )/[(Γ10 + iΔ3 )(α21 + Δ21 )Γ0 Γ1 ] and B4 ≈ 2α2 /[(Γ20 + iΔ3 )(α22 + Δ22 )Γ0 ]. According to Eq. (3.39), when the dispersion is balanced (i.e., δτ = 0), the SFPB signal exhibits a damping oscillation with frequency Ω2 + Ω1 and a damping rate Γ10 + Γ20 . Therefore, the modulation frequency, corresponding directly to the sum of the resonant frequencies of the three-level V system, can be measured with an accuracy approximately given by π(Γ10 + Γ20 ). To get an analytical expression, the total polarization beat signal can be calculated from a different way. Under the extreme Doppler-broadening limit (i.e., k3 u → ∞), we have 

+∞

−∞



+∞

−∞



+∞

−∞



+∞

−∞

dvw(v) exp[−iθI (v)] ≈

dvw(v) exp[−iθII (v)] ≈

dvw(v) exp[−iθIII (v)] ≈ dvw(v) exp[−iθIV (v)] ≈

√ 2 π δ(t3 − ξ1 t1 ) k3 u √ 2 π δ(t3 + ξ1 t1 ) k3 u √ 2 π δ(t3 − ξ2 t1 ) k3 u √ 2 π δ(t3 + ξ2 t1 ) k3 u

(3.40)

(3.41)

(3.42)

(3.43)

When Eqs. (3.22), (3.40)–(3.43) are substituted into Eqs. (3.23)–(3.28), we obtain the stochastic average of the total third-order polarization as P (3)  = P (I) + P (III ) + P (V ) . Notice that because of the δ functions in Eqs. (3.41) and (3.43), we have P (II) = P (IV ) = P (VI) = 0 since t1 , t3 > 0. Let’s first consider the situation when laser linewidths are much narrower than the

3.3 Second-order Stochastic Correlation of SFPB

79

homogeneous linewidths of the transitions (i.e., α1 << Γa10 and α2 << Γa20 ). Under the approximations of Γa10 |τ | >> 1 and Γa20 |τ | >> 1, we have, for τ > −δτ , −δτ > τ > 0 and τ < 0, ! √ exp(−α1 |τ | − iω1 τ ) 1 2 π 1 (3) S1 (r) exp(−iω3 t) ( + )+ P  = a a k3 u (Γ10 + iΔ1 ) Γ0 Γ1 " η exp(−α2 |τ + δτ | − iΔk · r + iω2 τ + iω2 δτ ) (3.44) (Γa20 + iΔa2 )Γ0 where, Γa10 = Γ10 + ξ1 Γ10 , Δa1 = Δ1 + ξ1 Δ3 , Γa20 = Γ20 + ξ2 Γ10 , and Δa2 = Δ2 +ξ2 Δ3 . Equation (3.44) shows that for the laser frequency ω1 , the DFWM spectrum has a linewidth 2Γa10 centered at Δ1 = 0. Furthermore, by fixing the frequency of beam 3 to Ω1 , the maximum signal intensity of the NDFWM (with twin beam 1 and beam 2 consisting of only the ω2 frequency component) occurs at Δ2 = 0 with linewidth 2Γa20 . Therefore, although the modulation frequency of the SFPB signal can be measured with a high precision when twin beams 1 and 2 are nearly monochromatic, the overall accuracy of the SFPB in a Doppler-broadened system is determined by the homogeneous linewidths of the optical transitions. The second-order correlation SFPB signal intensity is calculated to be I (τ, r) ∝ |P (3) |2 ∝

2

2

(Γ0 + Γ1 ) exp (−2α1 |τ |) |η| exp(−2α2 |τ + δτ |) + + [(Γa10 )2 + (Δa1 )2 ]Γ20 Γ21 [(Γa20 )2 + (Δa2 )2 ]Γ20

exp (−α1 |τ | − α2 |τ + δτ |) {B5 exp[−i(Δk · r − (ω2 + ω1 ) τ − (3.45) ω2 δτ )] + B5∗ exp[i(Δk · r − (ω2 + ω1 ) τ − ω2 δτ )]} where B5 = η(Γ0 + Γ1 )/[(Γa10 − iΔa1 )(Γa20 + iΔa2 )Γ20 Γ1 ]. The one-photon DFWM signal (the first term in Eq. (3.45)) has a resonance at Δa1 = 0 with linewidth 2Γa10 . Similarly, the one-photon NDFWM signal [the second term in Eq. (3.45)] has a resonance at Δa2 = 0 with linewidth 2Γa20 . SFPB with narrow band lights is again a Doppler-free precision spectroscopy. This result is consistent with Eq. (3.38). When dispersion is balanced (i.e., δτ = 0), the second-order correlation SFPB signal intensity is also modulated with a sum frequency ω2 + ω1 as τ is varied. We now consider temporal behaviors of the second-order SFPB signal intensity when laser beams are broadband so that α1 >> Γa10 and α2 >> Γa20 . In this case, the SFPB signal rises to its maximum quickly and then decays with a time constant mainly determined by the transverse relaxation time of the system. At the tail of the signal (i.e., α1 |τ | >> 1 or α2 |τ | >> 1), the modulated SFPB signal intensity can be approximately written as: (i) 0 < τ < −δτ % √ 2α1 exp[−Γa10 |τ | − i(Ω1 + ξ1 Δ3 )τ ] 2 π S1 (r) exp(−iω3 t) P (3)  = × k3 u α21 + (Δa1 )2  &  1 1 η exp[−Γa20 |τ + δτ | − iΔk · r + i(Ω2 + ξ2 Δ3 )(τ + δτ )] + Γ0 Γ1 (α2 + iΔa2 )Γ0

80

3 Attosecond Polarization Beats

I (τ, r) ∝ |P (3) |2 ∝

4α21 (Γ0 + Γ1 )2 exp(−2Γa10 |τ |) + [α21 + (Δa1 )2 ]2 Γ20 Γ21

2

|η| exp(−2Γa20 |τ + δτ |) + exp (−Γa10 |τ | − Γa20 |τ + δτ |) × [α22 + (Δa2 )2 ]Γ20 {B6 exp[−i(Δk · r − (Ω2 + Ω1 + (ξ1 + ξ2 )Δ3 ) τ − (Ω2 + ξ2 Δ3 )δτ )] + B6∗ exp[i(Δk · r − (Ω2 + Ω1 + (ξ1 + ξ2 )Δ3 )τ − (Ω2 + ξ2 Δ3 )δτ )]} where B6 = (ii) τ < 0

[α21

(3.46)

2α1 η(Γ0 + Γ1 ) . + (Δa1 )2 ](α2 + iΔa2 )Γ20 Γ1

I (τ, r) ∝ |P (3) |2 ∝

4α22 exp(−2Γa20 |τ + δτ |) [α22 + (Δa2 )2 ]2 Γ20

(3.47)

(iii) τ > −δτ I (τ, r) ∝ |P (3) |2 ∝ 

4α21 (Γ0 + Γ1 )2 exp (−2Γa10 |τ |) 2

[ξ12 Γ210 − (Δa2 )2 − α21 ] + 4ξ12 Γ210 (Δa2 )2 Γ20 Γ21 (3.48)

Equation (3.46) is basically consistent with Eq. (3.39). Equation (3.47) only shows the term with one-photon resonant NDFWM auto-correlation with an asymmetric factor exp(−2Γa20 |τ + δτ |). Due to the approximation α1 |τ | >> 1 and α2 |τ | >> 1, the nonresonant DFWM auto-correlation term (with a symmetric factor exp(−2α1 |τ |)), and the nonresonant DFWM and resonant NDFWM cross-correlation terms (with the factor exp(−2α1 |τ | − 2Γa20 |τ + δτ |)) are neglected. By contrast, Eq. (3.48) only shows the onephoton resonant DFWM auto-correlation term with a symmetric factor exp(−2Γa10 |τ |). Note that this cross interference between resonant and nonresonant processes is dramatically different from that of the femtosecond DFPB described in Section 2.1 of the Chapter 2 [30 – 38]. Ulness, et al. [18] directly observed the resonant–nonresonant cross-term contribution to the coherent Raman scattering of a quasi-cw noisy light in molecular liquids. It is important to understand the underlying physics in SFPB with broadband nontransform limited quasi-cw (noisy) lights [13]. For the phase- matching conditions k1 − k1 + k3 and k2 − k2 + k3 , the three-pulse stimulated photon-echo exists for the perturbation chains (I), (III), and (V) [12]. The broadband limit (with laser coherence time τc ≈ 0) corresponds to “white” noise, i.e. noise can be characterized by a δ-function correlation time or, alternatively, it possesses a constant spectrum. So, the second-order correlation function Eq. (3.22) can be approximated by a δ-function ui (t1 ) u∗i (t2 ) = exp (−αi |t1 − t2 |) ≈

2 δ(t1 − t2 ) αi

(3.49)

3.3 Second-order Stochastic Correlation of SFPB

81

Under the large Doppler-broadening limit (i.e., k3 u → ∞) and laser broadband limit (i.e., αi → ∞) approximations, substituting Eqs. (3.40)– (3.43), and (3.49) into Eqs. (3.23)–(3.28) gives √  ∞  ∞  ∞ 4 π P (I)  = S1 (r) exp [−i (ω3 t + ω1 τ )] dt3 dt2 dt1 × k3 uα1 0 0 0 δ(t3 − ξ1 t1 )δ(t1 − τ ) exp[−(Γ10 + iΔ3 )t3 − (3.50) Γ0 t2 − (Γ10 + iΔ1 )t1 ] √  ∞  ∞  ∞ 4 π P (III )  = S1 (r) exp [−i (ω3 t + ω1 τ )] dt3 dt2 dt1 × k3 uα1 0 0 0 δ(t3 − ξ1 t1 )δ(t1 − τ ) exp[−(Γ10 + iΔ3 )t3 − (3.51) Γ1 t2 − (Γ10 + iΔ1 )t1 ] √  ∞  ∞  ∞ 4 π P (V )  = S2 (r) exp{−i[ω3 t − ω2 (τ + δτ )]} dt3 dt2 dt1 × k3 uα2 0 0 0 δ(t3 − ξ2 t1 )δ(t1 + τ + δτ ) exp[−(Γ10 + iΔ3 )t3 − Γ0 t2 − (Γ20 + iΔ2 )t1 ]

(3.52)

Since the one-photon DFWM of attosecond SFPB is similar to the case of femtosecond DFPB as described in Section 2. 1 of Chapter 2[30 – 40], here we will mainly concentrate on the one-photon NDFWM in SFPB. Equation (3.52) can be explained as follows [see Eqs. (3.4), (2.6), and (2.8) also]: The optical polarization induced by A2 at time t − t1 − t2 − t3 exhibits a damping oscillation in a time interval t1 with decay rate Γ20 and frequency Δ2 in a rotating frame. It then interacts with A2 at time t − t2 − t3 ; as a result, (0) A

 ∗ (1) (A )

(2)

1 ρ10 − −−1− → ρ00 ) is induced. The population a population grating (ρ00 −−→ grating undergoes a damping oscillation with decay rate Γ0 . After time t2 , beam 3 probes the population grating of state |0 at time t − t3 and induces a polarization that is responsible for the one-photon NDFWM signal. We are interested in the NDFWM signal at time t, which is t3 after the application of beam 3. Here, again the polarization exhibits a damping oscillation in the time interval t3 with decay rate Γ10 and frequency Δ1 . As a result of the distribution of resonant frequencies in an inhomogeneously broadened system, the dipoles induced by A2 at t − t1 − t2 − t3 will soon run out of phase with one another. The dephased dipoles can join phrases again after the application of beam 2, and the three-pulse stimulated photon-echo will then appear [12]. From Eq. (3.52), the three-pulse stimulated photon-echo occurs at time t when t3 = ξ2 t1 . We consider the case when twin beams 1 and 2 have broadband linewidths, so beams can be modeled as a sequence of short, phase-incoherent subpulses of duration τc , where τc is the laser coherence time. Although the NDFWM signal can be generated by any pair of subpulses in twin beams 1 and 2, only those pairs that are phase-

82

3 Attosecond Polarization Beats

correlated give rise to the τ -dependence of the NDFWM signal. When the ω2 component of beam 1 is delayed with respect to the ω2 component of beam 2 by τ , the time duration t1 between the phase-correlated subpulses in beam 1 and beam 2 should equal to −τ −δτ for the NDFWM. The total third-order polarization is the accumulation of the polarizations induced at the different time. The maximum of the resonant one-photon NDFWM signal occurs at τ ≈ −δτ . The total SFPB signal intensity is: (i) 0 < τ < −δτ % √ Γ 0 + Γ1 4 π (3) S1 (r) exp(−iω3 t) exp[−Γa10 |τ | − i(Ω1 + ξ1 Δ3 )τ ] + P  = k3 u α1 Γ0 Γ1 & η exp[−Γa20 |τ + δτ | − iΔk · r + i(Ω2 + ξ2 Δ3 )(τ + δτ )] α2 Γ0 I (τ, r) ∝ |P (3) |2 ∝

2

(Γ0 + Γ1 )2 exp(−2Γa10 |τ |) |η| exp(−2Γa20 |τ + δτ |) + + α21 Γ20 Γ21 α22 Γ20

(Γ0 + Γ1 ) exp(−Γa10 |τ | − Γa20 |τ + δτ |){η exp[−i(Δk · r − Γ20 Γ1 α2 α1 (Ω2 + Ω1 + (ξ1 + ξ2 )Δ3 ) τ − (Ω2 + ξ2 Δ3 )δτ )] + η ∗ exp[i(Δk · r − (Ω2 + Ω1 + (ξ1 + ξ2 )Δ3 ) τ − (Ω2 + ξ2 Δ3 )δτ )]}

(3.53)

(ii) τ < 0 I (τ, r) ∝ |P (3) |2 ∝

1 exp(−2Γa20 |τ + δτ |) Γ20

(3.54)

(Γ0 + Γ1 )2 exp(−2Γa10 |τ |) Γ20 Γ21

(3.55)

(iii) τ > −δτ I (τ, r) ∝ |P (3) |2 ∝

Equations (3.53)–(3.55) are basically consistent with the results obtained in Eqs. (3.46)–(3.48). The second-order correlation SFPB signal can be calculated under balanced and controllable dispersion (i.e., δτ = 0). Under the large Dopplerbroadening limit (i.e., k3 u → ∞) and laser broadband limit (i.e., αi → ∞) approximations, when Eqs. (3.22), (3.40)–(3.43) are substituted into Eqs. (3.23)–(3.28), the total third-order polarization and the SFPB signal intensity are given as follows: (i) τ < 0 and α2 |τ | >> 1 √ 2 π S1 (r) exp(−iω3 t){B7 exp(−α1 |τ | − iω1 τ )+ P (3)  = k3 u

3.3 Second-order Stochastic Correlation of SFPB

83

η exp(−α2 |τ | + iω2 τ − iΔk · r) + (Γa20 − α2 + iΔa2 )Γ0 ηB8 exp[−Γa20 |τ | − iΔk · r + i(Ω2 + ξ2 Δ3 )τ ]}

(3.56)

I (τ, r) ∝ |P (3) |2 ∝ |B7 |2 exp (−2α1 |τ |) + |ηB8 |2 exp(−2Γa20 |τ |) + exp[−(α1 + Γa20 ) |τ |]{B7∗ ηB8 × exp[−i(Δk · r − (Ω2 + ω1 + ξ2 Δ3 ) τ )] + B7 η ∗ B8∗ exp[i(Δk · r − (Ω2 + ω1 + ξ2 Δ3 ) τ )]}

(3.57)

(ii) τ > 0 and α1 |τ | >> 1 √ 2 π P (3)  = S1 (r) exp(−iω3 t){ηB10 exp(−α2 |τ | + iω2 τ − iΔk · r) + k3 u (Γ0 + Γ1 ) exp(−α1 |τ | − iω1 τ ) + (iΔa1 + Γa10 − α1 )Γ0 Γ1 B9 exp[−Γa10 |τ | − i(Ω1 + ξ1 Δ3 )τ ]} 2

(3.58) 2

I (τ, r) ∝ |P (3) |2 ∝ |B9 | exp(−2Γa10 |τ |) + |ηB10 | exp(−2α2 |τ |) + exp[−(Γa10 + α2 ) |τ |]{B9∗ ηB10 × exp[−i(Δk · r − ∗ (ω2 + Ω1 + ξ1 Δ3 ) τ )] + B9 η ∗ B10 exp[i(Δk · r −

(ω2 + Ω1 + ξ1 Δ3 ) τ )]}

(3.59)

where B7 = (Γ0 + Γ1 )/[(α1 + Γa10 + iΔa1 )Γ0 Γ1 ]  B8 = 2α2 {[α22 − (Γa20 + iΔa2 )2 ]Γ0 }  B9 = 2α1 (Γ0 + Γ1 ) {[α21 − (Γa10 + iΔa1 )2 ]Γ0 Γ1 }  B10 = 1 [(α2 + Γa20 + iΔa2 )Γ0 ] In general, both homogeneous and inhomogeneous broadenings are included in the material response functions in the standard fashion. Understanding the mechanisms involved in coherence decay (T2 ) and population decay (T1 ) is of great theoretical and practical interests. In the absence of inhomogeneous broadening, the experimental linewidth contains contributions from the population decays and the pure dephasing processes, and the total width (the dephasing time T2 ) is related to the pure dephasing  time T2 and the population relaxation time T1 as 1/T2 = 1/T2 + 1 T1 . In the inhomogeneously broadened system, the dephasing time T2 is  related to  the pure  dephasing  ∗ time T2 and the inhomogeneous dephasing time ∗  T2 as 1 T2 = 1 T2 + 1 T2 . Specifically, the loss of quantum mechanical coherence, a T2 process, in gas phase samples is caused by both homogeneous (T2 , collisions) and inhomogeneous (T2∗ , for example Doppler broadening) broadenings. The characteristics of the interferogram are a result of

84

3 Attosecond Polarization Beats

two main components: the material response and the light response along with the interplay between the two responses. In general, the two-sided time symmetry of the noisy light is seen experimentally in the autocorrelation [25]. By contrast, in this same sense the material response function of the chromophore has one-sided time symmetry. Figures 3.3 and 3.4 present spectra for the cases of one-photon DFWM (τ > 0) and one-photon NDFWM (τ < 0), respectively. In the Fig 3.3,  a a a −4 Γ /α for (a) α = 1, Γ , DFWM signal intensity versus Δ 1 1 0 /Γ10 = 1  a 10 a  10 −4 a −4 a Γ1 Γ10 = 3.35 × 10 and (b) α1 Γ10 = 100, Γ0 Γ10 = 10 , Γ1 Γ10 = 3.35 × 10−4 . Γa10 τ = 0 (dash-dotted curve), 1.5 (dotted curve), 3 (dashed curve), 10 (solid curve). The curves of Fig. 3.3 are normalized within each set. It is interesting to know how the peak values of different curves are related to one another in Fig.3.4 where NDFWM signal intensity versus    = 10−4 , Γ1 Γa10 = 3.35 × 10−4 and (b) Δa2 α2 for (a) α2 Γa20 = 1, Γ0 Γa10  α2 Γa20 = 100, Γ0 Γa10 = 10−4 , Γ1 Γa10 = 3.35 × 10−4 . Γa20 τ = 0 (dashdotted curve), 1.5 (dotted curve), 3 (dashed curve), 10 (solid curve). The four curve signal sizes of Fig. 3.4 (a) are scaled by a factor of 1.0, 1.2, 7.1 and 8.3 × 105 , while the four curve signal sizes of Fig. 3.4 (b) are scaled by a factor of 2.5 × 103 , 1 × 104 , 1.3 × 105 , and 6.3 × 1010 , respectively. Such information is lost in the individual normalization (see Fig. 3.3). We can see that, both in Figs. 3.3 and 3.4, the line shape at the α2 /Γa20 = 1 case is much sensitive to the Γa10 τ (or Γa20 τ ) variation for the twin-field correlations than that at α1 /Γa10 = 100 (or α2 /Γa20 = 100) broadband case. More specifically, the four curves in Fig. 3.3(b) are partially overlapping. When the twin fields become almost uncorrelated (at large Γa10 τ and Γa20 τ cases), aweak damping oscillation in frequency space with a modulation period of 2π τ is showed in Fig. 3.3 (a) [or 3.4 (a)]. According to Eqs. (3.56) and (3.58), the one-photon DFWM depends on the laser coherence time when τ < 0, and it depends on both the laser coherence time and the transverse relaxation time of the population grating when τ > 0. In contrast, the one-photon NDFWM signal depends on the laser coherence time when τ > 0, while depends on both the laser coherence time and the transverse relaxation time of the population grating when τ < 0. In general, for noisy light the nonresonant frequency component has just as much to do with the material as the resonant one. When the broadband laser frequency is “off resonance” from the atomic transition (some frequencies within the bandwidth of the noisy light may still act ), the DFWM and NDFWM signals exhibit hybrid radiation-matter detuning terahertz damping oscillations, as shown in Figs. 3.5 and Here, the DFWM  3.6, respectively.  signal intensity versus Γa10 τ for (a) α1 Γa10 = 1, Γ0 Γa10 = 10−4 , Γ1 /Γa10 = 3.35 × 10−4 and (b) α1 /Γa10 = 100, Γ0 Γa10 = 10−4 , Γ1 /Γa10 = 3.35 × 10−4 . Δa1 /α1 = 3 (dash-dotted curve), 7 (dotted curve), 10 (dashed curve), 20 (solid curve) in the Fig. 3.5. Similarly, the NDFWM signal intensityversus Γa20 τ for (a) α2 /Γa20 = 1, Γ0 Γa10 = 10−4 and (b) α2 Γa20 = 100, Γ0 Γa10 =

3.3 Second-order Stochastic Correlation of SFPB

85

‹ Fig. 3.3. Normalized one-photon DFWM signal intensity versus Δa1 α1 . Adopted from Ref. [45].

‹ Fig. 3.4. Normalized one-photon NDFWM signal intensity versus Δa2 α2 . Adopted from Ref. [45].

 10−4 . Δa2 α2 = 3 (dash-dotted curve), 7 (dotted curve), 10 (dashed curve), 20 (solid curve) in the Fig. 3.6. This is similar to the radiation difference oscillation or Rabi detuning oscillation [24]. The Rabi detuning oscillations and the radiation difference oscillation are exactly synonymous [25]. More specifically, the last two parts in Eq. (3.56) or Eq. (3.58) interfere and give rise to a modulation of the signal intensity versus Γa10 τ when τ > 0 for the DFWM as shown in Fig. 3.5, or versus Γa20 τ when τ < 0 for NDFWM as shown in Fig. 3.6, respectively. The complicated modulation frequency of the radiation-matter detuning beats depends sensitively on Δa1 (see Fig. 3.5) or Δa2 (see Fig. 3.6). Based on the phase-conjugate polarization interference between two onephoton processes and at the large Doppler-broadening limit, we have obtained an analytic and closed form for the second-order stochastic correlation of V-type three-level SFPB in attosecond time scale. Such treatment for the phase coherent control of the SFPB might be enough. According to Eqs. (3.56) and (3.58), a novel interferometric oscillatory behavior is exposed in terms of radiation-matter detuning beats and radiation-radiation, radiationmatter and matter-matter polarization beats. As shown in the Fig. 3.7,

86

3 Attosecond Polarization Beats

Fig. 3.5. Normalized one-photon DFWM signal intensity versus Γa10 τ .

Fig. 3.6. Normalized one-photon NDFWM signal intensity versus Γa20 τ .

the second-order correlation SFPB signal intensity versus time delay τ for Γ0 /Γa20 = 10−4 , Γ1 /Γa20 = 3.35 × 10−4 , Γa10 /Γa20 = 0.9, ω1 /Γa20 = 18 210.6, ω2 /Γa20 = 18 229, η = 1. Δa1 /α1 = Δa2 /α2 = 0, α1 /Γa20 = α2 /Γa20 = 5 000 (dash-dotted curve), 6000  (dotted curve), 7 500 (dashed curve), 9 000 (solid curve) for (a); while α1 Γa20 = α2 Γa20 = 6000, Δa1 /α1 = Δa2 /α2 = 0 (dashdotted curve), 3.5 (dotted curve), 4 (dashed curve), 5.5 (solid curve) for (b). The interferograms of Figs. 3.7(a) and 3.7(b) come from the closed combination of the analytic forms given in Eqs. (3.56) and (3.58). When Δa1 /α1 = Δa2 /α2 = 0, the second-order SFPB signals, with attosecond time scale beats, show significant differences for different laser bandwidths in Fig. 3.7(a). On the other hand, the attosecond modulation of the second order SFPB signal is strongly distorted by the Δa1 α1 = Δa2 /α2 factor in Fig. 3.7(a). The phase coherent control of light beams in SFPB is subtle. We consider the case when pump lasers have narrow bandwidths. The nonlinear polarization, responsible for the one-photon DFWM from the ω1 -frequency component of beam 1 and beam 2, is given by P1 (t) = χ(3) (ω3 ; ω1 , −ω1 , ω3 )E1 (t)E2∗ (t)E3 (t) where χ(3) (ω3 ; ω1 , −ω1 , ω3 ) is the nonlinear susceptibility. If time delay τ be-

3.3 Second-order Stochastic Correlation of SFPB

87

Fig. 3.7. Normalized second-order correlation SFPB signal intensity versus time delay τ .

tween beam 1 and beam 2, which come from a single light source, is shorter than the coherence time τc of the laser (i.e., αi |τ | << 1), the relative phase between E1 (t) and E2 (t) will be given by P1 (t) ∝ exp(−i ω3 t) exp(−i ω1 τ ). Similarly, the nonlinear polarization from ω2 -frequency components of beam 1 and beam 2 is given by P2 (t) ∝ exp(−iω3 t) exp(iω2 τ ). The interference between P1 (t) and P2 (t) causes the signal intensity to modulate with frequency ω1 + ω2 as τ is varied. However, the relative phase between E1 (t) and E2 (t) will fluctuate randomly if τ >> τc (i.e., αi |τ | >> 1). In this case, the modulation of FWM signal intensity disappears. Now, we consider the case when pump lasers have broadband linewidths and the FWM signal come from the ω1 -frequency components of beam 1 and beam 2. The time-delayed FWM, which is related to the three-pulse stimulated photonecho, originates from the interaction of atoms with the phase-correlated subpulses in beam 1 and beam 2, separated by time delay τ . For incoherent pump beams the factor exp(−α1 |t1 − τ |) in Eqs. (3.23) and (3.25) can be taken as a delta function, i.e., exp(−α1 |t1 − τ |) ≈ (2/α1 )δ(t1 − τ ). Then, in the extreme Doppler-broadening limit, we have the expression, from Eqs. (3.23) and (3.25), P1 (t) ∝ exp(−iω3 t) exp[−Γa10 |τ | − i(Ω1 + ξ1 Δ3 )τ ] for τ > 0. The phase factor exp[−i(Ω1 + ξ1 Δ3 )τ ] reflects the free evolution of the one-photon coherence during the time interval τ . Similarly, the nonlinear polarization from the ω2 -frequency components of beam 1 and beam 2 is P2 (t) ∝ exp(−iω3 t) exp(−α2 |τ | + iω2 τ ) under the balanced dispersion condition (i.e., δτ = 0). Then, the interference between P1 (t) and P2 (t) causes the signal intensity to modulate with frequency (ω2 + Ω1 ) + ξ1 Δ3 . Thus, the subtle polarization interference between two FWM pathways can be used to control the outcome of laser-matter interactions.

88

3 Attosecond Polarization Beats

3.4 Fourth-order Stochastic Correlation of SFPB For a macroscopic system where phase matching takes place the SFPB signal must be drawn from P (3) developed on one chromophore with one time variable multiplied by (P (3) )∗ developed on another chromophore with another time (located elsewhere in space), and then with summation over all such pairs [16 – 23]. In general, the signal is detected by homodyne technique. The signal at the detector is derived from the squared modulus of the sum over all the fields generated from the huge number of polarized chromophores in the interaction volume. The sum over these chromophores leads to the phase-matching condition at the signal level and its square modulus is dominated by bichromophoric cross terms. Thus, this detected quadrature signal is effectively built from the products of all polarization fields derived from all pairs of chromophores. This bichromophoric model is particularly important for the noisy light spectroscopy where the stochastic averaging at the signal level must be carried out [16 – 23]. Basically, the characteristics of the interferogram are a result of two main components: the material response and the light response along with the interplay between two responses [25]. Note that the three types of Markovian stochastic fields (i.e. chaotic field, phase-diffusion field, and Gaussian-amplitude field) can have the same spectral density and thus the same second-order coherence function. The fundamental differences in these fields are only manifested in the higherorder coherence functions. In this section, we will treat different stochastic fields up to the fourth-order coherence function. Physically, the intensity correlation of the chaotic field has the property of photon bunching, which can affect multi-photon processes when the fourth-order correlation function of the field plays an important role. The SFPB signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic process and is given by I (τ, r) ∝ |P (3) |2  = P (3) (P (3) )∗  = (P (I) + P (III ) + P (V ) )[(P (I) )∗ + (P (III) )∗ + (P (V ) )∗ ] which contains 3×3 = 9 different terms depending on the fourth- and secondorder coherence functions of ui (t) in the phase-conjugation geometry. The SFPB signal intensity in the self-diffraction geometry is related to the sixthorder coherence functions of the incident fields [26, 27]. We first consider the chaotic field, which is used to describe a multimode laser source and is characterized by the fluctuations in both the amplitude and the phase of the field. The stochastic functions ui (t)of the complex noisy fields are taken to obey complex circular Gaussian statistics with its fourth-order coherence function to satisfy [7, 8] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 ) + ui (t1 )u∗i (t4 )ui (t2 )u∗i (t3 )

(3.60)

3.4 Fourth-order Stochastic Correlation of SFPB

89

For simplicity, we neglect the Doppler effect in the following calculation. We first consider the case when laser sources are taken to be narrow- band so that α1 , α2 << Γ10 , Γ20 . The noisy beam 1 (beam 2) with two frequency components is treated as a spectrum with a sum of two Lorentzians. According to Eq. (3.60) and under the approximations of Γ10 |τ | >> 1 and Γ20 |τ | >> 1, the SFPB signal intensity in fourth-order stochastic correlations then becomes (for both τ > 0 and τ < 0), I (τ, r) ∝ |P (3) |2  ∝ B7 + |η|2 B8 + |B9 |2 exp(−2α1 |τ |) + 2

|ηB10 | exp(−2α2 |τ |) + exp[−(α1 + α2 ) |τ |] × {ηB9∗ B10 exp[−i(Δk · r − (ω2 + ω1 ) τ )] + ∗ η ∗ B9 B10 exp[i(Δk · r − (ω2 + ω1 ) τ )]}

where

(3.61)

% (Γ0 + Γ1 )2 − 16α21 Γ210 − (Γ210 + Δ21 )2 (Γ20 − 4α21 ) (Γ210 + Δ23 )(Γ21 − 4α21 ) & 2α1 (2Γ0 + Γ1 ) Γ0 (Γ0 + Γ1 )(Γ10 + iΔ3 )(Γ0 + Γ10 − iΔ3 ) % Γ220 1 − B8 = 2 2 2 2 2 2 (Γ20 + Δ2 ) (Γ0 − 4α2 ) (Γ10 + Δ23 ) & 2α2 Γ0 (Γ10 + iΔ3 )(Γ0 + Γ10 − iΔ3 ) Γ10 (Γ0 + Γ1 ) B9 = (Γ10 + iΔ3 )(Γ210 + Δ21 )Γ0 Γ1 Γ20 B10 = . (Γ10 + iΔ3 )(Γ220 + Δ22 )Γ0 B7 =

These constants (Bi ) mainly depend on typical parameters, i.e. the short correlation time of the light and rapid dephasing rate. Equation (3.61) consists of five terms. The first and third terms depend on the fourth-order coherence functions of u1 (t) for one-photon nonresonant DFWM, while the second and fourth terms depend on the fourth-order coherence functions of u2 (t) for one-photon nonresonant NDFWM. The first and second terms, which originate from the amplitude fluctuation of the chaotic field, are independent of the relative time-delay τ . The third and fourth terms include an exponential decay of the SFPB signal as |τ | increases. The fifth term, depending on the second-order coherence functions of u1 (t) or u2 (t), gives the cross interference between one-photon nonresonant DFWM and one-photon nonresonant NDFWM with a modulation frequency ω2 + ω1 and a decay rate α1 + α2 . Equation (3.61) indicates that the SFPB signal oscillates not only temporally but also spatially with a period 2π/Δk along the direction Δk (which is almost perpendicular to the propagation direction of the beat signal), where

90

3 Attosecond Polarization Beats

Δk ≈ 2π|λ1 − λ2 |θ/λ2 λ1 and θ is the angle between beam 1 and beam 2 [see Fig. 3.1 (b)]. The above polarization-beat model assumes that both pump beams are plane waves. Therefore, DFWM and NDFWM signals, which propagate along the directions of ks1 = k1 − k1 + k3 [see Fig. 3.2 (a)] and ks2 = k2 − k2 + k3 [see Fig. 3.2 (b)], respectively, are planewaves also. Since DFWM and NDFWM signals propagate along two slightly different directions, a spatial interference pattern is expected, which can be detected by a pinhole detector. Equation (3.61) also indicates that the SFPB signal has a temporal modulation with a frequency of ω2 + ω1 as τ is varied. In this case when ω1 and ω2 are tuned to the resonant frequencies of transitions (i.e. from |0 to |1 and from |0 to |2), then the modulation frequency equals to Ω2 + Ω1 . This means that we can obtain the sum-frequency beating between the resonant frequencies of the three-level V-type system. A Doppler-free precision can be achieved in the measurement of Ω2 + Ω1 [35, 36]. We now consider the case when the laser sources are broadband (α1 , α2 >> Γ10 , Γ20 ). Under the conditions of α1 |τ | >> 1 and α2 |τ | >> 1, the SFPB signal rises to its maximum value quickly and then decays with a time constant mainly determined by the transverse relaxation rates of the atomic system. 2

2

I (τ, r) ∝ |P (3) |2  ∝ B11 + |η| B12 + |B13 | exp (−2Γ10 |τ |) + 2

|ηB14 | exp (−2Γ20 |τ |) + exp[−(Γ10 + Γ20 ) |τ |] × ∗ {ηB13 B14 exp[−i(Δk · r − (Ω2 + Ω1 ) τ )] + ∗ ∗ exp[i(Δk · r − (Ω2 + Ω1 ) τ )]} η B13 B14

(3.62)

where B11 = B12 =

2(Γ210

+

Δ23 )(α21

α21 (α21 + Δ21 + Δ23 )(Γ20 + 6Γ0 Γ1 + Γ21 ) + Δ21 )[α21 + (Δ3 − Δ1 )2 ][α21 + (Δ3 + Δ1 )2 ]Γ0 Γ1 Γ10 (Γ0 + Γ1 )

α22 (α22 + Δ22 + Δ23 ) 2(Γ210 + Δ23 )(α22 + Δ22 )[α22 + (Δ3 − Δ2 )2 ][α22 + (Δ3 + Δ2 )2 ]Γ0 Γ20

α1 (Γ0 + Γ1 ) (Γ10 + iΔ3 )(α21 + Δ21 )Γ0 Γ1 α2 = (Γ10 + iΔ3 )(α22 + Δ22 )Γ0

B13 = B14

The SFPB signal can be calculated in the extreme Doppler broadening limit. By substituting Eqs. (3.22), (3.40)–(3.43) into Eqs. (3.16)–(3.21), we can obtain the stochastic average of the mod square of the total third-order polarization I(τ, r) ∝ |P (3) |2  = |P (I) + P (III) + (P (V ) )|2 

(3.63)

Let’s consider the case when the laser sources are taken to be narrow band, so that α1 , α2 << Γ10 , Γ20 , Γ10 , Γ20 >> Γ0 , Γ1 . Under the approximations of

3.4 Fourth-order Stochastic Correlation of SFPB

91

Γ10 |τ | >> 1 and Γ20 |τ | >> 1, we have, for both τ > 0 and τ < 0, 2

2

I (τ, r) ∝ |P (3) |2  ∝ B15 + |η| B16 + |B17 | exp (−2α1 |τ |) + 2

|ηB18 | exp (−2α2 |τ |) + exp[−(α1 + α2 ) |τ |] × ∗ B18 exp[−i(Δk · r − (ω2 + ω1 ) τ )] + {ηB17 ∗ η ∗ B17 B18 exp[i(Δk · r − (ω2 + ω1 ) τ )]}

where B15 =

B16 = B17 = B18 =

(3.64)

! 1 1 1 + + a a 2 2 (Γ10 ) + (Δ1 ) Γ0 (Γ0 + 2α1 ) Γ1 (Γ1 + 2α1 ) " 2(Γ0 Γ1 − 4α21 ) 4α1 (Γ20 + Γ21 − 8α21 ) − (Γ20 − 4α21 )(Γ21 − 4α21 ) (Γ0 + Γ1 )(Γ20 − 4α21 )(Γ21 − 4α21 ) 1 Γ0 (Γ0 + 2α2 )[(Γa20 )2 + (Δa2 )2 ] Γ 0 + Γ1 (Γa10 + iΔa1 )Γ0 Γ1 1 a (Γ20 + iΔa2 )Γ0

This equation is consistent with Eq. (3.61). We now consider the case when the laser sources are broadband with α1 >> Γa10 and α2 >> Γa20 . The SFPB signal rises to its maximum value quickly and then decays with a time constant mainly determined by the transverse relaxation rates of the system. The general expression of the modulated SFPB signal is complicated. However, under the conditions of α1 |τ | >> 1 or α2 |τ | >> 1, the simplified result of the SFPB can be written as: (i) τ < 0 and α2 |τ | >> 1 2

2

I (τ, r) ∝ |P (3) |2  ∝ B19 + |η| B20 + |B21 | exp (−2α1 |τ |) + 2

|η| B22 exp (−2Γa20 |τ |) + exp[−(α1 + Γa20 ) |τ |] × {ηB23 exp[−i(Δk · r − (Ω2 + ω1 + ξ2 Δ3 ) τ )]

(3.65)

(ii) τ > 0 and α1 |τ | >> 1 I (τ, r) ∝ |P (3) |2  ∝ B19 + |η|2 B20 + |B24 |2 exp (−2Γa10 |τ |) + |η|2 B25 exp (−2α2 |τ |) + exp[−(Γa10 + α2 ) |τ |] × {ηB25 exp[−i(Δk · r − (ω2 + Ω1 + ξ1 Δ3 ) τ )] + ∗ exp[i(Δk · r − (ω2 + Ω1 + ξ1 Δ3 ) τ )]} η ∗ B25 where B19 = α21 (6Γ0 Γ1 + Γ21 + Γ20 )/{[α21 + (Δa1 )2 ]2 (Γa10 )2 (Γ0 + Γ1 )Γ0 } B20 = α21 /{[α21 + (Δa1 )2 ]2 Γa10 Γ0 }

(3.66)

92

3 Attosecond Polarization Beats

B21 = (Γ0 + Γ1 )/[(α1 + iΔa1 )Γ0 Γ1 ] B22 = 4α22 /{[(α22 + (Δa2 )2 ]2 Γ20 } B23 = 2α2 (Γ0 + Γ1 )/{(α1 + iΔa1 )[α22 + (Δa2 )2 ]Γ20 Γ1 } B24 = 4α21 /{[(α21 + (Δa1 )2 ]2 Γ0 Γ1 } B25 = 2α1 (Γ0 + Γ1 ){(α2 − iΔa2 )[α21 + (Δa1 )2 ]Γ20 Γ1 } Equation (3.65) [or Equation. (3.66)] consists of five terms. The first and third terms depend on the fourth-order coherence functions of u1 (t) for DFWM, while the second and fourth terms depend on the fourth-order coherence functions of u2 (t) for NDFWM. The first and second terms come from the amplitude fluctuation of the chaotic field and do not depend on the relative time-delay τ . The third and fourth terms have an exponential decay as |τ | increases. The fifth term includes the second-order coherence functions of u1 (t) or u2 (t) and has the sum-frequency modulation of the SFPB signal. Equation (3.65) shows the nonresonant DFWM and the resonant NDFWM cross interference with a modulation frequency of Ω2 +ω1 , while Eq. (3.66) shows the resonant DFWM and the nonresonant NDFWM cross interference with a modulation frequency of ω2 + Ω1 . In this case the precision of using SFPB to measure Ω2 + Ω1 is determined by how well ω1 or ω2 can be tuned to the transition frequency Ω1 or Ω2 , respectively. The measurement of SFPB with broadband lights is again a Doppler-free precision spectroscopy. In Fig. 3.8(a), SFPB signal intensity versus time delay τ for Ω2 + Ω1 = 6.393 (fs)−1 , r = 0, η = 0.5, ξi = 1, Δi = 0, Γa10 = 1.74 × 10−4 (fs)−1 , Γa20 = 1.754 × 10−4 (fs)−1 , Γ0 = 5.7 × 10−8 (fs)−1 , Γ1 = 5.882×10−8 (fs)−1 . δτ = 3.3 fs, α1 = 0.271 (fs)−1 and α2 = 0.272 (fs)−1 (dotted curve), α1 = 0.406 (fs)−1 and α2 = 0.407 (fs)−1 (dashed curve), α1 = 0.542 (fs)−1 and α2 = 0.543 (fs)−1 (solid curve) for (a); while α1 = 0.406 (fs)−1 , α2 = 0.407 (fs)−1 , δτ = 0 fs (dotted curve), δτ = 1.5 fs (dashed curve), δτ = 3.3 fs (solid curve) for (b). The cross correlation with a decay factor exp[−(α1 + Γ20 )|τ |] or exp[−(Γ10 + α2 )|τ |] (produced from twin composite stochastic fields) displays features on a time scale significantly shorter than the autocorrelation of the one-photon resonant NDFWM or resonant DFWM with a decay factor exp(−2Γ20 |τ |) or exp(−2Γ10 |τ |) (produced from the single-color stochastic fields in Eq. (3.65) or (3.66)). The fourth-order correlation SFPB signal for chaotic fields can be calculated under Doppler and broadband limit. The broadband limit (τc ≈ 0) corresponds to the so called “white” noise limit with a constant spectrum. Under the extreme Doppler-broadening limit (i.e., k3 u → ∞) and laser broadband limit (i.e., αi → ∞), and substituting Eqs. (3.40)–(3.43), (3.49), and (3.60) into Eqs. (3.16)–(3.21), we can analytically obtain the stochastic averaging of the mod square of the total third-order polarization as:

3.4 Fourth-order Stochastic Correlation of SFPB

93

Fig. 3.8. SFPB signal intensity versus time delay τ .

(i) τ < 0 and α2 → ∞ I (τ, r) ∝ |P (3) |2  ∝

2

(Γ20 + 6Γ0 Γ1 )2 |η| + + [4Γ0 Γ1 (Γ0 + Γ1 )Γa10 ]2 16(Γa10 )2 Γ20

(Γ0 + Γ1 )2 exp(−2α1 |τ |) |η|2 exp(−2Γa20 |τ |) + + Γ20 Γ21 [α21 + (Δa1 )2 ] α22 Γ20 % η Γ 0 + Γ1 a exp[−(α + Γ ) |τ |] exp[−i(Δk · r − 1 20 α2 Γ20 Γ1 α1 − iΔa1 (Ω2 + ω1 + ξ2 Δ3 ) τ )] +

& η∗ exp[i(Δk · r − (Ω + ω + ξ Δ ) τ )] 2 1 2 3 α1 + iΔa1

(3.67)

(ii) τ > 0 and α1 → ∞ I (τ, r) ∝ |P (3) |2  ∝

2

(Γ20 + 6Γ0 Γ1 )2 |η| + + a 2 [4Γ0 Γ1 (Γ0 + Γ1 )Γ10 ] 16(Γa10 )2 Γ20 2

exp(−2Γa10 |τ |) |η| exp(−2α2 |τ |) + + 2 2 2 2 α1 Γ0 Γ0 [α2 + (Δa2 )2 ] % exp[−(Γa10 + α2 ) |τ |] η exp[−i(Δk · r − α2 Γ20 α2 + iΔa2 (ω2 + Ω1 + ξ1 Δ3 ) τ )] +

& η∗ exp[i(Δk · r − (ω2 + Ω1 + ξ1 Δ3 ) τ )] α2 + iΔa2

(3.68)

The cross correlation term with a decay factor exp[−(α1 + Γa20 )|τ |], produced from the twin stochastic fields of beam 1 and beam 2, shows features on a time scale that is significantly shorter than the autocorrelation term with a decay factor exp(−2Γa20 |τ |), which is produced from the single-color stochastic fields in Eq. (3.67). Similarly, the cross correlation term with a decay

94

3 Attosecond Polarization Beats

factor exp[−(Γa10 + α2 )|τ |], produced from the twin stochastic fields of beam 1 and beam 2, displays certain features on a time scale that is significantly shorter than the autocorrelation term with a decay factor exp(−2Γa10 |τ |), produced from the single-color stochastic fields in Eq. (3.68). We have assumed that the laser sources are chaotic fields in the above calculations. As we have discussed in before, the chaotic field can be used to describe a multimode laser source with both amplitude and the phase fluctuations. Another commonly used stochastic model is phase-diffusion model, which is used to describe an amplitude-stabilized laser source with a random phase fluctuation caused by spontaneous emission. In this case, the complex ergodic stochastic function ui (t) can be written as ui (t) = exp[iφi (t)] with φ˙ i (t1 )φ˙ i (t2 ) = 2αi δ(t1 − t2 ), φ˙ i (t1 ) = 0, and φ˙ i (t1 )φ˙ j (t2 ) = 0. If the laser has a Lorentzian line shape, the fourth-order coherence function is given by [7,8] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = exp[−αi (|t1 − t3 | + |t1 − t4 | + |t2 − t3 | + |t2 − t4 |)] × exp[αi (|t1 − t2 | + |t3 − t4 |)]

(3.69)

The fourth-order correlation SFPB signal under this phase-diffusion model can be analytically calculated in large Doppler (i.e., k3 u → ∞) and broadband (i.e., αi → ∞) limits. After substituting Eqs. (3.40)–(3.43), (3.49), and (3.69) into Eqs. (3.16)–(3.21), the stochastic averaging of the mod square of the total third-order polarization can be explicitly calculated to be: (i) τ < 0 and α2 → ∞ 2

(Γ0 + Γ1 )2 exp(−2α1 |τ |) |η| exp(−2Γa20 |τ |) + + Γ20 Γ21 [α21 + (Δa1 )2 ] α22 Γ20 % η Γ 0 + Γ1 a exp [− (α1 + Γ20 ) |τ |] exp[−i(Δk · r − 2 α2 Γ0 Γ1 α1 − iΔa1

I (τ, r) ∝ |P (3) |2  ∝

(Ω2 + ω1 + ξ2 Δ3 ) τ )] +

& η∗ exp[i(Δk · r − (Ω2 + ω1 + ξ2 Δ3 ) τ )] α1 + iΔa1

(3.70)

(ii) τ > 0 and α1 → ∞ ! 2 " Γ0 + Γ21 α21 + (Γa10 )2 (3) 2 + 2 a 2 I (τ, r) ∝ |P |  ∝ exp(−2Γa10 |τ |) + α21 Γ20 Γ21 α1 (Γ10 ) Γ0 Γ1 |η|2 exp(−2α2 |τ |) exp[−(Γa10 + α2 ) |τ |] + × Γ20 [α22 + (Δa2 )2 ] α2 Γ20 % η exp[−i(Δk · r − (ω2 + Ω1 + ξ1 Δ3 ) τ )] + α2 + iΔa2 & η∗ exp[i(Δk · r − (ω + Ω + ξ Δ ) τ )] 2 1 1 3 α2 − iΔa2

(3.71)

3.4 Fourth-order Stochastic Correlation of SFPB

95

Equation (3.70) [or Eq, (3.71)] consists of three terms. The first term depends on the fourth-order coherence functions of u1 (t) for DFWM, while the second term depends on the fourth-order coherence functions of u2 (t) for NDFWM. The first and second terms show an exponential decay of the beat signal as |τ | increases. The third term, depending on the second-order coherence functions of u1 (t) and u2 (t), gives rise to the sum-frequency modulation of the SFPB signal. This case is similar to the results of the secondorder stochastic correlations of SFPB, in which the part independent of τ has been ignored. Therefore, the fourth-order stochastic correlations of chaotic fields are of vital importance in SFPB. Equation (3.70) shows the interference between the nonresonant DFWM and the resonant NDFWM processes and generates a modulation frequency of Ω2 + ω1 , while Eq. (3.71) shows the interference between the resonant DFWM and the nonresonant NDFWM processes and gives rise to a modulation frequency of ω2 + Ω1 . The Gaussian-amplitude field has a constant phase but its real amplitude undergoes Gaussian fluctuations. If the laser has a Lorentzian line shape, the fourth-order coherence function can be written as [7, 8] ui (t1 )ui (t2 )ui (t3 )ui (t4 ) = ui (t1 )ui (t3 )ui (t2 )ui (t4 ) + ui (t1 )ui (t4 )ui (t2 )ui (t3 ) + ui (t1 )ui (t2 )ui (t3 )ui (t4 )

(3.72)

The fourth-order correlation SFPB signal of the Gaussian-amplitude model can also be analytically calculated in the large Doppler (i.e., k3 u → ∞) and broadband (i.e., αi → ∞) limits. By substituting Eqs. (3.40)–(3.43), (3.49), and (3.72) into Eqs. (3.16)–(3.21), the stochastic averaging of the mod square of the total third-order polarization is found to be: (i) τ < 0 and α2 → ∞   (Γ20 + 6Γ0 Γ1 )2 |η|2 |η|2 (3) 2 + + a 2 2 + I (τ, r) ∝ |P |  ∝ [4Γ0 Γ1 (Γ0 + Γ1 )Γa10 ]2 16(Γa10 )2 Γ20 (Γ20 ) Γ0 (Γ0 + Γ1 )2 exp(−2α1 |τ |) |η|2 exp(−2Γa20 |τ |) + + Γ20 Γ21 [α21 + (Δa1 )2 ] α22 Γ20 Γ 0 + Γ1 exp[−(α1 + Γa20 ) |τ |] × α2 Γ20 Γ1 % η exp[−i(Δk · r − (Ω2 + ω1 + ξ2 Δ3 ) τ )] + α1 − iΔa1 & η∗ exp[i(Δk · r − (Ω2 + ω1 + ξ2 Δ3 ) τ )] α1 + iΔa1 (ii) τ > 0 and α1 → ∞ I (τ, r) ∝ |P

(3.73)

  2 2 (Γ20 + 6Γ0 Γ1 )2 |η| |η| | ∝ + + a 2 2 + [4Γ0 Γ1 (Γ0 + Γ1 )Γa10 ]2 16(Γa10 )2 Γ20 (Γ20 ) Γ0

(3) 2

96

3 Attosecond Polarization Beats 2

exp(−2Γa10 |τ |) |η| exp(−2α2 |τ |) + + 2 2 α21 Γ20 Γ0 [α2 + (Δa2 )2 ] % η exp[−(Γa10 + α2 ) |τ |] exp[−i(Δk · r − α2 Γ20 α2 + iΔa2 (ω2 + Ω1 + ξ1 Δ3 ) τ )] +

& η∗ exp[i(Δk · r − (ω + Ω + ξ Δ ) τ )] 2 1 1 3 α2 + iΔa2

(3.74)

Both Eq. (3.73) and Eq. (3.74) consist of five terms. The first and third terms depend on the fourth-order coherence functions of u1 (t) for DFWM, while the second and fourth terms depend on the fourth-order coherence functions of u2 (t) for NDFWM. The first and second terms come from the amplitude fluctuation of the Gaussian-amplitude field and are independent of the relative time-delay τ . The third and fourth terms have an exponential decay of the SFPB signal as |τ | increases. The fifth term depends on the second-order coherence functions of u1 (t) and u2 (t) and gives rise to the sum-frequency modulation of the SFPB signal. Again, equation (3.73) shows the cross interference between the nonresonant DFWM and the resonant NDFWM with a modulation frequency of Ω2 + ω1 , while Eq. (3.74) shows the cross interference between the resonant DFWM and the nonresonant NDFWM with a modulation frequency of ω2 + Ω1 . The overall accuracy of using the SFPB with broadband lights to measure the sum-frequency of the energy-levels (Ω2 + Ω1 ) is limited by the homogeneous linewidths of atoms [35, 36].

3.5 Discussion and Conclusion The asymmetric temporal behaviors of the SFPB signals (due to the unbalanced dispersions of the optical components between two arms of the Michelson interferometer) show dramatic evolutions in Fig. 3.8(b). Note that as δτ increases, the peak-to-background contrast of interferogram diminishes. Two autocorrelation processes corresponding to population gratings ρ00 (ρ11 ) from low frequency component ω1 and ρ00 from high frequency component ω2 , respectively, are differently stretched in τ because each color component between twin beams 1 and 2 is maximally correlated at different delay-times. Owing to the difference between the zero time-delays for the frequency components ω1 and ω2 , the optical paths between these twin beams 1 and 2 will be different by cδτ for the ω2 component. As a result, there is an extra phase factor ω2 δτ for the ω2 frequency component. For the optical elements made from fused silica, the refractive index at λ2 = 588.996 nm is larger than that at λ1 = 589.593 nm by approximately 0.000 066. A 3.3 fs time delay between the ω1 and ω2 components corresponds to a propagation through

3.5 Discussion and Conclusion

97

one 0.5 cm beamsplitter and one 0.5 cm dispersion compensating optical flat for three times (about 1.5 cm). This controllable dispersion effect needs to be taken into account in the SFPB in attosecond time scale. The broadband multimode lights are tailored in a controllable fashion by dispersion [21]. The dispersion effects of the polarization beat signal can also be exactly balanced between the two arms [38, 40]. By contrast, coherent ultrashort pulses of equivalent bandwidth are not immune to such dispersive effects (even when balanced) because the transform limited ultrashort pulse is temporally broadened (it is chirped) and this has drastic effects on its time resolution. Experimentally, the difficulties inherent to the noisy light techniques are on par with the cw methods. These techniques are significantly easier to setup and perform than the ultrashort pulse experiments. In this sense the SFPB technique with twin Markovian stochastic fields has an advantage. The characteristic features of the interferogram are caused by two main components, i.e. the material response and the light response along with the interplay between these two responses. Figure 3.9 shows that the homodyne detected SFPB signal oscillates not only temporally with an ultrafast period of 2π/|Ω2 + Ω1 | = 982.837 as for sodium atoms, but also spatially with a period of 2π/Δk = 22.219 mm along the direction of Δk, which is almost perpendicular to the propagation direction of the SFPB signal. Here, the intensity (displayed on the vertical axis) has been normalized. The parame−1 ters are Ω2 +Ω1 = 6.393 (fs) , Δk = 22.219 mm−1 , α1 = 2.709×10−3(fs)−1 , −3 −1 a α2 = 2.715 × 10 (fs) , Γ10 = 1.74 × 10−4 (fs)−1 , Γa20 = 1.754 × 10−4 (fs)−1 , Γ0 = 5.7 × 10−8(fs)−1 , Γ1 = 5.882 × 10−8(fs)−1 , η = 0.5, ξi = 1, Δi = 0. The three-dimensional plot (temporal-spatial interferogram) of the SFPB signal intensity I(τ, r) versus time delay τ and transverse position r has a larger constant background caused by the intensity fluctuation of the chaotic field. At zero relative time delay (τ = 0), the twin beams (originating from the same light source) can have a perfect overlap in their noise patterns, which gives the maximum interferometric contrast [defined by the peak to background  contrast ratio I(τ = 0) I(τ = ∞)]. As |τ | is increased, the interferometric contrast diminishes on the time scale that reflects material no frictional memory, usually much longer than the correlation time of the noisy light [22]. One advantage of the SFPB technique is that the ultrafast modulation period 2π/ |Ω2 + Ω1 | = 982.837as with a Doppler-free precision can still be decreased, since the resonant frequencies Ω2 and Ω1 can be very large in principle. It is important to note that these three types of Markovian stochastic fields can have the same spectral density and thus the same secondorder coherence function which does not involve the bichromophoric model. The fundamental differences between these different statistical fields are in their higher-order coherence functions. According to Gaussian statistics a chaotic field can be completely described by its second-order coherence function. However, to fully describe the phase-diffusion field and the Gaussianamplitude field all-order coherence functions are required [7, 8]. Here, we

98

3 Attosecond Polarization Beats

Fig. 3.9. A three dimensional plot (temporal-spatial interferogram) of the SFPB signal intensity I(τ, r) versus time delay τ and transverse position r for the chaotic field. Adopted from Ref. [45].

have shown that different Markovian stochastic models of the laser fields only affect the fourth-order coherence functions. Figure 3.10 presents the SFPB signal intensity versus relative time delay for different stochastic fields. The four curves are for the chaotic field (dashed line), phase-diffusion field (dotted line), Gaussian-amplitude field (solid line), and the second-order correlation curve (dash-dotted line), respectively. The parameters are taken as Ω2 + Ω1 = 6.393(fs)−1 , r = 0, η = 0.5, ξi = 1, Δi = 0, Γa10 = 1.74 × 10−4 (fs)−1 , Γa20 = 1.754 × 10−4 (fs)−1 , Γ0 = 5.7 × 10−8 (fs)−1 , Γ1 = 5.882 × 10−8 (fs)−1 , α1 = 0.542(fs)−1 , and α2 = 0.543(fs)−1 . The SFPB signal is shown to be particularly sensitive to the statistical properties of the Markovian stochastic fields with arbitrary bandwidth. This is quite different from the fourth-order partial-coherence effects in the formation of integratedintensity gratings with pulsed light sources [14], which are insensitive to the specific radiation field models. The τ -independent contribution to the SFPB signal for a Gaussian-amplitude field or a chaotic field is much larger than for a phase-diffusion field, as shown in Fig. 3.10. This is due to the stronger intensity fluctuation in the Gaussian-amplitude field in comparison to the chaotic field, which in turn has a much larger intensity (amplitude) fluctuation than the phase-diffusion field (which has only pure phase fluctuation). The field correlation has a weak influence on the SFPB signal when the laser has a narrow bandwidth (such as in cw case). In contrast, the sensitivity of the SFPB signal intensity to the three Markovian stochastic models increases as time delay is increased when the laser has a broadband linewidth (noisy light).  The even-order correlation functions consist of (2n!) (2n · n!) distinct terms, where 2n is the order of the correlation functions. u3 (t) of Eq. (3.3) must be neglected in |P (3) |2 . Due to ui (t) = 0 and u∗i (t) = 0, the absolute square of the stochastic average of the polarization |P (3) |2 , which involves the second-order coherence functions of ui (t), cannot fully describe the temporal behaviors of the SFPB [30-32]. The second-order

3.5 Discussion and Conclusion

99

Fig. 3.10. SFPB signal intensity versus relative time delay.

coherence function theory, which does not involve the bichromophoric model, is valid when one is only interested in the τ -dependent part of the attosecond sum-frequency beating signal. For this specific point, it is similar to the fourth-order stochastic correlation (intensity correlation) of the phasediffusion model in Fig. 3.10. The theoretical results with higher-order stochastic correlations fit the DFPB experimental data much better than the expressions involving only the second-order coherence functions [34, 38]. Next, we discuss the difference of chromophore P (3) between the SFPB and the sum-frequency UMS [27] in the self-diffraction geometry. The frequencies and wave vectors of the sum-frequency UMS signals are ωs1 = 2ω1 − ω1 , ωs2 = 2ω2 − ω2 and ks1 = 2k1 − k1 , ks2 = 2k2 − k2 , respectively, which means that one photon is absorbed from each of the two correlated fluctuating pump beams. On the other hand, the frequencies and wave vectors of the SFPB signals [Fig. 3.2] are ωs1 = (ω1 − ω1 ) + ω3 , ωs2 = (ω2 − ω2 ) + ω3 and ks1 = k1 − k1 + k3 , ks2 = k2 − k2 + k3 , respectively, which indicates that photons are absorbed from and emitted to the mutually correlated fluctuating twin beams 1 and 2, respectively. This difference between the SFPB and the UMS has profound impact on the field-correlation effects. Note that roles of beam 1 and beam 2 are interchangeable in the UMS; this interchangeable feature also makes the second-order coherence function theory fail in the UMS. Due to u(t1 )u(t2 ) = 0, the absolute square of the stochastic average of the polarization |P (3) |2 cannot be used to describe the temporal behavior of the sum-frequency UMS [30, 31]. So, the fourth-order correlation (intensity correlation) treatment presented here should also be very importance in the sum-frequency UMS. In practice, noisy light has been described as a random train of short subpulses. However, one must use this description with a great caution, since the short pulses imply at least some sort of phase locking, which further implies that color locking does not hold. This will make the theoretical calculations be not consistent with the real experiments [25], since color locking is a fundamental property of the noisy light. Color locking from stationarity and the Wiener-Khintchine theorem is fundamental to understand noisy light signals. From above frequency mixing, one can expect the spectrum of the signals to be centered at ωs1 = (ω1 − ω1 ) + ω3 for DFWM or ωs2 = (ω2 − ω2 ) + ω3 for NDFWM and to have the widths of the

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noisy fields. In the broadband case (noisy light), color locking forbids this however since the out-of-phase actions of twin fields from beam 1 and beam 2 are correlated and hence on any given chromophore only the precise subpulse from both beams 1 and 2 may act. The out-of-phase action of the single subpulse cancels. This holds for all subpulses, so the noises, in fact, cancels, and one is left with simply the bandwidth of the narrowband probe field [25]. By contrast, the in-phase DFWM (or NDFWM) signal is relatively much weaker. Since the in-phase twin noisy field actions are not pair-correlated and hence not color-locked, which results in no noise cancellation. The stochastic correlation spectroscopy using broad-band quasi-cw (nontransform limited) noisy lights can be considered as an intermediate technique between the steady-state spectroscopy (pure frequency domain) and the pure time-resolved techniques [16 – 23]. For the frequency-domain techniques (such as saturated-absorption spectroscopy and two-photon absorption spectroscopy), the spectral resolution is determined by the laser linewidth, therefore, narrow-band cw laser sources are usually required. A common feature in the time-domain techniques is that the temporal resolution is determined by the laser’s pulse width. More specifically, the excitation laser’s pulses in quantum beat must have a spectral width larger than the energy-level splitting, so energy sublevels can be excited simultaneously. The disadvantage of this quantum beat technique is that it is not efficient and, therefore, not practical to excite two transitions simultaneously with an extremely broadband light when the sum (Ω2 +Ω1 ) of the energy-level resonant frequencies is large. However, the SFPB technique can overcome this difficulty because we can excite the two transitions separately with two lasers that have bandwidths much narrower than the energy-level splitting. The phase coherence control of the light beams in SFPB is subtle. We first consider the twin fields to have narrow bandwidths. If the time delay τ between the twin noisy fields (which come from a single light source) is shorter than the coherence time τc of the laser (i.e., αi |τ | << 1), the relative phase between the twin fields will be well defined. The interference between the one-photon nonresonant DFWM and the one-photon nonresonant NDFWM causes the SFPB signal intensity to modulate with a frequency ω2 + ω1 as τ is varied. In contrast, the relative phase between the twin fields will fluctuate randomly if αi |τ | >> 1. In this case, the temporal, as well as spatial, modulations of the SFPB signal intensity will disappear. Now, we consider the case with the twin beams to have broadband linewidths. Population gratings, generated by coherent pairings, have a common phase and therefore grow up coherently. These coherent gratings determine the temporal behavior of the Bragg reflection signals, which can be generated only when the separation τ between pairings is less (2) (2) than the dephasing time of the optical coherence ρ10 , ρ20 (i.e., Γa10 |τ | < 1 a or Γ20 |τ | < 1). On the other hand, gratings generated by incoherent pairings have random spatial phases. This incoherent grating contributes to the Bragg reflection signal intensity with a constant background. Consider the DFWM and NDFWM processes from ω1 or ω2 frequency component of twin beam

3.5 Discussion and Conclusion

101

1 and beam 2, respectively. The DFWM and NDFWM, which are related to the three-pulse stimulated photon-echo, originate from the interaction of atoms with the phase-correlated subpulses in twin beams 1 and 2, which are separated by the time-delay |τ |. The homodyne detected SFPB signal interestingly shows the cross interference between the resonant and nonresonant terms with broadband linewidths (noisy light). The subtle polarization interference between the two FWM pathways allows one to control the outcome of the laser-matter interactions. The Doppler-free one-photon DFWM or NDFWM occurs when two overlapping counter-propagating beams are both on resonance with the same atomic velocity group [37]. If the beams 1, 2, and 3 [see Fig. 3.1 (b)] are from the same laser source with the frequency ω1 , they will only satisfy the condition of being simultaneously on resonance with the gas when they are on resonant with the zero-velocity group (except in crossover resonance case). Only those atoms whose velocities are centered at k1 · v ≈ 0 are effective in generating the conjugate signal. Therefore, as in the case of saturated absorption spectroscopy [46], we have a one-photon Doppler-free DFWM spectrum with a peak located at Δ1 = 0 [44]. We then fix the frequency of beam 3 and perform the one-photon NDFWM experiment with beam 1 and beam 2 only consisting of the ω2 frequency component [37]. Since only atoms with velocities near k1 · v ≈ 0 interact with beam 3, the condition for beam 1 and beam 2 to interact with the same group of atoms is Δ2 = 0. Again, the onephoton NDFWM signal is Doppler free, since only atoms in a specific velocity group will contribute to the NDFWM signal. The one-photon Doppler free NDFWM spectrum is somehow similar to the general saturated absorption, but they are actually different. In the saturated absorption spectroscopy, if there is just a tiny frequency offset between the pump beam and the probe beam, then they can be considered to be simultaneously on resonance with a nonzero velocity group whose first-order Doppler shift is half the tiny offset in frequency. This means that a moving atom can only be resonance with both lasers if its resonant frequency is halfway between the two lasers [47]. Finally, the Doppler-free absorption of two photons can be illustrated as follows. If the atom has a velocity component in the direction of the laser beams, the resonance condition is changed to (Ef − Ei )/ = ω1 + ω2 − v · (k1 + k2 ). If both fields have the same frequency ω and counterpropagate through the medium, their wave vectors are opposite to each other. This implies that the velocity-dependent term in the resonance condition vanishes. All atoms with (Ef − Ei )/ = 2ω are on resonance. This more restrictive condition results in the line narrowing, and the fact that all atoms can be excited at the same frequency, gives the enhancement in the cross section over the Doppler-broadened single-beam case. If the laser is detuned from resonance by |Δ| = (Ef − Ei )/ − 2ω, then the zero-velocity group is out of resonance with any combinations of photons from the two lasers. If Δ > 0, the laser propagating in zaxis direction is on resonance with an atomic velocity group having v = −c|Δ|/2ω and the laser in −z direction

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is on resonance with an atomic velocity group having v = c|Δ|/2ω. By contrast, if Δ < 0, the laser propagating in z direction is on resonance with atoms having v = c|Δ|/2ω and the laser in −z direction is on resonance with atoms having v = −c|Δ|/2ω. Thus, when the laser is detuned off resonance, only two particular velocity groups are on resonance. Whereas when the laser is on the exact resonance frequency, all the velocity groups are on resonance for absorbing one photon from the pump and another from the probe. The line shape of the same frequency two-photon absorption is the  sum of the broad Doppler-broadened background (I12 + I22 )δωi (δωD ) (here, δωD Doppler width and Ii laser intensity) from the two-photon absorption in each of the beams and the sharp Doppler-free line (I1 +I2 )2 N . If the two lasers have the same intensity, the integral of the Doppler-free signal should be at least twice that of the Doppler-broadened one. Since the enhancement of the Doppler-free signal increases proportionally to the uncollimated helium beam Doppler width, the observed enhancement of the Doppler-free signal over the Doppler-broadened one was a few hundred to one [43]. As we have mentioned in Section 3.1.1, one-photon resonant DFWM can provide a Doppler-free spectrum with peak located at Δ1 = 0. When the ω1 is set to the center of the Doppler profile, then only atomic velocity group with velocities near k1 · v ≈ 0 interact with beam 3. This group of atoms will interact with beam 1 of frequency ω2 and contribute to the different-frequency two-photon NDFWM signal. Since only atoms in a specific velocity group contribute to the signal, the different-frequency two-photon NDFWM is also Doppler-free. A similar situation in the two-photon absorption with a resonant or nearly resonant intermediate state has been discussed in Ref. [44]. Combining the capability of the high accuracy in measuring the SFPB modulation frequency, a Doppler-free precision can be achieved in the measurement of Ω2 + Ω1 . Figure 3.11 shows the Fourier spectrum in which τ is varied for a range of 900 fs. The paramters are α1 = 2.709 × 10−5 (fs)−1 , α2 = 2.715 × 10−5 (fs)−1 , Γa10 = 1.74 × 10−4 (fs)−1 , Γa20 = 1.754 × 10−4 (fs)−1 , Γ0 = 5.7 × 10−8 (fs)−1 , Γ1 = 5.882 × 10−8 (fs)−1 , ω2 + ω1 = 6.392 9(fs)−1 , r = 0, ξi = 1, |τ |  450 fs, and η = 0.5. The modulation frequency of Ω2 + Ω1 = 6.39291 × 1015 s−1 can be obtained which corresponds to the sum of the resonant frequencies of the transitions from 3S1/2 to nP1/2 and from 3S1/2 to nP3/2 , respectively.

Fig. 3.11. Fourier spectrum of SFPB signal given by Eq. (3.45).

References

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In this chapter, the new phenomenon of sum-frequency attosecond polarization beats by twin composite stochastic fields has been demonstrated and its origins explained. The homodyne detected SFPB signal shows the resonant-nonresonant cross correlation. Significant differences are identified for the three different Markovian stochastic driving fields using fourthorder coherence functions. The DFWM (or NDFWM) signal exhibits hybrid radiation-matter detuning terahertz damping oscillation. This technique can achieve Doppler-free precision in measuring the frequency sum for the two dipole-allowed transitions in the system. It has also been found that the asymmetric time-domain behaviors of the polarization beat signals due to the unbalanced dispersion effect between the two arms of interferometer do not affect the overall accuracy in case of using SFPB to measure the sum of transition frequencies. As an attosecond ultrafast modulation technique, in principle it can be extended to measure sum-frequency of energy-level transitions with very large energy differences, which can give an even faster time scale.

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4 Heterodyne/Homodyne Detection of MWM

In this chapter, phase-sensitive heterodyne (or homodyne) detection technique is developed in investigating real (dispersion) and imaginary (absorption) parts of high-order nonlinear susceptibilities using color-locked twin noisy fields. In a three-level system, the complex third-order nonlinear susceptibility is determined by heterodyning signals from the two-photon NDFWM with the reference signal from another one-photon DFWM process in the same system, which propagate along the same optical path and have the same frequency. By controlling the relative phase between these two co-existing nonlinear wave-mixing processes, the third-order nonlinear absorption and dispersion of such ultrafast polarization beat signals can be obtained. Using such phase-sensitive heterodyne detection technique, real and imaginary parts of the fifth-order (χ(5) ) and seventh-order (χ(7) ) nonlinear susceptibilities can be determined through beating between the SWM signal and a FWM reference (local oscillator) beam and between the EWM signal and a SWM reference (local oscillator) beam, respectively, in specially designed energy-level configurations. The greatly enhanced third- fifth- and seventh-order nonlinear responses with different signs can be modified and controlled through the color-locked correlations of twin noisy fields. Determining and controlling real and imaginary parts of the high-order nonlinear susceptibilities is very important in understanding the propagation of high-intensity pulses and solitons, and can lead to many other interesting applications.

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order Polarization Beats Recently, studies of nonlinear optical effects in multi-level atomic systems have received renewed interests due to greatly enhanced nonlinear optical processes and, at the same time, reduced linear absorption caused by lightinduced atomic coherence among the energy levels [1 – 3]. By carefully choosing the atomic energy levels and laser fields configurations, the efficiencies of wave mixing processes can be greatly increased near optimal atomic coherence conditions. Through directly measuring the third-order nonlinear optical co-

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

107

efficients in a three-level atomic system [4], one can see that the nonlinearity depends sensitively on various experimental parameters. The large enhancement of third-order nonlinear index in a four-level atomic system was also demonstrated [5]. In order to optimize certain nonlinear optical processes, it is beneficial to have the exact knowledge of nonlinear coefficients and their dependences on various experimental parameters. However, due to the residual linear absorption and dispersion of probe and signal beams, it is usually difficult to measure nonlinear coefficients, especially both real and imaginary parts under same conditions. One of early experiments to measure the Kerr nonlinear coefficient in a three-level atomic system used an optical cavity to eliminate linear contributions [4], which directly gives the Kerr nonlinear refractive index n2 . In this section, we present a phase-sensitive detection technique to obtain the third-order complex susceptibility in a three-level gas medium. In detecting the two-photon NDFWM signal in attosecond polarization beats (ASPB), the two-photon NDFWM signal can beat with a reference (local oscillator) signal from a one-photon DFWM process, which propagates in the direction very close to the NDFWM signal and has the same frequency. One can adjust the relative phase between the local oscillator field (the one-photon DFWM process) and the two-photon NDFWM signal by changing the relative time delay (τ ) between two pump beams for the co-existing DFWM and NDFWM processes through a Michelson interferometer. As theτ -dependent phase dif ference approaches to 2nπ or (2n + 1 2)π, the ASPB signal evolves into the nonlinear dispersion or absorption of the two-photon NDFWM, respectively. Here, the reference beam comes from a coexisting one-photon DFWM process, which is introduced by adding an additional frequency component to the pump beams of the NDFWM scheme. The two-photon NDFWM signal beam and the one-photon DFWM reference beam then beat directly at the detector. This method is based on attosecond polarization interference between two FWM processes in the pure homogeneously-broadened [6,7] or Doppler-broadened three-level ladder-type system [8 – 10]. This technique is a good way to measure the third-order susceptibility directly, especially to determine its real and imaginary parts at the same time. Different calculations are used to treat nonlinear responses in the pure homogeneously-broadened and the extremely Doppler-broadened media, respectively, in the three-level ladder-type system. We proceed in the standard manner by first calculating the expression for the density-matrix element (3) (third-order response functions ρ10 ), then finding the complex susceptibility, and finally breaking it down into real and imaginary parts. The modified third-order nonlinear absorption and dispersion coefficients can be controlled by the noisy light color-locking bandwidth, the frequency detuning, and the time delay. Another advantage of this system is the use of a two-photon Doppler-free counter-propagation configuration [9, 10], which allows us to observe such interesting effects in long atomic vapor cells.

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4.1.1 Liouville Pathways Nonlinear optical properties of the atomic medium can be modified and controlled through color-locked correlations of twin noisy driving fields. To illustrate such controllable third-order nonlinearity, we consider a three-level ladder-type atomic system in the bare-state picture. The polarization interference between two excitation pathways, |0 → |1 → |0 → |1 (one-photon DFWM between the two lower levels |0 and |1) and |0 → |1 → |2 → |1 (two-photon NDFWM in the three-level ladder system), leads to a third-order ASPB phenomenon [7, 8, 10]. This polarization beat is based on the interference at the detector between FWM signals which originate from macroscopic polarizations excited simultaneously in the homogeneously- [6, 7] or inhomogeneously- [8 – 10] broadened samples. It requires that all the generated third-order polarizations have the same frequency. The three-level ladder-type ASPB comes from the sum-frequency polarization interference between one-photon and two-photon optical processes in the attosecond time scale (see Chapter 3), while the femtosecond polarization beat (FSPB) corresponds to the difference-frequency polarization interference in the femtosecond time scale [9] (see Chapter 2). A three-level ladder-type atomic system [see Fig. 4.1 (a)] consists of the ground state |0, an intermediate state |1, and an excited state |2. In Fig. 4.1 (a), the solid, dashed and dash-dotted vertical lines correspond to the ket interaction, bra interaction, and FWM signal, respectively The time evolves from left to right. States between |0 and |1 and between |1 and |2 are coupled by dipole transitions with resonant frequencies Ω1 and Ω2 , respectively, while transition between |0 and |2 is dipole forbidden. Let’s consider a two-color time-delayed FWM configuration in which beams 1 and 2 consist of two frequency components ω1 and ω2 , while beam 3 has frequency ω3 [see Fig. 4.1 (b)]. As defined in Chpater 2 and Chapter 3, the primed k vectors indicate a time delay relative to the unprimed k vectors. We further assume that (ω3 = ω1 ≈ Ω1 ) and ω2 ≈ Ω2 , therefore ω1 (ω3 ) and ω2 will drive transitions from |0 to |1 and from |1 to |2, respectively. There are two distinct nonlinear processes involved in this two-color ASPB. First, the ω1 frequency components of twin composite beam 1 and beam 2 induce a population grating bet ween states |0 and |1, which is probed by frequency ω3 of beam 3. This is a one-photon resonant DFWM ω1 −ω1 ω3 |1 −−−→ |0 −−→ |1) (with one typical pathway (I) in Fig. 4.1 (a): |0 −−→ and the generated signal (beam 4) has frequency ω3 . More specifically, the typical one-photon DFWM process indicates that one pump photon ω1 is absorbed and one dump photon ω1 is emitted first (both the ket and bra of the density operator would be promoted to |0 (or |1) to create the ground-state (or an intermediate-state)) population, one probe photon ω3 is then absorbed to generate a phase-matched coherent photon ω3 , emitted along beam 4 [see Fig. 4.1 (b)]. This process occurs only between state |0 and state |1. Second, beam 3 and the ω2 frequency component of beam 1 induce a two-photon atomic coherence between levels |0 and |2 (only the

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109

ket is being promoted to create an atomic coherence rather than a population), which is then probed by the ω2 frequency component of beam 2. This ω3 ω2 −ω2 is a two-photon NDFWM process (|0 −−→ |1 −−→ |2 −−−→ |1) with the resonant intermediate state and the frequency of the generated signal equals to ω3 again. The two-photon NDFWM process shows that the ket is promoted to the excited state |2 by the two-field action, one probe photon with frequency ω2 is emitted, and then one phase-matched coherent photon with frequency ω3 is finally emitted along the beam 4 direction [see Fig. 4.1 (b)]. Thus, the first two-field action would imply that both the ket and bra would be promoted to |0 (or |1) to create the ground-state (or an intermediatestate) population in chains (I)–(IV); Only the ket is being promoted to create an atomic coherence (between the ground state and the excited state) rather than the population in chain (V).

Fig. 4.1. (a) Energy-level diagrams of two-state one-photon DFWM and threestate two-photon NDFWM processes for perturbation chains (I)–(V). (b) Phaseconjugation geometries of ASPB and FSPB.

In the ASPB case, to accomplish this arrangement the ω1 and ω2 frequency components of the lights are split and then recombined to provide two double-frequency pulses in such a way that the ω1 component is delayed by τ in beam 2 and the ω2 component delayed by the same amount in beam 1 [see Fig. 4.1 (b)]. The twin composite stochastic fields of beam 1 (Ep1 ) and beam 2 (Ep2 ) can be written as Ep1 = E1 + E2 = ε1 u1 (t) exp[i(k1 · r − ω1 t)] + ε2 u2 (t − τ ) exp[i(k2 · r − ω2 t + ω2 τ )]

(4.1)

Ep2 = E1 + E2 = ε1 u1 (t − τ ) exp[i(k1 · r − ω1 t + ω1 τ )] + ε2 u2 (t) exp[i(k2 · r − ω2 t)]

(4.2)

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4 Heterodyne/Homodyne Detection of MWM

Here, εi ,ki (εi ,ki ) are the constant field amplitude and the wave vector of the ωi component in beam 1 (beam 2). ui (t) is the dimensionless statistical factor that contains phase and amplitude fluctuations. It is taken to be a complex ergodic stochastic function of t, which obeys complex circular Gaussian statistics in the chaotic field. τ is the relative time delay between the prompt (unprime) and delayed (prime) fields. The time delay τ is introduced in both beams, which is quite different from that of the FSPB scheme studies in Chapter 2 [9]. Beam 3 is assumed to be a quasi-monochromatic light [u3 (t) ≈ 1], so the complex electric field of beam 3 can be written as E3 = A3 (r, t) exp(−iω3 t) = ε3 u3 (t) exp[i(k3 · r − ω3 t)]. Here, ω3 ,ε3 and k3 are the frequency, the field amplitude, and the wave vector, respectively. In the bare-state picture, equations of motion for the atomic polarization and population (atomic response) are considered up to different orders of Liouville pathways. To proceed further, and to simplify the mathematics, (0) we will neglect the ground-state depletion (ρ00 ≈ 1) and not consider propagation characteristics of the pulsed pump, probe and FWM fields here. The approximation of no ground-state depletion is valid for the case of a weak probe beam. Also, we only retain the resonant dipole interaction terms in deriving the complex susceptibility, known as the rotating-wave approximation (RWA). Because of the selectivity imposed by the RWA, each pulse interaction contributes in a unique way to the phase-matching direction of the nonlinear signal. Pulse sequence control of the third-order response functions representing the Liouville pathways for P1 , P2 , P3 , P4 , and P5 , respectively, as shown in Fig. 4.2. The left and right vertical lines represent the ket and bra, respectively; applied electric fields are indicated with arrows oriented toward the left if propagating with a negative wave vector and vice versa for a positive wave vector. Time evolves from the bottom to the top of the diagram. We shall employ the perturbation theory to calculate density-matrix elements by the following perturbation chains (see Fig. 4.2) [10], (0) E

 ∗ (1) (E )

(2) E

(3)

1 3 1 ρ10 −−− → ρ00 −−→ ρ10 (I) ρ00 −−→

(E1 )∗

E

(2) E

1 3 (II) ρ00 −−−→ (ρ10 )∗ −−→ ρ00 −−→ ρ10

(0)

(0) E

(1)

 ∗ (1) (E )

(2) E

(3)

(3)

1 3 1 (III) ρ00 −−→ ρ10 −−− → ρ11 −−→ ρ10  ∗ (0) (E )

1 3 1 (IV) ρ00 −−− → (ρ10 )∗ −−→ ρ11 −−→ ρ10

(0) E

(1)

(1)

E2

E

(2) E

(2) (E2 )

∗

(3)

(3)

3 (V) ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ10

Chains (I)–(IV) correspond to the one-photon DFWM processes with the same phase-matching condition ks1 = k1 −k1 +k3 , while the chain (V) corresponds to the two-photon NDFWM process with phase-matching condition ks2 = k2 − k2 + k3 , as shown in Fig. 4.1(a). Since DFWM and NDFWM signals propagate along slightly different directions, the interference between them leads to a spatial oscillation [6 – 10]. The DFWM signal is the sum of two grating diffraction contributions (the small-angle static grating induced

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

111

by ω1 and −ω1 & a large-angle static grating induced by ω3 (≈ ω1 ) and −ω1 ), while the NDFWM signal comes from the sum of other two grating diffraction contributions (the small-angle static grating induced by ω2 and −ω2 & a large-angle moving grating induced by ω3 and −ω2 ).

Fig. 4.2. (a) One-photon DFWM of ASPB and FSPB; (b) Two-photon NDFWM of ASPB and FSPB. (3)

The third-order response functions (ρ10 ) of the perturbation chains (I)– (V) (relevant to the three-pulse FWM) are given using double-sided Feynman diagrams (DSFD) as shown in Fig. 4.2 [7,10]. Time evolutions of the densitymatrix elements of the optically driven atoms or molecules can be represented schematically by either the Liouville space coupling representation [the chains (I)–(V)], the DSFD (see Fig. 4.2), or the ladder energy-level diagrams [see Fig. 4.1 (a)]. Each diagram represents a distinct Liouville space pathway. We show diagrammatic representations corresponding to the lowest three orders of the resonant dipole interactions for a system with two electronic states or with three electronic states. In the Liouville space coupling representation [chains (I)–(V)], states of the system is designated by a position in Liouville space with indices corresponding to the ket-bra “axis”. Up and down transitions on the ket are excited by positive and negative frequency fields, whereas negative and positive frequency fields induce up and down transitions on the bra. The DSFD (as shown in Fig. 4.2) can be described as follows: vertical left and right lines of the diagram represent the time evolution (bottom to

112

4 Heterodyne/Homodyne Detection of MWM

top) of the ket and bra, respectively; the applied electric fields are indicated with arrows oriented toward the left if propagating with a negative wave vector and toward the right for the positive wave vector. Each interaction with the electric field produces a transition between the two electronic states of either the bra or the ket. In the ladder diagrams, as shown in Fig. 4.1 (a), the solid and dashed lines correspond to ket and bra interactions, respectively, and its time evolves from left to right. The ability to track the evolution of the bra and ket simultaneously makes the density-matrix representation a most appropriate tool for the description of many dynamical phenomena in nonlinear optical processes. Generally, there are 48 terms in the third-order density operators for the (j) given FWM process. In time evolution of the density-matrix element ρab , each specified field action transforms either the “ket” or the “bra” side of the density-matrix element. Thus, for any specified j-th order generator, there can be 2j detailed paths of the evolution. In addition, evolution for each of the j! generators corresponding to all possible fields ordering must be considered. One then has total 2j j! paths of evolution. Thus, at third order, where beams 1, 2, and 3 are distinct, there are 2j j! = 48 (j=3) different Liouville pathways at the polarization level. Often the experimental constraints reduce the number of diagrams to a significantly smaller subset which dominates the behavior of the signal. Under RWA, phase-matching and frequency selections of the FWM signals along ks greatly restrict the number of third-order perturbative pathways (see Fig. 4.2). Moreover, polarization beat is based on the interference at the detector between multi-FWM signals, which originate from macroscopic polarizations excited simultaneously in the sample. It requires that all the polarizations have the same frequency [18]. Now, we consider the other possible density-operator pathways: (0) E

 ∗ (1) (E )

(2) E

(3)

3 1 1 ρ10 −−− → ρ00 −−→ ρ10 (VI) ρ00 −−→

(E1 )∗

E

(2) E

3 1 (VII) ρ00 −−−→ (ρ10 )∗ −−→ ρ00 −−→ ρ10

(0)

(0) E

(1)

 ∗ (1) (E )

(2) E

(3)

(3)

3 1 1 (VIII) ρ00 −−→ ρ10 −−− → ρ11 −−→ ρ10

(E1 )∗

3 1 (IX) ρ00 −−−→ (ρ10 )∗ −−→ ρ11 −−→ ρ10

(0)

(1)

E2

(0) E

(1)

 (0) E

 (1) E

E

(2) E

(2) (E2 )

∗

(3)

(3)

1 (X) ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ10 ∗ (2) (E2 )

(3)

1 2 (XI) ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ10

Here, the population grating (a large-angle static grating) induced by beam 3 and ω1 frequency component of beam 2 is responsible for the generation of the FWM signal. These large-angle static gratings have much smaller fringe spaces which equal to approximately one half of wavelengths of the    θ≈180◦ incident lights (λi [2 sin(θ 2)] −−−−−→ λi 2). For a Doppler-broadened system, these gratings will be washed out by the atomic motion. In addition, the wave vectors k2 − k2 + k1 and k2 − k2 + k1 of the density-operator pathways

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

113

(X) and (XI) propagate along dramatically different directions compared with ks1 = k1 − k1 + k3 and ks2 = k2 − k2 + k3 for the interested signals. Therefore, it is appropriate to neglect contributions to the FWM signals from these density-operator pathways. The stricter requirements on phase matching also make certain processes (VI–XI) unimportant (see Chapter 3).

4.1.2 Color-locking Stochastic Correlations Lasers are inherently noisy devices, in which both phases and amplitudes of fields can fluctuate. Noisy laser beams can be used to probe atomic and molecular dynamics, and it offers a unique alternative to more conventional frequency-domain cw spectroscopy and ultrashort pulse time-domain spectroscopy [11]. Typical Markovian noisy fields include chaotic fields, phasediffusion fields, and Gaussian-amplitude field [7, 10]. Color-locking technique results in complete cancellation of the spectrally broad noise carried by the noisy light beams [11]. The fundamental difference is that the transformlimited femtosecond laser pulse is phase coherent (phase-locked) while noisy light has random phase and is non-transform limited. For the “biatomic” model [11] of the macroscopic system where phase-matching takes place, the FWM signal must be drawn from the third-order polarization P (3) (having t time variable) developed on one “atom” multiplied by the (P (3) )∗ (having s time variable) that is developed on another “atom” which must be located elsewhere in space (with summation over all such pairs). The third-order response functions (P (3) ) relevant to three-pulse FWM are given using doublesided Feynman diagrams as shown in Fig. 4.2. The homodyne detected ASPB signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic process |P (3) |2  (having both t and s time variables), which involves fourth- or sixth-order coherence functions of ui (t) in phase-conjugation geometry. Unlike the second-order coherence function at the field-level averaging, expansions of fourth- and sixth-order coherence functions at the intensity-level averaging strongly depend on statistical properties of Markovian noisy fields [7, 10]. The characteristics of the ASPB interferogram are a result of two main components: the material response and the light response along with the interplay between the two responses. In general, the ASPB (at the intensity level) can be viewed as the sum of five contributions: (i) the resonantresonant, nonresonant-nonresonant, or resonant-nonresonant types of τ -independent auto-correlation terms; (ii) the purely resonant τ -dependent autocorrelation terms; (iii) the purely nonresonant τ -dependent auto-correlation terms; (iv) the resonant-nonresonant τ -dependent auto-correlation terms; (v) resonant-resonant, nonresonant-nonresonant, or resonant-nonresonant types of τ -dependent cross-correlation terms. When the ASPB signal is dominated by the case that u1 (t) and u2 (t) (field 3 is quasi-monochromatic) field actions

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4 Heterodyne/Homodyne Detection of MWM

on single “atom” are correlated, and the cross “atom” correlated actions are not important (which is usually true for atomic vapor systems), the ASPB signal intensity can be approximated by the absolute square of the non-trivial stochastic average of the polarization |P (3) |2 (averaging at the field level), which only involves second-order coherence functions of ui (t). This works because for this particular spectroscopic technique the τ -dependent terms only have “intra-atomic” correlations and no “inter-atomic” correlations in these terms. In dealing with the gas-phase atomic medium, an approximation can be made by averaging at the field level, which only needs second-order correlation functions of noise fields, which are given by ui (t1 )u∗i (t2 ) = exp(−αi |t1 − t2 |) for a Lorentzian line shape, and  √ ui (t1 )u∗i (t2 ) = exp{−[αi (t1 − t2 ) 2 ln 2]2 } 1 δωi (with δωi being the linewidth 2 of the ωi frequency component) is the decay rate for the autocorrelation function of noisy fields. Noisy light is color-locked, because each color is coherent only with itself. Such color-locking is a consequence of the WienerKhintchine theorem, which is expressed mathematically (most conveniently) by examining the second-order coherence function in frequency domain, i.e., for a Gaussian line shape. Here, αi =

u∗i (ωk ) = δ(ωj − ωk )Ji (ωj ) ˜ ui (ωj )˜ where u ˜i is the Fourier transform of the broadband light field envelope and Ji is the spectral density of the stochastic function ui [11]. The form of the second-order coherence function, which is determined by the laser line shape as given above, is a general feature of three different Markovian stochastic models: chaotic field, phase-diffusion field, and Gaussian-amplitude field [7, 10]. In the cw limit (ui (t) ≈ 1), the ASPB signal can then be written as I ∝ |P (3) |2 = |P (3) |2 = |P (3) |2  The nonlinear polarization Pn (responsible for the phase-conjugate FWM signal) is given by stochastic averaging over the velocity distribution function W (v). Thus  +∞ (3) Pn = N μ1 dvw(v)ρ10 (v) −∞

where v is the atomic velocity and N is the atomic density. For a Dopplerbroadened system, the  velocity distribution function w(v) =  atomic √ exp[−(v u)2 ] πu. Here, u = 2kB T /m with m being the mass of an atom. kB is the Boltzmann constant and T is the absolute temperature. Polarizations of the DFWM (PA = P1 + P2 + P3 + P4 ) and NDFWM (PB = P5 )

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

115

contributions are given in the bare-state basis. P1 , P2 , P3 , P4 , and P5 correspond to the third-order polarizations of the perturbation chains (I), (II), (III), (IV), and (V), respectively. The formulas below show how the initial densitymatrix elements can be transformed into higher-order elements through interactions with the electric fields.  P1 = S1 (r) exp[−i(ω3 t + ω1 τ )] w(v) exp[−iθI (v)]H1 (t1 )H2 (t2 )H3 (t3 ) × u1 (t − t1 − t2 − t3 )u∗1 (t − t2 − t3 − τ )dΣ (4.3)  P2 = S1 (r) exp[−i(ω3 t + ω1 τ )] w(v) exp[−iθII (v)]H1∗ (t1 )H2 (t2 )H3 (t3 ) × u1 (t − t2 − t3 )u∗1 (t − t1 − t2 − t3 − τ )dΣ (4.4)  P3 = S1 (r) exp[−i(ω3 t + ω1 τ )] w(v) exp[−iθI (v)]H1 (t1 )H4 (t2 )H3 (t3 ) × u1 (t − t1 − t2 − t3 )u∗1 (t − t2 − t3 − τ )dΣ (4.5)  P4 = S1 (r) exp[−i(ω3 t + ω1 τ )] w(v) exp[−iθII (v)]H1∗ (t1 )H4 (t2 )H3 (t3 ) × u1 (t − t2 − t3 )u∗1 (t − t1 − t2 − t3 − τ )dΣ (4.6)  P5 = S2 (r) exp[−i(ω3 t − ω2 τ )] w(v) exp[−iθIII (v)]H3 (t1 )H5 (t2 )H3 (t3 ) × u2 (t − t2 − t3 − τ )u∗2 (t − t3 )dΣ

(4.7)

where  S1 (r) = −iN μ41 ε1 (ε1 )∗ ε3 exp[i(k1 − k1 + k3 ) · r] 4  S2 (r) = −iN μ21 μ22 ε2 (ε2 )∗ ε3 exp[i(k2 − k2 + k3 ) · r] 4 θI (v) = v · [k1 (t1 + t2 + t3 ) − k1 (t2 + t3 ) + k3 t3 ]

θII (v) = v · [−k1 (t1 + t2 + t3 ) + k1 (t2 + t3 ) + k3 t3 ] θIII (v) = v · [k3 (t1 + t2 + t3 ) + k2 (t2 + t3 ) − k2 t3 ] H1 (t) = exp[−(Γ10 + iΔ1 )t] H2 (t) = exp(−Γ0 t) H3 (t) = exp[−(Γ10 + iΔ3 )t] H4 (t) = exp(−Γ1 t) H5 (t) = exp[−(Γ20 + iΔ2 + iΔ3 )t]

μ1 (μ2 ) is the dipole-moment matrix element between |0 and |1 (|1 and |2); Γ0 (Γ1 ) is the population relaxation rate of state |0 (|1). By considering contributions of nonradiative processes in such gas-phase medium, we assume Γ0 to be small, but nonzero. Γ10 (Γ20 ) is the transverse relaxation rate of the transition from |0 to |1 (|0 to |2), which contains material dephasing dynamics; Frequency detunings are Δ1 = Ω1 − ω1 , Δ2 = Ω2 − ω2 , and

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4 Heterodyne/Homodyne Detection of MWM

Δ3 = Ω1 − ω3 ;



dΣ denotes the four-fold integrations 





+∞

dΣ =

dv −∞



∞

dt3 0



∞

∞

dt2 0

dt1 0

4.1.3 Purely Homogeneously-broadened Medium For lifetime-broadened three-level atoms (Doppler-free approximation with ki · v ≈ 0 and ki · v ≈ 0) and τ > 0, Eqs. (4.3)–(4.7) can be written as P1 = S1 (r) exp[−i(ω3 t + ω1 τ )] 1 Γ0

%

1 × Γ10 + iΔ3

e−α1 τ 2α1 e−(Γ10 +iΔ1 )τ − Γ10 − α1 + iΔ1 (Γ10 + iΔ1 )2 − α21

& (4.8)

P2 = S1 (r) exp[−i(ω3 t + ω1 τ )]

1 e−α1 τ 1 Γ10 + iΔ3 Γ0 Γ10 + α1 − iΔ1

P3 = S1 (r) exp[−i(ω3 t + ω1 τ )]

1 × Γ10 + iΔ3

1 Γ1

%

e−α1 τ 2α1 e−(Γ10 +iΔ1 )τ − Γ10 − α1 + iΔ1 (Γ10 + iΔ1 )2 − α21

(4.9)

&

P4 = S1 (r) exp[−i(ω3 t + ω1 τ )]

1 e−α1 τ 1 Γ10 + iΔ3 Γ1 Γ10 + α1 − iΔ1

P5 = S2 (r) exp[−i(ω3 t − ω2 τ )]

1 1 × Γ10 + iΔ3 Γ10 + α2 + iΔ3

e−α2 τ Γ20 + i(Δ3 + Δ2 )

(4.10) (4.11)

(4.12)

Thus, P1 and P3 (with a radiation-matter detuning oscillation (RDO)) show both atomic and light responses together, but P2 , P4 , and P5 (without RDO) show light response only (without the τ -dependent decay factors like e−(Γ10 +iΔ1 )τ or e−(Γ20 +iΔ2 +iΔ3 )τ ). In the limit of weak noisy fields and under the condition of zero correlation time for the noisy lights, the decay of the DFWM signal yields the dephasing time Γ10 for the atomic medium. The onephoton DFWM and two-photon NDFWM complex susceptibilities χA and χB at frequency ω3 (ω3 ≈ Ω1 ) are obtained from third-order polarizations PA and PB , respectively, as follows: χA (τ, Δi , α1 ) =

PA ψ1 (Γ0 + Γ1 ) = ×  ∗ ε0 E1 (E1 ) E3 (Γ10 + iΔ3 )Γ0 Γ1

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

117

%

1 1 + − Γ10 + α1 − iΔ1 Γ10 − α1 + iΔ1 & 2α1 e−(Γ10 +iΔ1 −α1 )τ (Γ10 + iΔ1 )2 − α21

χB (Δi , α2 ) =

(4.13)

PB ε0 E2∗ E2 E3

ψ2 1 1 (4.14) Γ10 + iΔ3 Γ10 + α2 + iΔ3 Γ20 + i(Δ3 + Δ2 )   where ψ1 = −iN μ41 ε0 3 and ψ2 = −iN μ21 μ22 ε0 3 . These complex susceptibilities are greatly modified by color-locked noisy fields. Specifically, χA and χB strongly depend on the linewidth αi and time delay τ in broadband case, while it becomes independent of αi and time delay τ under the narrow band condition. In the cw limit (αi = 0), the imaginary part and real part of χA or χB can correspond to the non-modified nonlinear absorption and dispersion. In absorption plots, a positive (negative) value indicates gain (absorption). The signal response of the FWM process has been calculated previously in the system of four-level double-Λ cold atoms [3]. The anomalous dispersion generally corresponds to the strong absorption  a of the medium. Close inspection  a of Eq. (4.14) shows that when α2 Γ10 decreases to the cw case (α2 Γ10 = 0), the absorption will increase, and the slope  of normal dispersion curve will dramatically increase at near  resonance Δ2 Γ10 = 0. According to Eq. (4.14), the parameters are Γ20 Γ10 = 1.3, Δ3 Γ10 = −0.001, and for α2 Γ10 = 0(cw case, dash-dotted curve), 700 (dotted curve), 1200 (dashed curve), and 2000 (solid curve) in the Fig. 4.3. Moreover, near the two-photon transition, there is a large induced nonlinear amplification (ImχB negative) and ReχB is large and can have either sign. =

‹ Fig. 4.3. Nonlinear dispersion (a) and absorption (b) versus Δ2 Γ10 .

The expression of χB , which is responsible for the modification of the

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4 Heterodyne/Homodyne Detection of MWM

non-modified nonlinear absorption-dispersion profile, arises in the bare-state formalism from a two-photon induced coherence between levels |0 and |2, (2) i.e. a nonzero matrix element ρ20 . Clearly, even for an atom at rest (v = 0), the color-locking feather of the noisy light modifies the normal susceptibility in a nontrivial way which can not be simply characterized either as a mere level shift or as a broadening of the resonance. The phase dispersion θB (Δ2 , Δ3 , α2 ) =

tg−1 {[(Γ210 − Δ23 )(α2 + Γ20 ) − 2Γ10 Δ3 (Δ2 + Δ3 )] [2Γ10 Δ3 (α2 + Γ20 ) + (Γ210 − Δ23 )(Δ2 + Δ3 )]}

for χB = |χB |eiθB can be easily obtained from Eq. (4.14). There are two ways to measure the nonlinear susceptibility of the NDFWM process experimentally. One is the conventional detection method in which the NDFWM polarization |PB | is measured at its own absolute square, PB (PB )∗ . The two-photon NDFWM signal intensity is proportional to |χB |2 and all phase information of χB have been lost in such detection. The second way to measure χB is to introduce another polarization PA (called a reference signal or a “local oscillator”), designed to conjugate in frequency and wave vector in its complex representation with the PB polarization of interest. Thus, in this heterodyne (or homodyne) case, the signal photons are derived from (PA + PB )[(PA )∗ + (PB )∗ ]. In heterodyne detected FWM signal, the phase information is well preserved and one can take a full measure of the complex susceptibility, including its phase. The phase-sensitive detection of this two-photon NDFWM signal is based on the polarization interference between two FWM processes. Since optical fields oscillate too quickly for direct detection, they must be measured by beating with another optical field with the similar frequency. The phase of induced complex polarization, P (3) = PA + PB , determines how its energy will partition between the absorbed or emitted active spectroscopy and the passive spectroscopy with a new launched field spectroscopy [11]. The ASPB signal intensity (beating between one-photon DFWM (PA ) and two-photon NDFWM (PB ) in the three-level system, as shown in Fig. 4.1.) can be obtained as, I(τ, Δi , αi ) ∝ PA PA∗ + PB PB∗ + PA PB∗ + PA∗ PB = η1 |χA |2 e−2α1 τ + η2 |χB |2 e−2α2 τ + 2η12 |χA | |χB | e−(α1 +α2 )τ cos(θA − θB + θR )

(4.15)

where χA = |χA |eiθA = |χA | cos θA +i|χA | sin θA , χB = |χB |eiθB = |χB | cos θB + i|χB | sin θB ; θR = Δk · r − (ω1 + ω2 )τ ; Δk = (k1 − k1 ) − (k2 − k2 ); η1 = ε20 ε21 (ε1 )2 ε23 , η2 = ε20 ε22 (ε2 )2 ε23 , η12 = ε20 ε1 (ε1 )∗ ε2 ε∗2 ε23 . Although the complex susceptibilities (nonlinear responses) are greatly modified by the color-locked noisy fields, they can still be obtained effectively in certain limiting cases. In the heterodyne detection, we usually assume that |PA |2 >> |PB |2 at intensity level (or |χA | >> |χB |at field level), so the reference signal (one-photon DFWM), originated from the ω1 frequency

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119

components of twin beams 1 and 2, is much larger than two-photon NDFWM signal (generated by the ω2 frequency components of the twin beams 1 and 2), which gives I(τ, Δi , αi ) ∝ η1 |χA | e−2α1 τ + 2η12 |χB | |χA | e−(α1 +α2 )τ cos(θA − θB + θR ) (4.16) 2

Equation (4.16) shows that the sum-frequency ASPB signal from heterodyne detection is modulated with a frequency ω1 + ω2 as τ is varied. The phase coherence control of light beams in ASPB is subtle. The phase of the ASPB signal strongly depends on the phase θB of χB . Such, the ASPB can be effectively employed via the optical heterodyne detection to yield the real and imaginary parts of χB . If we adjust thetime delay τ or r to make θA + θR = 2nπ (i.e., τ = [θA (τ ) + Δk · r − 2nπ] (ω1 + ω2 ), Δk · r = 0, the value of integer n depends sensitively on the sign of τ ), then I(Δ2 ) ∝ η1 |χA | e−2α1 τ + 2η12 e−(α1 +α2 )τ Re[χB (Δ2 )]

(4.17)

However, if θA +θR = (2n+1/2)π (i.e., τ = [θA (τ )+Δk·r −2nπ−π/2]/(ω1 + ω2 ), Δk · r = 0), we then have I(Δ2 ) ∝ η1 |χA | e−2α1 τ + 2η12 e−(α1 +α2 )τ Im[χB (Δ2 )]

(4.18)

In other words, by changing the time delay τ between different frequency components ω1 and ω2 , we can selectively obtain real and the imaginary parts of χB (Δ2 , Δ3 ). The subtle value of τ is generally determined by the gradual approaching method from θA (τ ) + θR (τ ) = 2nπ or (2n + 1/2)π. Due to Δ1 ≈ Δ3 , this procedure can not be used for determining χA (Δ1 , Δ3 ). In a homogeneously-broadened three-level ladder-type atomic system, when τ < 0, Eqs. (4.3)–(4.7) can be reduced to P1 = S1 (r) exp[−i(ω3 t + ω1 τ )]

1 eα1 τ 1 Γ10 + iΔ3 Γ0 Γ10 + α1 + iΔ1

P2 = S1 (r) exp[−i(ω3 t + ω1 τ )]

1 × Γ10 + iΔ3

! " 2α1 e(Γ10 −iΔ1 )τ 1 eα1 τ − Γ0 Γ10 − α1 − iΔ1 (Γ10 − iΔ1 )2 − α21 P3 = S1 (r) exp[−i(ω3 t + ω1 τ )]

1 eα1 τ 1 Γ10 + iΔ3 Γ1 Γ10 + α1 + iΔ1

P4 = S1 (r) exp[−i(ω3 t + ω1 τ )]

1 × Γ10 + iΔ3

! " eα1 τ 1 2α1 e(Γ10 −iΔ1 )τ − Γ1 Γ10 − α1 − iΔ1 (Γ10 − iΔ1 )2 − α21

(4.19)

(4.20) (4.21)

(4.22)

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4 Heterodyne/Homodyne Detection of MWM

P5 = S2 (r) exp[−i(ω3 t − ω2 τ )] !

1 × (Γ10 + iΔ3 )2

2α2 eτ (Γ20 +iΔ2 +iΔ3 ) eα2 τ − α22 − (Γ20 + iΔ2 + iΔ3 )2 −Γ20 + α2 − iΔ2 − iΔ3

" (4.23)

Thus, P2 , P4 and P5 can show atomic and light responses together, but P1 and P3 only show the light response. It is then straightforward to explicitly obtain χA and χB , as follows: χA (τ, Δi , α1 ) PA ψ1 (Γ0 + Γ1 ) = = ×  ∗ ε0 E1 (E1 ) E3 (Γ10 + iΔ3 )Γ0 Γ1 ! " 1 1 2α1 e(Γ10 −iΔ1 −α1 )τ + − (4.24) Γ10 + α1 + iΔ1 Γ10 − α1 − iΔ1 (Γ10 − iΔ1 )2 − α21 χB (τ, Δi , α2 ) =

PB ψ2 = × ε0 E2∗ E2 E3 (Γ10 + iΔ3 )2 ! " 2α2 e(Γ20 +iΔ2 +iΔ3 −α2 )τ 1 − α22 − (Γ20 + iΔ2 + iΔ3 )2 −Γ20 + α2 − iΔ2 − iΔ3

(4.25)

The one-photon DFWM and two-photon NDFWM complex susceptibili ties χA and χB all show atomic and light responses together. When α2 Γa10 decreases [12] in Eq. (4.24) for τ < 0, the slope of the NDFWM dispersion curve will increase at the near resonance, this term (the controllable  slope of the normal dispersion d[ReχB (Δ2 )] dω2 |Δ2 =0 > 0) can lead to slow propagation of the phase-matched coherent NDFWM fields and therefore the longer effective interaction length, which makes the NDFWM process more efficient [see Figs. 4.3 (a) and  4.4 (a)]. Moreover, the NDFWM absorption curve becomes deeper as α2 Γa10 decreases [see Figs. 4.3 (b) and 4.4 (b)]. The RDO contrast of the NDFWM dispersion curve dramatically improves ver sus α2 Γa10 increasing. The RDO periods in the NDFWM dispersion and absorption curves also increase versus decrease in Γ10 |τ | [see Fig. 4.4 (c)]. Although the complex susceptibilities are greatly modified by the colorlocked noisy fields, they can still be obtained effectively in the ideal limit by employing the heterodyne detection as: 2

2

I(τ, Δi , αi ) ∝ η1 |χA | e2α1 τ + η2 |χB | e2α2 τ + 2η12 |χA | |χB | e(α1 +α2 )τ cos(θA − θB + θR )

(4.26)

If |χA | >> |χB | at the field level, we then obtain I(τ, Δi , αi ) ∝ η1 |χA | e2α1 τ + 2η12 |χB | e(α1 +α2 )τ cos(θA − θB + θR ) (4.27)

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

121

Fig. 4.4. χB (τ, Δi , α2 ) response for the NDFWM signal and its Fourier transform in homogeneously-broadened medium for τ < 0. Nonlinear dispersion (a) and ‹ absorption (b) versus Δ2 Γ10 .

If we adjust the time  delay τ and r such that θA + θR = 2nπ (i.e., τ = [θA (τ ) + Δk · r − 2nπ] (ω1 + ω2 )), then I(Δ2 ) ∝ η1 |χA | e2α1 τ + 2η12 e(α1 +α2 )τ Re[χB (Δ2 )]

(4.28)

However, if θA + θR = (2n + 1/2)π (i.e., τ = [θA (τ ) + Δk · r − 2nπ −  π/2] (ω1 + ω2 )), we have I(Δ2 ) ∝ η1 |χA | e2α1 τ + 2η12 e(α1 +α2 )τ Im[χB (Δ2 )]

(4.29)

The one-photon DFWM |PA |2 exhibits the hybridii radiation-matter terahertz detuning damping oscillation at τ > 0 and τ < 0, while the twophoton NDFWM |PB |2 shows RDO at τ < 0 only. In the narrow band limit (α1 << Γ10 , α2 << Γ20 ), and tail approximation (Γ10 |τ | >> 1), it is then straightforward to obtain χA = |χA | eiθA = 2Γ10 ψ1 (Γ10 + iΔ3 )(Γ0 + 2 2 2 Γ1 ) [Γ0 Γ1 (Γ210 + Δ  3 )(Γ10 + Δ21 )] from Eqs. (4.13) and (4.23) and χB = iθB = ψ2 (Γ10 + iΔ3) [Γ20 + i(Δ3 + Δ2 )] from Eqs. (4.14) and |χB | e (4.24), where θA = tan−1 (Γ10 Δ3 ). The real and imaginary parts of χA (Δ1 ,

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4 Heterodyne/Homodyne Detection of MWM

Δ3 ) or χB (Δ2 , Δ3 ) are completely independent of αi of the color-locked noisy lights and the time delay τ , and correspond to the non-modified nonlinear dispersion   – absorption expressions. If n = 0, we can obtain τ = ω2 ) for ReχB (Δ2 ) in Eqs. (4.17) and (4.27), and τ = tg−1 (Γ10 Δ3 ) (ω1 +  tg −1 [(Γ10 Δ3 ) − π/2] (ω1 + ω2 ) for ImχB (Δ2 ) in Eqs. (4.18) and (4.28). As shownin the Figs. 4.4, parameters are Γ20 Γ10 = 1.3, Δ3 Γ10 = −0.001, for α2 Γ10 = 800 and Γ10 τ = −0.000 644 2 (dash-dotted curve), 2 000 and −0.000 644 2 (dotted curve), 5000 and −0.000 644 2 (dashed curve), 2 000 and −0.001 5 (solid curve); (c) FFT of dispersion and absorption curves, Γ10 τ = −0.000 644 2 (high peak), −0.001 5 (low peak). Since the phase-matched coherent NDFWM signal intensity is given by IB ∝ |EB |2 ≈ |iχB |2 = (ReχB )2 + (ImχB )2 [2], it makes propagation characteristics of the two-photon NDFWM pulse more complicated. Three key contributions are involved in the propagation characteristics of the NDFWM pulse [13]: linear response term, cross-Kerr (or self-Kerr) nonlinear term, and the phase-matched coherent NDFWM term (the dominant term). This means that the propagation characteristics of the NDFWM pulse is determined by all three of these contributions together. The ReχB and ImχB may correspond to the phase-matched quasi cross-Kerr nonlinear index and the quasi two-photon absorption coefficient, respectively [18].

4.1.4 Extremely Doppler-broadened Limit When the atomic velocity distribution cannot be neglected, a straightforward semiclassical analysis shows that the contribution of atoms with velocity v to the complex susceptibility of NDFWM is given by the heterodyne-detected ASPB. Under the extremely Doppler-broadened limit (i.e., k3 u → ∞, in the limit of pure inhomogeneous broadening), we have 

+∞

−∞



+∞

−∞



+∞

−∞

 √ dvw(v) exp[−iθI (v)] ≈ 2 πδ(t3 − ξ1 t1 ) k3 u  √ dvw(v) exp[−iθII (v)] ≈ 2 πδ(t3 + ξ1 t1 ) k3 u  √ dvw(v) exp[−iθIII (v)] ≈ 2 πδ[t3 + t1 − (ξ2 − 1)t2 ] k3 u

  where, ξ1 = k1 k3 , ξ2 = k2 k3 > 1 for two-photon coherence effect. It is then straightforward to obtain third-order polarizations of DFWM and NDFWM as follows: √ 2 π P1 = S1 (r) exp[−i(ω3 t + ω1 τ )]× k3 u

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

%

a −(Γa 10 +iΔ1 )τ

123

&

e−α1 τ 1 2α1 e + 2 a Γ0 Γ10 − α1 + iΔa1 α1 − (Γa10 + iΔa1 )2 √ 2 π S1 (r) exp[−i(ω3 t + ω1 τ )] × P3 = k3 u % & a a e−α1 τ 1 2α1 e−(Γ10 +iΔ1 )τ + 2 Γ1 Γa10 − α1 + iΔa1 α1 − (Γa10 + iΔa1 )2 √ 2 π (ξ − 1)e−α2 τ S2 (r) exp[−i(ω3 t − ω2 τ )] a 2 P5 = k3 u (Γ20 + α2 + iΔa2 )2 P2 = P4 = 0.

(4.30)

(4.31)

(4.32)

Here, Γa10 = Γ10 + ξ1 Γ10 , Δa1 = Δ1 + ξ1 Δ3 ; Γa20 = Γ20 + (ξ2 − 1)Γ10 , and Δa2 = Δ2 + ξ2 Δ3 .

Fig. 4.5. (a) ASPB and RDO versus Γa10 τ , (b) FFT of ASPB and RDO.

Thus P1 and P3 with both atom and light responses lead to one-photon DFWM |P1 + P3 |2 exhibiting hybrid radiation-matter terahertz detuning damping oscillation, while P5 with the light response alone cannot cause the RDO in the two-photon NDFWM |P5 |2 . As shown in the Fig. 4.5, attosecond polarization beat, RDO, and their Fourier transform in the a extremely Doppler τ with the  a broadened media, ASPB  a and RDO  a versus Γ10 a a Γ Γ Γ = 0.6, ξ = 2, Γ = Γ = 0.5, Γ parameters α 2 2 0 1 10 10 20 Γ10  a  a 10  = 1,  a a a (ω2 + ω1 ) Γ10 = 37 104.535. α1 Γ10 = 0.3, Δ1 Γ10 = 2 000, and Δ2 Γa10 = 3 000 (dash-dotted curve); 0.6, 2 000 and 3 000 (dotted curve); 0.6, 1 000 and 3 000 (dashed curve); 0.6, 2 000 and 1 500 (solid curve). FFT of ASPB and RDO for dash-dotted and  line positions δi (i =1, 2, 3, 4)   dotted curves, corresponds toΔa1 Γa10 , Δa2 Γa10 ,(ω2 + ω1 ) Γa10 , (ω2 + ω1 + Δa1 ) Γa10 and (ω2 + ω1 + Δa2 ) Γa10 values, respectively. Corresponding complex susceptibilities χA and χB are obtained from third-order polarizations PA = P1 + P3 and PB = P5 , respectively, as follows: √ PA 2 πψ1 Γ0 + Γ1 χA (τ, Δa1 , α1 ) = = × ε0 E1 (E1 )∗ E3 k3 u Γ0 Γ1

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4 Heterodyne/Homodyne Detection of MWM

%

& a a 1 2α1 e−(Γ10 +iΔ1 −α1 )τ + Γa10 − α1 + iΔa1 α21 − (Γa10 + iΔa1 )2 √ ξ2 − 1 PB 2 πψ2 = χB (Δa2 , α2 ) = ε0 E2∗ E2 E3 k3 u (Γa20 + α2 + iΔa2 )2

(4.33) (4.34)

The χB (Δa2 , α2 ) = |χB |eiθB is completely independent of τ , and close to the non-modified nonlinear dispersion–absorption  a expression, where the a −1 a phase dispersion θ (Δ , α ) = tg [(Γ + α ) Δ2 ] and modulus |χB | = B 2 2 2 20  iψ2 (ξ2 − 1) [(Γa20 + α2 )2 + (Δa2 )2 ]. Real and imaginary parts of χB (Δa2 , α2 ) are given by odd function (on Δa2 ) √ 2 πψ2 (ξ2 − 1)Δa2 ReχB = {ik3 u[(Γa20 + α2 )2 + (Δa2 )2 ]} and even function (on Δa2 ) ImχB =

√ 2 πψ2 (ξ2 − 1)(Γa20 + α2 ) , respec{ik3 u[(Γa20 + α2 )2 + (Δa2 )2 ]}

tively. However, in general case, χA and χB will strongly depend on linewidth αi and time delay τ in broadband (That is to say that complex susceptibilities are greatly modified by color-locked noisy fields), while it becomes independent of αi and time delay τ in narrowband. In the cw limit (αi = 0), the real part and imaginary part of χA or χB can correspond to the non-modified nonlinear dispersion and absorption, respectively. In the extreme Doppler-broadened ASPB, using Eqs. (4.30)–(4.34), we obtain I(τ, Δai , αi ) ∝ |PA + PB |2 = |PA |2 + |PB |2 + PA PB∗ + PA∗ PB

(4.35)

where √ PA = 2 πS1 (r) exp[−i(ω3 t + ω1 τ )](Γ0 + Γ1 ) × a

a

e−α1 τ 2α1 e−(Γ10 +iΔ1 )τ + (Γa10 − α1 + iΔa1 ) [α21 − (Γa10 + iΔa1 )2 ] k3 uΓ0 Γ1 (DFWM at field level) √ 2 πS2 (r) exp[−i(ω3 t − ω2 τ )](ξ2 − 1)2 e−α2 τ PB = [k3 u(Γa20 + α2 + iΔa2 )2 ] (NDFWM at field level) Generally, the ASPB can be viewed as the sum of three contributions: |PA |2 = η1 |χA |2 e−2α1 τ (one-photon DFWM signal at intensity level), |PB |2 = η2 |χB |2 e−2α2 τ (two-photon NDFWM signal at intensity level), PA PB∗ + PA∗ PB = 2η12 |χA ||χB |e−(α1 +α2 )τ cos(θA − θB + θR ) (cross term between DFWM and NDFWM at intensity level).

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

125

Similar to the previous discussions in Eqs. (4.15)–(4.18), we can write I(τ, Δai , αi ) ∝ |PA | + PA PB∗ + PA∗ PB = η1 |χA | e−2α1 τ + 2

2

2η12 |χA | |χB | e−(α1 +α2 )τ cos(θA − θB + θR )

(4.36)

I(Δa2 ) ∝ η1 |χA | e−2α1 τ + 2η12 |χA | e−(α1 +α2 )τ |χB (Δa2 )| cos[θB (Δa2 )] (4.37) 2

I(Δa2 ) ∝ η1 |χA | e−2α1 τ + 2η12 |χA | e−(α1 +α2 )τ |χB (Δa2 )| sin[θB (Δa2 )] (4.38) 2

Due to adding the local oscillator intensity in Eqs. (4.37) and (4.38), dispersion and absorption profiles only show positive values compared with Fig. 4.3. After one subtracts the local oscillator background |PA |2 from them, they then become in good agreement with Fig. 4.3. In other words, by changing the time delay τ of the heterodyne detected ASPB signal we can obtain the real partsof χB (Δa2 ). In the Fig.  aand theimaginary  a4.6, the parama 1, (ω + ω ) Γ10 = 37104.535, eters are Γ0 Γ10 = Γ1 Γ10 = 0.5, Γa20 Γa10 = 2 1  ξ2 = 2, Δa1 Γa10 = −0.001. α1 Γa10 = α2 Γa20 = 0 cw case (dash-dotted curve), 1 (dotted curve), 2 (dashed curve), 5 (solid curve), the heterodyne  detection spectra versus Δa2 Γa10 of the ASPB (for the Doppler extremely  broadened three-state atoms τ > 0) with (a) τ = [θA (τ )+ Δk ·r − 2nπ] (ω1 + ω Γa10 τ = 0.0000589 for the real part and (b) τ = [θA (τ ) + Δk · r − 2nπ − 2 ),  π 2] (ω1 + ω2 ), Γa10 τ = 0.0000393 for the imaginary part.

‹ Fig. 4.6. Heterodyne detection spectra versus Δa2 Γa10 .

The broadband limit (noisy field coherence time τc ≈ 0, or αi → ∞) corresponds to “white” noise, characterized by a δ-function time correlation or, alternatively, it possesses a constant spectral density. Under the large Doppler-broadening limit (i.e., k3 u → ∞, inhomogeneous broadening limit) and broadband (α√1 >> Γa10 and α2 >> Γa20 ) approximation, we can obtain a a 1 )τ [α2 − χA (Δa1 , α1 , τ ) = 2 πψ1 (Γ0 + Γ1 ){1 (iΔa1 − α1 ) + 2α1 e−(Γ10 −α1 +iΔ 1   √ a a 2 a 2 (Γ10 + iΔ1 ) ]} (k3 uΓ0 Γ1 ) and χB (Δ2 , α2 , τ ) = 2 πψ2 (ξ2 − 1) [k3 u(α2 + iΔa2 )2 ] from Eqs. (4.33) and (4.34). These complex susceptibilities are greatly

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4 Heterodyne/Homodyne Detection of MWM

modified by the color-locked noisy field. That is to say that χA and χB strongly depend on noisy field parameter αi and time delay τ in broadband. In a inhomogeneously-broadened three-level ladder-type system, we can obtain √ eα1 τ 2 π 1 P1 = S1 (r) exp[−i(ω3 t + ω1 τ )] (4.39) a k3 u Γ0 Γ10 + α1 + iΔa1 √ eα1 τ 2 π 1 S1 (r) exp[−i(ω3 t + ω1 τ )] (4.40) P3 = k3 u Γ1 Γa10 + α1 + iΔa1 % √ (ξ2 − 1)2 eα2 τ 2 π S2 (r) exp[−i(ω3 t − ω2 τ )] P5 = + k3 u [α2 − (Γa20 + iΔa2 )]2 τ [(Γa20 + iΔa2 )2 − α22 ] − 2(Γa20 + iΔa2 ) × [α22 − (Γa20 + iΔa2 )2 ]2 & a a 2α2 (ξ2 − 1)2 e(Γ20 +iΔ2 )τ

(4.41)

P2 = P4 = 0 Thus P1 and P3 with light response alone cannot cause the RDO of onephoton DFWM |PA |2 , while P5 with both atom and light responses leads to two-photon NDFWM |PB |2 exhibiting the hybrid radiation-matter terahertz detuning damping oscillation (Fig. 4.5). Due to effects of noisy field colorlocking, in the limit of zero correlation time of the noisy light, the decay of the NDFWM signal yields a dephasing time Γa20 of the atomic medium. The maximum of Doppler-broadened NDFWM |PB |2 is shifted from zero timedelay compared with DFWM |PA |2 . Close inspection of |PB |2 shows that  a the maximum of NDFWM signal occurs at τ = 1 Γ20 . More specifically, the NDFWM profile becomes asymmetric due to the τ -dependent coefficient of the second term in Eq. (4.41) and the degree of asymmetry is determined by (Γa20 )−1 . It is then straightforward to obtain χA and χB as follows: χA (Δa1 , α1 )

√ 1 PA 2 πψ1 Γ0 + Γ1 = = ε0 E1 (E1 )∗ E3 k3 u Γ0 Γ1 Γa10 + α1 + iΔa1

(4.42)

PB ε0 E2∗ E2 E3 % √ (ξ2 − 1)2 2 πψ2 = + k3 u [α2 − (Γa20 + iΔa2 )]2

χB (τ, Δa2 , α2 ) =

τ [(Γa20 + iΔa2 )2 − α22 ] − 2(Γa20 + iΔa2 ) × [α22 − (Γa20 + iΔa2 )2 ]2 & a a 2α2 (ξ2 − 1)2 e(Γ20 −α2 +iΔ2 )τ

(4.43)

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

127

One-photon DFWM χA (Δa1 , α1 ) = |χA | eiθA shows the light response alone, while two-photon NDFWM χB (τ, Δa2 , α2 ) shows atom and light responses together, where √ 2 πiψ1 (Γ0 + Γ1 )  |χA | = k3 uΓ0 Γ1 (Γa10 + α1 )2 + (Δa1 )2  and θA = tg−1 [(Γa10 + α1 ) Δa1 ]. The nonlinear dispersion slope decreases and the absorption dip gets deeper, and their RDOs show strong competition versus the bandwidth increase of the color-locked noisy fields. In Fig. 4.7, NDFWM χB (τ, Δai , α2 ) response and its Fourier transform in the extremely Doppler broadened three-state atoms for τ < 0 with parameters  (Γa20 Γa10 = 1, ξ2 = 2, Γa10 τ = −0.000 594, α2 Γ10 = 1000 (dotted curve), 3000 (dashed curve), 4000 (solid curve); (c) FFT of dispersion and absorption curves, Γa10 τ = −0.000 594). The strong RDO can wash out the slope reduction effect, or change its variation direction [see Fig. 4.7 (a)] and suppress the nonlinear absorption [see Fig. 4.7 (b)]. The color-locked noisy effects of incoherent fields can lead to controllable change for the third-order nonlinear response and obvious hybrid radiation-matter terahertz oscillation. Figure 4.8 shows the phase dispersion of the two-photon NDFWM  aincluding the a Γ10 = 1, ξ2 = 2, influence of the color-locked noisy field. Parameters are Γ 20  Γa10 τ = −0.0000196. α2 Γa10 = 15 (dashed curve), 5 (solid curve). Similar to Eqs. (4.26)–(4.29), we can obtain 2

2

I(τ, Δai , αi ) ∝ η1 |χA | e2α1 τ + η2 |χB | e2α2 τ + 2η12 |χA | |χB | e(α1 +α2 )τ cos(θA − θB + θR ) (4.44) I(τ, Δai , αi )

2 2α1 τ

∝ η1 |χA | e

+

2η12 |χA | |χB | e(α1 +α2 )τ cos(θA − θB + θR ) (4.45) I(Δa2 )

2 2α1 τ

2 2α1 τ

∝ η1 |χA | e

+ 2η12 |χA | e

(α1 +α2 )τ

|χB (Δa2 )| cos[θB (Δa2 )]

+ 2η12 |χA | e

(α1 +α2 )τ

|χB (Δa2 )| sin[θB (Δa2 )]

(4.46) I(Δa2 )

∝ η1 |χA | e

(4.47) Because of the local oscillator intensity of in Eqs. (4.46) and (4.47), dispersion and absorption profiles (see Fig. 4.9) have been up-shifted by an amount of |PA |2 compared with Fig.  4.7. As shown in the Fig. 4.9, the heterodyne detection spectra versus Δa2 Γa10 of the ASPB (for the Doppler extremely  broadened three-state atoms τ < 0) with (a) τ = [θA (τ ) + Δk · r − 2nπ] (ω1 + ω2 ), Γa10 τ= −0.000 594 for the real part and (b) τ = [θA (τ ) + Δk · r − 2nπ − a π/2] part. Parameters:   a(ω1 + ω2 ),a Γ10 τ = −0.000  039 3 for the imaginary Γ0 Γ10 = Γ1 Γ10 = 0.5, Γa20 Γa10 = 1, (ω2 + ω1 ) Γa10 = 37 104.535, ξ2 = 2,

128

4 Heterodyne/Homodyne Detection of MWM

‹ Fig. 4.7. Nonlinear dispersion (a) and absorption (b) versus Δa2 Γa10 .

Fig. 4.8. Phase dispersion of two-photon NDFWM, θB (Δa2 , α2 , τ ) ∼ Δa2 .

   Δa1 Γa10 = −0.001, α1 Γa10 = 1 000. α2 Γa10 = 1 000 (dotted curve), 3 000 (dashed curve), 4 000 (solid curve). The dotted curve has been scaled by a factor 0.03. After subtracting the local oscillator background from them, they become in good agreement with Fig. 4.7 [18]. The one-photon DFWM |P1 + P3 |2 exhibits hybrid radiation-matter terahertz detuning damping oscillation at τ > 0, while two-photon NDFWM |P5 |2 shows RDO at τ < 0 [Table (I)]. The modified two-photon third-order absorption and dispersion can be controlled coherently by the noisy light

4.1 Modified Two-photon Absorption and Dispersion of Ultrafast Third-order · · ·

129

‹ Fig. 4.9. The heterodyne detection spectra versus Δa2 Γa10 .

color-locking bandwidth, frequency detuning, and time delay. In the narrowband and tail approximation (α1 << Γa10 , α2 << Γa20 and Γa10 |τ | >> 1), it is then straightforward to obtain √ 2 πψ1 (Γ0 + Γ1 ) a iθA χA (Δ1 ) = |χA | e = k3 uΓ0 Γ1 (Γa10 + iΔa1 ) from Eqs. (4.33) and (4.42), χB (Δa2 )

= |χB | e

iθB

√ 2 πψ2 (ξ2 − 1)2 = k3 u(Γa20 + iΔa2 )2

from Eqs. (4.34) and (4.43), where θA = tg−1 (Γa10 /Δa1 ) and θB = tg−1 {[(Γa20 )2 + (Δa2 )2 ]/2Γa20 Δa2 } The real part and imaginary part of χA (Δa1 ) or χB (Δa2 ) are independent of αi and τ , and correspond to the non-modified quasi cross-Kerr nonlinear index and two-photon absorption coefficient. Due to the transcendental functions, the precise value of τ is generally determined by the gradual approaching method from θA (τ ) + θR (τ ) = 2nπ or (2n + 1/2)π. If n = 0, we can readily obtain τ = tg−1 [(Γa10 /Δa1 )/(ω1 + ω2 )] for ReχB (Δa2 ) in Eqs. (4.37) and (4.46), τ = tg−1 {[(Γa10 /Δa1 ) − π/2]/(ω1 + ω2 )} for ImχB (Δa2 ) in Eqs. (4.38) and (4.47). Equations (34) and (43) contain rich dynamics of the color-locked noisy field correlation effects [7, 10, 11], and the competition between attosecond ultrafast modulation and hybrid terahertz RDO. Close inspection of Eqs. (4.34) observations (see Fig. 4.5): (a) When  and (4.43) reveals four interesting  α1 Γa10 of local oscillator and α2 Γa10 of two-photon NDFWM increase, the

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4 Heterodyne/Homodyne Detection of MWM

a RDO contrast of DFWM and NDFWM will  a dramatically improve for Γ10 τ > a a 0 and Γ10 τ < 0, respectively; (b) if Δ2 Γ10 increases, theattosecond beat modulation contrast will dramatically improve; (c) when Δa1 Γa10 and Δa2 Γa10 increase, the RDO oscillation periods of DFWM and NDFWM will decrease for Γa10 τ > 0 and Γa10 τ < 0, respectively; (d) the frequencies of RDO and ASPB can be read as [see Fig. 4.5 (c)]    Δa1 Γa10 , Δa2 Γa10 , (ω2 + ω1 ) Γa10 ,   (ω2 + ω1 + Δa1 ) Γa10 , (ω2 + ω1 + Δa2 ) Γa10

which is the combination of Γa10 τ > 0 and Γa10 τ < 0 results [Table (II)]. The detail results of frequency analysis of RDO and ASPB are listed in Tables (I) and (II).

4.1.5 Discussion and Conclusion As a time-domain technique, the main advantage of the ASPB technique over the conventional quantum beat technique is that the temporal resolution is not limited by the laser pulse width. With the laser pulse of ns timescale duration, femto- or atto-second timescale modulations were observed [10]. We reported studies of atto-second polarization beats induced by the thirdorder susceptibility χ(3) . This effect was exploited to consider the phase of χ(3) measurements in lifetime-broadened and extremely Doppler-broadened three-level atoms. The method presented here is simple to employ and can be applied to a large variety of materials in which backward FWM (phase conjugation) can be observed [9]. Specifically, let us consider Na atom as a possible FWM system. We take, for instance, |0 = |3S1/2 , |1 = |3P3/2 , and |2 = |4D3/2,5/2 . The respective transitions are |0 → |1 at 588.996 nm (Γ−1 ≈ 16.9 ns, Γ−1 1 10 ≈ 5.7 ps), and |1 → |2 at 568.822 nm, all accessible with non-transform-limited pulsed dye lasers operated in multi-longitudinal modes (typical color-locked chaotic fields). The ASPB signal at present three-level atomic system not only exhibits 965 as the ultrafast modulation [7, 10, 14], but also shows the hybrid radiation-matter detuning damping oscillation in THz scale [Tables (I) and (II)]. The maximum of the two-photon NDFWM is shifted from zero time-delay, and the signal also exhibits damping oscillation when the laser frequency is off resonant from the two-photon transition. This method can be useful for directly measuring χ(3) . The two-color FWM of attosecond polarization beat has been employed for studying the phase dispersion of χ(3) . This is a good way to measure the third-order susceptibility directly, especially its real and imaginary parts separately. Although our method is somewhat similar to the femtosecond polarization beats done in a solid by Ma et al. [6], we have shown that for two-photon resonance in a three-level atomic system one can obtain the phase

4.2 Color-locking Phase Control of Fifth-order Nonlinear Response

131

dispersion of χ(3) by simply measuring the phase change of the NDFWM signal modulation as ω2 detuning is varied. Moreover, the technique of using attosecond polarization beats to measure the third-order susceptibility has advantages over other (such as Z-scan) methods for atomic systems, because it can work with long atomic cells [2]. Generally speaking, our method can also be applied to study the phase dispersion of χ(3) of femtosecond polarization beats in the gas-phase media. In summary, we demonstrated a phase-sensitive technique to study the NDFWM in a three-level ladder-type atomic system. The reference signal is another DFWM signal, which propagates along the same optical path as the NDFWM signal. This point is very important since the reference signal always travels in basically the same direction, such it is much easier for mode matching and reducing background (all other fields, linear processes, scattering, etc). This method was used to investigate the phase dispersion of the third-order susceptibility and optical heterodyne detection of the twophoton NDFWM signal under various limits and conditions.

4.2 Color-locking Phase Control of Fifth-order Nonlinear Response We consider a folded four-level system (Fig. 4.10), in which states between |0 and |1, |1 and |2, and |2 and |3 are dipole allowed transitions with resonant frequencies Ω1 , Ω2 and Ω3 and dipole moments μ1 , μ2 and μ3 , respectively. As shown in Fig. 4.10 (b), beam 2 includes three color-locked fields, E2 (ω2 , k2 , and Rabi frequency G2 ), E3 (ω3 , k3 , and Rabi frequency G3 ), and E3 (ω3 , k3 , and Rabi frequency G3 ), and beam 3 has one color-locked field, E2 (ω2 , k2 , and Rabi frequency G2 ). A small angle exists between these two beams. Beam 1 is a monochromatic field E1 (ω1 , k1 , and Rabi frequency G1 ) which propagates along the opposite direction of beam 2. Assuming near resonance so that E1 drives the transition from |0 to |1 while E2 drives the transition from |1 to |2 simultaneously, which induce atomic coherence between |0 and |2 through two-photon excitation [14]. This established atomic coherence is probed by E2 in beam 3 and, as a result, a FWM signal of frequency ω1 (beam 4) is generated almost opposite to the direction of (0) ω1 (1) ω2 (2) −ω2 (3) beam 3, i.e., ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ10 (I). Next, we apply two coupling laser fields with the same frequency ω3 (≈ Ω3 ), both of which propagate along beam 2, to drive the transition |2 to |3.   The Rabi frequencies are defined as Gi = εi μi , Gi = εi μi , α = G1 G2 (G2 )∗ and β = G1 G2 (G3 )∗ G3 (G2 )∗ (G3 ≈ G3 ≈ G); while the noisy  fields are Ei = εi ui (t)eiki ·r−iωi t and Ei = εi ui (t − τ )ei(ki ·r−ωi t+ωi τ ) (τ is   a time delay). εi , ki (εi ,ki ) are the constant field amplitude and the wave vector. ui (t) (u1 ≈ 1) is a dimensionless statistical factor that contains phase and amplitude fluctuations (i.e., u1 ≈ 1, here εi (t) contains pure amplitude

132

4 Heterodyne/Homodyne Detection of MWM

Fig. 4.10. Ladder diagrams representing the dressed FWM evolution pathways, and the interplay between SWM and FWM via atomic coherence: (a) the dressed FWM, (b) Phase-conjugation geometry.

fluctuation, while θi (t) contains pure phase fluctuation). The ui (t) is taken to be a complex ergodic stochastic function of t, which obeys complex circular Gaussian statistics in the chaotic field. Γ10 , Γ20 and Γ30 are the transverse relaxation rates between states |0 and |1, |0 and |2, |0 and |3, respectively. Detuning factors are defined as Δ1 = Ω1 − ω1 , Δ2 = Ω2 − ω2 , Δ3 = Ω3 − ω3 , Δa = Δ1 +Δ2 , Δb = Δa −Δ3 , d1 = Γ10 +iΔ1 , d2 = Γ20 +iΔa , d3 = Γ30 +iΔb . The FWM and SWM phase matching conditions are kf = k1 + k2 − k2 and ks = k1 + k2 − k3 + k3 − k2 , respectively. Let us now consider PB between FWM and SWM in the time domain (non-steady state analysis). As mentioned before, in the weak coupling field limit (ζ << 1) the dressed FWM signal can be represented as a coherent su(3) (5) perposition of the signals from FWM and SWM, i.e., ρ10 ≈ ρ10 + ρ10 . FWM and SWM signals propagate along the same direction, which are indistin(3) (5) guishable. Since |ρ10 | >> |ρ10 | in general, this technique can be regarded as a heterodyne detection of the SWM with the FWM signal as the optical local oscillator. The heterodyne beat signal is proportional to the real or imaginary part of the SWM complex susceptibility at the particular controllable time delay. That is to say, in heterodyne detected SWM, phase information is retained and one can take a full measure of the fifth-order complex susceptibility χS , including its phase. Physically, the heterodyne beat of the dressed FWM signal comes from PB between FWM and SWM Liouville pathways: (0) ω1 (1) ω2 (2) −ω2 (3) (0) ω1 (1) ω2 (2) −ω3 (3) ω3 ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ10 and ρ00 −−→ ρ10 −−→ ρ20 −−−→ ρ30 −−→ (4) −ω

(5)

2 ρ10 . ρ20 −−−→

The polarization P (n) responsible for FWM or SWM is given by averaging over the velocity distribution function w(v). Since the dressed FWM τ -dependent behavior is dominated by terms which contain only intra-atomic correlation. There is no τ -dependent interaction between inter-atomic correlations. Under these circumstances, nonlinear polarizations on each of the two generic atom groups in the biatomic group model become independent. As such, the approximation |P (n) |2 = |P (n) |2  can be made. In other words, the correct τ -dependent form of the signal intensity follows from the absolute square of the non-trivial stochastic average of the polarization |P (3) +P (5) |2 (averaging at the field level). Of course, this relation does not hold for noisy light spectroscopies in general [19].

4.2 Color-locking Phase Control of Fifth-order Nonlinear Response

133

In dealing with the gas-phase atomic medium, we made an approximation by averaging at the field level, which only involves second-order correlation functions for the noise fields. We can obtain FWM and SWM polarizations for the non-steady state case as:  ∞  ∞  ∞ dt3 dt2 dt1 e−d1 (t1 +t3 )−d2 t2 × PF (t) = A 0

0

0

u2 (t − t2 − t3 )u∗2 (t − t3 − τ )wF  ∞  ∞  ∞  ∞  dt5 dt4 dt3 dt2 PS (t) = B 0

0

0

0

(4.48) ∞

dt1 ×

0

e−d1 (t1 +t5 )−d2 (t2 +t4 )−d3 t3 u2 (t − t2 − t3 − t4 − t5 ) × u∗3 (t − t3 − t4 − t5 − τ )u3 (t − t4 − t5 )u∗2 (t − t5 − τ )wS

(4.49)

Here  wF =  wS =

+∞

dvw(v)e−iθ

(I)

dvw(v)e−iθ

(II)

(v)

−∞ +∞

(v)

−∞

A = S1 (r)e−i(ω1 t+ω2 τ ) B = S2 (r)e−i(ω1 t+ω2 τ +ω3 τ ) The factors S1 (r), S2 (r), θ(I) (v), and θ(II) (v) are provided in Ref.[10]. (i) τ < 0 In the lifetime-broadened limit, by virtue of PF = ε0 χF E2 (E2 )∗ E1 = ε0 χF ε1 ε2 (ε2 )∗ eα2 τ eikf ·r−i(ω1 t+ω2 τ ) and PS = ε0 χS E2 (E3 )∗ E3 (E2 )∗ E1 = ε0 χS ε1 ε2 (ε3 )∗ ε3 (ε2 )∗ e(α2 +α3 )τ eiks ·r−i[ω1 t+(ω2 +ω3 )τ ] , we can obtain nonlinear responses at field level to be χF = |χF |eiθF = |χS |e χS =

iθS

χF 0 (d21 α2 )

= χS0 e

−α3 τ







ed 3 τ (eα3 τ − ed3 τ ) +  (d3 − α3 ) (α2 + α3 )

[d21 (α2 )2 ]



  Here, χF 0 = −iN μ21 μ22 ε0 3 , χS0 = iN μ21 μ22 μ23 ε0 5 ; d2 = Γ20 − iΔa , d4 = α3 − Γ20 + Γ30 − iΔ3 , α = α3 − α2 , α2 = d2 − α2 , α2 = α2 + d2 , α3 = α3 + d3 , d3 = α2 + d3 .N is the atomic density. Complex susceptibilities are greatly modified by color-locked noisy fields (see Fig. 4.11), which also show hybrid terahertz the Rabi detuning oscillation (RDO). Specifically, the giant χF and χS with opposite signs strongly depend on linewidth αi and time delay τ in broadband case, while it generally becomes independent of αi and time delay

134

4 Heterodyne/Homodyne Detection of MWM

τ in the narrowband [19]. One can obtain the fifth-order susceptibility by phase control (at particular τ ). In the homodyne beat detection, I(Δi ) ∝ |PF + PS |2 ∝ e2α2 τ [|χF |2 + |ηχS |2 e2α3 τ + χF (ηχS )∗ eα3 τ eiω3 τ + χ∗F ηχS eα3 τ e−iω3 τ ]

(4.50)

Fig. 4.11. SWM response χS [(a) for Re χS and (b) for Im χS ] and heterodyne detection signal [(c) θF + θ = 2nπ for Re χS or (d) (2n + 1/2)π for Im χS ] versus Δ3 /Γ10 .

If η0 is real number, I(Δi ) ∝ e2α2 τ [|χF |2 + η02 |χS |2 e2α3 τ + 2η0 |χF ||χS |eα3 τ cos(θF − θS + θ)] μ2 (ε )∗ ε3 eiΔk·r Here, θ = ω3 τ − Δk · r, η = SS21 ≡ − 3 3 2 = η0 eiΔk·r , the spatial  dependence in η can be neglected in a typical experiment. Although the complex susceptibilities (nonlinear responses) are greatly modified by the color-locked noisy fields, they can still be obtained effectively in the ideal limit. In the heterodyne detection, we assume that |PF |2 >> |PS |2 at the intensity level (|χF | >> |χS |at field level), so the reference signal (FWM) originated from the ω2 frequency components of the twin noisy beams 2 and 3 is much larger than the SWM signal originated from the ω2 and ω3 frequency components of the twin noisy beams 2 and 3. So 2

I(Δi ) ∝ e2α2 τ [|χF | + 2η0 |χF | |χS | eα3 τ cos(θF − θS + θ)]

(4.51)

The subtle phase coherence control can effectively be employed to yield the real and imaginary parts of χS . If we adjust the time delay τ and r

4.2 Color-locking Phase Control of Fifth-order Nonlinear Response

135

 such that θF + θ = 2nπ (i.e., τ = [2nπ + Δk · r − θF (τ )] ω3 , the value of integer n depends on the sign of τ sensitively), then I(Δ3 ) ∝ e2α2 τ [|χF |2 + 2η0 |χF |eα3 τ Re[χS (Δ3 )]. However, if θF + θ = (2n + 1/2)π (i.e., τ = [(2n + 1/2)π + Δk · r − θF (τ )]/ω3 ), we will have I(Δ3 ) ∝ e2α2 τ [|χF |2 + 2η0 |χF |eα3 τ Im[χS (Δ3 )]. In other words, by changing the time delay τ we can obtain real and imaginary parts of χS (Δ1 , Δ2 , Δ3 ). As shown in the Fig. 4.11, SWM response χS [(a) for ReχS and (b) for ImχS ] and the heterodyne detection signal  [(c) θF + θ = 2nπ for Re χS or (d) (2n + 1/2)π for ImχS ] versus Δ3 Γ10 for τ < 0 case, and their Fourier transform versus Γ10 |τ | (e). The parameters are Γ20 = Γ30 = 0.1Γ10 , α2 = 2Γ10 = 2α3 , Δ2 = 0.02Γ10 = 2Δ1 , ω3 = 4000Γ10 and Γ10 τ = −1. (ii) τ > 0 We can obtain % 2 −α2 τ  − e−α3 τ ) α2 τ d4 (e χS = χS0 e + [d4 α (α2 )2 ] α e−d2 τ [d4 (e−d4 τ − 1) − α2 (e−d4 τ − 1 + d4 τ )] + [d4 α (α2 )2 ] 

χF =



[d4 e−d2 τ − α2 (e−α3 τ + e−d2 τ )] [e−α3 τ + e−d2 τ (d4 τ − 1)] − [(α2 )2 (α2 + α3 )] (d4 α2 )     (1 − e−α2 τ ) e(α2 −d2 )τ + χF 0 α2 α2

&' d21 d4

d21

In the heterodyne beat detection, I(Δi ) ∝ e−2α2 τ [|χF |2 + 2η0 |χF | |χS | e−α3 τ cos(θF − θS + θ)]

(4.52)

In the limit of narrowband (α2 , α3 << Γ20 , Γ30 ) and tail (Γ20 |τ|, Γ30 |τ | >> 2 1) approximation,  2 2 it is then straightforward to obtain χF = χF 0 (d1 d2 ) and χS = χS0 (d1 d2 d3 ) for both τ > 0 and τ < 0. Real and imaginary parts of χF (Δ1 , Δ2 ) or χS (Δ1 , Δ2 , Δ3 ) are completely independent of αi of the colorlocked noisy lights and the time delay τ , and correspond to the non-modified nonlinear dispersion – absorption expressions. Close inspection of Eqs. (4.50) and (4.52) reveals rich dynamics of the color-locked noisy field correlation effects, the dramatic competition between sub-femtosecond ultrafast PB and hybrid terahertz RDO, and the τ -dependent asymmetry behavior of the SWM signal. As shown in the Fig. 4.12, the  parameters are Γ20 = Γ30 = 0.1Γ10 , α2 = 2Γ10 = 2α3 , Δ3 = 2Δ2 = 3Δ1 2 = 2Γ10 and ω3 = 50Γ10 for (a) and (b); α2 = 20Γ10 = 2α3 5 and Δ3 = 0.9Δ2 = 0.4Δ1 = 180Γ10 for (c) and (d).

136

4 Heterodyne/Homodyne Detection of MWM

Fig. 4.12. (a) SWM signal |PS |2 and (c) homodyne beat detection signal versus Γ10 τ ; (b) and (d) their Fourier transforms (including 12 oscillation periods of subfemtosecond PB and terahertz RDO).

4.3 Seventh-order Nonlinear Response With the basic system (in Fig. 4.13 (a)) of three energy levels (|0, |1, and |2) and three laser fields (ε2 , ε2 , and ε1 ), a FWM signal at frequency ω1 will be generated. By adding another energy level (either |3 or |4) and another laser field (ε3 or ε3 for level |3, or ε4 or ε4 for level |4), the original energy level (|1 or |2) will be dressed to produce two dressed states. Such four-level system with a dressing field will modify the original FWM process (called (3) (5) singly-dressed FWM with notation ρF iSj ) and generate SWM signals (ρSi ). If two energy levels (|3 and |4) are both added with two additional fields, (3) the original FWM system is said to be doubly dressed (denoted as ρF ij ), (5)

(7)

which can generate not only SWM signals (ρSi ), but also EWM (ρEi ). One can consider such system first as a four-level system (|0, |1, |2 and |3) which generates SWM, and then by adding another level (|4) and a field ε4 , this four-level atomic system is (singly) dressed again to give a singly-dressed (5) SWM signal (ρSij ), which will have contributions from EWM under certain (3)

(5)

conditions. Note that ρF ij and ρSij are not purely third-order and fifth-order nonlinearities, instead they are the doubly-dressed FWM and singly-dressed SWM, respectively, including higher-order nonlinear responses. The main purpose of using such doubly-dressed schemes is to generate efficient EWM and, at the same time, to allow us to control the relative strengths of various wave-mixing processes, so such high-order nonlinear optical processes can be enhanced, manipulated, and studied in detail [19].

4.3 Seventh-order Nonlinear Response

137

For a five-level atomic system as shown in Fig. 4.13 (a), states |i − 1 to |i (i = 1, 2, 3, 4) are coupled by laser fields εi and εi [ωi , ki (ki ) with Rabi frequency Gi (Gi )]. The Rabi frequencies are defined as Gi = εi μij / and Gi = εi μij /, respectively, where μij is the transition dipole moment between level i and level j. Fields εi and εi with the same frequency and different time delays (εi is delayed by time τ ) propagate along beams 2 and 3 with a small angle [see Fig. 4.13 (b)], while the weak probe field ε1 (beam 1) propagates along the opposite direction of beam 2. The nonlinear polarizations, responsible for multi-wave mixing signals, are proportional to the (n) off-diagonal density matrix elements ρ10 . We will assume, as usual, that G1 is weak, whereas the laser fields G2 , G2 , G3 , G3 , G4 , and G4 can be of (n) arbitrary magnitudes. Thus, ρ10 needs to be calculated to the lowest-order in G1 , but to all orders in other fields under various conditions.

Fig. 4.13. (a) Energy-level diagram for co-existing FWM, SWM and EWM in an open five-level system. (b) Phase-conjugate schematic diagram of phase-matched multi-wave mixing.

The coexistence of these nonlinear wave-mixing processes in this open fivelevel system can be used to evaluate the high-order nonlinear susceptibilities. Susceptibilities χ(3) , χ(5) and χ(7) can be obtained by beating between FWM (F1 ) and FWM (F2 ), FWM (F1 ) and SWM (S2 , S3 ), SWM (S1 ) and EWM (E1 , E2 ) processes, respectively, since signals of each beating pair propagate along the same direction. First, we evaluate the susceptibility χ(7) by the blocking k2 of beam 3, overlapping k4 and k4 along beam 2 [see Figs. 4.13 (2) (a) and (b)]. In such case, only SWM (S1 of kS1 ) and EWM (E1 , E2 of (3) (5) (7) kE1,2 ) processes exist in beam 4. Since |ρ10 | >> |ρ10 | is generally true under normal condition and the SWM (S1 ) and EWM (E1 , E2 ) signals are diffracted in the same direction with the same frequency, real and imaginary parts of χ(7) can be measured by the homodyne detection with the SWM (S1 ) signal (which is assumed to be known here and can be determined by beating with FWM signal) as the strong local oscillator. Under this circumstance, the polarization beat signal is proportional to the real or imaginary part of the complex susceptibility χ(7) at a particular controllable time delay [19]. The polarization beat is based on the interference at the detector between multi-wave mixing signals which originate from macroscopic polarizations excited simultaneously in the homogeneously- or inhomogeneously-broadened sample. It critically requires that all the fifth- and seventh-order polariza-

138

4 Heterodyne/Homodyne Detection of MWM

tions have the same frequency. The polarization P (n) responsible for multiwave mixing is given by averaging over the velocity distribution function w(v). For the particular spectroscopic technique the τ -dependent terms only have “intra-atomic” correlations and no “inter-atomic” correlations in the τ -dependent terms (which is usually true for atomic vapor systems). Nonlinear polarizations on each of two generic atom groups in the biatomic group model become independent. As such, the approximation |P (n) |2 = |P (n) |2  can be made. In other words, the correct τ -dependent form of the signal intensity follows from the absolute square of the non-trivial stochastic average of the polarization |P (5) + P (7) |2 (averaging at the field level), which reduces to |P (5) + P (7) |2 for the CW light approximation. Of course, this relation does not hold for the noisy light spectroscopy in general. Under the extremely Doppler-broadened limit, we can obtain SWM and EWM polarizations for the non-steady-state CW light approximation case as √  ∞  ∞  ∞  ∞  ∞ 2 π (5) B dt5 dt4 dt3 dt2 dt1 × PS1 = k1 u 0 0 0 0 0 H1 (t1 )H2 (t2 )H3 (t3 )H2 (t4 )H1 (t5 ) ×

(7)

PE1

(4.53) δ[t1 + (1 − ξ1 )t2 + (1 − ξ1 + ξ2 )t3 + (1 − ξ1 )t4 + t5 ] √  ∞  ∞  ∞  ∞  ∞  ∞  ∞ 2 π C = dt7 dt6 dt5 dt4 dt3 dt2 dt1 × k1 u 0 0 0 0 0 0 0 H1 (t1 )H2 (t2 )H3 (t3 )H2 (t4 )H1 (t5 ) × H4 (t6 )H1 (t7 )δ(t1 − (ξ1 − 1)t2 − (ξ1 − ξ2 − 1)t3 −

(7)

PE2

(4.54) (ξ1 − 1)t4 + t5 − (ξ3 − 1)t6 + t7 ) √  ∞  ∞  ∞  ∞  ∞  ∞  ∞ 2 π C = dt7 dt6 dt5 dt4 dt3 dt2 dt1 × k1 u 0 0 0 0 0 0 0 H1 (t1 )H4 (t2 )H1 (t3 )H2 (t4 )H3 (t5 )H2 (t6 )H1 (t7 )δ(t1 − (ξ3 − 1)t2 + t3 − (ξ1 − 1)t4 − (ξ1 − ξ2 − 1)t5 − (ξ1 − 1)t6 + t7 )

(4.55)

Where H1 (t) = exp[−(Γ10 + iΔ1 )t] H2 (t) = exp[−(Γ20 + iΔ1 + iΔ2 )t] H3 (t) = exp[−(Γ30 + iΔ1 + iΔ2 − iΔ3 )t] H4 (t) = exp[−(Γ40 + iΔ1 + iΔ4 )t] B = (iN μ21 μ22 μ23 /5 )ε1 ε2 ε∗2 ε3 (ε3 )∗ exp[ikS1 · r − i(ω1 t + ω3 τ )] (2)

C = (−iN μ21 μ22 μ23 μ24 /7 )ε1 ε2 ε∗2 ε3 (ε3 )∗ ε4 (ε4 )∗ × (3)

exp{ikE1,2 · r − i[ω1 t + (ω3 + ω4 )τ ]}    ξ1 = k2 k1 , ξ2 = k3 k1 , ξ3 = k4 k1

4.3 Seventh-order Nonlinear Response

139

N is the atomic density. Nonlinear responses χ(5) = |χ(5) | exp(iθS ) and χ(7) = |χ(7) | exp(iθE ) [see Figs. 4.14 (a1)–(a2)] at the field level are obtained from PS1 = ε0 χ(5) ε1 ε2 (ε3 )∗ ε3 ε∗2 exp[ikS1 · r − i(ω1 t + ω3 τ )] (5)

(2)

PE = ε0 χ(7) ε1 ε2 (ε3 )∗ ε3 ε∗2 ε4 (ε4 )∗ exp[ikE1,2 · r − i(ω1 t + ω3 τ + ω4 τ )] (7)

(3)

One can experimentally obtain the seventh-order susceptibility χ(7) by phase coherent control (at a particular τ ). In the homodyne beat detection, the ultrafast polarization beat at sub-femtosecond scale is ( (5) ( (2 ( (2 (7) (7) (2 I(Δi ) ∝ (PS1 + PE1 + PE2 ( ∝ (χ(5) ( + (ηχ(7) ( + χ(5) (ηχ(7) )∗ exp(iω4 τ ) + (χ(5) )∗ ηχ(7) exp(−iω4 τ )

(4.56)

If the reference signal (SWM) is much larger than the EWM signal (i.e., (7) (7) >> |PE1 + PE2 |2 at intensity level or |χ(5) | >> |χ(7) | at field level), we have ( ( (2 ( (2 (( ( I(Δi ) ∝ (χ(5) ( + η 2 (χ(7) ( + 2η (χ(5) ((χ(7) ( cos[θS − θE + θ(τ )] (4.57) (5) |PS1 |2

Here θ = ω4 τ , η = −μ24 (ε4 )∗ ε4 /2 . The subtle phase coherent control can effectively be employed to yield real and imaginary parts of χ(7) . If we adjust the time delay such that θS + θ(τ ) = 2nπ then I(Δ4 ) ∝ |χ(5) |2 + 2η|χ(5) |Re[χ(7) (Δ4 )] However, if

 θS + θ(τ ) = (2n + 1 2)π

we will have I(Δ4 ) ∝ |χ(5) |2 + 2η|χ(5) |Im[χ(7) (Δ4 )] In other words, by properly choosing the time delay τ we can obtain either the real or the imaginary part of χ(7) (Δ4 ) [see Fig. 4.14 (a3) – (a4)]. Similarly, we can obtain third- (χ(3) (Δ2 , Δ4 )and fifth-order (χ(5) (Δ4 ) nonlinear responses as indicated in Table 1. The χ(3) (Δ2 , Δ4 ) comes from polarization beat between FWM (F1 of kF1 ) and FWM (F2 of kF2 ). χ(5) (Δ4 ) can be determined from the polarization beat of FWM (F1 of kF1 ) and SWM (1) (S2 , S3 of kS2,3 ) by overlapping k4 and k4 along beam 2 [see Fig. 4.13 (b)]. There exists an optical gain effect in the imaginary part of χ(3) . By comparing with the real parts of the coexisting susceptibilities χ(3) , χ(5) , and χ(7) , we find that Re(χ(3) ), Re(χ(5) ) and Re(χ(7) ) are always with alternating signs to each other. As shown in the Fig. 4.14, Co-existing FWM, SWM and EWM nonlinear responses [(a1) for Reχ(7) and (a2) for Imχ(7) ; (b1) for Reχ(5) and (b2) for Imχ(5) ; (c1) for Reχ(3) and (c2) for Imχ(3) ], and polarization beat

140

4 Heterodyne/Homodyne Detection of MWM

Fig. 4.14. Co-existing FWM, SWM, and EWM nonlinear responses.

References

141

 signals [(a3) 2nπ for Reχ(7) or (a4) θF + θ = (2n + 1 2)π for Imχ(7) ; (b3) for Reχ(5) or (b4) for Imχ(5) ; (c3) for Reχ(3) or (c4) for Imχ(3) ] versus Δ4 /Γ10 (Note that the polarization beat signals have been scaled by a factor 10). The parameters are Γ20 /Γ10 = 0.3, Γ30 /Γ10 = 0.9, Γ40 /Γ10 = 0.1, α2 /Γ10 = 2, α3 /Γ10 = 1, Δ1 /Γ10 = 0.01, Δ2 /Γ10 = 0.02, and ω3 /Γ10 = 4000. We have a balance of the competitive contribution from FWM, SWM and EWM depending on the dressing field amplitude and detuning. This type of competition can be called cubic-quintic-septimal type nonlinearity in media [15]. Giant χ(3) , χ(5) , and χ(7) susceptibilities (cubic-quintic-septimal type) with alternating signs can be measured by the phase coherent control of the polarization beat. Such high-order nonlinearities have attracted a lot of theoretical attentions recently [15]. The complex susceptibilities χ(3) , χ(5) , and χ(7) are generally modified by the color-locked noisy fields, which will show a damped Rabi oscillation 2π/Δ. Superfluid-like and liquidlike light condensates, as well as multi-dimensional solitons have been predicted in four-level systems with competitive and giant χ(3) and χ(5) of opposite signs [15], which correspond to the well known nonlinearity of cubic-quintic competition type (nonlinear refractive index n2 > 0 and n4 < 0) in media. Finally, note that there are two kinds of SWM processes (S2 , S3 ) and two EWM processes (E1 , E2 ) which all constructively contribute to fifth-order and seventh-order nonlinear susceptibilities, respectively. This is in contrast to a two-level system in which there are two kinds of FWM processes which destructively contribute to the third-order nonlinear susceptibility, and the phase-matched biphoton temporal correlation shows a damped Rabi oscillation and photon antibunching [20].

References [1]

[2]

[3]

[4]

Jain M, Xia H, Yin G Y, et al. Efficient nonlinear frequency conversion with maximal atomic coherence. Phys. Rev. Lett., 1996, 77: 4326 – 4329. Braje D A, Balic V, Goda S, et al. Frequency mixing using electromagnetically induced transparency in cold atoms. Phys. Rev. Lett., 2004, 93: 183601 – 183604. Lu B, Burkett W H, Xiao M. Nondegenerate four-wave mixing in a doubleLambda system under the influence of coherent population trapping. Opt. Lett., 1998, 23: 804-806. Li Y Q, Xiao M. Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms. Opt. Lett., 1996, 21: 1064 – 1066. Kang H, Hernandez G, Zhu Y F. Resonant four-wave mixing with slow light. Phys. Rev. A, 2004, 70: 061804(R). Kang H, Hernandez G, Zhu Y F. Superluminal and slow light propagation in cold atoms. Phys. Rev. A, 2004, 70, 011801(R). Wang H, Goorskey D, Xiao M. Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system. Phys. Rev. Lett., 2001 87: 073601. Wang H, Goorskey D J, Xiao M. Dependence of enhanced Kerr nonlinearity on coupling power in a three-level atomic system. Opt. Lett., 2002, 27:

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4 Heterodyne/Homodyne Detection of MWM

258 – 260. Kang H S, Zhu Y F. Observation of large Kerr nonlinearity at low light intensities. Phys. Rev. Lett., 2003, 91: 093601. Ma H, Acioli L H, Gomes A S L, et al. Method to determine the phase dispersion of the 3rd-order susceptibility. Opt. Lett., 1991, 16: 630; Ma H, Gomes A S L, De Araujo C B. Raman-assisted polarization beats in timedelayed four-wave mixing. Opt. Lett., 1992, 17: 1052 – 1054; Ma H, De Araujo C B. Interference between 3rd-order and 5th-order polarizations in semiconductor-doped glasses. Phys. Rev. Lett., 1993, 71: 3649. Zhang Y P, Gan C L, Li L, et al. Rayleigh-enhanced attosecond sumfrequency polarization beats via twin color-locking noisy lights. Phys. Rev. A, 2005, 72: 013812; Zhang Y P, Gan C L, Song J P, et al. Attosecond sum-frequency Raman-enhanced polarization beating by use of twin phasesensitive color locking noisy light beams. J. Opt. Soc. Am. B, 2005, 22: 694 – 711. DeBeer D, Usadi E, Hartmann S R. Attosecond beats in sodium vapor. Phys. Rev. Lett., 1988, 60: 1262 – 1266. Fu P M, Mi X, Yu Z H, et al. Ultrafast modulation spectroscopy in a cascade three-level system. Phys. Rev. A, 1995, 52: 4867 – 4870. Zhang Y P, De Araujo C B, Eyler E E. Higher-order correlation on polarization beats in Markovian stochastic fields. Phys. Rev. A, 2001, 63: 043802; Zhang Y P, Gan C L, Song J P, et al. Coherent laser control in attosecond sum-frequency polarization beats using twin noisy driving fields. Phys. Rev. A, 2005, 71: 023802. Morita N, Yajima T. Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light. Phys. Rev. A, 1984, 30: 2525 – 2536; Ulness D J. On the role of classical field time correlations in noisy light spectroscopy: color locking and a spectral filter analogy. J. Phys. Chem. A, 2003, 107: 8111-8123; Schulz T F, Aung P P, Weisel L R, et al. Complete cancellation of noise by means of color-locking in nearly degenerate, four-wave mixing of noisy light. J. Opt. Soc. Am. B, 2005, 22: 1052 – 1061; Kirkwood J C, Ulness D J, Albrecht A C. On the classification of the electric field spectroscopies: Application to Raman scattering. J. Phys. Chem. A, 2000, 104: 4167 – 4173. Burkett W H, Lu B, Xiao M. Influence of injection current noise on the spectral characteristics of semiconductor lasers. IEEE J. Quantum Electron., 1997, 33: 2111 – 2118. Drescher M, Hentschel M, Kienberger R, et al. Time-resolved atomic innershell spectroscopy. Nature, 2002, 419: 803 – 807. Garrett W R, Moore M A, Hart R C, et al. Suppression effects in stimulated hyperRaman emissions and parametric four-wave mixing in sodium vapor. Phys. Rev. A, 1992, 45: 6687 – 6709. Paz-Alonso M J, Michinel H. Superfluidlike motion of vortices in light condensates. Phys. Rev. Lett., 2005, 94: 093901-093904; Michinel H, PazAlonso M J. Turning light into a liquid via atomic coherence. Phys. Rev. Lett., 2006, 96: 023903 – 023906. Ulness D J, Kirkwood J C, Albrecht A C. Competitive events in fifth order time resolved coherent Raman scattering: Direct versus sequential processes. J. Chem. Phys., 1998, 108: 3897; Moll K D, Homoelle D, Gaeta A L, et al. Conical harmonic generation in isotropic materials Phys. Rev. Lett., 2002, 88: 153901 – 153904.

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Ma H, De Araujo L E E, Gomes A S L, et al. Phase measurements of the fifth-order susceptibility of Cd(S, Se)-doped glasses. Opt. Commun., 1993, 102: 89 – 92. Zhang Y P, Gan C L, Xiao M. Modified two-photon absorption and dispersion of ultrafast third-order polarization beats via twin noisy driving fields. Phys. Rev. A, 2006, 73: 053801. Zhang Y P, Brown A. W, Gan C L, et al. Intermixing between four-wave mixing and six-wave mixing in a four-level atomic system. J. Phys. B, 2007, 40: 3319 – 3329. Zhang Y P, Anderson B, Xiao M. Coexistence of four-wave, six-wave and eight-wave mixing processes in multi-dressed atomic systems. J. Phys. B, 2008, 41: 045502.

5 Raman- and Rayleigh-enhanced Polarization Beats

In the previous chapters, we have mainly discussed ultrafast polarization beats with multi-level atomic systems, in which laser beams are always near resonance with the atomic transitions. In this chapter, we will turn our attention to some different problems, i.e. to investigate Raman- and Rayleighenhanced polarization beats in liquids and solid materials, in which thermal and reorientational gratings are important. In liquids and solid materials, both resonance and nonresonance excitations contribute significantly to the nonlinear optical processes. We present methods to study the Ramanenhanced polarization beats (REPB) with broadband noisy lights using chaotic field, phase-diffusion, and Gaussian-amplitude models. The interferometric contrast ratio of the detected polarization beat signal is shown to be particularly sensitive to amplitude and phase fluctuations of Markovian stochastic fields with arbitrary bandwidth. It is found that the beat signal oscillates not only temporally but also spatially. The overall accuracy of using the REPB to measure the Raman resonant frequency is determined by the relaxation rates of Raman modes and the molecular-reorientational grating. Another interesting feature in field correlations is Rayleigh-enhanced polarization beats. Rayleigh-enhanced four-wave mixing (RFWM) and Ramanenhanced four-wave mixing with color-locking noisy lights shows spectral symmetry and temporal asymmetry that no coherence spike exists at τ = 0. Due to the interference between the Raman-resonant (Rayleigh-resonant) signal and the nonresonant background, the FWM signal exhibits hybrid radiation-matter detuning THz damping oscillation. The high-order correlation effects in Markovian stochastic fields are investigated in the homodyne or heterodyne detected Rayleigh-enhanced attosecond sum-frequency polarization beats (RASPB) and Raman-enhanced attosecond sum-frequency polarization beats. Based on the polarization interference between two FWM processes, the phase-sensitive detection of polarization beats are used to obtain the real and imaginary parts of the nonlinear susceptibility at Raman and Rayleigh resonance.

5.1 Raman-enhanced Polarization Beats

145

5.1 Raman-enhanced Polarization Beats Atomic responses to Markovian stochastic optical fields are now largely well understood [1 – 6]. This section addresses the roles of noises in the incident fields on the generated nonlinear wave-mixing signals – particularly in the time domain. This important topic has been treated extensively in the literature including the introduction of a new diagrammatic technique (called factorized time correlator diagrams) [7 – 10], which treat the high-order noise correlators when circular Gaussian statistics apply. There are two classes of two-component laser beams (one laser beam with two frequency components, as used in earlier chapters). In one class frequency components are derived from two separate lasers and their mixed (cross) correlators should vanish. In the second case, two frequency components are from a single laser source whose spectral output is doubly peaked. This can be created from a single dye laser in which two different dyes in solution are amplified [7 – 9] or one component is shifted in frequency by an acoustic optical modulator. In this section, we deal only with the first case, i.e., the two colors in the twin-color beam come from separate broadband laser sources. The doublepeaked beams 1 and 2 (as shown in Fig. 5.1.) are paired and correlated, but each of peaks is uncorrelated with the other peak. Beam 3 is split from one of broadband laser sources used for beam 1 and beam 2, so it is correlated with beam 1 and beam 2. Such multi-color noisy lights have been used previously to generate FWM in self-diffraction geometry [7 – 10]. In this section, we describe a treatment of REPB in phase-conjugation geometry using three types of stochastic noise field models with beam 3 also as a noisy field. The chaotic field, the Brownian-motion phase-diffusion field, and the Gaussian-amplitude field are considered as the noisy fields to generate the REPB, respectively. A unified theory is developed which involves the sixthorder coherence function to study influences of the partial-coherence properties of light fields on polarization beats. Studies of polarization beats, which originate from the interference between the macroscopic polarizations, have attracted lots of attentions in recent years, as we have discussed in earlier chapters [11 – 13]. It is closely related to quantum beats, Raman quantum beats [12], and coherent Raman spectroscopy (CRS). CRS has become a powerful tool for studying the vibrational or rotational modes of molecules. The most commonly used coherent Raman spectroscopy includes coherent anti-Stokes Raman scattering (CARS) and Raman-induced Kerr effect spectroscopy. Recently, Raman-enhanced nondegenerate four-wave mixing (RENFWM) has attracted great attention [14, 15]. This RENFWM is a thirdorder nonlinear phenomenon with phase-conjugation geometry. It possesses several advantages over the conventional CARS including nonresonant background suppression, excellent spatial resolution even for the case of small Raman shifts, free choice of interaction volume, and simple optical alignment. Furthermore, since the phase-matching condition is not critical in RENFWM, it has a large frequency bandwidth and is therefore suitable for studying sub-

146

5 Raman- and Rayleigh-enhanced Polarization Beats

picosecond relaxation processes which have broad resonant linewidths. Fu, et al performed a time-delayed RENFWM with incoherent lights to measure the vibrational dephasing time [14]. They also found an enhancement of the ratio between the resonant and nonresonant RENFWM signal intensities as the time delay was increased when the laser had broadband linewidth [15]. Another relevant problem is the FWM with broadband noisy lights, which was proposed by Morita, et al [16] to achieve an ultrafast temporal resolution of relaxation processes. Since they assumed that the laser linewidth is much larger than the transverse relaxation rate, their theory cannot be used to study the effects of light bandwidth on the Bragg reflection signal. Asaka, et al [17] considered the finite linewidth effect. However, the constant background contribution has been ignored in that analysis. The model with higher-order correlation functions on polarization beats in the following includes the finite light bandwidth effect, constant background contribution, light field fluctuations, and controllable unbalance dispersive effects [7, 10]. These effects are of vital importance in the REPB. REPB is a third-order nonlinear optical process. The basic laser beam configuration is shown in Fig. 5.1. Beams 1 and 2 consist of two frequency components ω1 and ω3 each, with a small angle between them. Beam 3 with a frequency ω3 propagates almost along the opposite direction of beam 1. In the Kerr medium, the nonlinear interactions of beams 1 and 2 with the medium give rise to two molecular-reorientational gratings, i.e., ω1 and ω3 will induce their own static gratings Grating1 and Grating2, respectively. The FWM signal is the result of the diffraction of beam 3 by either Grating1 or Grating2.

Fig. 5.1. Schematic diagram of the geometry for REPB.

If |ω1 −ω3 | is near the Raman resonant frequency ΩR , a large-angle moving grating and two small-angle moving gratings (formed by the interferences between beams 2 and 3 and between beams 1 and 2, respectively) will excite the Raman-active vibrational mode of the medium and enhance the FWM signal. The generated beat signal (beam 4) is along the opposite direction of beam 2 approximately. The complex electric fields of beam 1 (Ep1 ) and beam 2 (Ep2 ) can be written as Ep1 = E1 (r, t) + E2 (r, t) = A1 (r, t) exp(−i ω1 t) + A2 (r, t) exp(−i ω3 t) = ε1 u1 (t) exp[i(k1 · r − ω1 t)] + ε2 u3 (t) exp[i(k2 · r − ω3 t)] Ep2 =

E1 (r, t)

+

E2 (r, t)

=

A1 (r, t) exp(−i ω1 t)

+

(5.1)

A2 (r, t) exp(−i ω3 t)

5.1 Raman-enhanced Polarization Beats

147

= ε1 u1 (t − τ + δτ ) exp[i(k1 · r − ω1 t + ω1 τ − ω1 δτ )] + ε2 u3 (t − τ ) exp[i(k2 · r − ω3 t + ω3 τ )]

(5.2)

Here, εi , ki (εi ,ki ) (i=1, 2) are the constant field amplitude and the wave vector of frequency components ω1 and ω3 in beam 1 (beam 2), respectively. ui (t) is a dimensionless statistical factor that contains phase and amplitude fluctuations. We assume that the ω1 (ω3 ) components of Ep1 and Ep2 come from a single laser source and τ is the time delay of beam 2 with respect to beam 1. δτ denotes the difference between two autocorrelation processes in the zero time delay (δτ > 0). The complex electric field of beam 3 can be written as Ep3 = A3 (r, t) exp(−iω3 t) = ε3 u3 (t) exp[i(k3 · r − ω3 t)]

(5.3)

where ω3 , ε3 , and k3 are the frequency, the field amplitude, and the wave vector of the field, respectively. Different colors correlate at different delay times because they have been delayed in the dispersed beam relative to the undispersed beam. This is analogous to the stretching of short pulses by transmission through a dispersive medium (chirp). In fact, identical physical processes are responsible for chirp in coherent short pulses and correlation functions of broadband fields. Considering the situation in which the double-frequency noisy field derived from two separate lasers with a finite bandwidth is split into twin replicas; then one of twin fields, Ep2 , is transmitted through a dispersive medium so that it is no longer identical to the other one (Ep1 ). Two autocorrelations (corresponding to the static gratings Grating1 and Grating2, respectively) are stretched differently in τ because each color component between beam 1 and beam 2 is maximally correlated at different delay times, whereas in beam 1 or beam 2 both color components are maximally correlated at the same delay time. Phases of chirped correlation functions exhibit a time dependence that is similar to the time-dependent phases of chirped coherent short pulses. Unchirped (transform-limited) correlation functions and short pulses have phases that are independent of time. An important practical distinction between short pulses and noisy-light correlation functions is that the chirping of correlation functions in double-frequency noisy-light interferometry can occur only after the double-frequency noisy field is split into beam 1 and beam 2, and then only if there is a difference between the dispersion in the paths traveled by beam 1 and beam 2. However, a short pulse can be chirped as it propagates through any dispersive medium between the source and the sample. That is to say, ultrashort pulses of equivalent bandwidth are not immune to such dispersive effects (even when balanced) because the transform-limited light pulse is in fact temporally broadened (chirped) and this has drastic effects on its time resolution (auto-correlation). In this sense the REPB with double-frequency noisy lights has an advantage [7]. Order parameters Q1 and Q2 of the two static gratings induced by beam

148

5 Raman- and Rayleigh-enhanced Polarization Beats

1 and beam 2 satisfy the following equations [14, 15]: dQ1 + γQ1 = χγE1 (r, t)[E1 (r, t)]∗ dt dQ2 + γQ2 = χγE2 (r, t)[E2 (r, t)]∗ dt

(5.4) (5.5)

Here, γ and χ are the relaxation rate and the nonlinear susceptibility of the two static gratings, respectively. We consider a large-angle moving grating (with order parameter QR1 ) and two small-angle moving gratings (with order parameters QR2 and QR3 ) formed by the interferences between beam 2 and beam 3 and between beam 1 and beam 2, respectively. Order parameters (QR1 , QR2 , QR3 ) satisfy following equations: iαR [A1 (r, t)]∗ A3 (r, t) dQR1 + (γR − iΔ)QR1 = dt 4 dQR2 iαR A1 (r, t)[A2 (r, t)]∗ + (γR − iΔ)QR2 = dt 4 dQR3 iαR [A1 (r, t)]∗ A2 (r, t) + (γR − iΔ)QR3 = dt 4

(5.6) (5.7) (5.8)

Here, Δ = |ω1 − ω3 | − ΩR is the frequency detuning; ΩR and γR are the resonant frequency and the relaxation rate of the Raman mode, respectively. αR is a parameter to indicate the strength of the Raman interaction. Five induced third-order nonlinear polarizations which are responsible for FWM signals are P1 = Q1 (r, t)E3 (r, t) = χγε1 (ε1 )∗ ε3 exp{i[(k1 − k1 + k3 ) · r − ω3 t − ω1 (τ − δτ )]} ×  ∞ u1 (t − t )u∗1 (t − t − τ + δτ )u3 (t) exp(−γt )dt (5.9) 0

P2 = Q2 (r, t)E3 (r, t) = χγε2 (ε2 )∗ ε3 exp{i[(k2 − k2 + k3 ) · r − ω3 t − ω3 τ ]} ×  ∞ u3 (t − t )u∗3 (t − t − τ )u3 (t) exp(−γt )dt

(5.10)

0

 PR1 = N αR QR1 (r, t)E1 (r, t) exp[i(ω1 − ω3 )t − iω1 (τ − δτ )] 2 = iχR γR ε1 (ε1 )∗ ε3 exp{i[(k1 − k1 + k3 ) · r − ω3 t − ω1 (τ − δτ )]} ×  ∞ u1 (t)u∗1 (t − t − τ + δτ )u3 (t − t ) exp[−(γR − iΔ)]dt (5.11) 0

 PR2 = N αR QR2 (r, t)E3 (r, t) exp[i(ω3 − ω1 )t − iω3 τ ] 2 = iχR γR ε1 (ε2 )∗ ε3 exp{i[(k1 − k2 + k3 ) · r − ω1 t − ω3 τ ]}×

5.1 Raman-enhanced Polarization Beats



∞

0

u1 (t − t )u∗3 (t − t − τ )u3 (t) exp[−(γR − iΔ)]dt

149

(5.12)

 PR3 = N αR QR3 (r, t)E3 (r, t) exp[i(ω1 − ω3 )t − iω1 (τ − δτ )] 2 = iχR γR (ε1 )∗ ε2 ε3 exp{i[(k2 − k1 + k3 ) · r − (2ω3 − ω1 )t − ω1 (τ − δτ )]} ×  ∞ u∗1 (t − t − τ + δτ )u3 (t − t )u3 (t) exp[−(γR − iΔ)]dt

(5.13)

0

with χR = N α2R /8γR . N is the density of molecules.

5.1.1 Chaotic Field The total third-order polarization is given by P (3) = P1 +P2 +PR1 +PR2 +PR3 . For the macroscopic system where phase matching takes place this signal comes from P (3) developed on one chromophore multiplied by (P (3) )∗ that is developed on another chromophore located elsewhere in space and summation needs to be made over all such pairs [7 – 10]. In general, the signal is homodyne/heterodyne (quadrature) detected. This means that the signal at the detector is derived from the squared modulus of the summation over all fields that are generated from the huge number of polarized chromophores in the interaction volume. The sum over chromophores leads to the phasematching condition at the signal level and its square modulus (the signal) is fully dominated by the bichromophoric cross terms. Thus, the detected quadrature signal is effectively built from the products of all polarization fields derived from all pairs of chromophores. This bichromophoric model is particularly important to the noisy light spectroscopy where the stochastic averaging at the signal level must be carried out [8, 9]. The FWM signal is proportional to the average of the absolute square of P (3) over the stochastic random variable, so that the signal intensity I(Δ, τ ) ∝ |P (3) |2  contains 5×5=25 different terms in sixth-, fourth- and second-order coherence functions of ui (t)in phase conjugation geometry. The ultrafast modulation spectroscopy (UMS) in self-diffraction geometry is also related to the sixth-order coherence function of incident fields [11]. We first consider the laser sources as chaotic fields, which are commonly used to describe the multimode lasers. Such chaotic field is characterized by both amplitude and phase fluctuations of the field. The random functions ui (t) for the complex noisy fields are taken to obey complex Gaussian statistics with sixth- and fourth-order coherence functions given by [1, 5] ui (t1 )ui (t2 )ui (t3 )u∗i (t4 )u∗i (t5 )u∗i (t6 ) = ui (t1 )u∗i (t4 )ui (t2 )ui (t3 )u∗i (t5 )u∗i (t6 ) + ui (t1 )u∗i (t5 )ui (t2 )ui (t3 )u∗i (t4 )u∗i (t6 )+

150

5 Raman- and Rayleigh-enhanced Polarization Beats

ui (t1 )u∗i (t6 )ui (t2 )ui (t3 )u∗i (t4 )u∗i (t5 )

i = 1, 3

(5.14)

ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 ) + ui (t1 )u∗i (t4 )ui (t2 )u∗i (t3 )

(5.15)

All higher-order coherence functions can be expressed in terms of products of second-order coherence functions. Thus, any given 2n-order coherence function may be decomposed into the sum of n! terms, each consisting of the products of n second-order coherence functions. By further assuming laser sources to have the Lorentzian line shape, then we can write the second-order coherence function as ui (t1 ) u∗i (t2 ) = exp (−αi |t1 − t2 |)

(5.16)

1 where αi = δωi , with δωi being the linewidth of the laser with frequency 2 ωi . We first consider the situation when linewidths of laser sources in beam 1, beam 2,and beam 3 are broadband (i.e., α1 , α3 >> γ, γR ), and treat the composite noisy beam 1 (beam 2) as a simple spectrum with a sum of two Lorentzians. Under such conditions and after performing the tedious integrations, we obtain, for: (i) τ > δτ I(Δ, τ ) ∝ (1 + η12 + η22 )

χ2R γR (α1 + α3 ) − (α1 + α3 )2 + Δ2

Δ[(5α1 + α3 )(α1 + α3 ) + Δ2 ] χ2 γ + (α1 η12 η22 + α3 ) + 2 2 2 α1 [(α1 + α3 ) + Δ ] 2α1 α3 % & 2 2χR χγR Δ (α1 + α3 ) χ2R γR 2 − + χ × α1 [(α1 + α3 )2 + Δ2 ] (α1 + α3 )2 + Δ2 % & 2 χ2R γR (α1 + α3 ) 2 2 2 + χ η1 × exp(−2α1 |τ − δτ |) + η2 α3 [(α1 + α3 )2 + Δ2 ] χR χγR γ

exp(−2α3 |τ |) + 4η1 η2 × exp(−α1 |τ − δτ | − α3 |τ |) × %! 2 " χ χχR γR Δ − cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − 2 2[(α1 + α3 )2 + Δ2 ] & χχR γR (α1 + α3 ) sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] (5.17) 2[(α1 + α3 )2 + Δ2 ] (ii) 0 < τ < δτ I(Δ, τ ) ∝ (1 + η12 + η22 )

χ2R γR (α1 + α3 ) 4χR χγR γΔ − + (α1 + α3 )2 + Δ2 α1 (α1 + α3 )2

χ2 γ (α1 η12 η22 + α3 ) + χ2 exp(−2α1 |τ − δτ |)+ 2α1 α3

5.1 Raman-enhanced Polarization Beats

2χ2 η12 η22 exp(−2α3 |τ |) +

151

2 χ2R γR exp(−2γR |τ − δτ |) + α1 (α1 + α3 )

χ2 η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) cos[Δk · r − (ω1 − ω3 )τ + 2 ! 1 ω1 δτ ] + χR χγR γ exp(−α1 |τ − δτ | − α3 |τ |) + γ(α1 + α3 ) " 1 × exp(−α3 |τ − δτ | − α3 |τ |) sin[Δk · r − 2α3 (α1 + α3 ) (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |] (5.18) (iii) τ < 0, the modulation of the beat signal is complicated in general, −1 however, at the tail of the signal (i.e., |τ | >> α−1 1 , |τ | >> α3 ), the result can be simplified to be χ2R γR (α1 + α3 ) − (α1 + α3 )2 + Δ2 Δ[(5α1 + α3 )(α1 + α3 ) + Δ2 ] χR χγR γ + α1 [(α1 + α3 )2 + Δ2 ]2 3χ2 η12 η22 γ 2 χ2 γ (α1 η12 η22 + α3 ) + exp(−2γ |τ |) + 2α1 α3 2α23 2 2χ2R γR [(1 + α3 ) exp(−2γR |τ |) + (α1 + α3 )2

I(Δ, τ ) ∝ (1 + η12 + η22 )

(1 + α1 )η22 exp(−2γR |τ − δτ |)] + 4η1 η2 χχR γγR exp(−γR |τ − δτ | − γ |τ |) × α3 (α1 + α3 )4 {(α1 + α3 )(2γR + γ)Δ cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |] − [(α1 + 2α3 )(α1 + α3 )2 + α1 Δ2 ] sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |]} ε2 /ε1 ,

k1 )

k2 ).

(5.19)

and Δk = (k1 − − (k2 − Here, η1 = ε2 /ε1 , η2 = Equation (5.17) consists of six terms. The last (sixth) term, depending on fourth- and second-order coherence functions of u1 (t) or u3 (t), is the crosscorrelation intensity between five third-order nonlinear polarizations, which gives rise to the modulation of the beat signal. The interferometric contrast ratio (which mainly determines the modulation depth) is sensitive to both amplitude and phase fluctuations of chaotic fields. The other five terms (the τ -independent and the decay terms), depending on sixth-, fourth- or secondorder coherence functions of u1 (t) or u3 (t), are a sum of the autocorrelation intensities of five third-order nonlinear polarizations. Different stochastic models of laser fields affect mainly on the sixth- and fourth-order coherence functions [1 – 6]. Constant terms which are independent of the relative timedelay between beam 1 and beam 2 in Eqs. (5.17), (5.18), and (5.19) mainly

152

5 Raman- and Rayleigh-enhanced Polarization Beats

come from the amplitude fluctuations of the chaotic fields. Fourth and fifth terms in Eq. (5.17), which are shown to be particularly sensitive to amplitude fluctuations of chaotic fields, show an exponential decay of the beat signal as |τ | increases. In general, REPB is different for τ > δτ , 0 < τ < δτ , and τ < 0 cases. However, as |τ | → ∞, Eq. (5.17) is identical to Eq. (5.18) or Eq. (5.19). Physically, when |τ | → ∞, beam 1 and beam 2 are mutually incoherent, therefore whether τ is positive or negative does not affect the detected REPB signal. Equation (5.17) indicates that when τ > δτ , the temporal behavior of the beat signal intensity reflects mainly the characteristics of the lasers, i.e., the frequency ω3 −ω1 and the damping rate α1 +α3 of modulation are determined by incident laser beams. If one employs the REPB to measure the modulation frequency ωd = ω3 − ω1 , the accuracy can be improved by measuring as many cycles of the modulation as possible. Since the amplitude of the modulation decays with a time constant (α1 +α3 )−1 as |τ | increases, the maximum domain of time-delay |τ | should equal approximately 2(α1 + α3 )−1 . The theoretical limit of the uncertainty in the modulation frequency measurement Δωd using this technique is Δωd ≈ π(α1 + α3 ), which is determined by the decay time constants of the beat signal modulation amplitudes. In this case, the precision of using REPB to measure the Raman resonant frequency is determined by how well ω3 − ω1 can be tuned to ΩR . When 0 < τ < δτ , Eq. (5.18) reflects not only characteristics of the lasers, but also the vibrational property of the molecules. When τ < 0, Eq. (5.19) shows that the beat signal modulates with a frequency (ω3 − ω1 ) − Δ = ΩR and has a damping rate γR + γ as τ is varied. Such, we can obtain resonant frequencies of the Raman vibrational modes with an accuracy given by π(γR + γ) approximately, which is mainly determined by the vibrational property of the molecules. To illustrate these properties, Fig. 5.2 depicts the interferograms of the beat signal intensity versus relative time delay for three different values of the reduced offset imbalance δτ . Parameters are ω1 = 3 200(ps)−1 , ω3 = 3 324(ps)−1 , Δk = 0, η1 = η2 = 1, Δ = 0, χ/χR = 1, γR = 0.05(ps)−1 , γ = 0.2(ps)−1 , α1 = 10.8(ps)−1 , α3 = 11.6(ps)−1 . The case of δτ = 0 fs is the dotted line, δτ = 43 fs is the dashed line and δτ = 100 fs is the solid line. As δτ increases, the peak-to-background contrast ratio of the interferograms diminishes. Interestingly, phases of the beating signals also change sensitively to produce a variety of interferograms including asymmetric ones. δτ gives the unbalanced dispersion effect between the two arms. A simple realistic example is an interferometer having an effective thickness of quartz or glass that differs significantly (many millimeters to few centimeters) between its two arms. Changing the thickness in one arm will control the degree of imbalance in the dispersion effect [7, 10]. Physically, δτ corresponds to the separation of the peaks in the fourth and fifth terms of Eq. (5.17), i.e. the separation between the ω1 only interferogram and the ω3 only interferogram. Equations (5.17)–(5.19) show that the beat signal oscillates not only  temporally but also spatially with a period of 2π Δk along the direction

5.1 Raman-enhanced Polarization Beats

153

Fig. 5.2. The beat signal intensity versus relative time delay.

Δk, which is almost perpendicular to the propagation direction of the beat signal. Here, Δk ≈ 2π|λ1 − λ3 |θ/λ3 λ1 , with θ being the angle between beam 1 and beam 2. Since the polarization-beat model assumes the pump beams to be plane waves, the generated FWM signals from the two static gratings, which propagate along k1 − k1 + k3 and k2 − k2 + k3 directions, respectively, are also plane waves. Since FWM signals propagate along slightly different directions, the interference between them leads to the spatial oscillation. Figure 5.3 presents the theoretical curve of the normalized polarization beat signal intensity versus transverse distance r with a fixed time delay and frequency detuning. Parameters are α1 = 2.7(ps)−1 , α3 = 2.9(ps)−1 , τ = 0ps, δτ = 83fs, θ = 2.62×10−2rad, λ1 = 589nm, λ3 = 567nm, χ/χR = 1, γR = 0.05(ps)−1 , γ = 0.2(ps)−1 , η1 = η2 = 1, and Δ = 0. The beat signal oscillates spatially with a period of 2π Δk = λ1 λ3 |λ1 − λ3 |θ ≈ 0.6mm. To observe this spatial modulation of the beat signal the dimension of the detector should be smaller than 0.6 mm.

Fig. 5.3. Theoretical curve of normalized polarization beat signal intensity versus transverse distance r.

Now, let us consider the situation when linewidths of laser sources in beams 1, 2, and 3 are narrow band (i.e., α1 , α3 << γ, γR and γR |τ |, γ|τ | >> 1). In this limit and after performing tedious integrations, we obtain I(Δ, τ ) ∝ (1 + 2η12 + η22 )

2 χ2R γR 2χR χγR Δ − 2 + χ2 (1 + 2η12 η22 )+ 2 2 γR + Δ γR + Δ2

154

5 Raman- and Rayleigh-enhanced Polarization Beats



 2 χ2R γR 2χR χγR Δ 2 − + χ exp(−2α1 |τ − δτ |) + 2 + Δ2 2 + Δ2 γR γR  2 2  χR γR 2 2 η22 + 4η χ exp(−2α3 |τ |) + 1 2 + Δ2 γR 4 exp(−α1 |τ − δτ | − α3 |τ |) × %! "  1 α3 χ2 + χχR γR Δ × + 2 + Δ2 γγR + γ 2 + Δ2 γR cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] + ! " α3 (γR + γ) γR η1 η2 χχR γR − 2 + Δ2 × γ(γR + γ)2 + Δ2 γR & sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ ]

(5.20)

This result with narrow band linewidths for pump lasers is similar to the one obtained earlier in Eq. (5.17)

5.1.2 Raman Echo It is interesting to understand the underlying physics in REPB with incoherent lights. Studies of various ultrafast phenomena have been carried out by using incoherent light sources [7 – 10, 16]. The REPB with incoherent lights is closely related to the phenomenon of three-pulse Raman echoes [14, 15]. It is different from the conventional true Raman echo which is a seventhorder process or the Raman pseudo-echo which is a fifth-order process. We now consider the case when the laser sources have broadband linewidths, with exp(−αi |t1 − t2 |) ≈ 2δ (t1 − t2 ) /αi

i = 1, 3

(5.21)

By substituting Eqs. (5.14)–(5.16) and (5.21) into I(Δ, τ ) ∝ |P (3) |2 , we obtain: (i) τ > δτ ! " (1 + η12 + η22 ) η12 γR I(Δ, τ ) ∝ χ2R γR + − α1 + α3 α1 (γR + α3 ) 4χR χγR γΔ + α1 [(γR + γ + α1 + α3 )2 + Δ2 ] ! " 1 η 2 η 2 (3γ + α3 ) χ2 γ + 1 2 + χ2 exp(−2α1 |τ − δτ |) + η12 η22 χ2 × 2α1 2α3 (γ + α3 ) γ 3 + 5γ 2 α3 + 5γα23 + 2α23 exp(−2α3 |τ |)+ α3 (γ + α3 )(γ + 2α3 )

5.1 Raman-enhanced Polarization Beats

155

2η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) × %! 2 2χ (γ + α3 ) χχR γR γΔ − − γ + 2α3 γ(γR + α1 + α3 )2 + γΔ2 " 2χχR γR γΔ × α3 (γR + γ + α1 + α3 )2 + α3 Δ2 ! χχR γR γ(γR + α1 + α3 ) cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − + γ(γR + α1 + α3 )2 + γΔ2 " 2χχR γR γ(γ + γR + α1 + α3 ) × α3 (γR + γ + α1 + α3 )2 + α3 Δ2 & sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] (5.22) (ii) 0 < τ < δτ ! " (1 + η12 + η22 ) η12 γR I(Δ, τ ) ∝ χ2R γR + − α1 + α3 α1 (γR + α3 ) ! " η12 η22 (3γ + α3 ) 4χR χγR γΔ 1 2 γ + + χ + α1 [(γR + γ + α1 + α3 )2 + Δ2 ] 2α1 2α3 (γ + α3 ) χ2 exp(−2α1 |τ − δτ |) + η12 η22 χ2 exp(−2α3 |τ |) +

5γ + 2 × 2α3

2 2χ2R γR exp(−2γR |τ − δτ |) + α1 α3

2χ2 η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) × cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − 4η1 η2 χR χγR γ

α3 + 2γ × γα1 α3

exp(−α3 |τ − δτ | − α3 |τ |) sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |]

(5.23)

(iii) τ < 0 !

" η12 γR (1 + η12 + η22 ) I(Δ, τ ) ∝ + − α1 + α3 α1 (γR + α3 ) ! " 1 η12 η22 (3γ + α3 ) 4χR χγR γΔ 2 + χ γ + + α1 [(γR + γ + α1 + α3 )2 + Δ2 ] 2α1 2α3 (γ + α3 ) χ2R γR

2χ2 γ 2 2χ2 η12 η22 γ 2 exp(−2γ |τ |) + R R [exp(−2γR |τ − δτ |) + 2 α3 α1 α3 η22 exp(−2γR |τ |)] + exp(−γR |τ − δτ | − γ |τ |) ×

156

5 Raman- and Rayleigh-enhanced Polarization Beats

8η1 η2 {cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |] − α3 (α1 + α3 ) χR χγR γ sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |]}

(5.24)

Equations (5.22)–(5.24) are analogous to (5.17)–(5.19), respectively. The total polarization [see Eqs. (5.9)–(5.13)], which involves integrations of t from 0 to ∞, is the accumulation of the polarization induced at a different time. We consider the case that the pump beam 1, 2, and 3 have broadband linewidths, so that they can be modeled as sequences of short, phaseincoherent subpulses of duration τc , where τc is the laser coherence time [16]. Although gratings can be induced by any pair of subpulses in beam 1, beam 2, and beam 3, only the pairs that are phase correlated in beams 1 and 2 give rise to the τ dependence in the FWM signal. Therefore, the requirement for the existence of a τ -dependent FWM signal for τ > 0 is for the phase-correlated subpulses in beams 1 and 2 to overlap temporally. Since beams 1 and 2 are mutually coherent, the temporal behavior of the REPB signal for τ > 0 should coincide with the case when pump beams are nearly monochromatic.

5.1.3 Phase-diffusion Field We have assumed that laser sources are chaotic fields in the above calculations. Another commonly used stochastic model is the phase-diffusion model, which is used to describe an amplitude-stabilized laser source. This model assumes that the amplitude of the laser field is a constant, while its phase fluctuates as random process. If the lasers have Lorentzian line shape, sixthand fourth-order coherence functions are [1,5] ui (t1 )ui (t2 )ui (t3 )u∗i (t4 )u∗i (t5 )u∗i (t6 ) = exp[−αi (|t1 − t4 | + |t1 − t5 | + |t1 − t6 | + |t2 − t4 | + |t2 − t5 | + |t2 − t6 | + |t3 − t4 | + |t3 − t5 | + |t3 − t6 |)] × exp[αi (|t1 − t2 | + |t1 − t3 | + |t2 − t3 | + |t4 − t5 | + |t4 − t6 | + |t5 − t6 |)]

(5.25)

ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = exp[−αi (|t1 − t3 | + |t1 − t4 | + |t2 − t3 | + |t2 − t4 |)] × exp[αi (|t1 − t2 | + |t3 − t4 |)]

(5.26)

We first consider the situation when the lasers in beam 1, beam 2, and beam 3 are broad-band (i.e., α1 , α3 >> γ, γR ) sources. In this limit, after substituting Eqs. (5.16), (5.25), and (5.26) into I(Δ, τ ) ∝ |P (3) |2 , we obtain, for:

5.1 Raman-enhanced Polarization Beats

157

(i) τ > δτ I(Δ, τ ) ∝ (1 + η12 + η22 )

χ2R γR (α1 + α3 ) − (α1 + α3 )2 + Δ2

Δ[(5α1 + α3 )(α1 + α3 ) + Δ2 ] χ2 γ + (α1 η12 η22 + α3 ) + 2α1 [(α1 + α3 )2 + Δ2 ]2 2α1 α3 ! " 2χχR γR Δ 2 χ − exp(−2α1 |τ − δτ |) + (α1 + α3 )2 + Δ2

χR χγR γ

η12 η22 χ2 exp(−2α3 |τ |) + 2η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) × " %! χχR γR Δ 2 cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − χ − (α1 + α3 )2 + Δ2 & χχR γR (α1 + α3 ) sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] (5.27) (α1 + α3 )2 + Δ2  (ii) 0 < τ < δτ 2 I(Δ, τ ) ∝ (1 + η12 + η22 )

χ2R γR (α1 + α3 ) − (α1 + α3 )2 + Δ2

Δ[(5α1 + α3 )(α1 + α3 ) + Δ2 ] χ2 γ + (α1 η12 η22 + α3 ) + 2α1 [(α1 + α3 )2 + Δ2 ]2 2α1 α3 ! " 2χχR γR Δ 2 χ − exp(−2α1 |τ − δτ |) + (α1 + α3 )2 + Δ2

χR χγR γ

η12 η22 χ2 exp(−2α3 |τ |) + 2χ2 η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) × cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − 2χχR γR γη1 η2 × %! " 2 2α1 + exp(−α3 |τ − δτ | − α3 |τ |)+ (α1 − α3 )2 + Δ2 γ(α21 − α23 ) & ! " 1 1 − exp (−α1 |τ − δτ | − α3 |τ |) × (α1 − 3α3 )2 γ(α1 − α3 ) sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ ]

(5.28)

(iii) δτ /2 < τ < δτ I(Δ, τ ) ∝ (1 + η12 + η22 )

χ2R γR (α1 + α3 ) − (α1 + α3 )2 + Δ2

Δ[(5α1 + α3 )(α1 + α3 ) + Δ2 ] χ2 γ + (α1 η12 η22 + α3 ) + 2 2 2 2α1 [(α1 + α3 ) + Δ ] 2α1 α3 ! " 2χχR γR Δ χ2 − exp(−2α1 |τ − δτ |)+ (α1 + α3 )2 + Δ2

χR χγR γ

158

5 Raman- and Rayleigh-enhanced Polarization Beats

η12 η22 χ2 exp(−2α3 |τ |) + 2χ2 η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) × cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − ! 2α1 exp(−α3 |τ − δτ | − α3 |τ |) − 2χχR γR γη1 η2 γ(α21 − α23 ) " 1 exp(−α |τ − δτ | − α |τ |) × 1 3 γ(α1 − α3 )2 + γΔ2 sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ ]

(5.29)

(iv) τ < 0 and α1 |τ |, α3 |τ | >> 1 I(Δ, τ ) ∝ (1 + η12 + η22 ) χR χγR γ

χ2R γR (α1 + α3 ) − (α1 + α3 )2 + Δ2

Δ[(5α1 + α3 )(α1 + α3 ) + Δ2 ] χ2 γ + (α1 η12 η22 + α3 ) − 2 2 2 2α1 [(α1 + α3 ) + Δ ] 2α1 α3

exp[−α1 |δτ | − (γR + γ) |τ |] × 4η1 η2 χχR γγR [(α1 + 2α3 )(α1 + α3 )2 + α1 Δ2 ] × α3 [(α1 + α3 )2 + Δ2 ] sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |]

(5.30)

Equations (5.27)–(5.29) show that when τ > 0, temporal behaviors of the beat signal intensity reflect mainly characteristics of the lasers. When τ < 0, Eq. (5.30) is mainly determined by the vibrational property of molecules. We then consider the situation when laser sources in beam 1, beam 2, and beam 3 are narrow band (i.e., α1 , α3 << γ, γR and γR |τ |, γ|τ | >> 1). Under this condition and after performing tedious integrations, we obtain I(Δ, τ ) ∝ (1 + η12 + η22 )

2 χ2R γR χR χγR Δ − 2 + χ2 (1 + η12 η22 ) + + Δ2 γR + Δ2

2 γR

2η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) %  χχR γR Δ 2 χ − 2 cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − γR + Δ2 & 2 χχR γR sin[Δk · r − (ω − ω )τ + ω δτ ] (5.31) 1 3 1 2 + Δ2 γR Equation (5.31) indicates that temporal behaviors of the beat signal intensity reflect mainly characteristics of the lasers. Based on the phase-diffusion model, we now consider the three-pulse Raman echo when the laser sources have broadband linewidths. Substituting Eqs. (5.16), (5.21), (5.25), and (5.26) into I(Δ, τ ) ∝ |P (3) |2 , we obtain, for:

5.1 Raman-enhanced Polarization Beats

159

(i) τ > 0 I(Δ, τ ) ∝ (1 + η12 + η22 )

χ2R γR 4χR χγR γΔ + − α1 + α3 α1 [(γR + γ + α1 + α3 )2 + Δ2 ]

χ2 γ (α1 η12 η22 + α3 ) + 2η1 η2 χ2 exp(−α1 |τ − δτ | − α3 |τ |) × 2α1 α3 cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ]

(5.32)

(ii) τ < 0 I(Δ, τ ) ∝ (1 + η12 + η22 )

χ2R γR 4χR χγR γΔ + − α1 + α3 α1 [(γR + γ + α1 + α3 )2 + Δ2 ]

χ2 γ (α1 η12 η22 + α3 ) + 2η1 η2 χ2 exp(−α1 |τ − δτ | − α3 |τ |) × 2α1 α3 cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − exp[−α1 |δτ | − (γR + γ) |τ |] × 8η1 η2 χχR γγR sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ|τ − δτ |] α23

(5.33)

Equations (5.32) and (5.33) are analogous to Eqs. (5.31) and (5.30), respectively. Equations (5.27)–(5.33) are different from the results obtained under the chaotic model. Equation (5.27) consists of six terms. The last (sixth) term depends on the fourth- and second-order coherence functions of u1 (t) or u3 (t) and is the cross-correlation intensity between five third-order nonlinear polarizations, which gives rise to the modulation of the beat signal. The other terms (the τ -independent and the decay terms) depend on sixth-, fourth- or second-order coherence functions of u1 (t) or u3 (t) and are the sum of the autocorrelation intensities between five third-order nonlinear polarizations. Different stochastic models of laser fields mainly affect the sixth- and fourthorder coherence functions. Equations (5.30) and (5.33) do not have decay terms including factors exp(−2γ|τ |) and exp(−2γR |τ |). Equations (5.31) and (5.32) also do not include decay terms with factors exp(−2α1 |τ |) and exp(−2α3 |τ |). These decay terms are shown to be particularly insensitive to the phase fluctuation of Markovian stochastic light fields [6]. The results are different in the fourth-order coherence on ultrafast modulation spectroscopy when these two different (chaotic and phase-diffusion) laser models are employed. The chaotic field has the property of photon bunching, which can affect any multi-photon processes when the higher-order correlation functions of the fields play important roles.

5.1.4 Gaussian-amplitude Field The Gaussian-amplitude field has a constant phase but its real amplitude undergoes Gaussian fluctuations. If the lasers have Lorentzian line shape,

160

5 Raman- and Rayleigh-enhanced Polarization Beats

the sixth- and fourth-order coherence functions are [1,5] ui (t1 )ui (t2 )ui (t3 )ui (t4 )ui (t5 )ui (t6 ) = ui (t1 )ui (t4 )ui (t2 )ui (t3 )ui (t5 )ui (t6 ) + ui (t1 )ui (t5 )ui (t2 )ui (t3 )ui (t4 )ui (t6 ) + ui (t1 )ui (t6 )ui (t2 )ui (t3 )ui (t4 )ui (t5 ) + ui (t1 )ui (t2 )ui (t3 )ui (t4 )ui (t5 )ui (t6 ) + ui (t1 )ui (t3 )ui (t2 )ui (t4 )ui (t5 )ui (t6 )

(5.34)

ui (t1 )ui (t2 )ui (t3 )ui (t4 ) = ui (t1 )ui (t3 )ui (t2 )ui (t4 ) + ui (t1 )ui (t4 )ui (t2 )ui (t3 ) + ui (t1 )ui (t2 )ui (t3 )ui (t4 )

(5.35)

For the Gaussian-amplitude field, we first consider the case when laser sources have a broadband linewidth. Substituting Eqs. (5.16), (5.34), and (5.35) into I(Δ, τ ) ∝ |P (3) |2 , we obtain: (i) τ > δτ ! 2 2 " η1 η2 (1 + α3 ) (α1 + α3 )(1 + η12 + η22 ) 1 2 2 I(Δ, τ ) ∝ χR γR +χ γ + + (α1 + α3 )2 + Δ2 2α23 2α1 % & 2 2χ2R γR (α1 + α3 ) 2χR χγR Δ 2 +χ − × α1 [(α1 + α3 )2 + Δ2 ] (α1 + α3 )2 + Δ2 % & 2 2 2χ2R γR η2 (α1 + α3 ) η12 η22 χ2 γ 2 2 + η1 χ + × exp(−2α1 |τ − δτ |) + α3 [(α1 + α3 )2 + Δ2 ] 2α23 exp(−2α3 |τ |) + 2η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) ×  %! 2 " χ γ χR χγR Δ γ 2 +χ − × 1+ × 2α3 (α1 + α3 )2 + Δ2 2α3 χR χγR (α1 + α3 ) × (α1 + α3 )2 + Δ2 &   γ 2+ sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − 2α3

cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] −

4η1 η2 χχR γR γα3 exp(−2α1 |τ − δτ |) × (α1 + α3 )(α21 − α23 ) sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |] (5.36)  (ii) 0 < τ < δτ 2 ! 2 2 " η1 η2 (1 + α3 ) 1 (α1 + α3 )(1 + η12 + η22 ) 2 I(Δ, τ ) ∝ χ2R γR + χ γ + + (α1 + α3 )2 + Δ2 2α23 2α1 & % 2 χ2R γR (α1 + α3 ) 2 + χ exp(−2α1 |τ − δτ |)+ α1 [(α3 − α1 )2 + Δ2 ]

5.1 Raman-enhanced Polarization Beats

%

161

&

2 2 χ2R γR η2 (α1 + α3 ) η 2 η 2 χ2 γ + χ2 + 1 2 exp(−2α3 |τ |) + 2 2 α1 [(α1 − α3 ) + Δ ] 2α3 % 2 χ2R γR + exp(−2γR |τ − δτ |) α1 (α1 + α3 ) & 2 2χ2R γR [α1 + α3 − (α1 + α3 )2 + Δ2 ] + η1 η2 × α1 [(α1 + α3 )2 + Δ2 ]   2 χχR γ χ + exp(−α1 |τ − δτ | − α3 |τ |) × 2 2α3

cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] + ! 1 exp(−α1 |τ − δτ | − α3 |τ |) + η1 η2 χR χγR γ γ(α1 + α3 ) " 2α1 exp(−α |τ − δτ | − α |τ |) × 3 3 γ(α21 − α23 ) sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |]

(5.37)

(iii) δτ /2 < τ < δτ ! 2 2 " η1 η2 (1 + α3 ) + α3 )(1 + η12 + η22 ) 1 2 +χ γ + I(Δ, τ ) ∝ + (α1 + α3 )2 + Δ2 2α23 2α1 % & 2 χ2R γR (α1 + α3 ) 2 + χ exp(−2α1 |τ − δτ |) + α1 [(α3 − α1 )2 + Δ2 ] & % 2 2 2 χR γR η2 (α1 + α3 ) η12 η22 χ2 γ 2 + χ + exp(−2α3 |τ |) + α1 [(α1 − α3 )2 + Δ2 ] 2α3 % 2 χ2R γR + exp(−2γR |τ − δτ |) α1 (α1 + α3 ) & 2 2χ2R γR [α1 + α3 − (α1 + α3 )2 + Δ2 ] + α1 [(α1 + α3 )2 + Δ2 ]  2  χχR γ χ + exp(−α1 |τ − δτ | − α3 |τ |) × η1 η2 2 2α3 (α1 χ2R γR

cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] + ! 1 exp(−α1 |τ − δτ | − α3 |τ |) + η1 η2 χR χγR γ γ(α1 + α3 ) " 1 exp(−α3 |τ − δτ | − α3 |τ |) × 2α3 (α1 + α3 ) sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |]

(5.38)

162

5 Raman- and Rayleigh-enhanced Polarization Beats

(iv) τ < 0 and α1 |τ |, α3 |τ | >> 1 I(Δ, τ ) ∝

! 2 2 " η1 η2 (1 + α3 ) + α3 )(1 + η12 + η22 ) 1 2 +χ γ + + (α1 + α3 )2 + Δ2 2α23 2α1

(α1 χ2R γR

2 χ2R γR 3χ2 η12 η22 γ 2 exp(−2γ |τ |) + × 2 2α3 (α1 + α3 )2 + Δ2 " %! 2α3 exp(−2γR |τ |) + α3 + (α1 + α3 )2 − Δ2 + α1 ! " & 2α1 2 2 2 × exp(−2γR |τ − δτ |) + η2 α1 + (α1 + α3 ) − Δ + α3

exp(−γR |τ − δτ | − γ |τ |)

4χR χγR γη1 η2 × α3 [(α1 + α3 )2 + Δ2 ]

{(α1 + α3 )(2γR + γ)Δ cos[Δk · r − (ω1 − ω3 )τ + ω1 δτ ] − [(α1 + 2α3 )(α1 + α3 )2 + α1 Δ2 ] sin[Δk · r − (ω1 − ω3 )τ + ω1 δτ + Δ |τ − δτ |]}

(5.39)

Equation (5.36) indicates that when τ > δτ , the temporal behavior of the beat signal intensity reflects mainly characteristics of the lasers. When 0 < τ < δτ /2 and δτ /2 < τ < δτ , Eqs. (5.37) and (5.38) reflect not only characteristics of the lasers, but also certain vibrational properties of the medium. When τ < 0, Eq. (5.39) is mainly determined by the vibrational property of the medium. Let us now consider the situation when laser sources in beams 1, 2, and 3 are narrow band (i.e., α1 , α3 << γ, γR and γR |τ |, γ|τ | >> 1). Under this limit and after performing tedious integrations, we obtain I(Δ, τ ) ∝

2 (1 + 3η12 + η22 ) 2χχR γR Δ χ2R γR + χ2 (1 + 3η12 η22 ) − 2 + 2 2 γR + Δ γR + Δ2  2 2  χR γR χχR γR Δ 2 2 2 + Δ2 + χ − γ 2 + Δ2 + χχR exp(−2α1 |τ − δτ |) + γR R  2 2 2  χR γR η2 2 2 2 2 2 + Δ2 + 6χ η1 η2 exp(−2α3 |τ |) + γR

2η1 η2 exp(−α1 |τ − δτ | − α3 |τ |) × %! " 2χχR γR Δα3 2χχR γR Δ 2 − 2 + 3χ cos[Δk · r − γ(γR + γ)2 + γΔ2 γR + Δ2 ! 2α3 (γ + γR ) + (ω1 − ω3 )τ + ω1 δτ ] + χR χγR γ(γR + γ)2 + γΔ2 & " 1 2γR − 2 − ω )τ + ω δτ ] (5.40) sin[Δk · r − (ω 1 3 1 γ γR + Δ2

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163

Equation (5.40) is analogous to Eq. (5.36), which indicates that the temporal behavior of the beat signal intensity reflects mainly characteristics of the lasers. Equation (5.36) consists of six terms. The fifth and sixth terms depend on fourth- and second-order coherence functions of u1 (t) or u3 (t) and are the cross-correlations between five third-order nonlinear polarizations, which give rise to the modulation of the beat signal. The interferometric contrast ratio (which mainly determines the modulation term) is equally sensitive to the amplitude and phase fluctuations of the Markovian stochastic light fields. The other four terms, depending on the sixth-, fourth- or secondorder coherence functions of u1 (t) or u3 (t), are a sum of the auto-correlation intensities between five third-order nonlinear polarizations. Different stochastic field models mainly affect the sixth- and fourth-order coherence functions. The constant terms in Eqs. (5.36)–(5.40), which are independent of the relative time-delay between beam 1 and beam 2, mainly originate from the amplitude fluctuations of the Gaussian-amplitude fields. The third and fourth terms in Eq. (5.36) are particularly sensitive to the amplitude fluctuations of the Gaussian-amplitude fields, which include an exponential decay of the beat signal as τ increases. The τ -independent terms of Eq. (5.36) are identical to those of Eqs. (5.37)–(5.39). When |τ | → ∞, beam 1 and beam 2 are mutually incoherent, therefore whether τ is positive or negative does not affect the REPB. Equations (5.36)–(5.40) also indicate that the beat signal oscillates not only temporally but also spatially along the direction of Δk, which is almost perpendicular to the propagation direction of the beat signal. Figure 5.4 presents three three-dimensional plots for (a) the beat signal intensity I(τ, Δ) versus the time delay τ and the frequency detuning Δ, (b) I(τ, r) versus the time delay τ and the transverse distance r and (c) I(Δ, r) versus the frequency detuning Δ and the transverse distance r, respectively. There are larger constant backgrounds caused by the intensity fluctuations of the Gaussian-amplitude fields in these plots. The parameters are: ω 1 = 3 200 (ps)−1 , ω 3 = 3 324 (ps)−1 , δτ = 83 fs, α1 = 10.8 (ps)−1 , α3 = 11.6 (ps)−1 , θ = 2.62 × 10−2 rad, χ/χR = 1, γR = 0.05 (ps)−1 , γ = 0.2 (ps)−1 , η1 = η2 = 1; and Δk = 0/mm for Fig. 5.4(a), Δk = 10.83/mm, Δ = 0 for (b) and Δk = 10.83/mm, τ = 0 ps for (c). At zero relative time delay (τ = 0 and δτ = 0), twin beams from the same laser source have a perfect overlap of their corresponding noise patterns at the sample in Figs. 5.4 (a) and (b). This gives a maximum interferometric contrast. As |τ | is increased, the interferometric contrast diminishes on the time scale that reflects material memory, which is usually much longer than the correlation time of the light source. As δτ increases, the contrast ratio is seen to diminish and the symmetry of the interferogram is destroyed in Figs. 5.4 (a) and (b). It is important to note that three types of stochastic fields (e.g., chaotic field, phase-diffusion field, and Gaussian-amplitude field) can have the same spectral density and thus the same second-order coherence function. The fun-

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5 Raman- and Rayleigh-enhanced Polarization Beats

Fig. 5.4. Three-dimensional plots of (a) beat signal intensity I(τ, Δ) versus time delay τ and frequency detuning Δ; (b) I(τ, r) versus time delay τ and transverse distance r; and (c) I(τ, r) versus frequency detuning Δ and transverse distance r, respectively. Adopted from Ref. [18].

Fig. 5.5. Beat signal intensity versus relative time delay. The three curves represent the chaotic field (dotted line), phase-diffusion field (dashed line), and Gaussianamplitude field (solid line), respectively.

damental differences in the statistics of these fields are manifest in higherorder (above the second-order) coherence functions [1 – 6]. In this section, we have shown how different stochastic models of the laser field affect sixthand fourth-order coherence functions. Figures 5.5 (a) and 5.5 (b) present the beat signal intensity versus relative time delay using laser linewidths of 1 nm and 4 nm, respectively. The parameters are: ω 1 = 3 200 (ps)−1 , ω 3 = 3 324 (ps)−1 , Δk = 0, δτ = 0 fs, η1 = η2 = 1, Δ = 0, χ/χR = 1, γR = 0.05 (ps)−1 , γ = 0.2 (ps)−1 ; and α1 = 2.7 (ps)−1 , α3 = 2.9 (ps)−1 for (a)

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165

and α1 = 10.8 (ps)−1 , α3 = 11.6 (ps)−1 for (b). The three curves in each figure represent the chaotic field (dotted line), phase-diffusion field (dashed line), and Gaussian-amplitude field (solid line), respectively. For clarity, the same constant intensity is subtracted from these three signals (since we are only interested in relative values in these three cases. The intensity goes “negative” for certain values due to the subtraction of a constant level in curves.). The peak-to-background contrast ratio of the chaotic field is much larger than that of the phase-diffusion field or the Gaussian-amplitude field, and the contrast ratio of the phase-diffusion field is slightly larger than that of the Gaussian-amplitude field. The physical explanation for this is that the signal contrast ratio is equally sensitive to the amplitude and phase fluctuations of Markovian stochastic fields. The polarization beat signal is shown to be particularly sensitive to the statistical properties of Markovian stochastic light fields with arbitrary bandwidths. This is quite different from the fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources, in which results are insensitive to specific radiation models. Figures 5.6 (a) and (b) show the interferogram of the beat signal intensity versus time delay and the spectrum of the beat signal intensity versus frequency detuning, respectively. The parameters are: ω1 = 3 200 (ps)−1 , ω3 = 3 324 (ps)−1 , Δk = 0, δτ = 0 fs, η1 = η2 = 1, χ/χR = 1, γR = 0.05 (ps)−1 , γ = 0.2 (ps)−1 ; and Δ = 0, α1 = 10.8 (ps)−1 , α3 = 11.6 (ps)−1 for (a) and τ = 0 fs, α1 = 2.7 (ps)−1 , α3 = 2.9 (ps)−1 for (b). The constant background of the beat signal for a Gaussian-amplitude field or a chaotic field is much larger than that of the signal for a phase-diffusion field in Figs. 5.6 (a) and (b). This is caused by the stronger intensity fluctuations in the Gaussian-amplitude field than in the chaotic field. Also, the intensity (amplitude) fluctuations of the Gaussian-amplitude field or the chaotic field are always much larger impact to the background than the pure phase fluctuations of the phase-diffusion field.

Fig. 5.6. (a) interferogram of beat signal intensity versus time delay and (b) spectrum of the beat signal intensity versus frequency detuning. The three curves represent the cases for the chaotic field (dotted line), phase-diffusion field (dashed line), and Gaussian-amplitude field (solid line), respectively.

Now, we discuss the major differences between REPB and UMS in self-

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diffraction geometry. The frequency and wave vector of the UMS are ωs1 = 2ω1 − ω1 , ωs2 = 2ω2 − ω2 and ks1 = 2k1 − k1 , ks2 = 2k2 − k2 , respectively, which indicate that one photon is absorbed from each of the two mutually correlated pump beams. However, the frequency and wave vector of the FWM signal in the REPB are ωs1 = ω1 − ω1 + ω3 , ωs2 = ω3 − ω3 + ω3 , and ks1 = k1 − k1 + k3 , ks2 = k2 − k2 + k3 . In this case, photons are absorbed from and emitted to the mutually correlated beam 1 and beam 2, respectively. This difference between the REPB and UMS has a profound influence on field-correlation effects. We note that roles of the two composite beams are interchangeable in the UMS, which makes the second-order coherence function theory fail in the UMS. In virtue of u(t1 )u(t2 ) = 0, the absolute square of the stochastic average of the polarization |P (3) |2 can not be used to describe the temporal behavior of the UMS [13]. The higher-order theory presented here is of vital importance in the UMS [6]. The above discussion reveals an important fact that the amplitude and phase fluctuations play critical roles in the temporal behaviors of the REPB signal. Furthermore, different roles of the phase fluctuation and amplitude fluctuation have been pointed out in the time domain and in the frequency domain. This is quite different from the time-delayed FWM with incoherent light in a two-level system [16]. For the latter case, the phase fluctuation of the light field is crucial. However, amplitude and phase fluctuations of Markovian stochastic light fields are equally crucial in the REPB. Also, because of ui (t) = 0 and u∗i (t) = 0, the absolute square of the stochastic average of the polarization |P (3) |2 , which involves only second-order coherence functions of ui (t), can not be used adequately to describe the temporal behaviors of the REPB. The sixth-order coherence function theory reduces to the second-order coherence function theory in the case when the laser pulse width is much longer than the laser coherence time. The second-order coherence function theory is valid when one is only interested in the τ -dependent parts of the beating signal [13]. Therefore, the second-order coherence function theory is of vital importance in REPB. Applications of the above theoretical results to the REPB experiments yielded much better fits to the data than the expressions involving only the second-order coherence functions [6]. In the next sub-section, we present the experimental results for the material response in REPB with a phase-conjugation geometry using broadband chaotic fields.

5.1.5 Experiment and Result Temporal behaviors of the REPB signal intensity were studied with a fixed frequency detuning using chaotic fields. The carbon disulfide (CS2 ) with 655.7 cm−1 vibrational mode was contained in a sample cell with a thickness of 9 mm. The second harmonic beam of a Quanta-Ray YAG laser was used

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167

to pump two dye lasers (DL1 and DL2). The broadband light sources DL1 and DL2 had linewidths of about 1 nm, pulse widths of 10 ns, and output energy of about 1 mJ. The first dye laser (DL1) had a wavelength of 589 nm. The wavelength of DL2 was tuned near 567 nm and could be scanned by a computer-controlled stepping motor. A beam splitter was used to combine frequency components of ω1 and ω3 derived from broadband sources DL1 and DL2, respectively, for beam 1 and beam 2, which intersected in the sample with a small angle of θ = 1.5◦ . The relative time-delay τ between beam 1 and beam 2 could be varied. Beam 3, which propagates along the direction opposite to that of beam 1, was also derived from the broadband source DL2. All the incident laser beams were linearly polarized in the same direction. The beat signal has the same polarization as the incident beams, and propagates along a direction almost opposite to that of beam 2. The signal was detected by a photodiode.

Fig. 5.7. RENFWM spectrum with a fixed time delay

First, a NDFWM experiment was performed in which beam 1 and beam 2 only consisted of the ω1 frequency component. The RENFWM spectrum was measured with a fixed time delay by scanning ω3 , as shown in Fig. 5.7. An asymmetric resonant profile appears due to the interference between the Raman resonant term and the nonresonant background originating solely from the molecular reorientational grating [14, 15]. From this spectrum ω3 was tuned to the resonant frequency (i.e., Δ = |ω1 − ω3 | − ΩR = 0). Then the REPB experiment was performed with a fixed frequency detuning by measuring the beat signal intensity as a function of the relative time delay when beams 1 and 2 consist of both frequencies ω1 and ω3 . Figure 5.8 presents the result of the polarization beat experiment. Filled squares denote experimental results and the solid curve is the theoretical calculation with ω 1 = 3 200 (ps)−1 , ω 3 = 3 324 (ps)−1 , Δk = 0, η1 = η2 = 1, δτ = 83 fs, Δ = 0, χ/χR = 1, γR = 0.05 (ps)−1 , γ = 0.2 (ps)−1 , α1 = 2.7 (ps)−1 , and α3 = 2.9 (ps)−1 . The beat signal intensity modulates sinusoidally with a period of 51 fs. The modulation frequency can be obtained more directly by making a Fourier transform of the REPB data, as shown in Fig. 5.9 with τ being varied for a range of 5 ps. The solid curve is the theoretical calculation with ω1 = 3 200 (ps)−1 , ω3 = 3 324 (ps)−1 , Δk = 0, η1 = η2 = 1, δτ =

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83 fs, Δ = 0, χ/χR = 1, γR = 0.05 (ps)−1 , γ = 0.2 (ps)−1 , α1 = 2.7 (ps)−1 , and α3 = 2.9 (ps)−1 . This gives the modulation frequency of 124 ps−1 , corresponding to the resonant frequency of the Raman vibrational mode of 655.7 cm−1 with a standard deviation of 0.107 cm−1 .

Fig. 5.8. The beat signal intensity versus relative time delay.

The overall accuracy of using the broadband REPB to measure the Raman resonant frequency can be determined. Polarization beats can be employed as a spectroscopic tool because the modulation frequency corresponds directly to the resonant frequency of the system [11]. For the Raman resonant system, the modulation frequency is just the frequency difference between the two incident lasers when the lasers have narrow bandwidths. The precision of using REPB to measure the Raman resonant frequency is then determined by how well ω3 − ω1 can be tuned to the resonant frequency ΩR . However, due to the interference between the Raman resonant signal and the nonresonant background, the Raman-enhanced FWM signal spectrum is asymmetric and the peak of the spectrum does not correspond to the exact Raman resonance. Since there is a small uncertainty in tuning ω3 − ω1 to ΩR , this is a disadvantage of using narrow band polarization beat to measure the resonant frequency of the Raman mode [6]. However, since the modulation frequency of the beat signal corresponds directly to the Raman resonant frequency when the lasers have broadband linewidths, there is a great advantage of using broadband polarization beat to measure the resonant frequency of the Raman mode. The above presented theoretical and experimental results for the material response in REPB using broadband chaotic fields showed a good way to measure the Raman frequencies, and provide an interesting way to study the stochastic properties of light. The previously studied noisy-light based CRS, often called I(2) CARS or I(2) CSRS (coherent Stokes Raman scattering), can yield both Raman frequencies via radiation difference oscillations (RDO) and dephasing times in the interferometric time domains. Unlike in the REPB, in that spectroscopy technique the presence of one monochromatic light beam is essential [7 – 10]. Physically the REPB presented here can be considered to be similar to the corresponding CSRS [14, 15].

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Fig. 5.9. Fourier spectrum of the experimentally measured beat signal data with τ being varied for a range of 5 ps.

The temporal behavior of the REPB signal is quite asymmetric with the maximum of the signal shifted from τ = 0. We attribute this asymmetry to the difference between two autocorrelation processes in the zero time delay. To confirm this, the FWM signal can be measured when beams 1 and 2 only consist of one frequency component. It is due to the large difference between wavelengths of broadband noisy light sources DL1 and DL2 so that dispersions of the optical components become important. In this experiment, correlation functions of static gratings Grating1 and Grating2 are chirped due to the unbalanced dispersions in the two arms of the interferometer generated by a dispersive material (the optical glass in the delay line). Owing to the difference in indices of refraction of optical glass and air, beam 2 is delayed by the optical glass relative to beam 1. Therefore, the interferometer must be adjusted (the path of beam 2 is made shorter) to compensate for this delay and thus reestablish the overlap. The difference between the dispersion of optical glass and air causes chirping of the correlation functions, so different colors are optimally correlated at different values of the interferometric delay. The bluer color ω3 correlates at a later delay time (when the path of beam 2 is made shorter) and the redder color ω1 correlates at an earlier delay time (when the path of beam 2 is made longer) [7, 10]. Consider the case that the optical paths between beam 1 and beam 2 are equal for the ω3 component. Owing to the difference between the zero time delays for the frequency components ω1 and ω3 , the optical paths between beam 1 and beam 2 will be different by cδτ for the ω1 component, which causes an extra phase factor ω1 δτ for this ω1 frequency component. For an optical glass with the typical refractive index of n ≈ 1.5, the refractive index at λ3 = 567 nm is larger than that at λ1 = 589 nm by approximately 0.001. An 83 fs time delay between ω1 and ω3 corresponds to the propagation of the beams in the glass (mainly the prism in the optical delay line) for a distance of about 2.5 cm. It is worth mentioning that the asymmetric behaviors of the polarization beat signals due to the unbalanced dispersion effects of the optical components between two arms of interferometer do not affect the overall accuracy in the case of using the REPB to measure the energy-level difference. By contrast,

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ultrashort pulses of equivalent bandwidth are not immune to such dispersion effects (even when balanced) because the transform-limited light pulse is in fact temporally broadened (e.g., chirped), which has drastic effects on its time resolution (the auto-correlation). In this sense the REPB technique with broadband noisy lights has an advantage [7]. In this section, we have presented results of higher-order field correlation effects on the REPB with phase-conjugation geometry in Raman resonant system using chaotic, phase-diffusion, and Gaussian-amplitude field models. The polarization beat signal is shown to be particularly sensitive to the field statistics. Different stochastic models of the laser field mainly affect sixth- and fourth-order coherence functions [1 – 6]. We considered three types of field models and cases of having either narrow band or broadband laser linewidths. It has been shown that the beat signal oscillates not only temporally with a period of 51 fs but also spatially with a period of 0.6 mm in the experimental system of carbon disulfide. The temporal period corresponds to the Raman frequency shift of 655.7 cm−1 . The overall accuracy of using REPB to measure the resonant frequency of the Raman-active mode is determined by relaxation rates of the Raman mode and the molecularreorientational grating. The asymmetric behavior of the polarization beat signals due to the unbalanced dispersion does not affect the overall accuracy in measuring the Raman resonant frequency using REPB, and different colors are optimally correlated at different values of the interferometric delay.

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats Rayleigh-type FWM is a nonresonant process and a frequency-domain nonlinear laser spectroscopy with high frequency resolution, which is determined by the laser linewidth. Since the relaxation time is deduced from the FWM spectrum, the measurement is not limited by the laser pulse width or the laser correlation time. Bogdanov, et al have demonstrated the attosecond beats between different light sources: an interference between the Rayleigh scattered field and the FWM field of the phase-locking ultrashort laser pulses. The technique of using Rayleigh-enhanced attosecond sum-frequency polarization beats (RASPB) is an interesting way to study the stochastic properties of light [6]. Previous noisy-light-based CRS (often called CARS or coherent Stokes Raman scattering) yields both Raman frequencies via radiation difference oscillations and dephasing times in the interferometric time domains. Unlike in RASPB, in those spectroscopic techniques the presence of one monochromatic beam is essential [7 – 10, 18 – 23]. Characteristics of the interferogram of RASPB result from two main components: the material response (resonant term) and the light response (nonresonant term) along with the interplay between these two responses.

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171

In the previously discussed Raman-enhanced FWM (one of the CRS techniques) [14, 15], the Raman vibration is excited by the simultaneous presence of two incident beams whose frequency difference equals to the Raman excitation frequency and the Raman-enhanced FWM signal is the result of this resonant excitation. In contrast, Rayleigh-enhanced FWM is a non-resonant process with no energy transfer between the lights and the medium when the frequency difference between two incident beams equals to zero. The resonant structure in Rayleigh-enhanced FWM spectrum is the result of an induced moving grating. The Raman- or Rayleigh-enhanced FWM may be superior to all other CRS techniques. They possess features of non-resonant background suppression, excellent spatial signal resolution, free choice of interaction volume and simple optical alignment. Moreover, phase matching can be achieved for a very wide frequency range from many hundreds to thousands of cm−1 . This section addresses roles of noises in the incident fields played on the wave-mixing signals in the time- and frequency-domains. This important topic has already been treated extensively in the literature [3]. Ulness, et al invented the factorized time correlator diagram, “synchronization” and “accumulation” analysis, for noisy light response [7 – 10, 18 – 23], instead of the double-sided Feynman diagrams. A fundamental principle of noisy light spectroscopy is color-locking which results as a consequence of the phaseincoherent nature of the light beam. Color-locking is responsible for the complete cancellation of the noise spectrum carried by the noisy light used to produce it [20]. Same as discussed in the last section, there should be two classes of such two-component light beams. In one class the components are derived from two separate lasers and their cross-correlators should vanish. In the second case the two components come from a single laser source whose output spectrum has double peaks, which can be created by using two dyes in a single dye laser [7 – 10]. We only use the first case in this section with two light components from two separate noisy light sources. The double-peaked beams 1 and 2 [see Figs. 5.19(a) and (b)] are paired and correlated between different beams, but peaks in each beam are uncorrelated. Beam 3, having one of peaks in the twin beams 1 and 2, is dependent and correlated with twin beams 1 and 2. The case with a single multi-colored noisy light source (the second class) for FWM has been previously explored, both experimentally and theoretically [7 – 10, 18 – 23], in difference-frequency self-diffraction geometry. Those works did not treat the RASPB with the sum-frequency phase-conjugation geometry using three types of noisy models. Also, beam 3 was not noisy (it was “monochromatic”) in those earlier works. In this section, based on field correlations of color-locking twin noisy lights, the homodyne detected RFWM and RASPB, as well as heterodyne detected RASPB, will be presented in detail. Analytical and closed forms of results are obtained. RASPB is a third-order nonlinear polarization beat phenomenon. The basic geometry of the light beams is shown in Fig. 5.10 (a). Twin beams 1 and

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2 consist of two central circular frequency components ω1 and ω2 , with a small angle between the two beams. Beam 3 with a central circular frequency ω3 propagates along the opposite direction of beam 1. In an optical Kerr medium (no thermal grating effects), nonlinear interactions of beam 1 and beam 2 with the medium give rise to two molecular-reorientational gratings, i.e., ω1 and ω2 frequency components will induce their own nonresonant static gratings Grating1 and Grating2, respectively. Two FWM signals are generated from the diffractions of beam 3 by Grating1 and Grating2. Now, if the frequency detuning Δ = ω3 − ω1 is much smaller than Δ = ω3 − ω2 (i.e., Δ << Δ and Δ ≈ 0 (see Fig. 5.10), a large-angle resonant moving grating, formed by the interference between the ω1 frequency component of beam 2 and the ω3 frequency component of beam 3, will excite the Rayleigh mode of the medium and enhance the FWM signal of Grating1 (i.e., RFWM). Then, a polarization beat signal will be generated from the interference between the macroscopic polarizations from the RFWM process and the NDFWM process. The generated beat signal (beam 4) propagates along the opposite direction of beam 2 approximately.

Fig. 5.10. Phase-conjugation geometries for (a) RASPB and (b) RFDPB. Doublesided Feynman diagrams representing the Liouville pathways for P1 , P3 and P2 for (c) RASPB and (d) RFDPB.

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In a typical experiment, the ω1 and ω2 two-color light source enters a dispersion-compensated Michelson interferometer to generate the identical twin-composite beams. Twin-composite stochastic fields of beam 1, Ep1 (r, t), and beam 2, Ep2 (r, t), for the homodyne detection scheme of attosecond sumfrequency RASPB, can be written as Ep1 = E1 (r, t) + E2 (r, t) = ε1 u1 (t) exp[i(k1 · r − ω1 t)] + ε2 u2 (t − τ ) exp[i(k2 · r − ω2 t + ω2 τ )]

(5.41)

Ep2 = E1 (r, t) + E2 (r, t) = ε1 u1 (t − τ ) exp[i(k1 · r − ω1 t + ω1 τ )] + ε2 u2 (t) exp[i(k2 · r − ω2 t)]

(5.42)

Here, εi and ki (εi ,ki ) are the constant field amplitude and the wave vector of the ωi component in beam 1 (beam 2), respectively. ui (t) is a dimensionless statistical factor that contains phase and amplitude fluctuations. ui (t) is taken to be a complex ergodic stochastic function of t, which obeys complex circular Gaussian statistics for a chaotic field. τ is a relative time delay between the prompt (unprime) and delayed (prime) fields. To accomplish this light sources for the frequency components ω1 and ω2 are split and recombined to provide two double-frequency pulses in such a way that the ω1 component is delayed by τ in beam 2 and the ω2 component is delayed by the same amount in beam 1 [see Fig. 5.10(a) and (c)]. The time-delay τ is introduced in both composite beams, which is quite different from the case of Rayleighenhanced femtosecond difference-frequency polarization beats (RFDPB) [see Fig. 5.10(b) and (d)] [6]. The complex electric field of beam 3 can be written as E3 (r, t) = ε3 u3 (t) exp[i(k3 · r − ω3 t)].

(5.43)

Here, ω3 , ε3 , and k3 are the frequency, the field amplitude and the wave vector of the field, respectively. Order parameters Q1 and Q2 of the two nonresonant static gratings (induced by beam 1 and beam 2) satisfy following equations [6]  dQ1 dt + γQ1 = χγE1 (r, t)[E1 (r, t)]∗ (5.44) dQ2 /dt + γQ2 = χγE2 (r, t)[E2 (r, t)]∗

(5.45)

γ and χ are the relaxation rate and the nonlinear susceptibility of two static gratings, respectively. The optical Kerr effect for the liquid CS2 has at least two components, i.e., a relatively long “Debye” component and a shorter “interaction-induced” component. We consider a large-angle resonant moving grating formed by the interference between beam 2 and beam 3, and the order parameter Q3 satisfies the following equation dQ3 /dt + γQ3 = χγ[E1 (r, t)]∗ E3 (r, t)

(5.46)

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Three induced third-order nonlinear polarizations which are responsible for the FWM signals are P1 = Q1 (r, t)E3 (r, t)  ∞ u1 (t − t )u∗1 (t − t − τ )u3 (t) exp(−γt )dt = S1 (r)

(5.47)

P2 = Q2 (r, t)E3 (r, t)  ∞ u∗2 (t − t )u2 (t − t − τ )u3 (t) exp(−γt )dt = S2 (r)

(5.48)

0

0

P3 = Q3 (r, t)E1 (r, t)  ∞ u∗1 (t − t − τ )u3 (t − t )u1 (t) exp[−(γ − iΔ)t ]dt (5.49) = S1 (r) 0

Here S1 (r) = χγε1 (ε1 )∗ ε3 exp{i[(k1 − k1 + k3 ) · r − ω3 t − ω1 τ ]}

S2 (r) = χγε2 (ε2 )∗ ε3 exp{i[(k2 − k2 + k3 ) · r − ω3 t + ω2 τ ]} In general, there are 48 terms in the third-order density operator for FWM processes. The number of the Feynman diagrams at jth order (for cw experiments) corresponds to the number of Liouville pathways, i.e. 2j multiplied by the number of distinct temporal field orderings (as many as j!), for the given process. Therefore, at the third order, where beam 1, beam 2, and beam 3 are distinct, there are 2j j! = 48(j=3) different Feynman diagrams at the polarization level. Typically experimental constraints reduce the number of diagrams to a significantly smaller subset of terms which dominate the behavior of the signal. Under the rotating-wave approximation, we can omit most of the 2j j! Liouville pathways formally in the three-color FWM. More specifically, phase-matching and frequency selection of the FWM signals along k4 direction greatly restrict the number of relevant third-order perturbative pathways (see Fig. 5.10). Moreover, polarization beat is based on the interference at the detector between FWM signals, which originate from macroscopic polarizations excited simultaneously in the sample. It prefers that all polarizations have the same frequency. Frequencies of P1 , P2 , and P3 are ω3 . Furthermore, due to the phase mismatching, FWM signals from PR2 and PR3 are usually much smaller than that from P1 , P2 , and P3 . So the third-order nonlinear polarizations P1 , P3 (with Lorentzian line shape for Rayleigh mode) and P2 have the same frequency ω3 . P1 + P3 and P2 correspond to RFWM process and NDFWM process which have wave vectors k1 − k1 + k3 and k2 − k2 + k3 , respectively [see Figs. 5.10(a) and (c)].

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5.2.1 Stochastic Correlation Effects of RFWM The total third-order polarization is given by P (3) = P1 + P3 for RFWM. Similar to the discussion of REPB, for a macroscopic system where phase matching takes place this signal must be drawn from the P (3) developed on one “atom” multiplied by the (P (3) )∗ that is developed on another “atom” located elsewhere in space (with summation over all such pairs). For homodyne detection the RFWM signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic process |P (3) |2 , which involves fourth- and second-order coherence functions of ui (t). Although the bichromophoric model is needed to fully capture the subtle feature of the RFWM spectroscopy (especially effects due to different Markovian noise models), it will be interesting to first consider the approximation in which the averaging is done at the polarization level instead of the intensity level. This reduces the problem to involve only the second-order field correlation functions and eliminates difference in the three Markovian models. This allows one to have greater insight into the importance of the higher-order field correlation functions with respect to different stochastic Markovian field models. In the case that we are only interested in the τ -dependent part to investigate the temporal asymmetry of the RFWM signal, the RFWM signal intensity can be well approximated by the absolute square of the non-trivial stochastic average of the polarization |P (3) |2 , which involves only second-order coherence functions of ui (t) [13]. As we have discussed before, the averaging at the field level can not possibly capture the whole behavior of this beat signal. Beam 3 is a noisy field, so u3 (t) must be a random function. In such case, the assumption of u3 (t) ≈ 1 is not valid anymore and P  = u1 (t1 )u∗1 (t2 )u3 (t3 ) ∼ = u1 (t1 )u∗1 (t2 )u3 (t3 ) = 0, since u3 (t3 ) = 0 and all pair-correlators involving u3 (t) and another function vanish because field 3 is uncorrelated. So if the second-order stochastic averaging procedure indeed gives a good approximation to the beat signal, it must be because the beat signal is dominated by cases when the u1 (t) and u2 (t) field actions on single chromophores are correlated, and the crosschromophore correlated actions are not important. In this case, the fact that field 3 is noisy is irrelevant and might just be considered as a monochromatic field. In fact, setting u3 (t) ≈ 1 makes field 3 monochromatic, which gives P  = u1 (t1 )u∗1 (t2 )u3 (t3 ) ∼ = u1 (t1 )u∗1 (t2 ) = 0. Considering the resonant term P3 only and assuming beam 3 to be a coherent light (u3 (t) ≈ 1), the RFWM signal intensity is proportional to ∞

I(Δ, τ ) ∝ |P3 |2 , where P3  = S1 (r) 0

u1 (t)u∗1 (t − t − τ ) exp[−(γ −

iΔ)t ]dt . Assuming the noisy light beam to be Lorentzian line shape, we have ui (t1 )u∗i (t2 ) = exp(−αi |t1 − t2 |)

(5.50)

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5 Raman- and Rayleigh-enhanced Polarization Beats

1 δωi (with δωi being the linewidth of the ωi frequency com2 ponent) is the autocorrelation decay of the noisy light. The form of the second-order coherence function, which is determined by the laser line shape as expressed in Eq. (5.50), is a general feature for the three different Markovian stochastic models [5]. Using Eq. (5.50), we have: (i) τ > 0 Here, αi =

2

I(Δ, τ ) ∝ |P3 | =

|S(r)|2 exp(−2α1 |τ |) (γ + α1 )2 + Δ2

(ii) τ < 0 % 2

I(Δ, τ ) ∝ |P3 | = |S(r)|

2

exp(−2α1 |τ |) − (α1 − γ)2 + Δ2

4α1 exp[−(γ + α1 ) |τ |] × [(γ − α1 )2 + Δ2 ][(γ + α1 )2 + Δ2 ] [(γ + α1 ) cos(Δ |τ |) − Δ sin(Δ |τ |)] + & 4α21 exp(−2γ |τ |) α41 − 2α21 (γ 2 − Δ2 ) + (γ 2 + Δ2 )2 One interesting feature in field-correlation effects is that the RFWM signal exhibits a temporal asymmetry (see Figs. 5.11 and 5.12). More specifically, I(Δ, τ ) is asymmetric in τ in general because it only depends on the laser coherence time when τ > 0, while depends on both the laser coherence time and the relaxation time of the grating when τ < 0. Physically, beam 1 is used to probe the moving grating Q3 , which decays with rate γ. The γ-dependence in the temporal behavior of I(Δ, τ ) for τ < 0 is the result of the amplitude correlation between beam 1 and Q3 .

Fig. 5.11. Schematic diagrams to describe the second-order coherence function theory of RFWM. f (t ) = exp(−γt ) for curve (a), f (t ) = exp(−α1 |t + τ |) for curve (b) at τ < 0 and curve (c) at τ > 0, respectively.

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‹ Fig. 5.12. Normalized RFWM signal intensity versus time delay γτ . Δ γ = 0, ‹ α1 γ = 1.1 (solid curve), 2 (dashed curve), 5.6 (dotted curve), 20 (dash-dotted ‹ ‹ curve) for (a); while α1 γ = 5.6, Δ γ = 10 (dotted curve), 15 (dashed curve), 55 (solid curve) for (b).

Next, we consider the resonant case of P3 . At Δ = 0, we have  ∞ exp(−α1 |t + τ |) exp(−γt )dt P3  = S1 (r)

(5.51)

0

After performing the integral in Eq. (5.51), we obtain the following: (i) τ > 0 P3  = −S1 (r) exp(−α1 |τ |)/(α1 + γ) (ii) τ < 0 P3  = S1 (r)[exp(−α1 |τ |) + 2α1 exp(−γ|τ |)/(α1 + γ)]/(α1 − γ) It can be easily shown that the RFWM signal intensity (deduced by taking the absolute square of P3 ) is consistent with |P3 |2  if we set α3 = 0. From the second-order coherence function theory, the stochastic average of the nonlinear polarization determines the basic features of the temporal behavior of the RFWM signal. As discussed before, the establishment of the nonlinear polarization consists of two steps. First, order parameter Q3 is induced through the nonlinear interaction between the medium and two incident beams. Since Q3 satisfies the first-order differential equation, the integration effect is involved in this process. The induced order parameter Q3 is then probed by another incident beam, which leads to the generation of the nonlinear polarization. In the case that ω1 is far off resonance, this step is an instantaneous process and the induced polarization is proportional to the direct production of the order parameter and the probe beam field. For the Rayleigh-active mode, the order parameter Q3 is induced by the ω1 frequency component of beam 2 and the coherent light beam 3. Due to the integration effect, Q3 at time t is the summation of the order parameter induced at times before t. Since Rayleigh-active mode decays with the rate γ, we have a weight factor exp(−γt ) in Eq. (5.51). Q3 is probed by beam 1 at time t instantaneously. If the ω1 frequency component of beam 2 is

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5 Raman- and Rayleigh-enhanced Polarization Beats

delayed with respect to that of beam 1 by τ , then only that part of the order parameter (induced at time approximately τ later) is correlated with the ω1 frequency component of the probe beam 1. The factor exp(−α1 |t + τ |) in Eq. (5.51) reflects this mutual correlation between the order parameter and the ω1 frequency component of the probe beam 1. Temporal behaviors of the time-delayed RFWM can be understood from Fig. 5.11 directly. Figure 5.11 plots functions exp(−γt ) and exp(−α1 |t + τ |), where exp(−γt ) describes the decay of the Rayleigh mode, while exp(−α1 |t + τ |) gives the mutual correlation between the Rayleigh mode and the probe beam. Due to the integration effect, the nonlinear polarization is proportional to the integral of the production of these two functions over t (from 0 to ∞). Now, we discuss the relative time delay when the maximum of the RFWM signal intensity occurs. The function exp(−α1 |t + τ |) can be divided into two parts: region I is for t < −τ and region II for t > −τ . When τ = 0, only region II has contribution to the integral. However, region I gives additional contributions to the integral when τ < 0. Therefore, the delay time for the RFWM signal intensity to be maximum −τmax should not occur at 0, but is shifted to −τmax > 0. Next, we consider the τ dependence of the RFWM signal when αi >> γ and τ < 0. In this case, the function exp(−α1 |t + τ |) is very sharp at t = −τ that provides an extremely good temporal resolution and P3  decays with the rate γ. More specifically, one can approximate  exp(−αi |t + τ |) by a δ-function, i.e., exp(−αi |t + τ |) ≈ 2δ(t + τ ) αi . So, Eq. (5.51) becomes  ∞   P3  = S1 (r) 2δ(t + τ ) exp(−γt ) α1 dt = 2S1 (r) exp(−γ |τ |) α1 0

This result indicates that the τ dependence of P3  decays with rate γ as predicted. We then consider the case that the ω1 frequency component of beam 2 is delayed from that of beam 1 (i.e., τ > 0). In this case, the integral in Eq. (5.51) involves only part of region II [see curve (c) of Fig. 5.11]. Furthermore, the area of region II involved in the integral decreases with decay rate α1 as time delay increases, therefore the τ dependence of the RFWM signal reflects the coherence time of the laser. Figure 5.11 can also be used to discuss the case when αi << γ. In this limit, exp(−γt ) decays so rapidly that the integral in Eq. (5.51) is determined mainly by the function exp(−α1 |t + τ |) at t = 0. In other words, P3  decays with rate α1 when τ increases. Physically, the relaxation time of the Rayleigh mode is so short that the Rayleigh mode, excited by the ω1 frequency component of beam 2 and the ω3 frequency component of beam 3, needs to be probed immediately by the ω1 frequency component of beam 1. Therefore, the temporal behavior of the RFWM signal reflects the mutual correlation between beam 1 and beam 2. Finally, according to above discussion, no coherence spike exists at τ = 0, which is drastically different from the corresponding CSRS with color-locking noisy light. In general, RFWM is similar to the corresponding CSRS. Unlike CSRS,

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no coherence spike appears at τ = 0 [see Fig. 5.12 (a)]. As the laser linewidth α1 /γ increases, the maximum is more closer to τ = 0. Moreover, the RFWM exhibits hybrid radiation-matter detuning THz damping oscillation with a frequency close to Δ [see Fig. 5.12 (b)], which originates from the sin(Δτ ) and cos(Δτ ) factors for τ < 0. The maximum of the temporal profile for the RFWM signal is shifted to τ = 0 as the frequency detuning Δ γ increases. Field correlation effects in a medium (with a single relaxation rate γ) can be considered in Markovian noisy field. In the limit  of γ >> α1 , α3 for the CFM, we have I(Δ, τ ) ∝ [1+exp(−2α1 |τ |)][1+3γ 2 (γ 2 +Δ2 )] for both τ > 0 and τ < 0, the RFWM spectrum is independent of τ . Moreover, if we define a parameter R = Ires. (Δ = 0) Inonres. as the ratio between the resonant signal at Δ = 0 and the nonresonant background, we can get R ≈ 3. We then consider the RFWM spectra in the limit of γ << α1 , α3 for the CFM. We have: I(Δ, τ ) ∝

γ(α1 + α3 ) 1 + 2γ(α1 + α3 ) γ + + exp(−2α1 |τ |) 2α1 (α1 + α3 )2 + Δ2 (α1 + α3 )2 + Δ2

for τ > 0, I(Δ, τ ) ∝

γ γ(α1 + α3 ) 1 + 2γ(α1 − α3 ) + + exp(−2α1 |τ |) + 2α1 (α1 + α3 )2 + Δ2 (α1 − α3 )2 + Δ2

4γα1 exp[−(α1 + α3 )|τ |] 4 2 2 2 2 [α1 −2α1 (α3 −Δ )+(α3 +Δ2 )2 ][(α21 −α23 +Δ2 ) cos(Δ|τ |)+2α3 Δ sin(Δ|τ |)] for τ < 0. The equation for τ > 0 indicates that when τ = 0 the nonresonant back ground is larger than the resonant signal by a factor of (α1 + α3 ) 3γ >> 1 at Δ = 0. However, when αi |τ | >> 1, the resonant signal and  the nonresonant background become comparable, and we have R = 2α1 (α1 + α3 ), which equals to 1 if α1 = α3 . Next, we consider the field correlation effect in a single relaxation rate medium for  phase-diffusion model (PDM). Figures 5.13 is the RFWM spectra curve), –0.5 (dotted  curve), –1 versus Δ γ for PDM when α1 τ =0 (dashed  (dot-dashed),–10 (solid curve), α3 α1 = 1, γ α1 = 100 for (a), γ α1 = 0.1 for (c); α1 τ = 0 (dashed  curve), 0.5 (dotted curve), 1 (dot-dashed), 10 (solid curve), α3 α1 = 1, γ α1 = 0.1 for (b). As can  be observed, in the limit of γ >> α1 , α3 [see Fig. 5.13 (a)], I(Δ) ∝ 1 + 3γ 2 (γ 2 + Δ2 ) for both τ > 0 and τ < 0, the RFWM spectrum becomes completely independent of τ , and R ≈ 3 at Δ = 0. In the limit of γ << α1 , α3 , I(Δ, τ ) ∝

γ(α1 + α3 ) γ + + 2α1 (α1 + α3 )2 + Δ2 ! " 1−γ 2γ(α1 + α3 ) + exp(−2α1 |τ |) 2α1 (α1 + α3 )2 + Δ2

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5 Raman- and Rayleigh-enhanced Polarization Beats

for τ > 0 [see Fig. 5.13 (b)], γ γ(α1 + α3 ) + + 2α1 (α1 + α3 )2 + Δ2 ! " 1−γ 2γ(α3 − α1 ) − exp(−2α1 |τ |) + 2α1 (α3 − α1 )2 + Δ2

I(Δ, τ ) ∝

4γα1 exp[−(α1 + α3 ) |τ |] [α41 −2α21 (α23 −Δ2 )+(α23 +Δ2 )2 ][(α21 −α23 +Δ2 ) cos(Δ|τ |)+2α3 Δ sin(Δ|τ |)] for τ < 0 [see Fig. 5.13 (c)]. The equation for τ > 0 indicates that when τ = 0 the nonresonant background is larger than the resonant signal by a factor of  signal (α1 + α3 ) 3γ >> 1 at Δ = 0. However, when αi |τ | >> 1, the resonant  becomes comparable to the nonresonant background with R = 2α1 (α1 +α3 ), which equals to 1 if α1 = α3 .

Fig. 5.13. RFWM spectra versus Δ/γ for PDM.

Finally, we consider the field correlation effect in a single relaxation rate medium for Gaussian-amplitude model (GAM). In the limit of γ >> α1 , α3 [see Fig. 5.14 (a)], I(Δ, τ ) ∝ [1 + exp(−2α1 |τ |)][1 + 3γ 2 (γ 2 + Δ2 )] for both τ > 0 and τ < 0, the RFWM spectrum is independent of τ , and R ≈ 3 at Δ = 0. In the limit of γ << α1 , α3 ,

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats

181

‹ Fig. 5.14. RFWM spectra versus Δ γ for GAM.

I(Δ, τ ) ∝

γ γ(α1 + α3 ) + + 2α1 (α1 + α3 )2 + Δ2 " ! 1+γ 2γ(α1 + α3 ) + exp(−2α1 |τ |) 2α1 (α1 + α3 )2 + Δ2

for τ > 0 [see Fig. 5.14 (b)] and I(Δ, τ ) ∝

γ γ(α1 + α3 ) + + 2α1 (α1 + α3 )2 + Δ2 ! " 2γ(α1 − α3 ) 1+γ − exp(−2α1 |τ |) + 2α1 (α1 − α3 )2 + Δ2 4γα1 exp[−(α1 + α3 )|τ |] × [α41 − 2α21 (α23 − Δ2 ) + (α23 + Δ2 )2 ] [(α21 − α23 + Δ2 ) × cos(Δ|τ |) + 2α3 Δ sin(Δ|τ |)]

for τ < 0 [see Fig. 5.14 (c)]. The equation for τ > 0 shows that when τ = 0 the nonresonant  background is larger than the resonant signal by a factor of (α1 + α3 ) 3γ >> 1 at Δ = 0. However, when αi |τ | >> 1 then the resonantsignal and the nonresonant background become comparable, and R = 2α1 (α1 + α3 ), which equals to 1 if α1 = α3 .

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5 Raman- and Rayleigh-enhanced Polarization Beats

The difference in the temporal behavior of the RFWM for γ2 >> α1 , α3 and γ1 << α1 , α3 can be employed for the suppression of nonresonant thermal background in an absorbing medium (with nonlinear susceptibilities χ1 and χ2 , the relaxation rates γ1 and γ2 for the thermal grating and the molecularreorientational grating, respectively). For simplicity, letus consider the case of γ2 >> α1 , α3 >> γ1 for CFM, for τ = 0, I(Δ) ∝ χ21 + 2χ22

1 + 3γ22 1 + γ22 + 2χ1 χ2 2 2 2 γ2 + Δ γ2 + Δ2

whereas in the limit of |τ | → ∞, ! " γ1 1 + 3γ22 γ1 (α1 + α3 ) + I(Δ) ∝ χ21 + χ22 2 2 2 2α1 (α1 + α3 ) + Δ γ2 + Δ2 Suppose that the thermal grating is more efficient than the molecularreorientational grating, so that χ21 >> χ22 , then we have I(Δ) ∝ χ21 at the zero time delay. Hence, the RFWM spectrum is dominated by the nonresonant thermal background. The contribution from the thermal grating can be reduced significantly as the time delay between beam 1 and beam 2 increases. When beams 1 and 2 become completely uncorrelated, the condition for suppressing the nonresonant thermal background is   2  χ1 γ1 << 1 χ2 2α1 In this case I(Δ) ∝ χ22

1 + 3γ22 γ22 + Δ2

which is exactly the RFWM spectrum of a sample with molecularreorientational grating alone. Next, we consider the case of γ2 >> α1 , α3 >> γ1 for PDM and GAM. Figures 5.15 and 5.16 show the RFWMspectra of PDM and GAM, respectively,   α α = 1, γ = 1 × 10−5 , γ2 α1 = 10, in an absorbing medium for (a) α 3 1 1 1  curve),350 (dot-dashed) α1 τ = −5, χ1 χ2 = 50 (dashedcurve), 250 (dotted  χ α = 50, α = 1, γ1 α1 = 1 × 10−5 , and 500 (solid curve); (b) χ 1 2 3 1  curve), –4 (dot-dashed) γ2 α1 = 10, α1 τ = 0 (dashed  curve), –3 (dotted  −5 and  –10 (solid curve); (c) χ1 χ2 = 500, α3 α1 = 1, γ1 α1 = 1 × 10 , γ2 α1 = 10, α1 τ = 0 (dashed curve), –4 (dotted curve), –5 (dot-dashed) and –10 (solid curve). I(Δ) ∝ χ21 + χ22

1 + 3γ22 1 + γ22 + 2χ1 χ2 2 2 2 γ2 + Δ γ2 + Δ2

at τ = 0 for PDM (see Fig. 5.15), and I(Δ) ∝ χ21 + 3χ22

1 + 3γ22 1 + γ22 + 2χ1 χ2 2 2 2 γ2 + Δ γ2 + Δ2

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183

at τ = 0 for GAM (see Fig. 5.16), respectively. Figures 5.17  shows the RFWM −5 α spectra in an absorbing medium for (a) α = 1, γ 3 1 1     α1 = 1 ×10 , γ2 α1 = 10,α1 τ = 0, χ1 χ2 = 5; (b) χ1 χ2 = 5, α3 α1 = 1, γ1 α1 = 1 × 10−5 , γ2 α1 = 10, α1 τ = −2; (c) χ1 χ2 = 50, α3 α1 = 1, γ1 α1 = 1 × 10−5 , γ2 /α1 = 10, α1 τ = 0. The three curves represent the chaotic field (dotted line), phase-diffusion field (solid line), and Gaussian-amplitude field (dashed line), respectively. The three curves in (c) have been decreased by a factor 2580. Again, when the thermal grating is more efficient than the molecular-reorientational grating so that χ21 >> χ22 , we have I(Δ) ∝ χ21 at the zero time delay. Hence, the RFWM spectrum is dominated by the nonresonant thermal background. At zero delay time, drastic differences can be seen for the three Markovian stochastic field under γ2 >> α1 , α3 >> γ1 approximation. Also, results for PDM and GAM are the same as for CFM in the limit of |τ | → ∞ (see Fig. 5.17). The contribution from the thermal grating can also be significantly reduced as the time delay between beam 1 and beam 2 is increased.

Fig. 5.15. RFWM spectra of PDM in an absorbing medium.

In the case of nonresonant RFWM, the thermal effect of an absorbing medium can be suppressed by a time-delayed method. The resonant signal and the nonresonant background originate from the order parameters Q3 (r, t) and Q1 (r, t), respectively. According to Eqs. (5.47) and (5.49), effects due to integration are involved in establishing the order parameters of the grating. In the broadband noisy light case (i.e., γ << α1 , α3 ), the integration washes

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5 Raman- and Rayleigh-enhanced Polarization Beats

Fig. 5.16. RFWM spectra of GAM in an absorbing medium.

Fig. 5.17. RFWM spectra in an absorbing medium.

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185

out the grating. At zero time delay no washout takes place in establishing Q1 (r, t), because phase factor φ1 in A1 (t − t )[A1 (t − t )]∗ is stationary. However, phase factor φ3 in A1 (t − t )[A3 (t − t )]∗ is a random variable which fluctuates with a characteristic time scale of (α1 + α3 )−1 . Because of the integration, the fast random fluctuation of φ3 leads to the reduction of the amplitude of Q3 . Therefore, the RFWM spectrum is dominated by a large nonresonant background when τ = 0. The RFWM spectrum in the limit of |τ | >> 1/α is quite different. Similar to Q3 , Q1 is now induced by mutually incoherent fields. If α1 = α3 , then the influences of the integration on Q1 and Q3 are equal. In the case of Δ = 0, signals from Q1 and Q3 are equal. Furthermore, the relative phase between P1 (r, t) and P3 (r, t) is a stochastic variable. Since there is no interference between them, we have R ≈ 1, which is defined as the ratio between the intensity of the resonant signal at Δ = 0 and the nonresonant background. Letus now consider case for narrow bandwidth with γ >> α1 , α3 . Under this condition, the material gratings have very short relaxation time. therefore, they can respond to the phase fluctuation of the field almost immediately. More specifically, A1 (t − t )[A1 (t − t )]∗ and [A1 (t − t )]∗ A3 (t − t ) in Eqs. (5.47) and (5.49) are slowly varying functions in comparison to exp(−γt ) with a peak at t = 0, and therefore can be approximated as A1 (t)[A1 (t)]∗ and [A1 (t)]∗ A3 (t), respectively. We can then write

P1 (r, t) ∝

χγA1 (t)[A1 (t)]∗ A3 (t)

P3 (r, t) ∝ χγA1 (t)[A1 (t)]∗ A3 (t)



∞

exp(−γt )dt

0



∞

exp[−(γ − iΔ)t ]dt

0

Above results indicate that the RFWM spectrum is independent of τ . Although phases of P1 and P3 fluctuate randomly, the relative phase between them is well defined and fixed; therefore, we have R ≈ 3 instead of 1 due to the interference between P1 and P3 . The effect of field correlation on order parameters Q1 and Q3 is different. In particular, the time delay is increased, the phase fluctuation of the interference pattern between beam 1 and beam 2 affects the establishment of Q1 directly. In contrast, since Q3 is induced by beam 2 and beam 3, the effect of integration will not directly lead to the τ dependence in Q3 . The fieldcorrelation effect here is due to the coincidence of the intensity spikes between Q3 and beam 1 instead. The RFWM is also influenced by the interference between signals originated from Q1 and Q3 . The degree of the interference is given in the parameter R = Ires. (Δ = 0) Inonres. . For example, signal intensities arising from Q1 and Q3 are equal in the limit of |τ  | → ∞ when α1 = α3 and Δ = 0. Since it can be shown thatR ≈ 1 + 2γ (γ + α1 ), the  contribution from the interference is given by 2γ (γ + α1 ), which varies from 0 for α1 >> γ to 2 for α1 << γ (see Figs. 5.13 and 5.14).

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5 Raman- and Rayleigh-enhanced Polarization Beats

Due to field correlation effects, the thermal grating can be suppressed by a time-delay method. This method employs the intrinsic incoherence of the laser beams and the large (by orders of magnitude) difference between the relaxation times of the molecular reorientational grating and the thermal grating. Consider RFWM in an absorbing sample. Because of the high efficiency of the thermal effect, then similar to Figs. 5.15 – 5.17, the Rayleightype FWM spectrum exhibits a large nonresonant background when τ = 0. This nonresonant background can be reduced if the relative time  delay is increased between beam 1 and beam 2. A reduction factor of 2α1 γT can be achieved when beam 1 and beam 2 become uncorrelated. Residue thermal effects can be reduced further by using pump beams with broader linewidths and/or longer pulse widths. In addition, this method will not affect the strict phase-matching condition, because beams 1 and 2 come from a single laser source, therefore, they have the same frequency even though they become completely uncorrelated. Typically, the relaxation time of a thermal grating is on the order of a microsecond. Letting α1 ≈ 0.1γM and assuming that the relaxation time of the molecular-reorientational grating is a few hundreds of femtosecond, the reduction factor can be about 10−6 . If one employs pulse lasers with pulse width smaller than the relaxation time of the thermal grating, then due to the finite interaction time between the laser and the material, the role of the relaxation time should be replaced by the laser pulse width. For a laser pulse width of 5 ns, the reduction factor becomes 10−5 . On the other hand, in contrast to the thermal grating, the field correlation has little influence on the RFWM spectrum when the grating has a fast relaxation time. Therefore, ultrafast longitudinal relaxation time can be measured even in an absorbing medium [19]. Using a similar idea, the hidden Raman resonance can be revealed from thermal background by a time-delayed technique in Raman-enhanced FWM experiments [6]. As a nonlinear spectroscopic technique, the most important question in the RFWM is the “frequency bandwidth,” or, equivalently, the avoidance of phase mismatch during wavelength tuning. The coherence length in the RFWM is given by   lc = 2c [n(ω1 ω3 )|ω1 − ω3 |θ2 ] where n is the refractive index. To ensure that the phase mismatch does not affect the experimental results, it requires that the coherence length lc is larger than the thickness of the sample cell. Frequency bandwidth can be increased further by reducing the angle between beam 1 and beam 2. So, in principle, relaxation time shorter than 10 fs can be measured. In the coherent Raman spectroscopy the spectrum usually exhibits an asymmetric line shape, due to the interference between the resonant signal and the nonresonant background. In contrast, the RFWM is a nonresonant process, and the line shape is always symmetric even though interference between signals from the two gratings exists. Finally, although several frequency-domain nonlinear laser spectroscopy techniques have been proposed for ultrafast measurements

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats

187

[19], the RFWM possesses following features. It involves three incident laser beams, therefore, different tensor components of the nonlinear susceptibility can be measured independently. The angle between beam 1 and beam 2 can be adjusted for individual experiment to optimize the tradeoff between better phase matching and larger interaction volume or better spatial resolution. The RFWM signal is a coherent light, which makes it easier to detect. One interesting feature in field correlation is that the RFWM exhibits a temporal asymmetry and a spectral symmetry. Beam 1 is used to probe the moving grating Q3 , which decays with rate γ. The γ-dependence in the temporal behavior of I(Δ, τ ) for τ < 0 results from the amplitude correlation between beam 1 and Q3 . More specifically, according to Eqs. (5.47) and (5.49), I(Δ, τ ) is asymmetric in τ in general because it only depends on the laser coherence time when τ > 0, while it depends on both the laser coherence time and the relaxation time of the grating when τ < 0 for chaotic fields (see Fig. 5.18). The maximum of the temporal profile for the RFWM is shifted from τ = 0. Maximum positions are almost the same in curve (a) for |P3 |2 and curve (b) for |P3 |2  (see Fig. 5.18). Main terms with τ -dependence are all shown in the |P3 |2 case. Physically, the RFWM is similar to the corresponding CSRS. However, unlike CSRS, no coherence spike appears at τ = 0. As the laser linewidth α1 /γ increases, maximum moves more closer to τ = 0, and the τ -independent nonresonant background is increased. Moreover, terms P3 P3∗  (interference between purely Rayleighresonant signals), P1 P3∗  and P1∗ P3  (interference between the Rayleighresonant signal and the nonresonant background) in the RFWM exhibit hybrid radiation-matter detuning THz damping oscillation. On the other hand, unlike the Raman-enhanced FWM spectrum, which is asymmetric due to the interference between the resonant signal and the nonresonant background [6], the line shape of the Rayleigh-type FWM is always symmetric. Specifically, in the Raman-enhanced FWM the Raman vibration is excited by the simultaneous presence of two incident beams whose frequency difference equals the Raman excitation frequency and the Raman-enhanced FWM signal is the result of this resonant excitation, as discussed in Section 5.1. In contrast, the Rayleigh-type FWM is a nonresonant process with no energy transfer between the light and the medium when the frequency difference between the two incident beam equals to zero. The resonant structure in the Rayleigh-type FWM spectrum is the result of induced moving grating. This difference is also reflected in their line shape. The RFWM spectra show smooth and symmetric curves when τ > 0 [see Figs. 5.15(b) and 5.16(b)], while exhibit a wave-like structure when ατ = −0.5 or ατ = −1 for three different Markovian stochastic fields in Figs. 5.15(c) and 5.16(c). The wave-like structure comes from sin(Δτ ) and cos(Δτ ) in Eq. (5.54), which describes the interference between signals from order parameters Q1 and Q3 . Comparing with CFM and GAM, due to the absence of the single decay factor exp(−2α1 |τ |) (which comes from the amplitude fluctuation) in PDM [6], the RFWM spectrum for PDM shows a broadest line shape in the

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5 Raman- and Rayleigh-enhanced Polarization Beats

2 Fig. 5.18. Comparison of RFWM signals of ‹(a) |P3 |2 and ‹ (b) |P3 |  for ‹ CFM, versus time delay γτ . The parameters are α1 γ = 5.6, α3 γ = 5.4, and Δ γ = 0.

narrow band case [see Fig. 5.19 (a)], and extremely narrow line shape in the broadband linewidth case, respectively [see Fig. 5.19 (b)].

‹ ‹ ‹ Fig. 5.19. RFWM spectra versus Δ γ. α3 α1 = 1, α1 τ = −2, γ α1 = 10 for (a) ‹ and γ α1 = 0.1 for (b). These three curves represent the chaotic field (dotted line), phase-diffusion field (solid line), and Gaussian-amplitude field (dashed line), respectively.

5.2.2 Homodyne Detection of Sum-frequency RASPB As we have discussed in Section 5.1, the total third-order polarization can be written as P (3) = P1 + P2 + P3 . For the macroscopic system where phase matching takes place this beat signal must be drawn from P (3) developed on one chromophore multiplied by (P (3) )∗ that is developed on another chromophore located elsewhere in space and sum over all such pairs [7 – 10, 18 – 23]. The bichromophoric model is particularly important to the noisy light spectroscopy in which the stochastic averaging at the signal level must be carried out. The sum-frequency RASPB signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic process, so that I (Δ, τ ) ∝ |P (3) |2  = P (3) (P (3) )∗  = (P1 + P2 + P3 )[(P1 )∗ + (P2 )∗ + (P3 )∗ ] contains 3 × 3 = 9 different terms

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189

in the fourth- and second-order coherence functions of ui (t) in phase conjugation geometry. In general, the homodyne detection of the RASPB signal (at the intensity level) can be viewed as from the sum of three contributions: (i) the τ -independent or τ -dependent nonresonant auto-correlation terms of ω2 molecular-reorientational grating, which include fourth-order in u2 (t) and second-order in u3 (t) Markovian stochastic correlation functions; (ii) the τ -independent or τ -dependent auto-correlation terms (i.e., the RFWM) of ω1 nonresonant molecular-reorientational grating and |ω3 − ω1 | = Δ Rayleigh resonant mode, which include fourth-order in u1 (t) and second-order in u3 (t) Markovian stochastic correlation functions; (iii) the τ -dependent cross-correlation terms between the RFWM and the NDFWM processes, which include second-order Markovian stochastic correlation functions in u1 (t), u2 (t), and u3 (t). Different Markovian stochastic models for the laser fields only affect the fourth-order, but not the second-order correlation functions. For the case with the laser sources to be chaotic fields, we have [5] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = ui (t1 )u∗i (t3 )ui (t2 )u∗i (t4 ) + ui (t1 )u∗i (t4 )ui (t2 )u∗i (t3 )

(5.52)

The composite noisy beam 1 (beam 2) is treated to have a spectrum with a sum of two Lorentzians. The high-order decay cross-correlation terms are neglected in our treatment, as in the case of Ref. [22, 23]. After substituting Eq. (5.52) into I(Δ, τ ) ∝ |P (3) |2  and performing tedious integrations, we obtain for: (i) τ > 0 % γ η2 γ γa γ 2 + + 2 + I(Δ, τ ) ∝ χ γ + 2α1 γ + 2α2 γa + Δ2 (γ + γa )γ 2 2[γa (γ + γa ) − Δ2 ]γ 2 + + (γ + α1 )[(γa + γ)2 + Δ2 ] (γa2 + Δ2 )[(γa + γ)2 + Δ2 ] ! " 2γa γ γa γ 2 1+ 2 + exp(−2α1 |τ |) + γa + Δ2 (γ + α1 )(γa2 + Δ2 )   A∗ γ Aγ 2 ∗ η exp(−2α2 |τ |) + η A + A + + × γa − iΔ γa + iΔ & (5.53) exp[−(α1 + α2 ) |τ |] (ii) τ < 0

%

I(Δ, τ ) ∝ χ2

γ + γ + 2α1

2γ 2 (2γ 2 − Δ2 + 3γα1 + α21 + 3γα3 + 2α1 α3 + α23 ) + (γa2 + Δ2 )[(γa + γ)2 + Δ2 ]

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5 Raman- and Rayleigh-enhanced Polarization Beats

γγa 2γ 2 (γ + γa ) η2 γ + 2 + − 2 2 2 (γ + 2α1 )[(γa + γ) + Δ ] γa + Δ γ + 2α2 2γα1 (γ + γc + iΔ) × exp[−(γa − iΔ) |τ |] − (γc + iΔ)(γb − iΔ)(γa − iΔ) 2γα1 (γ + γc − iΔ) exp[−(γa + iΔ) |τ |] + (γc − iΔ)(γb + iΔ)(γa + iΔ) 2γ 2 α1 (2γ 2 α1 − 2α31 + γ 2 α3 − Δ2 α3 − 5α21 α3 − 4α1 α23 − α33 ) × (γ 2 − α21 )(γc2 + Δ2 )(γa2 + Δ2 ) !   " γγb γ exp(−2γ |τ |) + 1 + 2 + exp(−2α1 |τ |) + γ − α1 γb2 + Δ2 η 2 exp(−2α2 |τ |) −

2γα1 Aη exp[−(γa − iΔ) |τ |] − (γb − iΔ)(γa − iΔ)

2γα1 A∗ η exp[−(γa + iΔ) |τ |] + (γb + iΔ)(γa + iΔ)  &  A∗ γ Aγ ∗ + + A + A exp[−(α1 + α2 ) |τ |] η γb + iΔ γb − iΔ

(5.54)

 Here, η = ε2 ε2 ε1 ε1 (with ε1 ≈ ε1 and ε2 ≈ ε2 ); Δk = (k1 − k1 ) − (k2 − k2 ); γa = γ + α1 + α3 , γb = γ − α1 + α3 , γc = γ − α1 − α3 ; and A = exp[iΔk · r − i(ω1 + ω2 )τ ] = exp(iθ). In general I(Δ, τ ) is asymmetric in τ because it only depends on the laser coherence time when τ > 0, while it depends on both the laser coherence time and the relaxation time of the grating when τ < 0. The RASPB is generally different for τ > 0 and τ < 0. Even when |τ | → ∞, Eq. (5.53) is still different with Eq. (5.54). Different Markovian stochastic models of the laser fields only affect the fourth-order, but not the second-order correlation function. The interferometric contrast ratio of the interferogram, which mainly determines the cross-correlation between the RFWM and the NDFWM processes, is equally sensitive to the amplitude and phase fluctua tions of the chaotic field. The constant term χ2 γγa (γa2 + Δ2 ) in Eqs. (5.53) and (5.54) is independent of the relative time-delay between the twin beam 1 and beam 2 and is caused by the phase fluctuations of the chaotic fields, while purely decay terms (including factors exp(−2α1 |τ |), exp(−2α2 |τ |), and exp(−2γ|τ |) in Eqs. (5.53) and (5.54)) come from the amplitude fluctuations of the chaotic field. Physically, the chaotic field has the property of photon bunching, which can affect any multiphoton processes when the higher-order correlation function of the field plays an important role. For the chaotic field, the resonant auto-correlation between Rayleigh-active modes (i.e., with the factor exp(−2γ|τ |) originated from the P3 P3∗  term) is shown in Eq. (5.54) for τ < 0. Equations (5.53) and (5.54) indicate not only characteristics of the twin

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats

191

laser fields, but also the molecule’s vibrational property. Specifically, temporal behavior of the sum-frequency RASPB intensities mainly reflect the characteristics of the twin-composite laser fields for τ > 0, and the molecule’s vibrational property for τ < 0. The sum-frequency RASPB signal versus τ typically shows the attosecond time scale modulation with the sum-frequency ω2 + ω1 and a damping rate of α1 + α2 . If this sum-frequency RASPB technique in attosecond time scale is used to measure the modulation frequency ωs = ω2 + ω1 , the measurement accuracy can be improved by measuring as many cycles of the attosecond modulation as possible. Since the amplitude of the attosecond modulation decays with a time constant (α1 + α2 )−1 as |τ | increases, the maximum range of the time-delay |τ | should equal approximately to 2(α1 +α2 )−1 . The theoretical limit of the uncertainty in the modulation frequency measurement Δωs can be estimated to be Δωs ≈ π(α1 +α2 ), which is related to the decay time constant for the amplitude of the beat signal modulation. Equations (5.53) and (5.54) also indicate that the beat signal oscillates  not only temporally but also spatially with a period of 2π Δk along the direction Δk, which is almost perpendicular to the propagation direction of the beat signal. Here, Δk ≈ 2π|λ1 − λ2 |θ λ2 λ1 , with θ being the angle between beam 1 and beam 2. Physically, the polarization beat model assumes that the twin-composite beams are plane waves. Therefore, RFWM and NDFWM signals, which propagate along k1 − k1 + k3 and k2 − k2 + k3 , respectively, are plane waves also. Since generated FWM signals propagate along slightly different directions, the interference between them leads to the spatial oscillation. To observe such spatial modulation in the beat signal the dimension of the detector (or spatial resolution) should be smaller than 0.6 mm, which should be detected by a pinhole detector or CCD array. The finite thickness of the sample has a detrimental effect on the correlation of the counter-propagating color-locked noisy fields. Although the transverse modulation of the attosecond RASPB signal is considered, the effect of signal integration in the longitudinal direction is reasonably neglected here. Third-order polarizations [see Eqs. (5.47)–(5.49)], which involve the integration of t from 0 to ∞, are the accumulation of the polarization induced at a different time. The RASPB signal exhibits ω2 + ω1 ultrafast modulation in attosecond time scale, and the symmetric line shape of RFWM is good for tuning ω3 − ω1 to zero frequency resonance. The relative phase between P1 and P3 is now a stochastic variable. Due to the randomization of the relative phase between the Rayleigh resonant term from P3 and the nonresonant background fromP1 , the interference between them disappears almost completely. As a result, the RFWM spectrum exhibits a symmetric line shape (see Figs. 5.13 – 5.17). Comparing with the broadband case for CFM, the maximum of the RASPB signal in time- [see Fig. 5.20(a)] or frequencydomain [see Fig. 5.20(b)] has been shifted back and forth from τ = 0 or Δ = 0 in the case of CFM with narrow bandwidth, respectively. We have assumed the laser sources to be chaotic fields in the above calcu-

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5 Raman- and Rayleigh-enhanced Polarization Beats

‹ ‹ ‹ Fig. 5.20. RASPB signal versus γτ for α1 γ = α2 γ = α3 γ = 0.1, η = 1, r = 0, curve), 20 (dashed Δ/α1 = 0 (dot-dashed), 8 (dotted ‹ ‹ ‹ curve) and 200 (solid curve); (b) RASPB spectra for α1 γ = α2 γ = 0.06, α3 γ = 0.05, η = 1, r = 0, γτ = 0 (dot-dashed), –1 (dotted curve), –1.5 (dashed curve) and –2.4 (solid curve).

lations. Another commonly used stochastic model is PDM. If the lasers have Lorentzian line shape, the fourth-order coherence function is given by [5] ui (t1 )ui (t2 )u∗i (t3 )u∗i (t4 ) = exp[−αi (|t1 − t3 | + |t1 − t4 | + |t2 − t3 | + |t2 − t4 |)] × exp[αi (|t1 − t2 | + |t3 − t4 |)]

(5.55)

After substituting Eq. (5.55) into I(Δ, τ ) ∝ |P (3) |2 , we obtain for: (i) τ > 0 % γa γ 2(γ + γa )γ 2 γ 2 + I(Δ, τ ) ∝ χ + 2 + γ + 2α1 γa + Δ2 (γ + 2α1 )[(γa + γ)2 + Δ2 ] η2 γ 2[γa (γ + γa ) − Δ2 ]γ 2 + + 2 γ + 2α2 (γa + Δ2 )[(γa + γ)2 + Δ2 ] 2α1 (3γ 2 + 4γγa + γa2 + Δ2 ) exp[−(γ + 2α1 ) |τ |] + (γ + 2α1 )(γa2 + Δ2 )  2η 2 α2 exp[−(γ + 2α2 ) |τ |] + η A + A∗ + γ + 2α2 &  Aγ A∗ γ + exp[−(α1 + α2 ) |τ |] γa − iΔ γa + iΔ

(5.56)

(ii) τ < 0 % I(Δ, τ ) ∝ χ

2

γ γγa 2γ 2 (γ + γa ) + + 2 + γ + 2α1 γa + Δ2 (γ + 2α1 )[(γa + γ)2 + Δ2 ]

2(γγa + γa2 − Δ2 ) η2 γ + + 2 2 2 + Δ )[(γa + γ) + Δ ] γ + 2α2 ! " 2α1 4γγb α1 + 2 exp[−(γ + 2α1 ) |τ |]+ γ + 2α1 (γb + Δ2 )(γ + 2α1 ) (γa2

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats



193



2γα1 Aη 2γα1 2η 2 α2 exp[−(γ + 2α2 ) |τ |] + − × γ + 2α2 γb − iΔ γ + 2α1 + γa − iΔ   2γα1 A∗ η 2γα1 exp[−(γa − iΔ) |τ |] − + × γa − iΔ γb + iΔ γ + 2α1 + γa + iΔ ! " 2 γ exp[−(2α1 + γa − iΔ) |τ |] exp[−(γa + iΔ) |τ |] − + × γa + iΔ γb − iΔ (γa − iΔ)(2α1 + γa + γ − iΔ) 4γα21 exp[−(γ + γa − iΔ) |τ |] − (γa + γ − iΔ)(2α1 + γa − iΔ) ! " 2 γ exp[−(2α1 + γa + iΔ) |τ |] + × γb + iΔ (γa + iΔ)(2α1 + γa + γ + iΔ) 4γα21 exp[−(γ + γa + iΔ) |τ |] + (γa + γ + iΔ)(2α1 + γa + iΔ)  &  A∗ γ Aγ ∗ + + A + A exp[−(α1 + α2 ) |τ |] η γb + iΔ γb − iΔ

(5.57)

Equations (5.56) and (5.57) depend on both the laser coherence time and the relaxation time of the grating. temporal behaviors of the beat signal reflect both characteristics of the laser and the molecule’s vibrational property. These results [Eqs. (5.56) and (5.57)] are quite different from the results based on the chaotic field model. Equations (5.56) and (5.57) do not have the purely auto-correlation decay terms including the factors of exp(−2α1 |τ |), exp(−2α2 |τ |), and exp(−2γ|τ |), which are particularly insensitive to the phase fluctuations of the Markovian stochastic fields. Drastic differences of results for different stochastic models also exist in the higherorder correlation functions on the RASPB when three different Markovian stochastic models are applied [18, 22, 23]. In the case of α1 , α2 << γ, the phase-diffusion model predicts a damping oscillation of the attosecond sum-frequency RASPB signal around a constant value, which can be understood as follows. The interference pattern due to the ω1 (ω2 ) components of the twin-composite beam 1 and beam 2 will be in constant motion with a characteristic time constant α−1 (α−1 1 2 ) when |τ | is much longer than the laser coherence time τc . With α1 , α2 << γ, the relaxation time of the molecular-reorientational grating is so short that the induced gratings Grating1 and Grating2 always follow the interference pattern, and therefore the beat signal will never decay. On the other hand, the relative phase between Grating1 and Grating2 fluctuates randomly, which makes spatial interference between them impossible. In this case the total beat signal intensity is simply the sum of the signal intensities originated from Grating1 and Grating2. In contrast, fringes of Grating1 and Grating2 are stable when |τ | < τc . The constructive or destructive interference between Grating1 and Grating2 can enhance or reduce the beat signal intensity and give rise to the oscillation in the beat signal intensity as τ varies. Note that

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5 Raman- and Rayleigh-enhanced Polarization Beats

the main difference between the phase-diffusion field model and the chaotic field model is that amplitude fluctuation exists in the latter case. When |τ | < τc , the coincidence of intensity spikes of the two composite beams gives an additional enhancement of the beat signal for the CFM. The Gaussian-amplitude field has a constant phase but its real amplitude undergoes Gaussian fluctuations. The fourth-order coherence function is given by [5] ui (t1 )ui (t2 )ui (t3 )ui (t4 ) = ui (t1 )ui (t3 )ui (t2 )ui (t4 ) + ui (t1 )ui (t4 )ui (t2 )ui (t3 ) + (5.58) ui (t1 )ui (t2 )ui (t3 )ui (t4 ) For this Gaussian-amplitude field, the high-order decay cross-correlation terms have been reasonably neglected. After substituting Eq. (5.58) into I(Δ, τ ) ∝ |P (3) |2 , we obtain the following: (i) τ > 0 % γ η2 γ 2γa γ 2 + + 2 + I(Δ, τ ) ∝ χ γ + 2α1 γ + 2α2 γa + Δ2 (γ + γa )γ 2 2[γa (γ + γa ) − Δ2 ]γ 2 + 2 + 2 2 (γ + α1 )[(γa + γ) + Δ ] (γa + Δ2 )[(γa + γ)2 + Δ2 ] % & γ 4(γ + γa )γα1 −1− × γ + 2α1 [(γ + γa )2 + Δ2 ](γ + 2α1 ) ! " η2 γ 2 − η exp[−(γ + 2α2 ) |τ |] + exp[−(γ + 2α1 ) |τ |] + γ + 2α2 ! " γ 2 γa 4γa γ + 2+ 2 exp(−2α1 |τ |) + γa + Δ2 (γ + α1 )(γa2 + Δ2 )  Aγ 2η 2 exp(−2α2 |τ |) + η A + A∗ + + γa − iΔ &  A∗ γ (5.59) exp[−(α1 + α2 ) |τ |] γa + iΔ (ii) τ < 0 % I(Δ, τ ) ∝ χ

2

γ + γ + 2α1

2γ 2 (2γ 2 − Δ2 + 3γα1 + α21 + 3γα3 + 2α1 α3 + α23 ) + (γa2 + Δ2 )[(γa + γ)2 + Δ2 ] 2γ 2 (γ + γa ) 2γγa η2 γ + 2 + − 2 2 2 (γ + 2α1 )[(γa + γ) + Δ ] γa + Δ γ + 2α2

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats

!

195

"

γ + γc + iΔ α1 2γ 2 exp[−(γa − iΔ) |τ |] + − γc + iΔ (γ + γb ) − iΔ (γb − iΔ)(γa − iΔ) ! " γ + γc − iΔ α1 2γ 2 exp[−(γa + iΔ) |τ |] + + × γc − iΔ (γ + γb ) + iΔ (γb + iΔ)(γa + iΔ) 4γ 2 α21 exp[−(3γ + 2α3 + 2iΔ) |τ |] + (γ + γb + iΔ)(γb + iΔ)(γa + iΔ)(γ + γa + iΔ) 2γ 2 α1 (2γ 2 α1 − 2α31 + γ 2 α3 − Δ2 α3 − 5α21 α3 − 4α1 α23 − α33 ) × (γ 2 − α21 )(γc2 + Δ2 )(γa2 + Δ2 )

4γ 2 α21 exp[−(3γ + 2α3 − 2iΔ) |τ |] + (γ + γb − iΔ)(γb − iΔ)(γa − iΔ)(γ + γa − iΔ) !   " γ γγb 2+ 4+ × 2 exp(−2α1 |τ |) + γ − α1 γb + Δ2 ! " γ 4γb γα1 2 −1− 2 2η exp(−2α2 |τ |) + × γ + 2α1 (γb + Δ2 )(γ + 2α1 ) ! " η2 γ 2 − η exp[−(γ + 2α2 ) |τ |] − exp[−(γ + 2α1 ) |τ |] + γ + 2α2 exp(−2γ |τ |) +

2γα1 Aη exp[−(γa − iΔ) |τ |] 2γα1 A∗ η exp[−(γa + iΔ) |τ |] − + (γb − iΔ)(γa − iΔ) (γb + iΔ)(γa + iΔ)  &  A∗ γ Aγ ∗ + + A + A exp[−(α1 + α2 ) |τ |] η (5.60) γb + iΔ γb − iΔ Equations (5.59) and (5.60) show not only characteristics of twin noisy laser fields, but also the molecule’s vibrational property. Particularly, the temporal behaviors of the sum-frequency RASPB intensities mainly reflect characteristics of the twin-composite laser fields for τ > 0, and the material’s vibrational property for τ < 0. For the Gaussian-amplitude field, the resonant auto-correlation between the Rayleigh-active modes (with the factor exp(−2γ|τ |) originated from the P3 P3∗  term) is shown in Eq. (6.60) for τ < 0. The τ -independent constant background with the Gaussian-amplitude fields is slightly larger than that of the chaotic fields, which come from the amplitude fluctuation of such Markovian stochastic field. The calculated attosecond sum-frequency RASPB intensity shows that the beat signal oscillates not only temporally but also spatially along the direction Δk, which is almost perpendicular to the propagation direction of the beat signal. Three normalized three-dimensional interferograms can be plotted, which are (a) signal intensity I(Δ, τ ) versus time delay τ and frequency detuning Δ; (b) I(τ, r) versus time delay τ and transverse position r; and (c) I(Δ, τ ) versus frequency detuning Δ and transverse position r. There are smaller constant backgrounds in these plots caused by the intensity fluctuations of the narrowband chaotic fields. At zero relative time delay

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5 Raman- and Rayleigh-enhanced Polarization Beats

(τ = 0), twin beams (from the same light source) have perfect overlap of their corresponding noise patterns at the sample and the interferometric contrast is maximum. As |τ | is increased, the interferometric contrast diminishes on the time scale that reflects material memory, usually much longer than the correlation time of the noisy light. The pure auto-correlation decay terms of the Rayleigh-active mode, the molecular-reorientational grating, and laser fields for the RASPB originate from the amplitude fluctuation of Markovian stochastic fields. Different stochastic models of the laser field, mainly affect the fourth-order coherence functions in frequency- and time-domains. Due to the interference effect caused by the cos[Δk·r−(ω1 +ω3 )τ ] and sin[Δk·r−(ω1 +ω3 )τ ] factors, the temporal behavior of the RASPB is asymmetric with the maximum of the beat signal shifted from τ = 0 [see Fig. 5.20(a)]. Whereas in the limit of Δ → ∞, the term with sin[Δk · r − (ω1 + ω3 )τ ] will disappear, then the beat signal exhibits a symmetric structure. The Δ- or τ -independent constant background of the beat signal for the Gaussian-amplitude fields or the chaotic field is much larger than that of the signal for the phase-diffusion field in Fig. 5.20 because the Gaussian-amplitude field and the chaotic field have the intensity fluctuations, and the Gaussian-amplitude field has a larger intensity fluctuation than the chaotic field. Difference between the RASPB and UMS has profound influence on the field-correlation effects [24]. Frequencies and wave vectors of the sumfrequency UMS signal are ωs1 = 2ω1 −ω1 , ωs2 = 2ω2 −ω2 and ks1 = 2k1 −k1 , ks2 = 2k2 − k2 , respectively. However, the frequencies and wave vectors of the sum-frequency RASPB signal are ωs1 = ω1 − ω1 + ω3 , ωs2 = ω3 − ω3 + ω3 and ks1 = k1 − k1 + k3 , ks2 = k2 − k2 + k3 , respectively, which indicate that photons are absorbed from and emitted to the mutually correlated fluctuating twin beams 1 and 2, respectively. In the UMS, roles of beam 1 and beam 2 are interchangeable, which makes the second-order coherence function theory fail. The absolute square of the stochastic average of the polarization |P (3) |2 can not be used to describe the temporal behavior of the sum-frequency UMS due to u(t1 )u(t2 ) = 0 [24]. The higher-order correlation (intensity correlation) treatment presented above is of vital importance in the sum-frequency UMS. Moreover, because of ui (t) = 0 and u∗i (t) = 0, the absolute square of the stochastic average of the polarization |P (3) |2 , which involves only the second-order coherence function of ui (t), can not be used to fully describe the temporal behavior of the RASPB. The sixth-order correlation treatment using |P (3) |2  reduces to the second-order correlation theory |P (3) |2 in the case that the laser pulse width is much longer than the laser coherence time. We have applied the second-order coherence function theory to investigate the τ -dependent part of the beating signal [13]. Apparently, the statistical nature of the Markovian field has a more drastic effect on the outcome of the experiment than the underlying molecular nonlinearity [7 – 10, 19, 20]. Since real laser fields are unlikely to behave exactly like one of the three pure field classes (or field models discussed above), complicated superpositions of

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197

various types of responses are expected in practical experimental situations.

5.2.3 Heterodyne Detection of the Sum-frequency RASPB The phase-sensitive detection of the RFWM signal is based on the polarization interference between two FWM processes. An easy way to study the phase-sensitive behavior is to measure the signal in quadrature. One scheme for such measurements is through the homodyne detection in which new polarizations are measured by (P1 + P3 )[(P1 )∗ + (P3 )∗ ]. Detected signals must be proportional to |χ(3) |2 . Thus, Ihom odyne ∝ |χ(3) |2 and all phase information in χ(3) is lost. The second way to achieve phase-sensitive measurements for quadratures is to introduce another polarization, P2 , (called a reference signal) designed in frequency and wave vector to conjugate (go into quadrature) in its complex representation with the new polarization of interest. Thus, in this heterodyne case, the signal is derived from (P1 + P2 + P3 )[(P1 )∗ +(P2 )∗ +(P3 )∗ ] or Iheterodyne ∝ χ(3) (the signal is linear rather than quadratic). In heterodyne detected FWM, phase information is retained and one can take a full measure of the complex susceptibility, including its phase. The phase of the induced complex polarization (P (3) ) determines how its energy will partition between Class I (the absorbed or emitted active spectroscopy) and Class II (the passive spectroscopy with a new launched field) spectroscopy [7 – 10, 18 – 23]. In the following, we show the phase-sensitive method for studying the RFWM process. The reference beam is another FWM signal, which propagates along the same optical path as the RFWM signal [22, 23]. This method is used for studying the phase dispersion of the third-order susceptibility χ(3) and for the optical heterodyne detection of the RFWM signal. For the three stochastic field models described above, the subtle Markovian field correlation effect is investigated in the heterodyne detection of the attosecond sum-frequency RASPB. The composite twin beams 1 and 2 for the heterodyne detection scheme of the attosecond sum-frequency RASPB come from the same color-locking noisy light. Frequencies of P1 , P2 , and P3 are the same (ω3 ), while PR2 and PR3 have frequencies ω1 − ω2 + ω3 and ω2 − ω1 + ω3 , respectively [6]. Furthermore, due to the phase mismatching, FWM signals from PR2 and PR3 are usually much smaller than that from P1 , P2 , and P3 . So, the total thirdorder polarization is given mainly by P (3) = P1 + P2 + P3 . The thirdorder nonlinear polarizations P1 + P3 and P2 correspond to the RFWM process and the NDFWM process which have wave vectors k1 − k1 + k3 and k2 − k2 + k3 , respectively. The sum-frequency RASPB signal is proportional to the average of the absolute square of P (3) over the random variable of the stochastic process, so that the signal intensity I (Δ, τ ) ∝ |P (3) |2  = P (3) (P (3) )∗  = (P1 + P2 + P3 )[(P1 )∗ + (P2 )∗ + (P3 )∗ ] contains

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5 Raman- and Rayleigh-enhanced Polarization Beats

3 × 3 = 9 different terms in the fourth- and second-order coherence functions of ui (t) in phase-conjugation geometry. The terms P1 P1∗ , P2 P2∗ , P3 P3∗ , P1 P3∗ , and P1∗ P3  include the fourth-order coherence functions of u1 (t) or u2 (t); while terms P1 P2∗ , P1∗ P2 , P2 P3∗ , and P2∗ P3  include the second-order coherence function of ui (t). In general, the heterodyne detected result (at the intensity level) of the RASPB can be viewed as from the sum of three contributions, i.e. (1) I(Δ, τ ) ∝ IP2 + IP1 ,P3 + IP2 ,P1 ,P3 , where IP2 = P2 P2∗ ; (2) IP1 ,P3 = P1 P1∗  + P3 P3∗  + P1 P3∗  + P1∗ P3 ; and (3) IP2 ,P1 ,P3 = P1 P2∗  + P1∗ P2  + P2 P3∗  + P2∗ P3 . (1) the nonresonant auto-correlation term IP2 from the ω2 molecularreorientational grating, which includes fourth-order [in u2 (t)] and secondorder [in u3 (t)] Markovian stochastic correlation functions; (2) the autocorrelation term IP1 ,P3 (i.e., RFWM) from the ω1 nonresonant molecularreorientational grating and Δ = ω3 − ω1 ≈ 0 Rayleigh resonant vibrational mode, which includes fourth-order [in u1 (t)] and second-order [in u3 (t)] Markovian stochastic correlation functions; (3) the cross-correlation term IP2 ,P1 ,P3 between IP2 and IP1 ,P3 , which includes second-order [in u1 (t), u2 (t) and u3 (t)] Markovian stochastic correlation functions. In the heterodyne detection scheme, we assume that IP2 >> IP1 ,P3 at the intensity level (or ηχ >> |χ(3) | at the field level), so the reference beam, from the ω2 frequency components of twin beam 1 and beam 2, is much stronger than the RNFWM signal that originates from the ω1 frequency components of twin beam 1 and beam 2. For the chaotic field, the sum-frequency RASPB signal in heterodyne detection is given by, for: (i) τ > 0 % η2 γ + η 2 exp(−2α2 |τ |) + I(Δ, τ ) ∝ χ2 γ + 2α2  &  A∗ γ Aγ η A + A∗ + + exp[−(α1 + α2 ) |τ |] γa − iΔ γa + iΔ   A∗ γ Aγ 2 ∗ + = IP2 + χ η A + A + × γa − iΔ γa + iΔ (5.61) exp[−(α1 + α2 ) |τ |] (ii) τ < 0

%

A exp[−(γa − iΔ) |τ |] + (γb − iΔ)(γa − iΔ)   & A∗ γ A∗ exp[−(γa + iΔ) |τ |] Aγ 2 ∗ + +A+A × +χ η (γb + iΔ)(γa + iΔ) γb + iΔ γb − iΔ

I(Δ, τ ) ∝ IP2 − 2γα1 χ η 2

exp[−(α1 + α2 ) |τ |]

(5.62)

Here, IP2 = χ [η γ/(γ + 2α2 ) + η exp(−2α2 |τ |)] for the chaotic field. The third-order susceptibility for the RFWM process consists of a Rayleigh2

2

2

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats

199

resonant term and a nonresonant term originated from the ω1 molecular reorientational grating, i.e., χ(3) = χ + χγ (Δ − iγ  ), (γ  = γa for τ > 0, (3) whereas γ  = γb for τ < 0). χ(3) can be expressed R) =   2as |χ 2 | exp(iθ −1 (3) (3)  |χ | cos θR + i|χ | sin θR , with θR = tan {γγ [(γ ) + Δ + Δγ  ]} (see Fig. 5.21). The nonlinear susceptibility χ(3) can then be decomposed into a real and an imaginary part, i.e., χ(3) = χ + iχ , with χ = χ +  part 2 χγΔ [Δ + (γ  )2 ] and χ = χγ  γ/[Δ2 + (γ  )2 ].

‹ Fig. 5.21. Phase angle θR versus frequency detuning Δ γ. Theoretical curve with ‹ ‹ α1 γ = α2 γ = 1.

Equations (5.61) and (5.62) show that the sum-frequency RASPB signal from heterodyne detection is modulated with a frequency ω1 + ω2 as τ is varied. The phase of the signal oscillation depends on the phase θR of the nonlinear susceptibility. The two-color sum-frequency RASPB signal can also be employed for optical heterodyne detection to yield the real and imaginary parts of the nonlinear susceptibility. From Eq. (5.61), we can write I(τ > 0) ∝ IP2 + B + 2|χ(3) |χη exp[−(α1 + α2 )|τ |] sin(θR − θ) with B = 2χ2 η(cos θ + sin θ) exp[−(α1 + α2 )|τ |] = B0 (cos θ + sin θ) If the time delay τ and spatial position r are adjusted such that the phase satisfies θ = Δk · r − (ω1 + ω2 )τ = 2nπ then

I(τ > 0) ∝ (IP2 + B0 ) + 2η exp[−(α1 + α2 ) |τ | χ χ

However, if θ = (2n + 1/2)π, we will have I(τ > 0) ∝ (IP2 + B0 ) + 2η exp[−(α1 + α2 )|τ |]χ χ

200

5 Raman- and Rayleigh-enhanced Polarization Beats

From Eq (5.62), the heterodyne intensity can be written as % i(θ + Δ |τ |) I(τ < 0) ∝ IP2 − 2γα1 χ η exp(−γa |τ |) exp + (γb − iΔ)(γa − iΔ) & exp[−i(θ + Δ |τ |)] +B+ (γb + iΔ)(γa + iΔ) ( ( ( ( 2 (χ(3) ( χη exp[−(α1 + α2 ) |τ |] sin(θR − θ) 2

By adjusting the time delay τ and r (if Δ|τ | ≈ 0) to have θ = 2nπ, then I(τ < 0) ∝

(IP2 + B0 ) 4γα1 χη(γa γb − Δ2 ) exp(−γa |τ |) − + χ (γa2 + Δ2 )(γb2 + Δ2 )

2η exp[−(α1 + α2 )|τ |χ

However, if phase is θ = (2n + 1/2)π, we then have I(τ < 0) ∝

4γα1 χηΔ(γa + γb ) exp(−γa |τ |) IP2 + B0 + + χ (γa2 + Δ2 )(γb2 + Δ2 ) 2η exp[−(α1 + α2 )|τ |]χ

Therefore, by simply changing the time delay τ between the twin beam 1 and beam 2 (if r = 0) we can obtain the real [see the dashed line of Fig. 5.22 (a)] and imaginary [see the dashed line of Fig. 5.22 (b)] parts of χ(3) , respectively.

Fig. 5.22. Heterodyne detection spectra of ‹ RASPB with (a) Δk·r −(ω1 +ω2 )τ = 0, γτ = 0 and (b) Δk · r − (ω1 + ω2 )τ = π 2, γτ = 5. Theoretical curves represent the chaotic field (dashed line), phase-diffusion field (dotted ‹ line), ‹and Gaussianamplitude field (solid line) with parameters η = 0.3, α 1 γ = α2 γ = 0.06 and ‹ α3 γ = 0.05.

For the phase-diffusion field, the sum-frequency RASPB signal in heterodyne detection is calculated to be, for:

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats

201

(i) τ > 0 %

γη 2 2α2 η 2 exp[−(γ + 2α2 ) |τ |] + + γ + 2α2 γ + 2α2  &  A∗ γ Aγ ∗ η A+A + + exp[−(α1 + α2 ) |τ |] γa − iΔ γa + iΔ   A∗ γ Aγ 2 ∗ = IP2 + χ η A + A + + × γa − iΔ γa + iΔ (5.63) exp[−(α1 + α2 ) |τ |]

I(Δ, τ ) ∝ χ

2

(ii) τ < 0 %

A exp[−(γa − iΔ) |τ |] + (γa − iΔ)(γb − iΔ)  & A∗ γ Aγ A∗ exp[−(γa + iΔ) |τ |] + + + χ2 η (γa + iΔ)(γb + iΔ) γb + iΔ γb − iΔ  A + A∗ exp[−(α1 + α2 ) |τ |] (5.64)

I(Δ, τ ) ∝ IP2 − 2γα1 χ2 η

Here

% 2 2

IP2 = χ η

γ 2α2 exp[−(γ + 2α2 )|τ |] + (γ + 2α2 ) (γ + 2α2 )

&

for the phase-diffusion field. High-order decay terms are neglected in the τ < 0 case. If IP2 >> IP1 ,P3 , Eq. (5.63) gives I(τ > 0) ∝ IP2 + B + 2|χ(3) |χη exp[−(α1 + α2 )|τ |] sin(θR − θ) % exp[i(θ + Δ|τ |)] 2 I(τ < 0) ∝ IP2 − 2γα1 χ η exp(−γa |τ |) + [(γb − iΔ)(γa − iΔ)] & exp[−i(θ + Δ|τ |)] +B+ [(γb + iΔ)(γa + iΔ)] 2|χ(3) |χη exp[−(α1 + α2 )|τ |] sin(θR − θ) results from Eq. (5.64). Except for the reference beam IP2 , the heterodyne detection signal for the phase-diffusion field case is same as that of the heterodyne detected chaotic field case discussed earlier. By changing the time delay τ between twin beams 1 and 2 (if r = 0), the real [see the dotted line of Fig. 5.22 (a)] and imaginary [see the dotted line of Fig. 5.22 (b)] parts of χ(3) for the phase-diffusion field can be obtained, respectively. Due to the absence of amplitude fluctuation, the solid curves in Fig. 5.22 have the lowest Δ-independent constant background. For the Gaussian-amplitude field, the sum-frequency RASPB signal in heterodyne detection is given by, for:

202

5 Raman- and Rayleigh-enhanced Polarization Beats

(i) τ > 0 ! " η2 γ η2 γ + − η 2 exp[−(γ + 2α2 ) |τ |] + γ + 2α2 γ + 2α2  Aγ + 2η 2 exp(−2α2 |τ |) + η A + A∗ + γa − iΔ &  A∗ γ exp[−(α1 + α2 ) |τ |] γa + iΔ  Aγ + = IP2 + χ2 η A + A∗ + γa − iΔ  A∗ γ (5.65) exp[−(α1 + α2 ) |τ |] γa + iΔ %

I(Δ, τ ) ∝ χ2

(ii) τ < 0

%

A exp[−(γa − iΔ) |τ |] + (γb − iΔ)(γa − iΔ)  & A∗ γ A∗ exp[−(γa + iΔ) |τ |] Aγ + + + χ2 η (γb + iΔ)(γa + iΔ) γb + iΔ γb − iΔ  A + A∗ exp[−(α1 + α2 ) |τ |] (5.66)

I(Δ, τ ) ∝ IP2 − 2γα1 χ2 η

Here

! " η2 γ η2 γ + =χ exp[−(γ + 2α2 )|τ |] + (γ + 2α2 ) (γ + 2α2 ) − η 2 & 2η 2 exp(−2α2 |τ |) %

IP2

2

for the Gaussian-amplitude field. The high-order decay term is neglected in τ < 0 case. If the reference beam is much stronger than the RFWM signal (i.e., IP2 >> IP1 ,P3 ), except for IP2 , the heterodyne detected signal for the Gaussian-amplitude field is also same as that of the heterodyne detected chaotic field case. By changing the time delay τ between the twin beam 1 and 2 beam(if r = 0), we can obtain the real [see the solid line of Fig. 5.22 (a)] and imaginary [see the solid line of Fig. 5.22 (b)] parts of χ(3) , respectively, for the Gaussian-amplitude field case. Since the Gaussian-amplitude field has the largest amplitude fluctuation among the three stochastic field model, dotted curves in Fig. 5.22 have the highest Δ-independent constant background.

5.2.4 Discussion and Conclusion Resonance-enhanced FWM techniques have been employed for many different purposes. For example, it has been used to study the vibrational dephasing

5.2 Rayleigh-enhanced Attosecond Sum-frequency Polarization Beats

203

in molecular materials both in the frequency-domain and in the time-domain [1 – 5]. One of the CRS techniques, which may be superior to other CRS techniques, is the Raman-enhanced NDFWM [6, 19]. The main advantage of this technique is that the phase-matching condition is not so stringent, so it can be achieved over a very wide frequency range from many hundreds to thousands of cm−1 . It also possesses the features of nonresonant background suppression, excellent spatial signal resolution, free choice of interaction volume, and simple optical alignment. Especially, in the Raman-enhanced FWM the Raman vibration is excited by the simultaneous presence of two incident beams whose frequency difference equals to the Raman excitation frequency, so the Raman-enhanced FWM signal is the result of this resonant excitation. In contrast, the Rayleigh-type FWM is a nonresonant process with no energy transfer between the lights and the medium when the frequency difference between the two incident laser beam equals to zero. The resonant structure in the Rayleigh-type FWM spectrum is resulted from the induced moving grating. This difference is also reflected in their line shapes. Unlike the Raman-enhanced FWM spectrum, which is asymmetric due to the interference between the resonant signal and the nonresonant background, the line shape of the Rayleigh-type FWM is always symmetric. The Rayleigh-type FWM can be used for studying ultrafast processes in matters. In contrast to the conventional ultrafast time-domain techniques, the Rayleigh-type FWM is a frequency-domain technique, therefore the time resolution is independent of the incident laser pulse width. This Rayleightype FWM technique can be employed for measuring the ultrafast longitudinal relaxation time in the frequency-domain [19]. Based on the fieldcorrelation effects, this technique can be applied even to an absorbing medium if a time-delayed method is used. Rayleigh-type FWM is a third-order nonlinear phenomenon which involves three incident beams [see Fig. 22(a)]. Beam 1 and beam 2 have the same frequency ω1 and a small angle θ exists between their propagation directions. Beam 3 with frequency ω3 propagates along the opposite direction of beam 1. The generated FWM signal (beam 4) propagates along the direction that is almost opposite to that of beam 2. There are two mechanisms involved in their interactions with the medium. First, the nonlinear interaction of beam 1 and beam 2 with the medium gives rise to a static molecular reorientational grating. The FWM signal is the result of the diffraction of beam 3 by this static grating. Second, beam 2 and beam 3 with different frequencies build up a moving grating. If the grating lifetime τg is larger than the  time it needs to move over one spatial period (which is in the order of 1 |ω1 − ω3 |), then a destructive interference occurs during engraving and erases the grating.In other words, the signal comes only from the spectral region |ω1 − ω3 | < 1 τg . Generally speaking, ultrashort pulses of equivalent bandwidth are very sensitive to dispersive effects (even when balanced) because the transformlimited light pulse is in fact temporally broadened (chirped) and this has drastic effects on its time resolution (the auto-correlation). In this sense the

204

5 Raman- and Rayleigh-enhanced Polarization Beats

RASPB with double-frequency color-locking noisy lights has an advantage. Using three stochastic field models, the subtle Markovian field correlation effects have been investigated in the homodyne or heterodyne detected RASPB. The different roles played by the amplitude and the phase fluctuations can be understood by comparing the result for different stochastic field models in time- and frequency-domains. The Gaussian-amplitude field has stronger intensity fluctuations than the chaotic field, which is much larger than the pure phase fluctuations in the phase-diffusion field. The large intensity fluctuation has a large contribution to the constant background in the detected RASPB signal. Based on the polarization interference between the nonresonant NDFWM and the Rayleigh resonant RFWM processes, the RASPB can be employed to obtain the real and imaginary parts of the Rayleigh resonance (see Fig. 5.22). In the heterodyne detection of the RASPB, the nonresonant NDFWM (reference) signal was purposely introduced by adding another frequency component of the noisy light with frequency ω2 to the twin composite beam 1 and beam 2. This nonresonant NDFWM signal propagates along the same optical path as the RFWM signal. The relative phase between the reference beam and the RFWM signal is determined by time-delay τ between twin composite beam 1 and beam 2. Compared with the optical heterodyne detection Rayleigh-induced Kerr effect method, because the polarizations of the incident beams can be adjusted independently, the RASPB is more convenient for studying various components of the fourth-rank tensor of third-order susceptibility. The RASPB has also been employed for studying the phase dispersion of χ(3) . Although this method is similar to the method used in Ref. [20], we have shown that for Rayleigh resonance one can obtain the phase dispersion of χ(3) by simply measuring the phase change of the FWM signal modulation as ω3 is varied (see Fig. 5.22). Generally speaking, this method can be applied to study the phase dispersion of χ(3) in the RASPB. In this section, a time-delayed method is shown to be able to suppress the thermal effect in detecting the RFWM signal, and the ultrafast longitudinal relaxation time can be measured even in an absorbing medium. One interesting feature in field-correlation effects is that the RFWM signal exhibits temporal asymmetry and spectral symmetry. Also, no coherence spike in the RFWM signal with color-locking noisy light exists at τ = 0. The RFWM signal exhibits hybrid radiation-matter detuning THz damping oscillation with a frequency close to Δ. Based on three stochastic models, the subtle Markovian field correlation effects have been investigated in the homodyne or heterodyne detected RASPB. Closed analytic forms of the fourth-order Markovian stochastic correlations are characterized for homodyne (quadratic) and heterodyne (linear) detections, respectively. The heterodyne detected signal of RASPB potentially offers rich dynamic information about the homogeneously-broadened material phase of the third-order nonlinear susceptibility.

References

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6 Coexistence of MWM Processes via EIT Windows

In Chapters 6 – 8, we will describe how different orders of nonlinear wavemixing processes can be generated to coexist in the same multi-level atomic systems and how these different nonlinear optical processes interact with each other. Typically higher-order nonlinear optical processes are much weaker than the lower-order ones, so only the lowest-order non-zero nonlinear optical process is considered. However, as we will show that under certain laser beam configurations and energy-level arrangements, highly efficient four-wave mixing (FWM) and six-wave mixing (SWM) processes can be made to coexist in the same multi-level atomic systems with similar signal intensities. Due to specially-designed interaction schemes between laser beams and multi-level atomic systems, the atomic coherence and the multi-photon quantum interference induced between different atomic transitions play important roles, and the generated FWM and SWM signals can be made to transmit through the same or dual electromagnetically induced transparency (EIT) windows. These coexisting multi-wave mixing (MWM) processes and their relative strengths can be controlled and tuned by the intensities and the frequency detuning of the pump (or dressing) laser beams. In this chapter, co-existing and enhanced FWM and SWM processes in several fourlevel atomic systems are presented and their underlying physical mechanisms are discussed.

6.1 Opening FWM and SWM Channels Although generating FWM in the atomic media has been an active research field for past forty years [1, 2], some recent experimental demonstrations of enhancing FWM processes by making use of the atomic coherence in multilevel atomic systems have raised new interests in this research direction [3 – 8]. The keys in such enhanced nonlinear optical processes include the enhanced nonlinear susceptibility due to the induced atomic coherence (as discussed in Section 1.4) and the slowed laser beam propagation in the atomic medium due to the sharp dispersion (so the effective interaction time is increased), as well as greatly reduced linear absorption of generated FWM signals due to

208

6 Coexistence of MWM Processes via EIT Windows

EIT [9, 10,12]. The generation of SWM was reported in a four-level closecycled N-type system in a cold atomic sample, when the lower-order FWM process is completely turned off in this system by specially arranging laser beams [13]. On the other hand, two-photon and three-photon destructive interferences have also been observed in various four-level Y-type [14], N-type [15], and double-Λ type [5, 8] atomic systems due to the multiple laser beams involved in the interactions. These multi-photon interferences and light-induced atomic coherence are very important in nonlinear wavemixing processes and might be used to open certain nonlinear optical processes in multi-level atomic systems that are otherwise closed due to high linear absorptions near atomic resonances [16]. Normally, in order to observe the weak higher-order nonlinear signals (such as SWM, proportional to χ(5) ), the lower-order nonlinear optical processes (such as FWM, proportional to χ(3) ) and the linear absorption need to be greatly suppressed or even turned off. Since the lower-order nonlinear susceptibility is typically several orders of magnitude larger than the higherorder ones (as discussed in Section 1.1), the higher-order nonlinear optical processes are not efficient, except at extremely large laser intensities (such as with very short optical pulses). This is the case for the early reported experiments of generating SWM and even eight-wave mixing (EWM) in multi-level atomic systems [13, 17]. However, as we will show below, by manipulating the atomic coherence and employing spatial laser beam configurations in fourlevel atomic systems, SWM channels can be opened even in the existence of the FWM processes. In the following, we show that the highly efficient FWM and SWM processes can be generated simultaneously in an open-cycled Y-type atomic system, in which the dual-EIT windows are used to transmit the generated FWM and SWM signals, respectively. For a simple four-level Y-type atomic system, as shown in Fig. 6.1(b), if two strong (coupling) laser beams drive two upper transitions (|1 to |2 and |1 to |3, respectively) and a weak laser beam probe the lower transition (|0 to |1), two ladder-type EIT sub-systems (i.e., |0 − |1 − |2 and |0 − |1 − |3) will form, which coupled together via the shared energy levels, and two EIT windows appear [14]. Depending on the frequency detuning of two coupling laser beams, these two EIT windows can either overlap or be separated in frequency on spectrums of the probe beam transmission signal. Now, if two pump fields, E3 (ω3 , k3 , and Rabi frequency G3 ) and E3 (ω3 , k3 , and Rabi frequency G3 ), drive the upper transition |1 to |3 and one strong field, E2 (ω2 , k2 , and Rabi frequency G2 ), drives the transition |1 to |2, as shown in Fig. 6.1(c), there will be multi-wave mixing (MWM) processes that generate signal fields at frequency ω1 . The Rabi frequencies are defined by Gi = εi μij /, Gi = εi μij /, where μij are the transition dipole moments between level i and level j. Now, let us analyze various wave-mixing processes involved in this system. First, considering the case without the strong field E2 , a simple FWM process (involving the probe beam E1 and two pump fields E3 and E3 ) from one of the ladder-type sub-

6.1 Opening FWM and SWM Channels

209

systems will generate a signal field Ef at frequency ω1 via the perturbation chain  ∗ (0) E1 (1) E3 (2) (E3 ) (3) (I)ρ00 −→ ρ10 −→ ρ30 −→ ρ10 When the coupling laser field E2 is turned on, it will dress the energy level |1 to create the dressed states |+ and |−, and dressed-state FWM processes will occur as studied theoretically in various multi-level atomic systems [16]. In this case, other than FWM processes, there are also possible SWM processes [as shown in Figs. 6.1(e) and (f)] in this system, where two photons from field E2 and one photon each from fields E3 and E3 participate in SWM processes to generate Es with different Liouville pathways [20]. These FWM and SWM processes can exist at the same time and generated FWM and SWM signals can be phase-matched to travel in the same direction. Similarly, when two pump fields, E2 and E2 (ω2 , k2 , and Rabi frequency G2 ), are used to drive the transition |1 to |2 and one strong coupling field E3 drives the transition |1 to |3, as shown in Fig. 6.1(d), there will be similar FWM and SWM processes.

Fig. 6.1. (a) Spatial beam geometry used in the experiment; (b) simple Y-type atomic system with dual ladder-type EIT; (c) and (d) FWM processes; (e) and (f) SWM processes. The bold double heading arrows denote the strong coupling beams.

The pump and the coupling laser beams are aligned spatially in the pattern as shown in Fig. 6.1(a), with four pump and coupling laser beams (E2 , E2 , E3 , E3 ) propagating through the atomic medium in the same direction with small angles (about 0.3◦ ) between them in a square-box pattern (the angles are exaggerated in the figure). During the present experiment, one of the pump beams (E2 ) is always blocked for simplicity, so we only need to consider the system as shown in Fig. 6.1(c). The probe beam (E1 ) propagates in the opposite direction with a small angle, as shown in Fig. 6.1(a).

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6 Coexistence of MWM Processes via EIT Windows

Since angles between the propagation directions are very small, this configuration satisfies the two-photon Doppler-free conditions for the two laddertype EIT sub-systems (i.e., the coupling beam propagates in the opposite direction as the probe beam through the atomic medium) [12], which produce two EIT windows. For simplicity, we will only consider the diffracted FWM and SWM signals relevant to our experimental measurements (there are other FWM and SWM signals generated in third- and fifth-order nonlinear processes, which are diffracted into different directions due to differences in phase-matching conditions). By setting the propagation direction of the probe beam E1 [as indicated in Fig. 6.1(a)] and blocking E2 , Diffracted FWM (Ef ) and SWM (Es ) signal beams will be in the same direction determined by phase-matching conditions: kf = k1 +k3 −k3 and ks = k1 +k2 −k2 +k3 −k3 , respectively. The total FWM process can be considered as due to the constructive or destructive interference between two dressed FWM channels  ∗ (0) E1 (1) E3 (2) (E3 ) (3) (ρ00 −→ ρ±0 −→ ρ30 −→ ρ±0 ), where |+ and |− are the two dressed states for level |1 due to E2 coupling. The perturbation chains of the two SWM processes [as indicated in Figs. 6.1(e) and (f)] can be written as: (0) E

(1) E

∗ (2) (E2 )

(3) E

 ∗ (4) (E )

(5)

(0) E

(1) E

 ∗ (2) (E )

(3) E

∗ (4) (E2 )

(5)

1 2 3 3 (II) ρ00 −→ ρ10 −→ ρ20 −→ ρ10 −→ ρ30 −→ ρ10 1 3 2 3 (III) ρ00 −→ ρ10 −→ ρ30 −→ ρ10 −→ ρ20 −→ ρ10

Of course, if E3 is blocked instead of E2 [see Fig. 6.1(d)], same results can be obtained just with the indices 2 and 3 switched. Thus, the perturbation chains should be (0) E

(1) E

∗ (2) (E3 )

(3) E

 ∗ (4) (E )

(5)

(0) E

(1) E

 ∗ (2) (E )

(3) E

∗ (4) (E3 )

(5)

1 3 2 2 ρ10 −→ ρ30 −→ ρ10 −→ ρ20 −→ ρ10 ρ00 −→ 1 2 3 2 ρ00 −→ ρ10 −→ ρ20 −→ ρ10 −→ ρ30 −→ ρ10

In general for arbitrary field strengths of E2 , E3 , and E3 , one needs to solve the eleven coupled density-matrix equations to higher orders to obtain (3) (5) ρ10 for the FWM and ρ10 for SWM processes, which have been done in simulating experimental results and will be presented later on. Since the general calculations are too complicated and in order to see the relations and interplays between these FWM and SWM processes, we only present the calculations of these nonlinear susceptibilities via appropriate perturbation chains here for simplicity. When both E2 and E2 are blocked, the simple FWM via the perturbation chain (I) gives (3)

ρ10 = −

iGa eikf ·r d21 d3

where Ga = G1 G3 (G∗ 3 ), d1 = Γ10 +iΔ1 , d3 = Γ30 +i(Δ1 +Δ3 ) with Δi = Ωi − ωi and Γij is the transverse relaxation rate between states |i and |j. Next, when the coupling field E2 is turned on, the above simple FWM process will

6.1 Opening FWM and SWM Channels

211

be dressed and a perturbation approach for such interaction can be described by following coupled equations:  (1) (1) (0) ∂ρ10 /∂t = −d1 ρ10 + iG1 eik1 ·r ρ00 + iG∗2 e−ik2 ·r ρ20 (6.1) (1) ∂ρ20 /∂t = −d2 ρ20 + iG2 eik2 ·r ρ10 

−ik3 ·r ∂ρ10 /∂t = −d1 ρ10 + iG∗2 e−ik2 ·r ρ20 + iG∗ ρ30 3 e (3)

(3)

(2)

∂ρ20 /∂t = −d2 ρ20 + iG2 eik2 ·r ρ10

(3)

(6.2)

where d2 = Γ20 + i(Δ1 + Δ2 ). Equations (6.1) and (6.2) (with the approx(0) imation of ρ00 ≈ 1) can be solved together with the perturbation chain (I) to give 2iGa eikf ·r d2 ρ10 = − d1 d3 (d1 d2 + |G2 |2 ) The expression ρ10 shows an interesting interplay between coexisting FWM and SWM processes in this system [10]. With the coupling field E2 on, the intermediate energy level |1 is dressed to become two split dressedstate levels |+ and |− with induced coherence between them. The FWM signals will have the quantum interference between two two-photon transition channels via these two different split intermediate levels |+ and |−, which can either enhance or suppress the total observed FWM signal, depending on the frequency detuning. When the coupling laser field G2 is very strong (i.e., G2 >> G3 (G3 ) >> G1 ), there exists a maximum suppression of the total FWM signal at the exact multiple-EIT condition of Δ1 = −Δ2 = −Δ3 . Also, (5) one can easily calculate the fifth-order nonlinear susceptibility ρ10 for SWM processes from perturbation chains (II) and (III) [as shown in Figs. 6.1(e) and (f)], which is given to be (5)

(II)

(III)

ρ10 = ρ10 + ρ10 (3)

(5)

=

2iGa |G2 |2 eiks ·r d31 d2 d3

Since ρ10 and ρ10 have opposite signs, some interesting effects can exist in such cubic-quintic type media. For example, multi-dimensional solitons and light condensates have been predicted to exist in the system with competitive and giant third- and fifth-order nonlinearities χ(3) and χ(5) with opposite signs [19]. Also, such cubic-quintic nonlinear media can have very important impacts in propagations of very high intensity optical pulses [19]. Two EIT windows are generated by the double-ladder EIT sub-systems in the Y-type four-level system with both pump fields (between |1 and |3) and coupling field (between |1 and |2) stronger than the probe beam (between |0 and |1), as shown in Fig. 6.1(c). Since the generated SWM signal falls into one ladder-type EIT window (|0 − |1 − |2 branch), the SWM processes can be very efficient, especially when the FWM signal is suppressed (since the FWM and SWM processes compete for the photon energies available in the

212

6 Coexistence of MWM Processes via EIT Windows

system). For finite frequency detunings Δ2 and Δ3 , the two EIT windows in the Y-type system will be separated in frequency, and the generated FWM and SWM signals in these two EIT windows are easily distinguishable in the spectrum. It is easy to see that there exist dual ladder-type EIT windows for the probe beam (E1 ), the generated FWM signal (Ef ), as well as the SWM signal (Es ). There are one three-photon interference pathway (i.e., interference between the three-photon path ω1 + ω3 − ω3 and the singlephoton ω1 ) for FWM process and two five-photon interference pathways (i.e., between the five-photon path ω1 + ω2 − ω2 + ω3 − ω3 and the single-photon ω1 , and between the five-photon path ω1 + ω3 − ω3 + ω2 − ω2 and the singlephoton ω1 ) for the two SWM processes. In this system, the three-photon and five-photon interferences are destructive ones, in which χ(3) and χ(5) are zeros at line centers. The coexisting SWM and FWM signal efficiencies and the amount of suppression of the FWM signal are most prominent under the multiple-EIT condition of Δ1 = −Δ2 = −Δ3 and G2 >> G3 (G3 ) >> G1 in Fig. 6.1(c) [and G3 >> G2 (G2 ) >> G1 in Fig. 6.1(d)]. The experimental demonstrations of the above described coexisting and controllable FWM and SWM processes have been carried out in the atomic vapor of 87 Rb. The energy levels of 5s1/2 (F = 2), 5p3/2 , 5d3/2 , and 5d5/2 form the four-level Y-type system, as shown in Fig. 6.1(b). Four laser beams were carefully aligned as indicated in Fig. 6.1(a) (without E2 ). The vapor cell is 5 cm long and its temperature is set at 60◦ C. The probe laser beam E1 (with wavelength of 780 nm from an external cavity diode laser (ECDL), connecting the transition between 5s1/2 −5p3/2 ) is horizontally polarized and has a power of about P1 ≈ 3.5 mW. The pump laser beams E3 and E3 (wavelength 775.98 nm connecting the transition between 5p3/2 − 5d5/2 ) are split from a CW Ti: Sapphire laser with equal power (P3 ≈ P3 ), each with a vertical polarization. The coupling laser beam E2 (with power P2 , and wavelength 776.16 nm connecting the transition between 5p3/2 − 5d3/2 ) is from another ECDL and is also vertically polarized. Great cares were taken in aligning the laser beams with spatial overlaps and wave-vector phase-matching conditions with small angles (about 0.3◦ ) between them going through the atomic vapor cell, as indicated in Fig. 6.1(a). The diameters (at the vapor cell center) for the pump and coupling laser beams are about 0.5 mm, respectively, and the diameter of the probe beam (E1 ) is about 0.3 mm. The diffracted FWM and SWM signals (with kf and ks satisfying the phase-matching conditions) with horizontal polarizations are in the direction of Ef & Es [which is indicated at the lower right corner of Fig. 6.1(a)] and are detected by an avalanche photodiode detector (APD). The spatial square-box configuration for the laser beams in combination with the phase-matching conditions for the relevant FWM and SWM processes allows generated FWM and SWM signals to propagate in the same direction and to be spatially separated the applied laser beams for easy detection. The transmitted probe beam is simultaneously detected by a silicon photodiode. The dual-EIT windows in the Y-type system are measured by setting the

6.1 Opening FWM and SWM Channels

213

frequency detuning at Δ2 = −112 MHz and Δ3 = 0 for the strong coupling and pump laser beams, with all three laser beams (E2 , E3 , and E3 ) on. These two modified EIT windows from the two ladder-type EIT sub-systems (at standard two-photon resonance Δ1 = −Δ2 and Δ1 = −Δ3 EIT positions in the probe transmission trace detected at the silicon photodiode) are depicted in Fig. 6.2 [peaks 4 and 5 of curve (b)]. Meanwhile, as the probe detuning Δ1 is scanned, several generated wave-mixing signals are observed on the APD [as shown in curve (a) of Fig. 6.2]. The experimental parameters are P1 = 3.6 mW, P2 = 0 mW, P2 = 33 mW, P3 = P3 = 130 mW, Δ2 = −112 MHz, and Δ3 = 0. Peak 2 is identified as a combination of the FWM signal (kf ) and a small amount of SWM signal (ks ). Peak 3 is the SWM signal (ks ) and peak 1 is another FWM signal (kf ) outside the two EIT windows. Since generated FWM and SWM signals are diffracted in the same spatial direction, it is important to definitely identify them individually. This can be done by selectively and systematically blocking different laser beams and detuning different laser frequencies carefully to observe the changes in the various peaks, as shown in Fig. 6.2. By intentionally setting large frequency detunings to separate the generated FWM and SWM signals in Fig. 6.2, the FWM and SWM signals, and the two corresponding EIT windows, are separated in frequency and can be identified with clarity. The SWM signal intensity (peak 3) is comparable to the FWM signal intensity (major part of peak 2), and actually under certain conditions, peak 3 can be even made higher than peak 2. When the difference between frequency detuning Δ2 and Δ3 is reduced, these two EIT windows start to merge into one and the FWM and SWM signals begin to interfere (since they now have the same frequency and propagate in the same direction). The dip in the middle of the FWM signal [peak 2 of curve (a) in Fig. 6.2] is due to three-photon destructive interference, which is clearly observed in this coherent signal detection. Also, the five-photon destructive interference is seen as the middle dip in peak 3 for the SWM signal.

Fig. 6.2. Curve (a): Measured SWM (peak 3) and FWM (peaks 1 and 2) signals. Curve (b): Probe beam transmission (peaks 4 and 5: two ladder-type EIT windows) versus probe frequency detuning Δ1 . Adopted from Ref, [3].

214

6 Coexistence of MWM Processes via EIT Windows

In Fig. 6.3(a), the detected FWM signal is presented as a function of coupling laser power P2 , corresponding to the peak 2 in Fig. 6.2. The experimental parameters are P1 = 3.4 mW, P2 = 0 mW, P2 = 0 mW, 20 mW,40 mW,62 mW,80 mW, P3 = 34 mW, P3 = 17 mW, Δ2 = −200 MHz, and Δ3 = −450 MHz, respectively. The top curve is for P2 = 0 mW, which is the case for a simple FWM signal. The dip in the middle of the spectrum is due to three-photon destructive interference, as mentioned earlier. As the coupling field E2 is turned on, SWM signals within the EIT windows (peak 2 and peak 3 in Fig. 6.2) start to appear and increase as P2 increases. Such increases of generated SWM signals are at the cost of the FWM process, since the same applied laser beams provide photon energies for both nonlinear processes. As shown in Figs. 6.3 (a) and (b), as P2 increases, the FWM signal decreases substantially (about 3 times in the P2 power range shown) due to the destructive interference between FWM+ and FWM− channels (i.e., the two FWM pathways via the two middle dressed states |+ and |−, respectively). So, the relative strengths between generated FWM and SWM signals in the two EIT windows can be adjusted by simply tuning the coupling beam power P2 [notice that the SWM component is still much smaller than the FWM component in the peak 2 of Fig. 6.2 and in Fig. 6.3(a)], which is very important for the interplay and control between FWM and SWM processes in this system. The maximal enhancement of the SWM signal and the largest suppression of the FWM signal can be achieved at multi-EIT condition of Δ1 = −Δ2 = −Δ3 with large value of P2 .

Fig. 6.3. (a) Measured FWM signal intensity for various coupling field powers versus probe detuning Δ1 ; (b) FWM amplitude versus ω2 coupling field power. Adopted from Ref.[3].

Figure 6.4 presents the changes of the SWM signal (corresponding to the peak 3 in Fig. 6.2) as a function of the coupling field frequency detuning. The experimental parameters are P1 = 3.4 mW, P2 = 0, P2 = 34 mW, P3 = P3 = 96 mW, Γ10 2π = 3 MHz, Γ20 = Γ30 = 2π × 0.5 MHz, Δ3 = −450 MHz, and Δ2 = −150 MHz,0 MHz,150 MHz. It is seen from Fig. 6.4(a) that, as the coupling frequency detuning Δ2 changes, the generated SWM signal changes from symmetric to asymmetric, which is due to two-photon [12] or three-photon [15] resonant emission enhancement. Such asymmetric SWM

6.1 Opening FWM and SWM Channels

215

spectra has been simulated by numerically solving the eleven coupled densitymatrix equations for the system at the steady state and plotted in Fig. 6.4(b). The dips at the line center of the SWM spectra are due to five-photon (one probe photon plus four pump and coupling photons) destructive interference with the generated signal photon, which looks like a multi-photon EIT phenomenon. However, it is actually a suppression of the generated SWM signal due to multi-photon destructive interference at the exact resonance. Such agreements between the experimentally measured SWM spectra [see Fig. 6.4(a)] and theoretically calculated results [see Fig. 6.4(b)] for different frequency detunings indicate a good understanding of the underlying physical mechanism for enhanced SWM processes.

Fig. 6.4. (a) Measured SWM signal spectra for different coupling field (E2 ) frequency detunings; (b) Theoretical plots of SWM intensities versus Δ1 for different Δ2 values. Adopted from Ref. [3].

The maximal achievable FWM and SWM efficiencies in this system are quite high (measured to be about 10% and 1%, respectively). The coexistence of these two different-order nonlinear wave-mixing processes in this system can be used to evaluate the fifth-order nonlinear susceptibility χ(5) by beat(3) (5) ing the SWM signals with the FWM signal. Since |ρ10 | >> |ρ10 | is generally true (one can always tune the experimental parameters to satisfy such condition), the real and imaginary parts of χ(5) can be measured by the homodyne detection with the FWM signal as the strong local oscillator. Such homodyne beating measurements between different-order nonlinear wave-mixing processes will be discussed later in Chapter 7. We have also measured the generations of FWM and SWM signals when all four strong laser beams (E2 , E2 , E3 , E3 ) are present. In this case, the four-level Y-type atomic system with co-existing FWM and SWM Consists of three conventional two-photon Doppler-free EIT subsystems, i.e., |0 − |1 − |2 (ladder-type), |0 − |1 − |3 (ladder-type) and |2 − |1 − |3 (V-type). More FWM and SWM channels are opened in this system, and the analysis becomes complicated to separate these nonlinear wave-mixing processes. In general, one can investigate the interesting interplays between two fundamental nonlinear wave-mixing processes, and identify ways to enhance the higher-order nonlinear optical processes through opening new nonlinear channels via the atomic coherence and quantum interference.

216

6 Coexistence of MWM Processes via EIT Windows

There are several distinctly different and advantageous features in working with this double-EIT system over four-level close-cycled systems used previously for generating SWM processes [13, 20]. First, FWM and SWM processes can be generated and observed simultaneously in this open-cycled Y-type atomic system, which is not the case in the close-cycled N-type system [13]. Such coexistence of FWM and SWM processes allows one to investigate the interplays between these two interesting nonlinear optical effects, and to obtain the beat signal between them to get the fifth-order nonlinear susceptibility χ(5) . Second, the generated FWM and SWM signals fall into two separate EIT windows in this four-level double-EIT system, so the linear absorptions for the generated FWM and SWM signals are both greatly suppressed. By individually controlling (or tuning) the EIT windows, the generated FWM and SWM signals can be clearly separated and distinguished or pull together (by frequency detunings) to observe interferences between them. Third, since the amplitude of the FWM signal can be controlled (enhanced or suppressed) by the coupling laser beam (via dressed states), the relative strengths of the generated FWM and SWM signals can be adjusted easily. So, the SWM signal can be made to be in the same order of magnitude as the FWM signal. Fourth, multi-photon destructive interference effects for both FWM (three-photon interference) and SWM (five-photon interference) are clearly observed in the experiment. Although such multi-photon interference effects in double- [14] and triple- [15, 21] resonance EIT spectroscopy were reported previously by detecting fluorescence, the current method of using multiwave mixing is a coherent phenomenon. Finally, by designing the propagation directions of (pump, coupling and probe) laser beams, the Dopplerfree configurations [12] have been achieved for all the EIT sub-systems in this Y-type atomic system. This makes the FWM and SWM processes very efficient even with relatively weak cw laser beams in an atomic vapor cell. This specially designed experimental scheme that simultaneously generates different nonlinear wave mixing processes opens a new research frontier in manipulating higher-order nonlinear optical processes with induced atomic coherence and quantum interference.

6.2 Enhancement of SWM by Atomic Coherence The four-level Y-type atomic system discussed in Section 6.1 has the advantage of having two separate EIT windows, which are individually tunable by frequency detunings of the coupling/pumping laser beams [10]. However, since only one of the hyperfine energy levels in the ground state of the alkali atomic systems (such as rubidium atoms) can be used in such Y-type system, certain population is trapped in the other hyperfine level, which reduces the efficiencies for generating FWM and SWM processes. In this section, we consider another four-level atomic system with inverted-Y type [11],

6.2 Enhancement of SWM by Atomic Coherence

217

which makes use of both hyperfine energy levels of the alkali atoms, as shown in Fig. 6.6(b). This inverted-Y system has a three-level ladder-type and a three-level Λ-type sub-systems. If we still use the square-box spatial configuration for the coupling and probe laser beams, [as shown in Fig. 6.5(a)], only the ladder-type sub-system satisfies the two-photon Doppler-free condition for EIT in the atomic vapor cell [12], and the Λ-type sub-system has no EIT window since the coupling and probe beams counter-propagate through the medium [10]. SWM processes cannot only exist in this inverted-Y atomic system, but also be significant enhanced with the special arrangement of the laser beams. Efficient SWM processes are shown to coexist with the FWM ones. More importantly, one can optimize the SWM processes via an opened EIT window in the ladder-type sub-system and optical pumping from the additional hyperfine energy level (|3) in the Λ-type sub-system. Compared with FWM signals in the system, the SWM signals have been greatly enhanced. Similar to the case for the Y-type system presented in the Section 6.1, four stronger coupling/pump laser beams can be applied to interact with the transitions of |1 to |2 and |1 to |3, respectively. A weak probe beam is used to probe the transition |0 to |1. The CW laser beams are aligned spatially in the square-box pattern as shown in Fig. 6.5(a), with four beams (E2 , E2 , E3 , E3 ) propagating through the atomic medium in the same direction with small angles (about 0.3◦ ) between them. A weak probe beam (E1 ) propagates in the opposite direction to the other beams with a small angle. For the four-level inverted Y-type atomic system, as shown in Figs. 6.5(b)–(g), if three strong laser beams (with either E2 or E3 beam blocked) drive the two transitions (|1 to |2 and |1 to |3) and the weak laser beam E1 probes the transition (|0 to |1), these configurations satisfy the two-photon Doppler-free condition for the |0 − |1 − |2 ladder-type EIT sub-system (it is not the case for the |0 − |1 − |3Λ-type EIT sub-system) [12]. If two coupling fields, E3 (ω3 , k3 , and Rabi frequency G3 ) and E3 (ω3 , k3 , and Rabi frequency G3 ), drive the upper transition |1 to |3 and one strong dressing field, E2 (ω2 , k2 , and Rabi frequency G2 ), drives the transition |1 to |2, as shown in Fig. 6.5(b), there will be coexisting FWM and SWM processes both generating signal fields at the same frequency ω1 and both falling in the same EIT window created for the probe transition in the three-level ladder-type sub-system (|0− |1− |2). First, without the strong dressing field E2 , a simple FWM process (involving the probe beam E1 and the two coupling fields E3 and E3 ) will generate a (0) E

1 signal field EF at the frequency ω1 via the perturbation chain (F1) ρ00 −→

(1)

(E3 )∗

(2)

E

(3)

3 ρ10 . When the dressing field E2 is on, it will dress the ρ10 −→ ρ30 −→ energy level |1 to create the dressed states |+ and |− [see Fig. 6.5(c)],

(0)

E

 ∗ (1) (E )

(2)

E

(3)

1 3 3 ρ±0 −→ ρ30 −→ ρ±0 in the which have the dressed FWM chain ρ00 −→ dressed-state picture. This is very similar to the Y-type system discussed in Section 6.1. The constructive and destructive interferences between the “+” and “–” FWM channels result in the enhancement and suppression of

218

6 Coexistence of MWM Processes via EIT Windows

the FWM signal, respectively. Other than the dressed FWM processes, there are two possible SWM processes as shown in Figs. 6.5(f) – (g), in which two photons from E2 field and one photon each from E3 and E3 fields participate in the SWM processes to generate ES with two different Liouville pathways, which are given by ∗ (2) (E2 )

(0) E

(1) E

(0) E

 ∗ (1) (E )

 ∗ (3) (E )

(4) E

(5)

∗ (4) (E2 )

(5)

1 2 3 3 (S1)ρ00 −→ ρ10 −→ ρ20 −→ ρ10 −→ ρ30 −→ ρ10

(2) E

(3) E

1 3 2 3 (S2)ρ00 −→ ρ10 −→ ρ30 −→ ρ10 −→ ρ20 −→ ρ10

These FWM (kF 1 = k1 + k3 − k3 ) and SWM (kS1 = k1 + k2 − k2 + k3 − k3 ) processes can exist at the same time in this system and their generated signals can be phase-matched to travel in the same propagation direction. However, when two coupling fields, E2 and E2 (ω2 , k2 , and Rabi frequency G2 ), are used to drive the transition |1 to |2 and one strong dressing field E3 drives the transition |1 to |3 as shown in Figs. 6.1(d) and (e), there will (0) E

(1) E

1 2 be a different FWM process with perturbation chain [(F 2) ρ00 −→ ρ10 −→

(2)

(E2 )∗

(0) E

(3)

(1) (E3 )

∗

(2) E

(3) E

 ∗ (4) (E )

1 3 2 2 ρ20 −→ ρ10 ] and two SWM (ρ00 −→ ρ10 −→ ρ30 −→ ρ10 −→ ρ20 −→

(5)

(0) E

(1) E

 ∗ (2) (E )

∗ (3) (E3 )

(4) E

(5)

1 2 3 2 ρ10 and ρ00 −→ ρ10 −→ ρ20 −→ ρ10 −→ ρ30 −→ ρ10 ) processes.

Fig. 6.5. Three-dimensional beam geometry; (b) Four-level atomic system for generating co-existing FWM and SWM processes; (c) FWM dressed-state picture; (f) and (g) two SWM schemes; (d) four-level atomic system for dressed FWM process and (e) Its dressed-state picture.

To quantitatively understand such phenomenon of interplays between the coexisting FWM and SWM processes, perturbation chain expresses are needed that involve all the third-order and fifth-order nonlinear wave-mixing processes for arbitrary field strengths of E2 , E2 , E3 , and E3 . When both

6.2 Enhancement of SWM by Atomic Coherence

219

E2 and E2 are blocked, the simple FWM process via Liouville pathway (3) (F 1) gives the steady-state solution ρF 1 = −iGa eikF 1 ·r /(d21 d2 ), where kF 1 = k1 + k3 − k3 , Ga = G1 G3 (G3 )∗ , d1 = Γ10 + iΔ1 , d2 = Γ30 + i(Δ1 − Δ3 ) with Δi = Ωi − ωi . Γij is the transverse relaxation rate between states |i and |j. Next, when the dressing field E2 is turned on, there exist two interesting physical mechanisms. First, the above main FWM process (F 1) will be dressed by the dressing field E2 and a perturbation approach for such interaction can be described by following two pairs of coupled equations ∂ρ10 /∂t = −d1 ρ10 + iG1 eik1 ·r ρ00 + iG∗2 e−ik2 ·r ρ20 (1)

(1)

(0)

∂ρ20 /∂t = −d3 ρ20 + iG2 eik2 ·r ρ10

(1)

(6.3)

∂ρ10 /∂t = −d1 ρ10 + iG∗2 e−ik2 ·r ρ20 + iG3 eik3 ·r ρ30 (3)

(3)

(2)

∂ρ20 /∂t = −d3 ρ20 + iG2 eik2 ·r ρ10

(3)

(6.4)

where d3 = Γ20 + i(Δ1 + Δ2 ). In the steady state, Eqs. (6.3) and (6.4) can be solved together with the perturbation chain (F1) to give 

2iGa eikF 1 ·r d3

(3)

ρF 1 = −

2

d1 d2 (d1 d3 + |G2 | ) (5)

Second, one can easily calculate the fifth-order nonlinear susceptibility ρ10 for SWM from the Liouville pathways (S1) and (S2) for arbitrary G2 to be (5)

ρ10 =

2iGa |G2 | eikS1 ·r d31 d2 d3 2

with kS1 = k1 + k2 − k2 + k3 − k3 . Under the condition of |G2 | << Γ10 Γ20 ,  (3) the effective third-order nonlinear susceptibility ρF 1 can be expanded to be 2



2

1 − |G2 | (d1 d3 ) (3) (5) = 2ρF 1 + ρ10 d21 d2

2iGa eikF 1 ·r (3)

ρF 1 ≈ −

and the dressed FWM process converts to a coherent superposition of signals from FWM and SWM in the weak coupling field limit. Similarly, if the field E3 is blocked instead of the field E2 [see Fig. 6.5(d)], the dressed FWM process will have the perturbation chain 

(3)

(F2)ρF 2 = −

2iGb eikF 2 ·r d2 2

d1 d3 (d1 d2 + |G3 | )

with kF 2 = k1 + k2 − k2 , and the perturbation chain for dressed state to be (0)

E

(1)

E

 ∗ (2) (E )

(3)

1 2 2 ρ±0 −→ ρ20 −→ ρ±0 with the perturbed sate |1 [see Fig. 6.5(e)]. ρ00 −→

220

6 Coexistence of MWM Processes via EIT Windows

Under the condition of |G3 |2 << Γ10 Γ30 , the effective third-order nonlinear  (3) susceptibility ρF 2 can be expanded to be



(3)

ρF 2 ≈ −

2

1 − |G3 | (d1 d2 ) (3) (5) = 2ρF 2 + ρ10 d21 d3

2iGb eikF 2 ·r

where Gb = G1 G2 (G2 )∗ Again, this is the coherent superposition of the FWM and SWM contributions. Of course, when all four laser fields (E2 , E2 , E3 , E3 ) are turned on simultaneously, the situation will be much more complicated and cross terms will appear. For the experimental demonstration, two relevant atomic systems are shown in Figs. 6.5(b) and (d), respectively. Four energy levels from 85 Rb atoms are involved in experimental schemes. All involved laser beams are CW lights, as described in the last section. In the first experiment, let us consider only the effects related to the dressed FWM processes. As shown in Fig. 6.5(d), energy levels of |0 (5S1/2 , F = 3), |1 (5P3/2 ), and |2 (5D3/2 ) form a ladder-type three-level atomic system. The atoms are in a 5 cm long vapor cell with temperature control. With the coupling laser beam E2 (or E2 ) (connecting the transition from state |1 to state |2) propagating in the opposite direction of the weak probe field E1 (connecting the transition from |0 to |1), as shown in Fig. 6.5(a), two-photon Doppler-free EIT condition is satisfied [12] and a narrow EIT window appears for the probe beam as the probe frequency detuning is scanned, which is well understood. As Fig. 6.5(a) indicates, the laser beams E2 , E2 , and E3 (with E3 blocked) propagate through the atomic cell with small angles (about 0.3◦ ) between them and the probe laser beam E1 goes through the medium in the opposite direction also with a small angle with respect to the other beams. From the phase-matching condition (kF 1 = k1 + k3 − k3 ), the generated FWM signal EF (due to one probe photon from E1 and one photon each from E2 and E2 , respectively) travels in a slightly different direction as the original probe beam [at the lower right corner in Fig. 6.5(a)]. When a dressing beam E3 (connecting transition between 5S1/2 (F = 2) − 5P3/2 ) is added [see Fig. 6.5(d)], the original FWM signal is greatly modified due to the constructive or destructive interference between the two dressed FWM channels (via the perturbation (0) E

(1) E

 ∗ (2) (E )

(3)

1 2 2 ρ±0 −→ ρ20 −→ ρ±0 ), as shown in Figs. 6.6(a) and (b), which chain ρ00 −→ are measured at a frequency detuning of Δ2 = 400 MHz. The other experimental parameters are P1 = 3.6 mW, P2 = P2 = 18 mW, Δ2 = 400 MHz, and Δ3 = 362 MHz. Such dressed FWM effect has an optimum efficiency region as P3 reaches about 20 mW [see Fig. 6.6(b)] [4,8]. The FWM signal rises very fast as the dressing field E3 is turned on. It reaches the maximal value quickly before it slowly decreases as the power of E3 gets larger. The generated FWM signal (with frequency ω1 ) is inside the EIT window of the ladder-type sub-system, so as the coupling or probe frequency detuning

6.2 Enhancement of SWM by Atomic Coherence

221

changes, the FWM signal will follow the EIT window (via two-photon transition condition) to appear at different frequencies in the probe spectrum. As one can see, even with E3 blocked, the FWM signal has a dip in the middle (at Δ1 = −400 MHz), which is a result of three-photon destructive interference, as discussed in last section. The three-photon (with one probe photon plus two coupling photons) transition path interferes with the generated signal photon destructively, similar to the phenomenon observed in the fluorescence[15], but with coherent signal detection here.

Fig. 6.6. (a) Measured EIT-assisted FWM signal intensity for selected dressing field powers (P3 = 0 mW, 0.5 mW, 20 mW, 80 mW, 120 mW) versus probe detuning Δ1 ; and (b) evolution of the dressed FWM amplitude versus the dressing field power P3 (Δ1 = −400 MHz). Adopted from Ref. [10].

In the second experiment, let us consider a different laser beam configuration, as shown in Fig. 6.5(b). It is important to note that, unlike the Y-type system, the two three-level sub-systems (ladder-type: |0 − |1 − |2 and Λ-type: |0 − |1 − |3) are not symmetric due to the propagation directions for the coupling and probe beams (i.e., Doppler-free for the ladder-subsystem, but Doppler-broadened for the Λ-subsystem). We will mainly consider coexisting SWM and FWM processes in this case. Two coupling laser beams (E3 and E3 ) interact with the transition 5S1/2 (F = 2) − 5P3/2 (|3 to |2) and one dressing laser beam E2 drives the upper transition from 5P3/2 (1) to 5D3/2 (|2), as shown in Fig. 6.5(b). In this laser configuration, both FWM and SWM processes can co-exist and be generated in the same direction satisfying phase-matching conditions of kF 1 (kF 1 = k1 + k3 − k3 ) and kS1 (kS1 = k1 + k2 − k2 + k3 − k3 ), respectively. The dressed FWM and two SWM processes can be understood in simple pictures as given in Fig. 6.5(c), and Figs. 6.5(f)–(g), respectively. The FWM signal EF is generated by interactions between three fields E1 , E3 , and E3 [see Fig. 6.5(b)], which can be considered as scattering of E1 field over the small-angle static grating formed by the coupling field pair of E3 and E3 . The FWM signal is at the frequency of ω1 . Since the probe field E1 propagates in the opposite direction of the coupling fields E3 and E3 , such generated FWM signal is Doppler broadened, and is identified to be the broad peak at the bottom curve (a) in Fig. 6.7 or the broad shoulder in Fig. 6.7(b). The parameters used in this measurement are

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6 Coexistence of MWM Processes via EIT Windows

P1 = 3.6 mW,P2 = 31 mW,P3 = P3 = 30 mW,Δ2 = 0 mW, and Δ3 = 362 MHz. This FWM signal is determined by the phase-matching condition, the frequency detuning, and by blocking each of the laser beams E1 , E3 , and E3 , respectively. In this experimental configuration, efficient SWM process (the signal ES ) can also exist at the same time with one photon each from E1 , E3 , E3 , and two photons from the same dressing beam E2 , as shown in Figs. 6.5(f) and (g). The generated SWM signal is shown as the sharp peak at the middle of curve (b) in Fig. 6.7, which falls in the same EIT window of the ladder-type system. Again, this SWM signal can be clearly identified by systematically blocking each of participating laser beams, by considering the phase-matching condition, and by controlling the frequency detuning of participating laser beams. Due to EIT (only for the SWM signal) and optical pumping (fields E3 and E3 to pump the population from the ground state |3), the enhanced SWM signal intensity gets to be more than 10 times larger than the coexisting FWM signal intensity at certain frequencies. The SWM signal has a much narrower spectral width due to the EIT window. The dip in the middle of the SWM signal is due to the five-photon destructive interference (i.e., between five-photon process ω1 + ω2 − ω2 − ω3 + ω3 and the generated signal photon ω1 , or between five-photon process ω1 − ω3 + ω3 + ω2 − ω2 and ω1 ), which is clearly observed in this coherent signal detection. Also, one can see the interference between the coexisting weak broad FWM signal and the strong narrow SWM signal in Fig. 6.7 [curve (b)].

Fig. 6.7. (a) Measured optical pumping-assisted pure FWM (bottom broad peak) signal intensity; (b) measured coexisting EIT-assisted SWM (middle narrow peak) and FWM (broad shoulder) signal intensities. Adopted from Ref. [10].

Figure 6.8 presents the amplitude changes of the EIT-assisted SWM signal versus the dressing field power (for P3 = P3 = 120 mW) and the coupling field powers (for P2 = 31 mW), respectively. The SWM signal intensity increases monotonically with the dressing field power P2 , but it shows a maximal value for certain coupling field power. Such optimal efficiency was previously observed for FWM process in the four-level atomic system [5], which was attributed to the matching of the maximal atomic coherence in this system. So, this observed optimal SWM efficiency at P3 about 120 mW [see

6.3 Observation of Interference between FWM and SWM

223

Fig. 6.8(b)] can be considered to correspond to the slow group velocity region of the SWM signal or the maximal atomic coherence matching in this system [4,8]. The other experimental parameters are P1 = 3.6 mW, Δ1 = Δ2 = 0 mW, and Δ3 = 362 MHz. Figure 6.8(b) shows that the measurements agree quantitatively well with the theoretical calculation.

Fig. 6.8. (a) Measured EIT-assisted SWM amplitude versus the dressing-field power P2 ; (b) evolution of the EIT-assisted SWM amplitude versus the coupling field power P3 = P3 . The points are experimental measurements and the solid curve is the theoretical calculation. Adopted from Ref. [10].

6.3 Observation of Interference between FWM and SWM In this section, we will describe yet another multi-level atomic system, as shown in Fig. 6.9. Actually this is a five-level atomic system, involving two ground-state hyperfine levels (F = 1 and F = 2 for rubidium 5S1/2 ) and three upper energy levels [26]. Although the two sub-systems, i.e., the threelevel ladder-type sub-system (5S1/2 , F = 2−5P3/2 −5D3/2 ) and the two-level sub-system (5S1/2 , F = 1 − 5P1/2 ), seem to be not connected, they actually couple to each other via induced optical gratings. In Fig. 6.9(a), the threelevel ladder-type sub-system generates FWM signal in the EIT window due to the two-photon Doppler-free configuration, with laser beams arranged in the square-box pattern as shown in Fig. 6.9(c) (or discussed in the last two sections). The strong laser beam E3 optically pumps the population from 5S1/2 , F = 1 state into the F = 2 state, and therefore greatly enhances the FWM efficiency. In the laser beam arrangement as shown in Fig. 6.9(b), the two strong laser beams E3 and E3 form a small-angle static grating in the atomic medium, the scattering of the weak Ep field gives a FWM signal Ef (as shown in Fig. 6.9(d) with kf = k1 + k2 − k2 ). It is interesting to note that FWM process will not occur in the ladder-type sub-system in Fig. 6.9(b), since only one laser beam E2 couples to the upper transition and the phase-matching condition for FWM can not be satisfied. However, the scattering of the two fields (Ep and E2 ) together will generate SWM, as shown in Fig. 6.9(e), satisfying the phase-matching condition (ks = k1 +

224

6 Coexistence of MWM Processes via EIT Windows

k2 − k2 + k3 − k3 ). In this case the FWM and SWM signals can both be generated simultaneously, and they can be tuned relative to each other by the frequency and power of field E2 since it only involves in the SWM process [17]. Controlled interference between the generated FWM and SWM signals can be described in this system.

Fig. 6.9. (a) Five-level atomic system for EIT- and optical pumping-assisted FWM process; (b) Five-level atomic system for generating co-existing FWM and SWM processes; (c) three-dimensional beam geometry; (d) and (e) typical FWM and SWM processes, respectively.

The two relevant experimental systems are shown in Figs. 6.9 (a) and (b). Five energy levels from 87 Rb atoms (in vapor cell) are involved in the experimental schemes. In the first experiment [see Fig. 6.9 (a)], energy levels of |0 (5s1/2 , F = 2), |1 (5p3/2 ), and |2 (5d3/2 ) form a cascade three-level atomic system. With coupling beam E2 (or E2 ) (connecting transition |1 to |2) propagating in the opposite direction of the weak probe field Ep (connecting transition |0 to |1), as shown in Fig. 6.9 (c), two-photon Doppler-free EIT condition is satisfied and a narrow EIT window is obtained for the probe beam, which is well understood. As Fig. 6.9 (c) indicates, E2 , E2 , and E3 (E3 blocked) propagate through the atomic cell with a small angle (about 0.3◦ ) and Ep goes in the opposite direction also with a small angle. From phase-matching condition kf = k1 + k2 − k2 , the generated FWM signal Ef (due to one probe photon from Ep and two coupling photons from E2 and E2 , respectively) falls in a slightly different direction [see lower right corner in Fig. 6.9 (c)]. When an additional pumping laser beam E3 (connecting the transition 5s1/2 (F = 1) − 5p1/2 ) is added, the FWM signal is greatly enhanced due to pumped population into the F = 2 ground state for the ladder-type system from the 5s1/2 (F = 1) state, as shown in Fig. 6.10(a), which is measured at a frequency detuning of Δ2 = −450 MHz (where Δi = Ω i − ωi , Ω i is the

6.3 Observation of Interference between FWM and SWM

225

atomic transition frequency and ωi is the corresponding laser frequency.). The experimental parameters are P1 = 3.6 mW, P2 = P2 = 20 mW, P3 = 0 mW, Δ2 = −450 MHz, and λ3 = 794.97 nm. As shown in Fig. 6.10(a), such enhanced FWM effect due to optical pumping rises very quickly and then saturates quickly as P3 reaches 15∼20 mW. The generated FWM signal is inside the EIT window of the three-level ladder system, so as the coupling or probe frequency detuning changes, the generated FWM signal will follow the EIT window (satisfying the two-photon condition) to appear at different probe frequencies. Figure 6.10(b) shows FWM signals as a function of the coupling beam frequency detuning for three different probe detuning values (with E3 blocked). The experimental parameters are P1 = 3.6 mW, P2 = P2 = 130 mW, P3 = P3 = 0 mW, and Δ1 = −250, 0, 250 MHz. For Δ1 = 0 MHz, the FWM signal spectrum is symmetric and has a dip in the middle (Δ2 = 0 MHz), which is a result of three-photon destructive interference (i.e., three-photon process interferes with the generated signal photon destructively, as discussed in the last two sections.). As the probe frequency detuning changes, the double-peak FWM signal becomes asymmetric, but still within the EIT window.

Fig. 6.10. (a) Change of the FWM amplitude versus the pumping field power P3 ; (b) measured FWM signal intensity versus the pump frequency detuning Δ2 for selected probe field detunings. Adopted from Ref. [26].

In the second experiment, we consider the laser configuration as shown in Fig. 6.9(b), in which two laser beams (E3 and E3 ) in the pumping transition (from 5s1/2 (F = 1) to 5p1/2 ) and one laser beam E2 in the coupling transition (from 5p3/2 to 5d3/2 ) are used. The two pumping beams (E3 and E3 ) have equal power of 70 mW each and the coupling beam power is 33 mW. The probe beam is still kept at 3.6 mW. In this configuration, both FWM and SWM processes can co-exist as discussed above and FWM and SWM signals are generated to propagate in the same direction, as shown in Fig. 6.9(c), satisfying the phase-matching conditions of kf = k1 + k3 − k3 and ks = k1 + k2 − k2 + k3 − k3 , respectively. These FWM and SWM processes can be understood in simple pictures as given in Fig. 6.9(d) and (e). The FWM signal Ef is generated by interaction of three fields Ep , E3 , and E3 (Fig. 6.9(d)), which can be considered as scattering of Ep field over the smallangle static grating formed by the conjugate field pair of E3 and E3 . Since the

226

6 Coexistence of MWM Processes via EIT Windows

probe field Ep propagates in the opposite direction of the pumping fields E3 and E3 , such generated FWM signal is Doppler broadened, and is identified to be the broad (right side) peak in the bottom curve of Fig. 6.11(a). This FWM signal is determined by the phase-matching condition, the frequency detuning, and by blocking each of the beams Ep , E3 , and E3 , respectively. In this experimental setup, SWM process (signal Es ) can also exist at the same time, with one photon each from Ep , E3 , E3 and two photons from the same coupling beam E2 (satisfying the phase-matching condition), as shown in Fig. 6.9(e). One can consider this SWM process as the scattering of the FWM signal (generated in the ladder-type sub-system) over the smallangle static grating formed by the strong E3 and E3 pumping fields, which involves five-photon process (χ(5) ). Since the FWM signal (from the threelevel ladder-type sub-system) is within the EIT window of the ladder system, this generated SWM signal also falls in the same EIT window, as shown in Fig. 6.11(a) (the narrow peak in the left of the bottom curve). Again, this SWM signal can be identified by blocking each of participating laser beams, the phase-matching condition, and the frequency detuning of the participating laser beams. First, the FWM and SWM signals are intentionally tuned apart by choosing different frequency detunings of the coupling and pumping beams (bottom curve in Fig. 6.11(a)). Then, as the coupling frequency detuning Δ2 is tuned with pump frequency detuning Δ3 fixed, the two peaks move closer towards each other. The coupling beam E2 only participates in the SWM process, but not the FWM process, so Δ2 only tunes the SWM peak. As Δ2 is tuned to satisfy the condition of Δ1 = −Δ2 = Δ3 , the generated FWM and SWM signals move together to overlap with each other, and start to interfere, as shown in the top curves of Fig. 6.11(a). The experimental parameters are P1 = 3.6 mW, P2 = 33 mW, P2 = 0 mW, P3 = P3 = 70 mW, Γ10 = Γ30 = 2π × 3 MHz, Γ20 /2π = 0.5 MHz, ζ = 0.98, Δ3 = 285 MHz, and Δ2 = −385, −285, −135, 0, 195, 315 MHz, respectively, for the different curves in Fig. 6.11 (a).

Fig. 6.11. (a) Measured coexisting SWM (left peak) and FWM (right peak) signal intensities for different coupling frequency detunings. (b) theoretical plots of coexisting SWM and FWM versus Δ1 for the parameters as shown in (a). Adopted from Ref. [26].

6.4 Controlling FWM and SWM Processes

227

To quantitatively understand such phenomenon of interference between FWM and SWM processes, we need to use perturbation chain expresses involving all the third-order and fifth-order nonlinear wave-mixing processes. Although the calculations are very tedious, the final result can be written in (3) (5) (3) (5) a simple form of ρ10 ≈ ρ10 + ρ10 . ρ10 and ρ10 are the nonlinear densitymatrix elements. The total detected intensity at the location of Ef and Es [as shown in Fig. 6.9 (c)] is given by I(Δ1 ) ∝ |χ(3) |2 + η 2 |χ(5) |2 + 2η|χ(3) ||χ(5) | cos(θF − θS )

(6.5)

where η = −μ230 E32 /2 and ζ = k3 /k1 . The third- and fifth-order susceptibilities are written as (3)

χ(3) = |χ(3) | eiθF =

N μ10 ρ10 ε0 Ep

χ(5) = |χ(5) | eiθS =

N μ10 ρ10 ε0 Ep

(5)

After the Taylor expansions for θF and θS (the phase angles of third- and fifth-order susceptibilities), we get:   2   ! π Γ10 − Δ22 Δ1 ζ + Δ3 − θF − θS = − − − tan−1 2 Γ30 + Γ10 ζ 2Γ10 Δ2 " (Γ210 + 2Γ10 Γ20 + Δ22 )(Δ1 + Δ2 ) (6.6) Γ20 (Γ210 + Δ22 ) where μi0 and Γi0 are the dipole moments and the transverse relaxation rates between states |0 and |i, respectively. Using parameters similar to the ones in our experiment, we can plot the detected total intensity as a function of the probe frequency detuning with different coupling frequency detunings, as shown in Fig. 6.11 (b). The Doppler effect is considered by integrating with an atomic Boltzman velocity distribution in the calculation. As can be seen that the theoretically calculated results match quite well with the experimentally measured curves presented in Fig. 6.11 (a), showing clear interference between the FWM and SWM signals as the sharp SWM signal merges into the Doppler-broadened FWM peak [top two curves in Fig. 6.11 (b)]. Such theoretical calculation confirms the co-existing and interference between the FWM and SWM processes.

6.4 Controlling FWM and SWM Processes In the close-cycled (or folded) four-level atomic system, as shown in Fig. 6.12 ω1 ω2 −ω ω3 −ω |1 −→ |2 −→3 |3 −→ |2 −→2 (a), the SWM process ( via the path |0 −→

228

6 Coexistence of MWM Processes via EIT Windows −ω

ω

ω

−ω

1 2 |1 −→1 |0) will have to go through the FWM path (|0 −→ |1 −→ |2 −→2 −ω |1 −→1 |0) [23, 25]. Therefore, when an efficient FWM process with one weak probe beam E1 (ω1 , k1 , and Rabi frequency G1 , connecting transition |0 and |1) and two coupling laser beams (E2 with ω2 , k2 , and Rabi frequency G2 and E2 with ω2 , k2 , and Rabi frequency G2 , connecting upper transition |1 and |2) exists, it will dominate the wave-mixing processes since the SWM process will be several orders of magnitude smaller than the FWM signal in this case even when the strong fields E3 (ω3 , k3 , and Rabi frequency G3 ) and E3 (ω3 , k3 , and Rabi frequency G3 ) (connecting transition |2 to |3) are present. However, one can turn off the dominant FWM process by blocking either E2 or E2 beam, in which case the system will promote and only generate SWM (0) ω1 (1) ω2 (2) −ω (3) ω3 (4) −ω (5) ρ10 −→ ρ20 −→3 ρ30 −→ ρ20 −→2 ρ10 ), as demonstrated processes (ρ00 −→ in Ref. [13].

Fig. 6.12. (a) Folded four-level atomic system for generating FWM and SWM processes (E2 << E2 ); (b) atomic system with an additional pumping beam E4 (the dashed arrow); (c) square box-pattern beam geometry. The dash-dotted arrows are the generated MWM signals.

We show that by using a strong E2 and a weaker E2 (E2 << E2 ) for the upper transition |1 and |2, together with E3 and E3 (for transition |2 and |3), both FWM and SWM processes can be generated simultaneously and, with appropriate conditions, made to be in similar magnitudes. To experimentally demonstrate different wave-mixing processes, such as pure FWM, pure SWM, co-existing FWM and SWM, as well as dressed FWM, different coupling laser beams will be blocked during the experiments. Also, when an additional pumping laser beam E4 (ω4 , connecting transition |4 to |3) is turned on, as shown in Fig. 6.12 (b), the dressed-FWM and SWM signals can be greatly enhanced, indicating optical pumping, as well as dressedSWM effect. One ladder-type EIT subsystem will form between transitions |0 → |1 → |2 and an EIT window appears [bottom curve of Fig. 6.13 (a), labeled as P] [22], which depends on the frequency detuning Δ2 (Δi = Ωi − ωi with atomic resonant frequency Ωi for the corresponding transition). The generated multi-wave mixing (MWM) signals at frequency ω1 all fall into this EIT window and can pass through the medium with reduced absorption. To spatially separate these generated MWM signals from the probe beam E1 and satisfy the phase-matching conditions and two-photon Doppler-free

6.4 Controlling FWM and SWM Processes

229

configurations [3, 22], the laser beams are aligned spatially in the square-box pattern shown in Fig. 6.12 (c), with five laser beams (E2 , E2 , E3 , E3 , E4 ) propagating through the atomic medium in the same direction with small angles (about 0.3◦ ) between them. The probe beam E1 propagates in the opposite direction. The generated MWM signals are then all propagating in one direction labeled as EM in Fig. 6.12 (c). The experimental demonstration of co-existing MWM processes through one EIT window was carried out in atomic vapor of 85 Rb. The energy levels of 5s1/2 (F = 3), 5p3/2 , 5d3/2 , and 5p1/2 form the folded four-level system as shown in Fig. 6.12 (a). The atomic vapor cell was heated to temperature of 60◦ C. The six laser beams were arranged as indicated in Fig. 6.12 (c). The probe laser beam E1 (with wavelength near 780 nm from an external cavity diode laser (ECDL), connecting transition 5s1/2 −5p3/2 ) is horizontally polarized and has a power of about 7 mW. The laser beams E2 and E2 (wavelength 776.16 nm connecting transition 5p3/2 − 5d3/2 ) are split from another ECDL, each with a vertical polarization. The laser beams E3 and E3 (wavelength 762.10 nm connecting transition 5d3/2 − 5p1/2 ) are split from a CW Ti: Sapphire laser, each with a vertical polarization. Great cares were taken in aligning the six laser beams with spatial overlaps and wave vector phase-matching conditions with small angles (about 0.3◦ ) between them, as indicated in Fig. 6.12 (c). Under certain conditions, one FWM (EF along kF = k1 +k2 −k2 ), two dressed FWM (ED1 along kD1 = k1 +k2 −k2 +k3 −k3 and ED2 along kD2 = k1 + k2 − k2 + k3 − k3 ) and two SWM signals (ES1 along kS1 = k1 + k2 − k3 + k3 − k2 and ES2 along kS2 = k1 + k2 − k3 + k3 − k2 ) with horizontal polarization are all in the direction of EM with a small angle 0.3◦ [at the right upper corner of Fig. 6.12 (c)] and are detected by an avalanche photodiode detector (APD). The transmitted probe beam is detected by a silicon photodiode as a reference. The pumping laser beam E4 (wavelength 794.97 nm connecting transition 5s1/2 (F = 2) − 5p1/2 ) is from yet another ECDL, with a vertical polarization. Other than optically pumping the population from 5s1/2 (F = 2) to 5s1/2 (F = 3), this pumping beam also acts as a dressing beam to the system. Without E3 and E3 beams, the strong coupling beam E2 (a power of 30 mW) and weak coupling beam E2 (a power of 3 mW) (i.e., E2 >> E2 ) together with the weak probe beam E1 generate a pure FWM signal EF (0)

E

(1)

(0)

E

(1)

E

 ∗ (2) (E )

(3)

1 2 2 (ρ00 −→ ρ10 −→ ρ20 −→ ρ10 with kF ) at frequency ω1 as shown in Fig. 6.13 (a) (top curve), which falls in the EIT window P (lower curve) and is very efficient. The parameters are The experimental parameters are G1 = 2π × 28 MHz, G2 = 2π × 49 MHz, G2 = 2π × 17 MHz, G3 = 2π × 59 MHz, G3 = 2π × 46 MHz, G4 = 0 MHz, and Δ2 = Δ3 = 62 MHz. Next, E3 is turned on (E3 still blocked), which will perturb the FWM process and create dressed states for level |2 [24]. The resulting dressed FWM signal

E

(2)

(E  )∗

(3)

1 2 2 ρ10 −→ ρ(G3 ±)0 −→ ρ10 with kD1 ) is generated (Fig. 6.13 (b) (ρ00 −→ labeled as D1) in the same direction as EF and in the same EIT window.

230

6 Coexistence of MWM Processes via EIT Windows

Similarly, when E3 is turned on (E3 blocked), then a similar dressed-FWM (0)

E

(1)

E

(E  )∗

(2)

(3)

1 2 2 ρ10 −→ ρ(G ±)0 −→ ρ10 with kD2 ) is generated [Fig. 6.13 signal (ρ00 −→ 3 (c) labeled as D2]. From phase-matching conditions, kD1 = kD2 = kF . Next, we consider the case with both E3 and E3 on, but with either E2 or E2 blocked. In these cases, the original FWM process is cut off. When E2 (which is weaker than E2 ) is blocked, this system generates a strong SWM

(0) E

(1) E

 ∗ (2) (E )

(3) E

∗ (4) (E2 )

(5)

1 2 3 3 ρ10 −→ ρ20 −→ ρ30 −→ ρ20 −→ ρ10 with kS1 ) as shown signal (ρ00 −→ in Fig. 6.13 (d) (labeled as S1). However, when E2 is blocked instead of E2 ,

(0) E

 (1) E

 ∗ (2) (E )

(3) E

 ∗ (4) (E )

(5)

1 3 2 3 2 ρ10 −→ ρ20 −→ ρ30 −→ ρ20 −→ ρ10 the produced SWM signal (ρ00 −→  with kS2 ) gets very small since E2 is much weaker than E2 (Fig. 6.13 (e) labeled as S2). Now, with all five laser beams (E1 , E2 , E2 , E3 , E3 ) on, especially with E2 << E2 , both FWM and SWM processes can exist and propagate in the same direction, which can be shown by the fact that the total generated MWM signal [as shown in Fig. 6.13 (f), labeled as M] is larger than the pure FWM signal F [as shown in Fig. 6.13 (a)]. These generated MWM signals in M all fall into the same EIT window. This system with all five laser beams on as shown in Fig. 1 (a) can be considered as dressedFWM for the three-level ladder system |0 → |1 → |2 by the two coupling beams E3 and E3 , which should only reduce the FWM signal at on-resonance condition [24]. So, the additional signal strength in M (compared to the FWM signal F) must be due to the additional SWM signals [proportional to S1 and S2, as shown in Figs. 6.13 (d) and (e)]. More strikingly, one can easily see that the ratio of SWM/FWM strengths can be adjusted by controlling the strength of the E2 beam. When E2 has the similar intensity as E2 , only a very small percentage of SWM signal exists since the efficient FWM process dominates. As the intensity of E2 decreases, the large FWM signal gets suppressed, as the percentage of SWM increases, until the FWM process is completely turned off when E2 → 0.

Fig. 6.13. (a) Measured pure FWM signal (upper curve F) and probe transmission (bottom curve P); (b) and (c) two dressed-FWM signals; (d) and (e) two SWM signals; (f) total MWM signals (the upper curve M) and probe transmission (the bottom curve P).

6.4 Controlling FWM and SWM Processes

231

In addition, when all six laser fields (E1 , E2 , E2 , E3 , E3 , E4 ) are all (0) E

1 turned on simultaneously, this system also generates another SWM (ρ00 −→

(1)

E

 ∗ (2) (E )

(3)

E

 ∗ (4) (E )

(5)

2 3 3 2 ρ10 −→ ρ20 −→ ρ30 −→ ρ20 −→ ρ10 ) and possible eight-wave mixing

(0)

E

(1)

E

(2)

(E3 )∗

(3) (E4 )

∗

(4)

E

(5)

E

 ∗ (6) (E )

(7)

1 2 4 3 2 (ρ00 −→ ρ10 −→ ρ20 −→ ρ30 −→ ρ40 −→ ρ30 −→ ρ20 −→ ρ10 ) signals falling into this EIT window, which propagate in a different direction due to phase matching conditions kS3 = k1 + k2 − k3 + k3 − k2 and kE = k1 + k2 − k3 − k4 + k4 + k3 − k2 along the direction with a small angle of 0.6◦ , respectively, as demonstrated in solid and liquid system [23 – 26]. In the following, we show how the dressed FWM and SWM signals can be enhanced by employing an additional pumping field E4 connecting energy levels |3 to |4 (another hyperfine ground state of the atom, 5s1/2 (F = 2)), as shown in Fig. 6.12 (b). In this case, the E3 is blocked. Without E4 , the dressed FWM signals are shown in Figs. 6.14 (a) and (b) with two different frequency detuning values for Δ3 . The experimental parameters are G1 = 2π × 22 MHz, G2 = 2π × 35 MHz, G2 = 2π × 39 MHz, G3 = 2π × 150 MHz, Δ2 = 0, (a) G3 = 0, G4 = 0, Δ3 = 0; (b) G3 = 2π × 150 MHz, G4 = 0, Δ3 = −20 MHz. Specifically, the dip at the line center (Δ1 = Δ2 = 0) of the FWM spectra (Fig. 6.14) is due to three-photon (one probe E1 photon plus one each from E2 and E2 photons) destructive interference with the generated signal EF photon [3, 10, 25, 26]. It is seen from Fig. 6.14 (b) that, as the frequency detuning Δ3 changes, the dressing field E3 starts to move into the right peak of dressed states at Δ3 = −Δ1 = Δ2 = −20 MHz of the FWM signal (D1), and suppress it. When E4 is turned on, the dressed-FWM signal gets much larger, as shown in Figs. 6.14 (c)–(e), which are for these different Δ3 values. The experimental parameters are G1 = 2π × 22 MHz, G2 = 2π × 35 MHz, G2 = 2π × 39 MHz, G3 = 2π × 150 MHz, Δ2 = 0, (c) G3 = 0, G4 = 2π × 79 MHz, Δ3 = 0; (d) G3 = 2π × 150 MHz, G4 = 2π × 79 MHz, Δ3 = −20 MHz; (e) G3 = 2π × 150 MHz, G4 = 2π × 79 MHz, Δ3 = 0. The large enhancement can have contributions from optical pumping of population from the ground-state |4, and from effect due to perturbed original dressed-FWM system (considered as doubly-dressed FWM [24]). It is seen from Fig. 6.14 (d) that, as the frequency detuning Δ3 changes, the dressing field E3 starts to emerge into the right peak of the dressed FWM signal (D1), and suppress it. Meanwhile, it can emerge into the center of the FWM signal (D1), and suppress the signal at this frequency, as shown in the curve [Δ3 = Δ1 = Δ2 = 0 of Fig. 6.14 (e)]. So, the generated FWM signal can be locally and selectively suppressed via controlling frequency detuning of the coupling beam E3 . Also, if only E2 is blocked, the original folded four-level system generates SWM only, as shown in Fig. 6.15 (a). The double-peak structure is due to the destructive interference between five-photon (one probe E1 photon, two E2 photons plus one each from the E3 and E3 photons) and the generated signal ES1 photon. When the additional pumping field E4 is turned on, this SWM signal is also greatly enhanced, as shown in Fig. 6.15 (b). These two

232

6 Coexistence of MWM Processes via EIT Windows

Fig. 6.14. Measured FWM signals versus Δ1 for different G3 and Δ3 values without and with E4 beam. E3 beam is blocked.

SWM signals (with and without E4 ) are plotted in Fig. 6.15 (c) as a function of the coupling beam Rabi frequency G3 . The experimental parameters are G1 = 2π × 28 MHz, G2 = 2π × 49 MHz, Δ2 = Δ3 = 0, (a) G4 = 0 and G3 = G3 = 2π × 40 MHz; (b) G4 = 2π × 78 MHz and G3 = G3 = 2π × 40 MHz; (c) SWM signals versus G3 for G4 = 2π × 78 MHz (the upper curve) and G4 = 0 (the lower curve). As one can see that with E4 beam, the SWM signal strength changes significantly, showing a clear maximal value at G3 ∼ 2π × 76 MHz, which indicates a matching between various coupling powers to achieve optimal atomic coherence in the system, as observed in FWM and other nonlinear systems [8]. Such dressed-SWM system is very interesting and deserves further studies both theoretically and experimentally.

Fig. 6.15. Measured SWM signals versus the probe detuning Δ1 for different G3 = G3 values, without and with E4 beam. E2 is blocked.

In summary, we have experimentally demonstrated selective generates of FWM and SWM in a close-cycled four-level atomic system. By employing a weak coupling beam (E2 ), the ratio of SWM to FWM signal strengths has been adjusted, and can be made to be any desired values for applications. With an additional pumping/dressing laser beam, the dressed-FWM and SWM signals can all be greatly enhanced. Generation and understanding

References

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higher-order nonlinearities in multi-level atomic systems can find important applications in studying interesting effects such as stable 2-D soliton formation and liquid-like surface tension [19].

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7 Interactions of MWM Processes

As described in the previous chapter, different multi-wave mixing (MWM) processes can be generated simultaneously in the same multi-level atomic systems. Interactions occur between these MWM processes since they share some of the same coupling laser beams. In this chapter, we concentrate on the competition and interactions between different wave-mixing processes, and show how such intermixing interactions can be controlled in frequency, spatial, and temporal domains. We will start by describing the competition between two four-wave mixing (FWM) processes in a four-level Y-type atomic system, which opens two independent electromagnetically induced transparency (EIT) windows. Interplays between these FWM channels by the frequency detuning of the coupling laser beams will be discussed in detail. As shown in the previous chapter, both FWM and six-wave mixing (SWM) can coexist in four-level atomic systems. By controlling the relative phase delay between these two nonlinear optical channels, efficient energy exchange can occur between the generated FWM and SWM signals during their propagations through the nonlinear medium. Using such phase delay, both spatial and temporal interferences between the FWM and SWM signals can be generated with fast time beating signals. These interactions are very important in fundamental understanding of the interplays between such coexisting nonlinear optical processes, and potential applications of such cubic-quintic nonlinearities in soliton formation, high-intensity pulse propagation, optical communication, and even quantum information processing.

7.1 Competition between Two FWM Channels As we have shown in the previous chapter, efficient FWM processes, and coexisting FWM and SWM processes can be generated in multi-level atomic systems due to induced atomic coherence [1 – 7]. Recently, interference between two FWM processes in a two-level atomic system has been studied , which can generate biphotons and entangled photon pairs. In Section 6.1 we have shown that the four-level Y-type atomic system can open two symmetric EIT windows due to the two ladder-type three-level sub-systems, as shown in Fig. 7.1(a). By suppressing the FWM processes with atomic coherence in

236

7 Interactions of MWM Processes

this open-cycled four-level system, SWM can be generated and made to be comparable with the FWM processes. The generated FWM and the SWM signals have the same frequency and propagate in the same direction, and they fall into two separate EIT windows. In those experiments (Section 6.1), one upper transition has two coupling laser beams and another upper transition has only one coupling beam to generate the desired coexisting FWM and SWM processes.

Fig. 7.1. (a) Four-level Y-type atomic system with four coupling laser beams and one probe beam. The dash-dotted lines are the two generated FWMs (ωf 1,2 ) signals; (b) square box-pattern beam geometry used in the experiment; (c) dressed-state picture for the Y-type atomic system.

In this section, a different situation is considered with four coupling laser beams (two in each of the upper transitions) applied to the system at the same time, as shown in Fig. 7.1(a). In this case, two highly efficient and competing FWM processes will be simultaneously generated and the dual-EIT windows from the two ladder-type sub-systems (satisfying the two-photon Doppler-free configurations) are used to transmit the two generated FWM signal beams, respectively [9]. The lower transition (|0 to |1) together with one of the upper-branches (either |1 to |2 or |1 to |3) form a simple ladder-type three-level atomic system. If only one coupling beam is used in this upper transition, the three-level ladder sub-system has a simple EIT peak [10], as shown in Sections 1.4 and 6.1. If two coupling beams interact with the two upper branches (with one for each upper transition), respectively, double EIT peaks will appear in the probe transmission spectrum [11]. However, if two coupling beams (with the same frequency) are applied to one upper transition (for example, |1 to |2), a FWM signal (ωf 1 ) will be generated at the same frequency as the probe beam (ω1 )[1, 2], which falls into the EIT window created in this ladder-type system (|0 − |1 − |2). Similarly, a FWM signal (ωf 2 ) can be generated if only the other upper transition (|1 to |3) is considered with two coupling beams. The generated FWM signal ωf 2 falls into a different EIT window (due to the |0 − |1 − |3 ladder subsystem). Now, when both upper-branches are used with all four coupling laser beams on, as shown in Fig. 7.1(a), the two generated FWM signals (ωf 1 and ωf 2 ) via the two different ladder sub-systems (or different upper-

7.1 Competition between Two FWM Channels

237

branches) will compete with each other [5]. These two FWM signals can be either distinguishable when the frequency detuning of the coupling beams are different (with two separate EIT windows) or “no-distinguishable” when the two EIT windows are tuned to overlap with each other. In the following, we describe the interplays and competition between these two efficient FWM processes under different frequency detunings and coupling laser intensities. The laser beams are aligned spatially in the square-box pattern, as shown in Fig. 7.1(b), which is the same as used in Section 6.1. The four coupling laser beams (E2 , E2 , E3 , E3 ) propagate through the atomic medium in the same direction with small angles (about 0.3◦ ) between them. Two strong coupling fields, E2 (ω2 , k2 , and Rabi frequency G2 ) and E2 (ω2 , k2 , and Rabi frequency G2 ), drive the upper transition |1 to |2 and other two strong laser fields, E3 (ω3 , k3 , and Rabi frequency G3 ) and E3 (ω3 , k3 , and Rabi frequency G3 ), drive the transition |1 to |3. A weak laser field, E1 (ω1 , k1 , and Rabi frequency G1 ), probes the lower transition (|0 to |1). With the phase-matching conditions kf 1 = k1 + k2 − k2 (labeled as FWM I for the sub-system |0 − |1 − |2) and kf 2 = k1 + k3 − k3 (labeled as FWM II for the sub-system |0 − |1−|3), the two generated FWM signals propagate in the exactly same direction [as shown in the right lower corner in Fig. 7.1(b)]. This system also generates SWM signals [12], which propagate in a different direction due to the phase matching (ks = k1 + k2 − k2 + k3 − k3 ) when all four coupling beams are on. However, if one of the coupling beams (either E2 or E3 ) is blocked, one of the FWM processes will be turned off and a SWM signal will be generated in its place in the same direction as the FWM signal (determined by the different phase-matching condition), as is the case for Section 6.1. We can use the dressed-state picture to describe this system. Let us consider the FWM process from ladder sub-system |0 − |1 − |2 to be perturbed or dressed by the coupling fields E3 and E3 (i.e., the middle level |1 is dressed to be |+ and |−, as shown in Fig. 7.1(c)). So, the perturbed (0) ω1 (1) ω2 (2) −ω (3) ρ10 −→ ρ20 −→2 ρ±0 and FWM processes can be described by ρ00 −→ (0)

ω

(1)

ω

(2) −ω

(3)

1 2 ρ00 −→ ρ±0 −→ ρ20 −→2 ρ10 . The third-order nonlinear susceptibility can be calculated via appropriate perturbation chains (as described in Section 1.3), which give the leading contributions to the nonlinear wave-mixing processes. When both E3 and E3 are blocked, the simple FWM via Liouville (0) ω1 (1) ω2 (2) −ω (3) pathway (I: ρ00 −→ ρ10 −→ ρ20 −→2 ρ10 ) gives

(3)

ρ10 = −

iGa eikI ·r d21 d2

where Ga = G1 G2 (G2 )∗ , d1 = Γ10 + iΔ1 , and d2 = Γ20 + i(Δ1 + Δ2 ) with Δi = Ω i − ωi . Γij is the transverse relaxation rate between states |i and |j. As the dressing fields E3 and E3 are turned on, the FWM process (I) is perturbed (dressed) by both fields. Using the perturbation technique and

238

7 Interactions of MWM Processes

appropriate coupling equations (as in Section 1.3), we obtain (3)

ρI = −

2iGa eikI ·r d3 [d1 d2 (d1 d3 + |G3 |2 )]

( ( ( (3) ( = (ρI ( eiθI (0) ω

(1) ω

(2) −ω

1 3 Similarly, we can consider the FWM process (II: ρ00 −→ ρ10 −→ ρ30 −→3 (3) ρ10 ) from the right ladder sub-system (|0 − |1 − |3) as perturbed by the coupling fields E2 and E2 , so the perturbed FWM can be described by the (0) ω1 (1) ω3 (2) −ω (3) (0) ω1 (1) ω3 ρ±0 −→ ρ30 −→3 ρ10 and ρ00 −→ ρ10 −→ perturbation chains ρ00 −→

(2) −ω

(3)

ρ30 −→3 ρ±0 . Using the same procedure as above, we can obtain the modified third-order nonlinear susceptibility as (3)

ρII = −

2iGb eikF 2 ·r d2 2

[d1 d3 (d1 d2 + |G2 | )]

( ( ( (3) ( = (ρII ( eiθII

where Gb = G1 G3 (G3 )∗ and d3 = Γ30 + i(Δ1 + Δ3 ). Since both of these mutual-dressing processes exist at the same time in the real experiment, and the two generated FWM signals co-propagate in the same direction, the total detected FWM signal is proportional to the (3) (3) mod-square of ρ(3) (Δ), where ρ(3) (Δ) = ρI + ρII , with Δ = Δ3 − Δ2 . Detail calculations of ρ(3) (Δ) show several interesting physical effects demonstrating the interactions between these two FWM processes. The first one is the mutual dressings of the two ladder sub-systems. Such dressing effects perturb both FWM processes and suppress or enhance the total FWM signal amplitude, especially when these two FWM signals are tuned together in frequency through adjusting frequency detunings. Second, when the two generated FWM signals overlap in frequency, constructive or destructive (3) (3) interference can result due to the sign change either in ρI or ρII under different frequency detuning conditions. However, in the current experimental system the competition between the two coexisting FWM channels is dominated by the contributions from the mutual dressing effects, which can be an order of magnitude lager than this interference effect, as described in the two-level case [8]. (3) (3) Close inspection of ρ(3) (Δ) = ρI + ρII shows that when the frequency difference Δ (between Δ2 and Δ3 ) is reduced, both the modified FWM signals start to be suppressed via dressed states. The intensities of the co-existing FWM (I) (inside the Δ1 + Δ2 = 0 EIT window) and FWM (II) (inside the Δ1 + Δ3 = 0 EIT window) signals can maximally suppress each other at Δ1 + Δ3 = 0 (for the dressing fields G3 and G3 ) and Δ1 + Δ2 = 0 (for the dressing fields G2 and G2 ), respectively [8, 11]. There exists a maximum suppression for both of the dressed FWM signals at the exact multiple-EIT condition Δ1 = −Δ2 = −Δ3 (or Δ = 0). Also, close inspection of ρ(3) (Δ) reveals that interference of both types (constructive and destructive) between the two co-existing third-order nonlinear susceptibilities can be realized with

7.1 Competition between Two FWM Channels

239

different Δ. As shown in the Fig. 7.2(a) Specifically, by varying the detuning difference Δ from 0 to very large positive or negative values, the phase difference (Δθ = θI − θII ) between the two modified FWM processes alters from in-phase to out-phase, near in-phase and back to out-phase again, and therefore the interference also switches back and forth between constructive and destructive values [see Fig. 7.2(b)]. However, in the current system the competition between the two co-existing FWM channels is dominated by the contributions of the mutual dressing effects, which can be an order of magnitude lager than the interference effect.

(3)

(3)

(3)

(3)

Fig. 7.2. (a) Intensity (|ρI |2 dotted curve, |ρII |2 dashed curve, |ρI |2 + |ρII |2 (3) (3) dash-dotted curve, |ρI + ρII |2 solid curve) and (b) phases (θI dotted curve, θII dashed curve, θI − θII solid curve). The parameters are G1 = 2π × 28 MHz, G2 = G2 = 2π × 36 MHz, G3 = G3 = 2π × 75 MHz, Γ10 = 2π × 3 MHz, and Γ20 = Γ30 = 2π × 0.4 MHz.

The experiment was done in a 5 cm long vapor cell containing 85 Rb atoms. The energy levels of 5s1/2 (F = 3), 5p3/2 , 5d3/2 , and 5d5/2 form the four-level Y-type system, as shown in Fig. 7.1(a). The five laser beams were carefully aligned in the square-box pattern as depicted in Fig. 7.1(b). The probe laser beam E1 has a wavelength of 780 nm from an external cavity diode laser (ECDL) (driving the transition 5s1/2 − 5p3/2 ) and is horizontally polarized with a power of 7 mW (corresponding to the Rabi frequency of 2π×28 MHz). The laser beams E2 and E2 with wavelength 776.16 nm (driving the transition 5p3/2 − 5d3/2 ) are from another ECDL, which are split with an equal power of 16 mW (corresponding to the Rabi frequency of 2π × 36 MHz), each with a vertical polarization. The laser beams E3 and E3 have the wavelength of 775.98 nm (for the transition 5p3/2 − 5d5/2 ) and are from a CW Ti:Sapphire laser split with equal power, each with a vertical polarization. Great care has to be taken in order to align the five laser beams with good spatial overlaps and to satisfy the stringent phase-matching conditions with small angles (about 0.3◦ ) between them, as indicated in Fig. 7.1(b). Two diffracted FWM signals (kf 1 and kf 2 satisfying the appropriate phase-matching conditions) with horizontal polarization appear in the direction of Ef 1 & Ef 2 [at the lower right corner of Fig. 7.1(b)] and are detected by an APD. The transmitted probe beam is detected by a silicon photodiode.

240

7 Interactions of MWM Processes

The first measurement was to look at the probe transmission. When all five laser beams (E1 , E2 , E2 , E3 , and E3 ) are on at the same time with giving coupling frequency detunings (with fixed Δ2 and different Δ3 values), the probe transmission spectra show either dual-EIT or electromagnetically induced absorption (EIA) windows, as shown in Fig. 7.3(a). There are two modified EIT/EIA windows from the two ladder-type EIT sub-systems (at two-photon resonance conditions of Δ1 = −Δ2 and Δ1 = −Δ3 in the probe transmission). The experimental parameters are G1 = 2π × 28 MHz, G2 = G2 = 2π × 36 MHz, G3 = G3 = 2π × 110 MHz, Δ2 = 0 MHz, and Δ3 = −662 MHz, –396 MHz, –123 MHz, 0 MHz, 154 MHz, 276 MHz, 596 MHz, respectively. The fixed peaks along the dotted line with Δ2 fixed are due to |0 − |1 − |2 ladder sub-system, and the moving peaks with different Δ3 detunings are from |0 − |1 − |3 ladder subsystem. Notice that, by varying the detuning Δ3 from –662 MHz to 596 MHz, the probe transmission for the ladder (|0 − |1 − |3) sub-system alters from EIA to EIT and then back to EIA again, as shown in Fig. 7.3(a). Meanwhile, as the probe detuning Δ1 is scanned, two generated FWM signals are observed on the APD (which has a small angle from the probe transmission), as given in Fig. 7.3(b). Since the two FWM signals are diffracted in the same spatial direction, they can be individually identified by selectively blocking different laser beams and detuning different laser frequencies. The fixed single-peak at Δ1 = 0 (with Δ2 = 0) is the modified FWM (I) signal (kf 1 ) and the shifting double-peak signal (with different Δ3 values) is the other modified FWM (II) signal (kf 2 ), respectively, which appear always in the two different EIT windows. The dip in the middle of the FWM (II) signal for stronger fields of E3 and E3 [G3 G2 G1 , see Fig. 7.3(b)] is due to the three-photon destructive interference [1 – 3]. When the frequency difference (Δ) between Δ2 and Δ3 is reduced, the two EIT/EIA windows start to merge together and the two generated FWM signals begin to overlap and compete with each other, as clearly shown in Fig. 7.3(b). The line shape of the double-peak FWM signal changes from asymmetric to symmetric at the exact resonance (Δ3 = 0). At Δ = 0 (with the centers of the two FWM signals fully overlapped), the FWM (II) signal is reduced by approximately 50% and the FWM (I) signal is completely suppressed [see Fig. 7.3(b)]. Figure 7.4 presents the changes of two single-peak FWM signals at relatively low coupling Rabi frequencies with parameters G1 = 2π × 28 MHz, G2 = G2 = 2π × 36 MHz, G3 = G3 = 2π × 73 MHz, Γ10 /2π = 3 MHz, Γ20 = Γ30 = 2π × 0.4 MHz, Δ3 = 176 MHz, and Δ2 =16 MHz, 53 MHz, 117 MHz, 176 MHz, 270 MHz, 283 MHz, 356 MHz, respectively. The fixed peaks (indicated by the dotted vertical line) correspond to the modified FWM (II) signal and the moving peaks are the modified FWM (I) signal for different Δ2 values. In Fig. 7.4(a), as the frequency detuning Δ2 changes, the amplitudes of the two generated FWM signals change from large to small (when they overlap), and then to large again, where the moving peak (FWM I signal) shifts from the left side to the right side. To quantitatively

7.1 Competition between Two FWM Channels

241

Fig. 7.3. (a) Probe transmission (with two ladder-type EIT or EIA windows) versus the probe detuning Δ1 for different Δ3 values; (b) measured two FWM signals versus Δ1 for different Δ3 values. Adopted from Ref. [5].

understand this interplay between these two coexisting FWM processes, the evolution paths of the two co-existing FWM signals need to be calculated using ρ(3) (Δ). The theoretically calculated results with parameters similar to the ones used in the experiment are plotted in Fig. 7.4(b). The modified FWM (I) and FWM (II) signal intensities show the maximum suppressions at resonance conditions of Δ1 + Δ3 = 0 (for the dressing fields G3 and G3 ) and Δ1 + Δ2 = 0 (for the dressing fields G2 and G2 ), respectively [1 – 3, 14]. At the exact multi-EIT condition of Δ1 = −Δ2 = −Δ3 , both the FWM signals suffer their maximum suppressions at the same time.

Fig. 7.4. (a) Measured evolution of the two FWM signals for different laser field (E2 and E2 ) frequencies (Δ2 ); (b) theoretical plots of two FWM intensities versus Δ1 for different Δ2 values. Adopted from Ref. [5].

Next, the two FWM signals from different ladder-type EIT windows of this Y-type system are measured by fixing the coupling beam frequency detuning Δ3 and setting different Δ2 values, with all five laser beams (E1 , E2 , E2 , E3 , and E3 ) on at the same time. The single-peak FWM (I) signal

242

7 Interactions of MWM Processes

is shown to compete with each individual peak of the double-peak FWM (II) signal, as shown in Fig. 7.5(a). Figure 7.5(b) shows the competitions between these two distinguishable FWM signals through changing the frequency detuning Δ2 in a broader range with different G3 and Δ3 values from Fig. 7.5(a). Other parameters in Fig. 7.5 are G1 = 2π × 28 MHz, G2 = G2 = 2π × 36 MHz. As the frequency detuning value Δ2 decreases, the single-peak FWM (I) signal starts to emerge into the left-peak of the FWM (II) signal and suppresses it, as shown in the middle curve (Δ2 = 260 MHz) of Fig. 7.5(a) or the curve with Δ2 = 180 MHz in Fig. 7.5(b). Similarly, the FWM (I) signal can emerge into the right-peak of FWM (II) and suppress it, as shown in the middle curve (Δ2 = 155 MHz) of Fig. 7.5(a) or the curve with Δ2 = 92 MHz in Fig. 7.5(b). When the centers of the two FWM signals completely overlap at Δ = 0, the FWM signals are suppressed by about 90% at the multiEIT condition of Δ1 = −Δ2 = −Δ3 , as indicated by the curve with Δ2 = 143 MHz in Fig. 7.5(b).

Fig. 7.5. Measured two FWM signals versus the probe detuning Δ1 for different Δ2 values, with (a) G3 = G3 = 2π × 91 MHz, Δ3 = 200 MHz, and Δ2 =155, 260, 480 MHz; (b) G3 = G3 = 2π × 76 MHz, Δ3 = 143 MHz, and Δ2 =–94 MHz, –18 MHz, 12 MHz, 92 MHz, 143 MHz, 180 MHz, 188 MHz, 262 MHz, 332 MHz, respectively. Adopted from Ref. [5].

7.2 Efficient Energy Transfer between FWM and SWM Processes So far in all our discussions above, no propagation effects for the optical fields in the MWM processes have been considered. Although neglecting the propagation effects are justified in many situations as we have done in the previous sections, certain physical effects in some coherently-prepared multilevel atomic systems are critically dependent on the propagation characteristics, such as in studying ultraslow propagation of matched optical pulses

7.2 Efficient Energy Transfer between FWM and SWM Processes

243

[15], in obtaining optimal FWM generation in forward and backward directions [16], in generating correlated photon-pairs in dense multi-level atomic medium [17], and in strong relative intensity squeezing by four-wave mixing [18]. These effects are determined by the interactions and couplings between the laser fields during their propagations, and, therefore, the interaction length (or propagation distance) can be important for the systems to reach their equilibrium states [6, 15, 19, 20]. For example, the group velocities of the probe and conjugate pulses only become matched after certain propagation distance [15 ] and the rate of photon-pair generation gets higher for larger optical density (OD) of the atomic medium [8, 17]. The propagating fields reach their equilibrium states as results of the competitions or interferences between different field-generating processes during their propagations through the medium [15,19,20]. In the experiment to generate matched FWM pulses, the balance is reached between the probe field and the generated conjugate field [15]; in the two-wave mixing experiment, it is the destructive interference between the source field and the internally generated two-wave mixing field [20]. For the experiments described in Chapter 6 and Section 7.1, no propagation effects were considered since the atomic media were normally kept at high temperatures, which make the atomic media to be in the large OD limit. In such cases, the FWM and SWM generation processes have already reached their equilibrium. In this section, we describe experimental and theoretical studies of how the generated FWM and SWM signal fields reach their steady-state values during propagation [6]. The previous sections have shown how the FWM and SWM processes can be generated to coexist and interact with each other in multi-level atomic systems by manipulating the phase-matching conditions and intensities, as well as frequency detunings, of the coupling/pumping laser beams. Here, we will show how the generated coexisting FWM and SWM signals interact and exchange energy during their propagations through the coherently-prepared atomic medium [6]. Theoretical calculations for the coupled atom-field equations including propagation dependence are presented to explain the observed experimental results. Since the propagation distance z always appears together with the atomic density in the solution, the propagation (z-dependence) effects can be studied experimentally by simply changing the OD (atomic density) of the medium. Studying such energy exchange during propagation between the generated nonlinear wave-mixing signals can help us to understand and control these higher-order nonlinear optical processes, and hopefully lead to interesting applications in opto-electronic devices, soliton formations, generations of entangled photons, and quantum information processing. Let us consider a four-level inverted Y-type atomic system [1 – 3,21] with five laser beams, as shown in Fig. 7.6(a) (also in Section 6.2). The probe beam Ep (ω1 , kp , and Rabi frequency Gp ) drives the transition from |0 to |1. Two coupling beams, E2 (ω2 , k2 , and Rabi frequency G2 ) and E2 (ω2 , k2 , G2 ), interact with the upper transition from |1 to |2. Another two pumping

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beams, E3 (ω3 , k3 , G3 ) and E3 (ω3 , k3 , G3 ), link the transition from |3 to |1. The five laser beams propagate in a square-box configuration through the atomic medium as shown in Fig. 7.1(b) [Fig. 7.6(b) is a projection of the three-dimensional square-box pattern on the plane for simplicity]. Laser fields k2 and k3 travel in the opposite direction of fields kp , and fields k2 and k3 propagate in the direction having a small angle θ from k2 and k3 . Under this laser configuation, several FWM and SWM signals are generated in the directions either backward from k2 (denoted as km with a small angle θ) or with an angle 2θ from kp (shown as ks ) according to the phase-matching conditions [1 – 3]. When laser beams E3 and E3 are blocked, the laddertype sub-system (|0 − |1 − |2) generates a FWM signal in the direction (0) ω1 (1) ω2 with angle θ from kp (shown as kf = kp + k2 − k2 , via ρ00 −→ ρ10 −→ (2) −ω (3) ρ20 −→2 ρ10 ). When all five laser beams are on, SWM signals are generated from different interaction processes. To simply the discussion, we will only consider the dominant high-order nonlinear wave-mixing processes. From the phase-matching condition of ks = kp +k2 −k2 +k3 −k3 , efficient SWM signals (0) ω1 (1) ω2 (2) −ω (3) −ω (4) ω3 (5) ρ10 −→ ρ20 −→2 ρ10 −→3 ρ30 −→ ρ10 can be generated via either ρ00 −→ (0) ω1 (1) −ω3 (2) ω3 (3) ω2 (4) −ω2 (5) or ρ00 −→ ρ10 −→ ρ30 −→ ρ10 −→ ρ20 −→ ρ10 , which are both in the direction having a 2θ angle from kp (denoted as ks and Es with frequency ω1 in Fig. 7.6). At the same time, other SWM processes (using one photon each from fields Ep , E3 , E3 , but two photons either from E2 or E2 ) can exist and propagate in the direction of km , which can produce interference between the co-linear FWM and SWM processes [1,2], as described in Section 6.2.

Fig. 7.6. (a) Four-level atomic system for generating co-existing FWM and SWM processes. The dash-dotted lines are the generated FWM (Ef ) and SWM (Es ) signals. (b) Projection of the spatial square-box pattern on a plane for the laser beams used in the experiment [6].

However, in the current experiment the main goal is to spatially separate the generated FWM and SWM signals, so the SWM and FWM signals can be measured individually, which is very different from the objective in Section 6.2. In this case, the SWM signals (propagating collinearly with the FWM in the km direction) are intentionally suppressed by using lower powers for the E2 & E2 beams in comparison to the E3 & E3 beams, and by carefully misaligning the E3 & E3 beams slightly, so in the km direction the FWM

7.2 Efficient Energy Transfer between FWM and SWM Processes

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signal dominates, and the condition of km ≈ kf can be satisfied. Although the generated FWM (kf ) and SWM (ks ) signals travel in different directions (with a small angle θ between them), they share the same fields Ep , E2 and E2 . Both of these generated FWM and SWM signals fall in the same EIT window created by the ladder sub-system (|0 → |1 → |2) in the twophoton Doppler-free configuration [9] and have the same frequency (ω1 ) as the probe beam. The coherence lengths of the generated FWM and SWM signal beams are given by lcf = 2c/[n(ω2 /ω1 ) |ω2 − ω1 | θ2 ] and lcs = 2c/{n[(ω2 + ω3 )/ω1 ] |ω2 + ω3 − ω1 | θ2 }, respectively, with n being the refractive index at the frequency ω1 . In the experiment, θ is very small (about 0.3◦ ) so that lcf and lcs are much larger than the interaction length L(length of the atomic medium), so the phase-mismatch can be neglected. To theoretically study this system, propagation equations for the dominant FWM and SWM signals, as well as the probe field, are explicitly written, which are coupled to the multi-level atoms. These coupled equations determine the propagation dynamics of the probe, FWM, and SWM signal fields. Under the experimental conditions of G3 , G3 > G2 , G2 >> Gp , these coupled equations are as the following ⎧ ∂Gp D1 D2 D3 ⎪ = iξp ρp10 = − Gp + Gf exp(−iΔkf · r) − Gs exp(−iΔks · r) ⎪ ⎪ ⎪ ∂z D D Ds l f ⎪ ⎪ ⎨ ∂G D4 D5 D6 f = iξf ρf10 = − Gf + Gp exp(iΔkf · r) + Gs exp(iΔks · r) ⎪ ∂z Dl Df Df ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂Gs = iξs ρs = − D7 Gs − D8 Gp exp(iΔkf · r) − D8 Gf exp(iΔks · r) 10 ∂z Dl Ds Ds (7.1) where ξp(f,s) ≡ 2kp(f,s) μ2 N/, Δkf = k1 − kf , and Δks = k1 − ks . When the laser beams are all on resonances, the coefficients are given by D1 = ξp Γ20 Γ30 D2 = ξp Γ30 G22 D3 = ξp G22 G23 kf D1 kp kf D2 D5 = kp D4 =

kf G23 D2 G22 kp ks D1 D7 = kp ks G23 D2 D8 = Γ30 kp D6 =

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Dl = Γ10 Γ20 Γ30 + Γ30 G22 + Γ20 G23 Df = Γ10 Γ20 (Γ10 Γ30 + G23 ) Ds = Γ310 Γ20 Γ30 N , Γmn , Δi = Ω i − ωi , and μ are the atomic density, decoherence rates, frequency detunings from the atomic resonant frequency Ω i , and the dipole moment of the relevant transition, respectively. Equation (7.1) are derived from the optical responses of the medium to the probe, the generated FWM and SWM fields, respectively. The first terms in these equations contain the linear susceptibilities including the EIT effect. The second and third terms in each equation are contributions from the third- and fifth-order nonlinear susceptibilities, respectively, which are kinds of the parametric conversion processes. The linear susceptibilities control the dispersion profiles and transmission spectra of the probe and the generated FWM and SWM fields, while the third- and fifth-order nonlinear susceptibilities give the interactions between the laser fields and the atomic medium, and play the essential roles in determining the features of energy transfer between these FWM and SWM processes. The above mutually coupled equations indicate that not only the probe beam can generate FWM and SWM fields, but these FWM and SWM fields can also affect each other, as well as the probe beam via reabsorption and backward nonlinear processes during their propagations through the atomic medium [6,15,19,20]. Strong coupling and competitions between these fields during propagation are the key to establish the equilibrium among them. The solutions of these coupled equations determine the propagation characteristics of the generated Ef and Es fields. To show the basic physics more clearly without giving the complicated solutions, Eqs. (7.1b) and (7.1c) can be rewritten as   D5 D4 D8 (Gf + Gs ) + − ∂(Gf + Gs )/∂z = − Gp + Dl Df Ds   D6 D8 Gs − Gf (7.2) Df Ds At large propagation distance (which giveD1 ≈ D4 ≈ D7 , D1 ≈ D4 ≈ D7 , and D3 ≈ D8 ), the solutions can be significantly simplified. Under this condition, some balance conditions are satisfied, i.e., D2 Gf /Df −D3 Gs /Ds = 0 in Eq. (7.1a) for the probe beam, and D6 Gs /Df − D8 Gf /Ds = 0 in Eq. (7.2) for the sum of the FWM and SWM signals. Quantum destructive interferences between three-photon (Gf ) and five-photon (Gs ) excitation pathways are the underlying mechanisms for the probe and the generated FWM + SWM signals to reach equilibrium after long propagation distance. For given initial conditions of Gp (z = 0) = Q0 , Gf (z = 0) = Gs (z = 0) = 0 at the entrance face of the atomic medium, Eqs. (7.1a) and (7.2) can be

7.2 Efficient Energy Transfer between FWM and SWM Processes

easily solved analytically to give  !  " D8 Q0 Dl D5 D4 z Gf + Gs = − 1 − exp − D1 − D4 Ds Df Dl

247

(7.3)

where D4 z/Dl ∝ N z. Equation (7.3) clearly indicates that for a sufficiently large N z with exp(−D4 z/Dl ) << 1, the solution reaches a constant value (independent of z). These balancing conditions indicate that after an initial propagation distance for the FWM and SWM signals to build up (while the probe intensity decreases accordingly), the probe beam and the total signal intensity of the generated FWM + SWM beams reach their respective equilibrium values. After that the FWM and SWM fields will only transfer energies between themselves and, eventually, after a much longer propagation distance, reach their individual steady states, so they will propagate in the medium without further absorption and distortion. The experiment was performed using 85 Rb atoms in an atomic vapor cell (length 5 cm). The relevant energy levels for the inverted-Y system are 5S1/2 , F = 3 (|0), 5S1/2 , F = 2 (|3), 5P3/2 (|1), and 5D5/2 (|2), as shown in Fig. 7.6(a). An extended cavity diode laser (ECDL) serves as the probe field Ep at 780.24 nm. Another ECDL at 775.98 nm is split to yield the two coupling beams E2 and E2 . A cw Ti:Sapphire laser at 780.23 nm is divided into the two pump beams E3 and E3 . Since the exponential factor describing the propagation in the solutions of Eqs. (7.1) and (7.2) is proportional to N z, an easier way to change the “effective propagation length (N z)” is to change the atomic density N by changing the temperature of the atomic cell, and keep the atomic cell length (z) fixed as L [6,20]. Three detectors were set at three different directions: Da in the direction of the probe beam to monitor the probe transmission; Db at θ angle to detect the dominant FWM signal (kf ); and Dc at 2θ angle to detect the SWM signal (ks ), respectively, as shown in Fig. 7.6(b). The detected probe transmissions, FWM and SWM signals at two different atomic densities are depicted in Fig. 7.7 with the experimental parameters of G1 = 2π × 5 MHz, G2 = G2 = 2π × 35 MHz, G3 = G3 = 2π × 80 MHz, Δ3 = 0, and Δ2 = 312 MHz. At low temperature, a ladder-type EIT window is clearly shown [see Fig. 7.7(a3)] due to the two-photon Doppler-free configuration in this experimental setup [9,22]. However, at high temperature, the EIT window becomes an absorption peak [see Fig. 7.7(b3)]. By carefully aligning all the laser beams in the square-box pattern [as shown in Fig. 6.1 and its projection as in Fig. 7.7 (b)] [1,2], and selectively blocking different laser beams, the generated FWM [curves (a2) and (b1)] and SWM [curves (a1) and (b2)] signals are identified and separately detected (by detectors Db and Dc , respectively). There is a small SWM component in curves (a2) and (b1) of Fig. 7.7, which is more than ten times smaller than the FWM signal of interest and, therefore, can be neglected in the current discussion. The powers of the coupling and pumping laser beams E2 (E2 ) and E3 (E3 ) are 15 mW and 65 mW, and their corresponding frequency detunings are 312 MHz and 0 MHz, respectively. The probe beam power is set at 0.7 mW

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to optimize the generated FWM and SWM signals.

Fig. 7.7. Measured FWM and SWM signal intensities, and the corresponding probe beam transmission spectrum for two atomic densities (0.2×1012 /cm3 [curves (a1)– (a3)] and 1.1×1012 /cm3 [curves (b1)–(b3)] versus probe detuning Δ1 . Adopted from Ref. [6].

The measured FWM and SWM signal intensities, as well as the probe transmission, are presented in Fig. 7.8 (square points) as a function of atomic density. The experimental parameters are G1 = 2π × 5 MHz, G2 = G2 = 2π × 35 MHz, G3 = G3 = 2π × 80 MHz, Δ3 = 0, −Δ1 = Δ2 = 312 MHz, Γ10 /2π = 3 MHz, Γ20 /2π = 0.4 MHz, and Γ30 /2π = 2 MHz. Both of the FWM and SWM signal intensities increase initially at low atomic density (or equivalently short propagation distance), as the probe beam intensity decreases (inset of Fig. 7.8). The SWM signal intensity grows much faster than the FWM intensity at the beginning. At certain atomic density (about N =0.3×1012/cm3 ), the probe and the FWM + SWM intensities reach their equilibrium values, after which energy exchange occurs only between the FWM and SWM signals as they propagate through the atomic medium. Eventually, at a higher atomic density (about N =1.2×1012/cm3 , or equivalently a long propagation distance), the FWM and SWM signals reach their respective steady-state values individually and do not change after that. The theoretically simulated results from Eq. (7.1) are plotted in Fig. 7.8 (solid lines) with similar parameter values as the experiment, which match quite well with the measured data. Such energy exchanges between the generated FWM and SWM, and the probe beam during their propagation can be influenced by several parameters. For instance, the intensities of E3 and E3 can greatly modify the FWM and SWM generating processes and their relative intensities. By fixing the atomic cell temperature to give N = 1.4 × 1012 /cm3 (indicated by the dashed line in Fig. 7.8), the relative FWM and SWM intensities can be tuned by varying

7.2 Efficient Energy Transfer between FWM and SWM Processes

249

Fig. 7.8. Square points are the measured FWM and SWM peak intensities, as well as the FWM+SWM intensity, versus the atomic density. Inset: Measured probe intensity at the EIT window. Solid lines are the theoretically calculated results with same parameters as used in the experiment. Adopted from Ref. [6].

the power of E3 (E3 ). Figure 7.9(a) presents the generated FWM and SWM signal intensities (as well as the FWM+SWM intensity) as a function of G3 (G3 ) with the experimental parameters G1 = 2π×5 MHz, G2 = G2 = 2π× 35 MHz, N =1.4×1012/cm3 ,Δ3 = 0, −Δ1 = Δ2 = 312 MHz, Γ10 /2π = 3 MHz, Γ20 /2π = 0.4 MHz, and Γ30 /2π = 2 MHz. Note that the observed behaviors presented in Fig. 7.9(a) are not really governed by Eq. (7.1), since those propagation equations assume large G3 and G3 (which is not true for lower G3 power portion used in the experiment). It is interesting to notice that as E3 ( & E3 ) is turned on from zero, the SWM intensity increases very rapidly to a maximal value, and then decreases steadily as the pump beam power further increases. At the same time, the FWM signal intensity increases slowly up to a quite large value before it decreases again. The FWM process in the ladder sub-system is dressed by the E3 and E3 fields, which significantly modifies the FWM process. In Fig. 7.9(a), the top curve is the sum intensity of the FWM and SWM signals, which has the same behavior as the calculated total atomic coherence of ρ20 + (ρ20 + ρ30 )α [see Fig. 7.9(b)] (where α is the ratio between the fifth-order and third-order nonlinear susceptibilities), which is calculated by employing the full density-matrix equations without the approximations used in obtaining Eq. 7.1. By looking at the inverted-Y energy level system [see Fig. 7.6(a) and the discussions on generating the FWM and SWM], one can see that the FWM and SWM processes correspond to the atomic coherences ρ20 and (ρ20 + ρ30 )α, respectively. Figure 7.8 takes the value of G3 = 2π×80 MHz [at which the FWM and SWM signals have the same intensity, as indicated by the dashed line in Fig. 7.9(a)]. The comparisons between the FWM & SWM signal intensities and the corresponding induced atomic coherences, as offered by Figs. 7.9(a) and (b), indicate that the atomic coherences induced between the energy levels play significant roles in the enhancements of high-order multi-wave mixing processes and in the process of establishing equilibrium for the generated FWM and SWM signals during their propagation.

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Fig. 7.9. (a) Measured FWM and SWM peak intensities, as well as the total FWM+SWM intensity, versus the pump field G3 (G3 ); (b) theoretically calculated atomic coherences [curves ρ20 , (ρ20 + ρ30 )α, and ρ20 + (ρ20 + ρ30 )α] versus pump field G3 (G3 ). Adopted from Ref. [6].

Some important issues are worth to mention here. First, if a pulsed laser is used as the probe beam, the generated FWM and SWM signals will be slowed down due to sharp dispersion in the EIT window, and eventually become pulse-matched with the probe pulses, as in the case for FWM [15]. This system can also be used to generate the entangled FWM (or SWM) photon pairs and even triplet photons for quantum information processing [23]. Second, since both the FWM and SWM processes described here share the same Ep , E2 , and E2 beams, strong competitions between these wave-mixing processes, as well as three-photon and five-photon interferences, exist in this system. The energy exchange and interactions between the generated FWM and SWM during their propagation are the manifestation of the strong coupling and competitions between these high-order nonlinear optical processes. Third, in order to investigate the FWM and SWM energy exchange, the typically efficient SWM channels in the km direction (as shown in Section 6.2 or [4]) are intentionally suppressed by using lower E2 and E2 powers and by slightly misaligning the E3 & E3 beams, so only the dominant FWM process needs to be considered in this direction. From the above experimental demonstration, one can see that by choosing appropriate propagation length (or atomic cell temperature) or different pump powers for the E3 and E3 beams, one can get desired relative strengths between the FWM and SWM signals from IFWM < ISWM to IFWM > ISWM . A comprehensive theoretical model has been developed to compare with the experimentally measured FWM and SWM generation processes during propagation with excellent agreements. Understanding and controlling the high-order nonlinear optical processes (such as χ(3) and χ(5) ), especially their propagation properties, can be very important in studying new physical phenomena (such as two-dimensional soliton formation, pulse matching, and entangled photon generations) and in designing new applications such as pulse shaping and propagations of high-intensity optical pulses in cubic-quintic media.

7.3 Spatial and Temporal Interferences between Coexisting FWM and SWM · · ·

251

7.3 Spatial and Temporal Interferences between Coexisting FWM and SWM Signals In Chapter 6 and previous sections in this chapter, coexisting FWM and SWM (or competing FWM) processes in several four-level atomic systems have been studied [1 – 5]. The relative strengths of the generated FWM and SWM signals can be controlled by adjusting the amplitude and the frequency detuning of the coupling/pump laser beams. When the two nonlinear wavemixing signals are pulled together in frequency by tuning the frequency detuning of the coupling beams (Section 6.3 and Section 7.1), these generated signals (from different-order nonlinear processes) can interference in the frequency domain [1]. Energy exchange occurs between these nonlinear signals during their propagation (Section 7.2 and Ref. [6]) and the relative intensities of the generated FWM and SWM signals can be adjusted by the coupling laser intensity. Notice that in the above discussed open-cycled, as well as close-cycled, four-level systems, the coexisting FWM and SWM processes always share some of the coupling laser beams, however, at the same time, some of the coupling fields only participate in one of the wave-mixing processes, but not the other one [4, 6, 7]. For example, in the close-cycled four-level folded atomic system, as shown in Fig. 6.12 (Section 6.4), the E2 field (for the upper transition in the ladder sub-system) participates in the dominant FWM process, but not in the dominant SWM generation, whose amplitude can then be used to adjust the relative efficiencies between the FWM and SWM processes [4]. At the same time, the E3 and E3 fields in that system only involve in the SWM process, but not the FWM one. Making use of the above mentioned property in generating the coexisting FWM and SWM processes in four-level atomic system, we show yet another way to control the coexisting FWM and SWM signals, i.e., by tuning the relative phase between these two nonlinear optical processes. As the relative phase between the generated FWM and SWM signals is changed, the constructive and destructive interferences in spatial and temporal domains are expected to occur. To illustrate such interesting spatial and temporal interferences between different order wave-mixing signals, two multi-level atomic systems are used in the following as examples. Let us consider the five-level atomic system, as shown in Fig. 7.10(b), which has been used to demonstrate interference between FWM and SWM signals in frequency domain in Section 6.3 [1]. This five-level system in 85 Rb atoms consists of two sub-systems. One three-level ladder-type sub-system involves levels 5S1/2 , F = 3, 5P3/2 (|1), and 5D3/2 (|2). Another sub-system has two energy levels 5S1/2 , F = 2 and 5P1/2 (|3). Two coupling laser beams (E2 and E2 with frequency ω2 ) drive the upper transition (|1 to |2) of the ladder sub-system and a weak probe laser beam E1 with frequency ω1 couples to the lower transition (5S1/2 , F =3 to |1). Two pumping laser beams E3 and E3 (with frequency ω3 ) interact with the two-level sub-system with transition from 5S1/2 F = 2 to |3. The five laser beams (E1 ,E2 ,E2 ,E3 , and E3 ) propa-

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gate in a spatial square-box pattern, as shown in Fig. 7.10(a), which has been used in previously described experiments. The four coupling/pumping beams (E2 ,E2 ,E3 , and E3 ) propagate in one direction with small angles (about 0.5◦ ) between them, and the probe beam (E1 ) propagates in the opposite direction with a small angle from those coupling/pumping beams. As shown in Section 6.3, FWM and SWM processes can coexist in this system and be made to interfere in frequency domain by adjusting the frequency detuning of the coupling beams [1]. Since the E2 coupling beam only involves in the dominant FWM process and not the dominant SWM process, as discussed in Section 6.3 [1], a time delay is introduced in this beam’s path [as indicated in Fig. 7.10(a)] to adjust the phase of the FWM channel relative to the SWM channel, which produces spatial and temporal interferences between these two nonlinear optical processes [7].

Fig. 7.10. (a) Spatial (square-box) beam geometry used in the experiment with a time delay in beam E2 ; (b) five-level atomic system in 85 Rb for generating coexisting FWM and SWM signals.

Without the pumping laser beams E3 and E3 the three energy levels (5S1/2 , F = 3 − |1 − |2) form a ladder-type system. Therefore, the probe beam (E1 ) together with one of the coupling beams (E2 or E2 ) exhibit an EIT window, since the probe beam E1 and the coupling beam E2 or E2 counterpropagate through the atomic medium satisfying the two-photon Dopplerfree (first-order) configuration [9]. When both E2 and E2 beams are present, efficient FWM process can be generated to produce a conjugate FWM signal at frequency ω1 , as discussed in Section 6.3. The generated FWM signal propagates in the Kf (= K1 + K2 − K2 ) direction owing to the phasematching condition. When the strong pumping beams E3 and E3 (with same frequency) are both turned on, they form a small-angle static grating in the atomic medium. The FWM signal in the ladder sub-system scatters off this static grating to generate an efficient SWM signal Ks (= K1 + K2 − K2 + K3 − K3 , as the dominant one) propagating in the direction determined by the phase-matching condition. With the spatial square-box pattern [as shown in Fig. 7.10(a)] for the coupling/pumping and the probe laser beams, the generated FWM (Kf ) and SWM (Ks ) signals not only have the same frequency (ω1 ), but also propagate in the same direction (Kf ||Ks ), as shown

7.3 Spatial and Temporal Interferences between Coexisting FWM and SWM · · ·

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in Section 6.3. Three laser sources (two ECDLs and a cw frequency-locked Ti:Sapphire laser) are used to interact with the five-level atomic system, as shown in Fig. 7.10(b). One of the ECDLs is split to create the coupling beams E2 and E2 , and the Ti:Sapphire laser is split to create the strong pumping beams E3 and E3 . The second ECDL is used as the probe beam E1 and has the orthogonal linear polarization relative to the coupling/pumping beams, which are all linearly polarized in the same direction. The atomic cell is 5 cm long and is heated to 65◦ C. Owing to the phase-matching conditions the generated FWM and SWM signals propagate at a small vertical angle (about 0.5◦ ) with respect to the transmitted probe beam. The probe beam transmission is monitored by a photodiode and the generated FWM and SWM signals are detected with an avalanche photodiode and a CCD at the same time. The strengths of the generated FWM and SWM signals are determined by the frequency detuning and power in each of the coupling/pumping beams. The frequency detuning of the beams are adjusted to maximize the nonlinear processes and to avoid and minimize the background Dopplerbroadened FWM signal created by the E1 , E3 , and E3 beams. To observe simultaneous FWM and SWM signals in this system the FWM process needs to be suppressed so that it does not completely dominate the SWM process. Since the most efficient SWM is achieved by having only one coupling beam (E2 ) in the upper transition of the ladder sub-system, and efficient FWM (in the ladder sub-system) requires to have both coupling beams (E2 and E2 ) at same time and with similar intensities, one can reduce the FWM signal and enhance the SWM signal by decreasing the power of E2 . For this purpose the power of E2 is always set to be greater than that of E2 . When E3 and E3 beams are blocked the only signal generated is from the FWM process from the ladder sub-system |0–|1–|2. The three-dimensional FWM profile and its cross sections (or two-dimensional projections) are shown in Fig. 7.11(a). If the E2 beam is blocked and the E3 and E3 pumping beams are turned on, only the SWM signal can be generated in the system (in this case the FWM signal from the ladder sub-system is completely turned off). The pure SWM profile and its appropriate cross sections are shown in Fig. 7.11(b). If all five beams are turned on at the same time, there are a couple of different nonlinear processes that can occur to generate beams in this direction. The two dominant ones are the FWM process using the beams E1 , E2 , and E2 in the ladder sub-system and the SWM process using the beams E1 , E2 , E2∗ , E3 , and E3 . The other SWM processes that can generate fields in this direction (e.g., E1 , E2 , E2∗ , E3 ,and E3 ) are all limited in their efficiencies because of the lower power contained in beam E2 . In this case, both the FWM and SWM signals are generated simultaneously and they interfere spatially with each other. The three-dimensional spatial interference pattern and its cross sections for the FWM and SWM signals are presented in Fig. 7.11(c). The interference fringe is clearly seen in the cross section of Fig. 7.11(c). The experimental parameters are Δ1 = 250 MHz, Δ2 = −250 MHz, Δ3 = 750 MHz, P1 =

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16.7 mW, P2 = 10.2 mW, P2 = 2.6 mW, P3 = 66 mW, and P3 = 72 mW. The frequency detuning of the lasers are defined as Δi = ωi − Ω i , where Ω i is the atomic transition frequency and ωi is the frequency of the laser.

Fig. 7.11. (a) FWM profile and its cross sections; (b) SWM profile and its cross sections; (c) interference profile and its cross sections. Adopted from Ref. [7].

The FWM and SWM signals that interfere in Fig. 7.11(c) are generated under the phase-matching conditions of Kf = K1 + K2 − K2 for the FWM and Ks = K1 + K2 − K2 + K3 − K3 for the SWM, respectively. By closely inspecting these phase-matching conditions one can see that the E2 laser beam only contributes to the FWM signal, and it does not enter the SWM process. However, the E2 beam contributes to both the nonlinear processes. Since the E2 and E2 coupling beams are split from the same ECDL, they have a well-defined phase relation. This gives us the ability to independently control the phase of the E2 beam relative to the E2 beam. A precision translation stage is placed in the E2 beam path (to tune the relative phase between E2 and E2 ), as shown in Fig. 7.10(a), so that the phase dependence of the interference can be examined to gain information about the interplay between the FWM and SWM processes. This added relative time delay modifies the phase-matching condition of the FWM signal relative to the SWM signal by adding a phase factor Φ to the FWM signal, where Φ is a phase controlled by the position of the translation stage (controlled time delay). As the translation stage moves, the phase Φ is scanned and the spatial profile is measured as a function of Φ. The interference profiles can be measured as a function of time delay and a time average can be made from the data sets. When the time-averaged component is subtracted from the measured interference data, the phase-dependent part can be extracted to better show the effect on interference due to phase change. The cross sections of the phase-dependent component of the spatial profile as a function of phase are shown in Fig. 7.12. As the phase changes from Φ = 0 to Φ = π, the interference goes from destructive to constructive and then back to destructive interference again at Φ = 2π. The choice of Φ = 0 is arbitrary but the number of profiles measured in each π/2 interval is a constant.

7.3 Spatial and Temporal Interferences between Coexisting FWM and SWM · · ·

255

The experimental parameters are Δ1 = 250 MHz, Δ2 = −250 MHz, Δ3 = 750 MHz, P1 = 16.7 mW, P2 = 10.2 mW, P2 = 2.6 mW, P3 = 66 mW, and P3 = 72 mW. Since the change of this spatial interference pattern is caused by the time delay in beam E2 , Fig. 7.12 can be considered as a spatial- temporal interference effect.

Fig. 7.12. Cross sections of the phase-dependent component in the spatial profile for different phases (time delays). Adopted from Ref. [7].

Another way to investigate the phase-dependent interference effect between the generated FWM and SWM processes is to include the time-averaged component in the profile but only looking at a small spatial region (at the center) [7]. In this way, the total spatial profile does not need to be considered and, therefore, it simplifies the analysis. The signal intensity on a small circular spot at the center of the beam (with a diameter of about 40 μm) is detected, and can be expressed, as a function of phase change, as  (7.4) I = If + Is + 2 If Is sin (Φ) where If and Is are the intensities of the FWM and SWM signals, respectively. Since the power of beam E2 determines the relative strengths of the generated FWM and SWM signal intensities, it also governs the contrast of this interference fringe. Measured interference patterns for two different powers of E2 are presented in Fig. 7.13 as a function of phase (time delay). The measured intensities are normalized to the maximum intensity observed in the two runs in order to better display the contrast between them. The experimental parameters used for Fig. 7.13 are Δ1 = 250 MHz, Δ2 = −250 MHz, Δ3 = 750 MHz, P1 = 7.7 mW, P2 = 15.8 mW, P3 = 82.5 mW, and P3 = 83.5 mW. As discussed in Chapter 4, through beating (or homodyne/heterodyne detecting) two signals from different orders of nonlinear optical processes, the higher-order nonlinear susceptibility can be determined [26, 27]. In the current case, if the SWM process is intentionally made to be weaker, this phase-dependent measurement, as shown in Fig. 7.13, can be used to determine the fifth-order nonlinear susceptibility χ(5) by interfering with the

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7 Interactions of MWM Processes

Fig. 7.13. Total signal intensity in a small region in the center of the beam for two different values of E2 power. (a) PE2 = 712 μW and (b) PE2 = 460 μW. Adopted from Ref. [7].

stronger third-order nonlinear process (proportional toχ(3) ). By varying the power of E2 (and keeps it smaller than the power of E2 ), interference curves, similar to those shown in Fig. 7.13, are measured. The visibilities of these interference curves are measured and plotted as a function of the power of E2 , as shown in Fig. 7.14. From the definitions of the FWM and SWM signal intensities If and Is , and using Eq. (7.4), the visibility of the interference pattern for the FWM and SWM signals can be written as ) χ(3) E2∗ 2 (5) χ E2∗ E3 E3∗ (7.5) V = χ(3) E2∗ 1 + (5) ∗ χ E2 E3 E3∗ which shows how the E2 beam can influence and control the relative strengths of the generated FWM and SWM signals. The measured visibilities in Fig. 7.14 can be fitted to the functional form of Eq. (7.5) (see solid curve in Fig. 7.14). From such fitting, the ratio of the nonlinear susceptibilities (χ(3) /χ(5) ) can be experimentally determined with measured intensities for the coupling and pumping fields. Since the third-order nonlinear susceptibility χ(3) can be measured in other experiments [28], then the fifth-order nonlinear susceptibility χ(5) can be determined. The experimental parameters used in Fig. 7.14 are Δ1 = 250 MHz, Δ2 = −250 MHz, Δ3 = 750 MHz, P1 = 7.7 mW, P2 = 15.8 mW, P3 = 82.5 mW, and P3 = 83.5 mW. The technique of controlling the relative phase (time delay) between coexisting FWM and SWM channels described above is not limited to the five-level atomic system, as shown in Fig. 7.10(b). Actually, the ability to tune the relative phase between the coexisting multi-wave mixing processes can also be applied to other four-level atomic systems such as the ones used in Chapter 6 and earlier sections of this chapter. In the following, we briefly describe the temporal-spatial interference between the FWM and SWM processes in a four-level inverted-Y system, used earlier in Section 6.2 for enhancing SWM signals [2] and in Section 7.2 for demonstrating energy exchanges between

7.3 Spatial and Temporal Interferences between Coexisting FWM and SWM · · ·

257

Fig. 7.14. Measured visibilities of the interference patterns as a function of the E2 power. The solid curve is a fit to Eq. (7.5). Adopted from Ref. [7].

the FWM and SWM signals [6]. To illustrate the basic scheme for phase control in this system, the fourlevel inverted-Y system is shown again in Figs. 7.15(a) and (b). As discussed in earlier sections (see Sections 6.2 and 7.2), efficient FWM and SWM signals can be generated to coexist in this system by using the spatial square-box pattern for the four coupling/pumping laser beams, as shown in Fig. 7.10(a). The dominant FWM process is generated in the ladder-type sub-system shown in Fig. 7.15(a), and the dominant SWM process is generated in the laser configuration of Fig. 7.15(b). As clearly seen from these figures, field E2 only involves in the dominant FWM channel, but not the SWM channel, so it can used to control the relative efficiency and the relative phase between these two nonlinear processes. A time delay τ is introduced in the beam path of field E2 . Again, since fields E2 and E2 are split from the same laser source, they maintain a well-defined phase relation. By keeping E2 much weaker than E2 , the FWM and SWM signal intensities can be made to be similar in strength. The phase-matching conditions (see Sections 6.2 and 7.2) for these dominant FWM and SWM processes let them propagate in the same direction and with same frequency, so they will interfere.

Fig. 7.15. Atomic energy levels for generating (a) FWM (a) and (b) SWM processes in the inverted-Y system; (c) temporal interference at r = 0 (square points are experimental data and the solid curve is the theoretically simulated result).

The calculation shows that the total signal intensity has not only the

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7 Interactions of MWM Processes

spatial interference with a period of 2π/Δk, but also an ultrafast time oscillation with a period of 2π/ω2 [29]. With a plane-wave approximation and the square-box configuration for the laser beams with small angles [see Fig. 7.10(a)], the spatial interference occurs in the plane perpendicular to the propagation direction. The spatial interference pattern is similar to the one shown in Fig. 7.11 for the five-level system. However, the temporal interference is quite interesting in this case, since its beating frequency is given by the optical frequency (ω2 ) of the E2 beam. The time period (2π/ω2 ) for the oscillation is very short, down to the time scale of few femtoseconds. The experiment was carried out in 85 Rb with energy levels as shown in 7.15(a). The measured time oscillation is presented in Fig. 7.15(c), as the time delay τ for the E2 beam is scanned. The frequency measured from the time oscillation data is close to the transition frequency between the 5P3/2 to 5D5/2 (|1 to |2) transition frequency in 85 Rb [29]. The solid curve in Fig. 7.15(c) is from the theoretical simulation. The field of coexisting MWM processes in coherently-prepared multi-level atomic media is still in its early stage. We have shown several schemes to manipulate these different wave-mixing processes by controlling the intensities and frequencies, as well as the relative phase (time delay), of the coupling/pumping laser beams. The propagation effects of the coexisting FWM and SWM processes are investigated to show the energy exchange between the generated signals. The temporal and spatial interferences controlled by the time delay in one laser beam are important and can be used for coherent controls of more complicated molecular systems, where certain desired nonlinear optical processes can be selectively enhanced or suppressed. The above experimental and theoretical studies show that optical nonlinearities, especially higher-order ones, in multi-level systems can be optimized and controlled. More research works are needed in this field to better implement these basic concepts and identify more practical applications for the coexisting high-order nonlinearities. The physics learned from studying multi-level atomic systems can be applied to solid and liquid materials, where more potential applications can be found.

References [1] [2] [3]

[4]

Zhang Y P, Brown A W, Xiao M. Observation of interference between fourwave mixing and six-wave mixing. Opt. Lett., 2007, 32: 1120 – 1122. Zhang Y P, Xiao M. Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system. Appl. Phys. Lett., 2007, 90: 111104. Zhang Y P, Brown A W, Xiao M. Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows. Phys. Rev. Lett., 2007, 99: 123603. Zhang Y P, Xiao M. Controlling four-wave and six-wave mixing processes in multilevel atomic systems. Appl. Phys. Lett., 2007, 91: 221108.

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Zhang Y P, Anderson B, Brown A W, Xiao M. Competition between two four-wave mixing channels via atomic coherence. Appl. Phys. Lett., 2007, 91: 061113. Zhang Y P, Anderson B, Xiao M. Efficient energy transfer between four-wavemixing and six-wave-mixing processes via atomic coherence. Phys. Rev. A, 2008, 77: 061801(R). Anderson B, Zhang Y P, Khadka U, Xiao M. Spatial interference between four- and six-wave mixing signals. Opt. Lett., 2008, 33: 2029 – 2031. Du S W, Wen J M, Rubin M H, G Y Yin. Four-wave mixing and biphoton generation in a two-level system. Phys. Rev. Lett., 2007, 98: 053601. Gea-Banacloche J, Li Y, Jin S, Xiao M. Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment. Phys. Rev. A, 1995, 51: 576 – 584. Xiao M, Li Y Q, Jin S, Gea-Banacloche J. Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms. Phys. Rev. Lett., 1995, 74: 666 – 669. Agarwal G S, Harshawardhan W. Inhibition and enhancement of two photon absorption. Phys. Rev. Lett., 1996, 77: 1039 – 1042. Ulness D J, Kirkwood J C, Albrecht A C. Competitive events in fifth order time resolved coherent Raman scattering: Direct versus sequential processes. J. Chem. Phys., 1998, 108: 3897 – 3902. Nie Z Q, Zheng H B, Li P Z, et al. Interacting multiwave mixing in a five-level atomic system. Phys. Rev. A, 2008, 77: 063829. Zhang Y P, Xiao M. Generalized dressed and doubly-dressed multiwave mixing. Opt. Exp., 2007, 15: 7182 – 7189. Boyer V, McCormick C F, Arimondo E, et al. Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor. Phys. Rev. Lett., 2007, 99: 143601. Kang H, Hernandez G, Zhu Y F. Superluminal and slow light propagation in cold atoms. Phys. Rev. A, 2004, 70: 061804. Braje D A, Bali´c V, Goda S, et al. Frequency mixing using electromagnetically induced transparency in cold atoms. Phys. Rev. Lett., 2004, 93: 183601. McCormick C F, Boyer V, Arimondo E, et al. Strong relative intensity squeezing by four-wave mixing in rubidium vapor. Opt. Lett., 2007, 32: 178. Kang H, Hernandez G, Zhang J P, et al. Phase-controlled light switching at low light levels. Phys. Rev. A, 2006, 73: 011802. Jiang K J, Deng L, Payne M G. Observation of quantum destructive interference in inelastic two-wave mixing. Phys. Rev. Lett., 2007, 98: 083604. Li Y, Xiao M. Enhancement of non-degenerate four-wave mixing using electromagnetically induced transparency in rubidium atoms. Opt. Lett., 1996, 21: 1064. Harris S E. Electromagnetically induced transparency. Phys. Today, 1997, 50: 36 – 42. Michinel H, Paz-Alonso M J, Perez-Garcia V M. Turning light into a liquid via atomic coherence. Phys. Rev. Lett., 2006, 96: 023903; Wu Y, Deng L. Ultraslow Optical Solitons in a Cold Four-State Medium. Phys. Rev. Lett., 2004, 93: 143904; Wen J M, Du S W, Zhang Y P, Xiao M, Rubin M H. Nonclassical light generation via a four-level inverted-Y system. Phys. Rev. A, 2008, 77: 033816.

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Lu B, Burkett W H, Xiao M. Nondegenerate four-wave mixing in a doubleLambda system under the influence of coherent population trapping. Opt. Lett., 1998, 23: 804. Hemmer P R, Katz D P, Donoghue J, et al. Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium. Opt. Lett., 1995, 20: 982. Zhang Y P, Brown A W, Gan C L, Xiao M. Intermixing between four-wave mixing and six-wave mixing via atomic coherence. J. Phys. B, 2007, 40: 3319 – 3329. Zhang Y P, Anderson B, Xiao M. Coexistence of four-wave, six-wave and eight-wave mixing processes in multi-dressed atomic systems. J. Phys. B, 2008, 41: 045502. Zhang Y P, Gan C L, Xiao M. Modified two-photon absorption and dispersion of ultrafast third-order polarization beats via twin noisy driving fields. Phys. Rev. A, 2006, 73: 053801. Zhang Y P, Khadka U, Anderson B, Xiao M. Temporal and spatial interference between four-wave mixing and six-wave mixing channels. unpublished.

8 Multi-dressed MWM Processes

In Chapters 7 – 8, experimental generations of the coexisting four-wave mixing (FWM) and six-wave mixing (SWM) processes in several simple multilevel atomic systems and their interactions/interplays have been presented. The previous discussions for those atomic systems were mainly on the basic physical concepts to generate the coexisting multi-wave mixing (MWM) processes and their experimental implementations, and very little theoretical details were given. In this Chapter, we present some detail theoretical calculations on the related topics of generating coexisting MWM processes in the coherently-prepared atomic media. The first topic (Section 8.1) is on the ultraslow propagations of the nondegenerate FWM (NDFWM) signal and the weak probe beam in a close-cycled four-level double-ladder atomic system. Under certain conditions (such as balanced laser beam powers or atomic coherence), matched pulses are achievable for the pulsed probe beam and the generated NDFWM signal, which can transmit through the atomic medium with little absorption. The second topic is on the multiple dressing MWM processes in multi-level atomic systems. Section 8.2 will present the theoretical treatments for the generalized dressed and the doubly-dressed MWM in a general (n+1)-level cascade atomic system. Higher-order nonlinear wavemixing processes can be generated in such a close-cycled cascade system. Then, using a five-level atomic system, we show how three doubly-dressing (i.e., nested-type, parallel-type, and sequential-cascade-type) schemes for the FWM process (in the three-level system) can be used to generate coexisting FWM, SWM, and eight-wave mixing (EWM) processes. The dressing fields provide energy for the large enhancements of the high-order nonlinear wave-mixing processes. Investigations of these multi-dressing mechanisms and interactions are very useful to understand and control the generated highorder nonlinear optical signals.

8.1 Matched Ultraslow Pulse Propagations in HighlyEfficient FWM As we have shown in Chapters 6 – 7, efficiencies of the nonlinear optical processes in multi-level atomic systems can be greatly enhanced through

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the light-induced atomic coherence. The third-order nonlinear susceptibility (or Kerr index) has been shown to be greatly modified and enhanced near the electromagnetically induced transparency (EIT) resonance in three-level atomic systems [1]. The sharp normal dispersion slope near the EIT resonance can significantly slow down the group velocity of optical pulses in the medium [2 – 4]. With four energy levels involved, more level configurations can be envisioned (as we have shown in the last two chapters), and various nonlinear optical processes can be optimized by suppressing linear absorption through EIT (destructive interference) and increasing the nonlinear susceptibilities through constructive interferences in three-photon processes [5 – 7]. One of such interesting nonlinear optical processes is the NDFWM, which normally has high efficiency in close-cycled multi-level systems, such as the four-level double-Λ [8 – 10], cascade [11], and double-ladder systems, as shown in Figs. 8.1(a) – (d).

Fig. 8.1. Closely-cycled four-level (a) double-Λ, (b) cascade, and (c)&(d) doubleladder systems. (e) Forward and (f) backward NDFWM schemes.

Some of the distinct features of the double-Λ system include its symmetry in laser frequencies and near degeneracy between the probe beam and the generated signal beam. For a chosen laser beam configuration [with two strong pump beams sharing one lower state, but connecting to different excited states, as shown in Fig. 8.1(a)], both the probe beam (ω1 ) and the generated signal beam (ωf ) can satisfy the EIT condition simultaneously to minimize the linear absorptions near the atomic resonance during their propagations through the medium At the same time, the NDFWM process can have very high efficiency. Recent studies have predicted 100% NDFWM efficiency in the backward double-Λ configuration, but the forward NDFWM efficiency can only reach 25% [12]. Since near two-photon Doppler-free conditions are easily satisfied in such double-Λ system [3],the hot atomic vapor cells can be used for experimental demonstrations of the predicted effects. In such a double-Λ atomic system, the generated signal in NDFWM has the similar frequency

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263

as the pump beams. Recent theoretical studies on cascade four-level ladder systems [see Fig. 8.1(b)] have predicted high efficiency (up to 75%) in the forward NDFWM geometry [see Fig. 8.1(e)], which can generate frequency up-converted light beam (with ωf > ω1 , ω2 , ω3 ). However, due to the large difference in the generated signal frequency from the applied laser beams, two-photon Doppler-free configuration cannot be satisfied in this system, so experiments in this configuration cannot be done in atomic vapors. The four-level double-ladder system [see Figs. 8.1(c)&(d)] lies between these two four-level (double-Λ and cascade) systems with some unique features, as will be presented below in this section. In this section, we mainly consider the propagation characteristics of the generated FWM signal beam in the double-ladder system. For this purpose, the probe laser beam is assumed to be optical pulses. An important issue in such nonlinear optical processes is the pulse matching between the weak probe beam E1 and the generated signal beam Ef during their propagations through the atomic medium. Since both the probe and signal beams are under EIT conditions (for double-Λ or double-ladder system with two midlevels close to each other), their group velocities of the optical pulses are reduced owing to the sharp slope changes accompanying EIT dips [4]. The high efficiency in the NDFWM is mainly due to the slowing down of the group velocities of these two pulses and pulse matching between them during propagation, so good overlap can always be achieved. In other theoretical studies using the atomic amplitude technique [10 – 12], a four-state system in double-Λ or cascade configuration interacting with long and short laser pulses in a weak probe beam approximation was investigated. With the three-photon destructive interference, the conversion efficiency can reach up to 25%. When the three-photon destructive interference does not occur, it was shown that the photon flux conversion efficiency is independent of the probe intensity and can be close to 100% [10]. A pulsed probe field and a pulsed NDFWM field of considerably different frequencies are shown to evolve into a pair of matched solitons with the same temporal shape and ultraslow group velocity [11]. In this section, we present the theoretical studies of the propagation behaviors of the generated optical pulses in the four-level double-ladder configuration, as shown in Figs. 8.1 (c) & (d). The parameter δ in the figure is the controllable frequency detuning factor. Such four-level double-ladder system can be easily realized in rubidium or sodium atoms. This system has some features that combine the advantages of the double-Λ [see Fig. 8.1(a)] and four-level cascade [see Fig. 8.1(b)] sconfigurations. For example, if the level |3 is chosen to be near level |1, the NDFWM efficiency and pulse matching behaviors will be similar to the double-Λ configuration (with near degenerate frequencies for the probe and generated NDFWM signal beams). The two-photon Doppler-free condition for the NDFWM signal and the probe beam can be satisfied. However, if level |3 is chosen to be near level |2, then the NDFWM efficiency, as well as pulse matching behaviors, will

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approach that of the four-level cascade system [see Fig. 8.1(b)]. In this case, the frequency up-conversion can be realized, but the two-photon Doppler-free condition for having the EIT in the system is then lost. Under the appropriate (especially power balance) conditions, the probe and phase-matched NDFWM pulses (after a characteristic propagation length) can evolve into a pair of amplitude and group velocity matched pulses. Dual EIT windows for the probe and NDFWM beams are opened owing to an efficient oneand three-photon destructive interference involving the NDFWM beam and its back-reaction to the probe beam. The efficienciy for the forward and backward NDFWM configurations are calculated and compared. In this double-ladder system, there are four usual single-photon interference EIT configurations (i.e., two (ω1 + ω2 & ωf + ω3 ) counte-propagating ladder-types, a co-propagating (ω2 + ω3 )Λ-type and a (ω1 + ωf ) V-type subsystems), and two three-photon interference EIT configurations (i.e. threephoton ω1 + ω2 − ω3 and ωf , three-photon ωf + ω3 − ω2 and ω1 ). The forward NDFWM scheme [see Fig. 8.1(e)] with a maximum 50% efficiency is good for the Doppler-free requirements of all four typical EIT sub-systems [3] and the two three-photon interference EIT sub-systems. Specifically, two Dopplerfree schemes of three-photon interference EIT are k1 v − k2 v + k3 v ≈ kf v and kf v − k3 v + k2 v ≈ k1 v (v is a atomic velocity). While the backward NDFWM scheme [see Fig. 8.1(f)] with a maximum 100% efficiency can only satisfy the Doppler-free requirements of the two ladder-type EITs. Parameters for certain desired objectives in this double-ladder system are calculated and optimized. Also, by considering the effect of “back action” [the NDFWM process due to the existence of the generated signal, as indicated in Fig. 8.1(d)], the interplay between the NDFWM and the cross-phase modulation (XPM) conditions is carefully analyzed to better understand the underlying mechanisms in achieving good pulse matching between the probe beam and the signal beam. Finally, we present fully time-dependent adiabatic solutions and steady-state density-matrix analysis of the forward and backward NDFWM schemes in an ultraslow propagation regime, respectively. The analytical expressions are obtained for the pulsed probe beam, the NDFWM-generated pulse, competitions from the different linear and nonlinear contributions, ultraslow group velocities, and NDFWM efficiencies. The matched ultraslow propagations of the probe and NDFWM pulses are linked to the back-and-forth population transfer and coherent coupling in this double-ladder EIT system. The balance condition is very important for the matched probe and the NDFWM pulse propagation. It is found that a larger coupling field is good for the NDFWM generation, while a smaller coupling field is good for the ultraslow propagation of the NDFWM pulses. On the other hand, a larger pump field is good for the probe conversion, while a smaller pump field is good for the ultraslow propagation of the probe pulse. Two destructive interferences via one- and three-photon excitation pathways connecting the ground and terminal states strongly compete with each other, leading to simultaneous attenuations of the probe and the NDFWM pulse in

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265

this double-ladder system. After a characteristic propagation distance, the nonlinear absorption and the dispersion contributions of the cross-Kerr terms and the NDFWM terms, all cancel out with each other. As a consequence a pair of ultraslow, temporally, as well as group-velocity, the matched probe and NDFWM pulses is generated. This unique type of one- and three-photon induced transparencies and their consequences are qualitatively different from the standard EIT situation where the destructive interference occurs between two single-photon transition channels. These properties have been studied in Ref. [10 – 12]. The closed-loop NDFWM process can be greatly enhanced by the induced atomic coherences involving two destructive interferences via one- and three-photon pathways [see Figs. 8.1(c) & (d)]. The nonlinear effects are particularly strong when the four-photon closed-loop paths with resonant energy levels are possible [7 – 12]. We first consider lifetime-broadened fourlevel double-ladder atoms interacting with two CW coupling (ω2 ) and pump (ω3 ) fields. When a weak probe (ω1 ) laser pulse is injected into the system, a pulsed NDFWM field (ωf , satisfying the appropriate phase-matching condition) is then generated. This NDFWM field can acquire the same ultraslow group velocity and pulse shape as the probe pulse, and the maximum NDFWM efficiency in the forward scheme can be greater than 50%. When the generated NDFWM field is sufficiently intense (due to the high efficiency), efficient feedback to the NDFWM generating state becomes important. The internally generated feedback NDFWM field can provide an efficient suppression to the loss of the probe field (probe EIT) [see Fig. 8.1(d)]. This feedback pathway also leads to a competitive multi-photon excitation of the NDFWM generating state by three applied fields and one internally generated field [see Fig. 8.1(c)]. The strong competition is destructive in nature, so it results in induced transparency due to multi-photon destructive interference that efficiently suppresses the amplitude of the states involved. In the four-level double-ladder system depicted, as shown in Figs. 8.1(c) & (d), a weak probe laser (driving the transition |0−|1 with Rabi frequency G1 ) and a coupling laser (driving the transition |1 − |2 with Rabi frequency G2 ) form a standard ladder-type EIT sub-system. A pump laser drives the transition |2 − |3 with Rabi frequency G3 and makes a closed coherent ω

ω

−ω

−ωf

1 2 |1 −→ |2 −→3 |3 −→ |0, which results in the NDFWM path, |0 −→ generation of photons with wave vector kf at the frequency ωf . The required phase-matching condition is given by kf = k1 + k2 − k3 . When the generated NDFWM field is sufficiently intense, efficient feedback [|0 → |3 → |2 → |1, as shown in Fig. 8.1(d)] becomes important, which forms the second closed coherent NDFWM path and generates photons with the wave vector k1 at frequency ω1 . The feedback phase-matching condition is given by k1 = kf + ωf ω3 −ω |2 −→2 k3 − k2 . This NDFWM feedback excitation pathway (|0 −→ |3 −→ −ω1 |1 −→ |0) leads to a multi-photon induced transparency of the probe field through one- (|0 − |1) and three-photon (|0 → |3 → |2 → |1) destructive

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interference at |1. Furthermore, the back-and-forth population transfer and coherent coupling in this system cause the phase-matched coherent NDFWM field to have the same group velocity and pulse shape as that of the slowed probe field. We start with the atomic density-matrix equations of motion. By using the transformations of ρ10 (t) = ρ10 e−iω1 t , ρ20 (t) = ρ20 e−iω1 t−iω2 t , ρ30 (t) = ρ30 e−iω1 t−iω2 t+iω3 t , ρ13 (t) = ρ13 eiω2 t−iω3 t , ρ21 (t) = ρ21 e−iω2 t and ρ23 (t) = ρ23 e−iω3 t , we can write: ⎧ ∂ρ10 ⎪ ⎪ = −(iΔ1 + Γ10 )ρ10 + iG1 eik1 ·r ρ00 + iG∗2 e−ik2 ·r ρ20 − ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ iG1 eik1 ·r ρ11 − iGf eikf ·r ρ13 ⎪ ⎪ ⎪ ⎪ ∂ρ20 ⎨ = −(iΔa2 + Γ20 )ρ20 + iG2 eik2 ·r ρ10 + iG3 eik3 ·r ρ30 − ∂t (8.1) ⎪ ik1 ·r ikf ·r ⎪ iG e ρ − iG e ρ 1 21 f 23 ⎪ ⎪ ⎪ ⎪ ∂ρ30 ⎪ ⎪ = −(iΔf + Γ30 )ρ30 + iG∗3 e−ik3 ·r ρ20 + iGf eikf ·r ρ00 − ⎪ ⎪ ∂t ⎪ ⎪ ⎩ iG1 eik1 ·r ρ31 − iGf eikf ·r ρ33 Rabi frequencies of the NDFWM signal field, the probe field, the coupling field and the pump field are Gf = εf μf /, G1 , G2 , and G3 (Gi = εi μi /), respectively. The actual laser and NDFWM fields are Ei = εi eiki ·r−iωi t and Ef = εf eikf ·r−iωf t . The decoherence rate of the polarization ρi0 is denoted by Γi0 . The dipole moment between states |i and |j is μij , where μ10 = μ01 = μ1 , μ21 = μ12 = μ2 , μ32 = μ23 = μ3 , μ30 = μ03 = μf . The frequency detuning is Δi = Ωi − ωi , where Ωi is the corresponding atomic transition frequency. The close-cycled condition requires Δf = Δa2 − Δ3 , where Δa2 = Δ1 +Δ2 . A controllable detuning factor is defined by δ ≡ Ω2 −Ω3 . Equations (8.1a) – (8.1c) can first be solved under the nondepleted groundstate approximation, i.e., ρ00  1, and the approximations with weak probe and NDFWM fields in the four-level double-ladder system. The last two terms in Eq. (8.1) are higher-order terms with small quantities G1 and Gf for the field and the small atomic state amplitudes. In the weak signal treatment, these higher-order terms can be neglected, only keeping the energy conserving terms in equations of motion. So, Eq. 8.1 can then be rewritten as ⎧ ∂ρ10 ⎪ ⎪ = −(iΔ1 + Γ10 )ρ10 + iG1 eik1 ·r ρ00 + iG∗2 e−ik2 ·r ρ20 ⎪ ⎪ ∂t ⎪ ⎨ ∂ρ20 (8.2) = −(iΔa2 + Γ20 )ρ20 + iG2 eik2 ·r ρ10 + iG3 eik3 ·r ρ30 ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎩ ∂ρ30 = −(iΔf + Γ30 )ρ30 + iG∗ e−ik3 ·r ρ20 + iGf eikf ·r ρ00 3 ∂t

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8.1.1 Time-dependent, Adiabatic Treatment for Matched Probe and NDFWM Signal Pulses Under the standard ladder-type EIT conditions of G2 >> G1 and G3 >> Gf [3], the depletions of the strong (cw) coupling and the pump fields can be neglected. For the forward NDFWM scheme [see Fig. 8.1(e)], the propagation equations for the probe beam and the generated NDFWM signal, under the slowly-varying amplitude approximation, are [10 – 12] ⎧ ∂ ⎪ ⎪ ⎪ ⎨ ∂z +  ⎪ ∂ ⎪ ⎪ + ⎩ ∂z



i4πω1 N μ21 ρ10 ≡ iξ10 ρ10 ch  i4πωf N μ2f 1 ∂ ρ30 ≡ iξ30 ρ30 Gf = c ∂t ch 1 ∂ c ∂t

G1 =

(8.3)

where ξ10(30) ≡ 4πω1(f ) N μ21(f ) /ch. Taking the Fourier transforms of Eqs. (8.2) and (8.3), and under the non-depleted ground state approximation (i.e., ρ00  1, which holds well for the weak probe field), the resulting equation are ⎧ M1 G3 G∗2 ⎪ ⎪ D10 = Q1 + Qf ⎪ ⎪ M M ⎪ ⎪ ⎪ ⎨ N3 G2 N1 G3 Q1 − Qf D20 = − ⎪ M M ⎪ ⎪ ⎪ ⎪ ⎪ Mf G2 G∗3 ⎪ ⎩ Qf + Q1 D30 = M M

(8.4)

Here, D10 , D20 , D30 , Q1 , and Qf are the Fourier transforms of ρ10 , ρ20 , ρ30 , G1 , and Gf , respectively. Other redefined parameters are M1 ≡ M1 (ω) = 2 2 2 N2 N3 − |G3 | , Mf ≡ Mf (ω) = N1 N2 − |G2 | , M ≡ M (ω) = |G2 | N3 + 2 |G3 | N1 −N1 N2 N3 , N1 = ω +d1 , N2 = ω +d2 , N3 = ω +d3 , d1 = −Δ1 +iΓ10 , d2 = −Δa2 + iΓ20 , and d3 = −Δf + iΓ30 . Then, the following results are obtained for the forward probe and NDFWM schemes [see Fig. 8.1(e)] (with the initial conditions Q1 (z = 0, ω) and Qf (z = 0, ω) = 0 at the entrance): ⎧ ∂ ⎪ ⎪ ⎪ ⎨ ∂z −  ⎪ ∂ ⎪ ⎪ − ⎩ ∂z

iω c iω c



 Q1 = iξ10 

 Qf = iξ30

M1 G3 G∗2 Q1 + Qf M M



Mf G2 G∗3 Qf + Q1 M M



(8.5)

This calculation is similar to the one used in Ref. [10] for the double-Λ system. For comparison, the field propagation equations for the forward probe and backward NDFWM signal configuration [as shown in Fig. 8.1(f)] (with

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initial conditions Q1 (z = 0, ω) and Qf (z = l, ω) = 0 at the entrance) are ⎧    ∂ iω G3 G∗2 M1 ⎪ ⎪ ⎪ ⎨ ∂z − c Q1 = iξ10 M Q1 + M Qf (8.6)     ⎪ ∂ Mf iω G2 G∗3 ⎪ ⎪ + Q Q = −iξ + Q ⎩ f 30 f 1 ∂z c M M The propagation is through a medium of length l. The second terms on the right-hand side of Eqs. (8.5) and (8.6) represent the back-and-forth coupling between the probe field and the generated NDFWM field, while the first terms denote both linear and nonlinear absorptions and dispersions of the atomic medium for the respective fields. For a given Q1 (z = 0, ω) with condition Qf (z = 0, ω) = 0 in the forward NDFWM configuration [see Fig. 8.1(e)], Eq. (8.5) can be solved analytically to yield  Q1 (z, ω) = Q1 (0, ω)(ψ+ eizk− − ψ− eizk+ )/(ψ+ − ψ− ) (8.7) Qf (z, ω) = Q1 (0, ω)ψ+ ψ− (eizk− − eizk+ )/(ψ+ − ψ− )  where ψ± ≡ ψ± (ω) = ψ± (0) + o(ω) and k± ≡ k± (ω) = k± (0) + k± ω+ 2 o(ω ) in the adiabatic regime. Interesting physical insight can be gained by seeking their inverse Fourier transforms under the approximation of neglecting both o(ω 2 ) terms in k± (ω) and o(ω) terms in ψ± (ω). This is the adiabatic approximation and can be well justified under the condition of 2 2 |G2 | , |G3 | > Max(|d1 d2 | , |d2 d3 |), which can be easily satisfied for typical parameters [10,11]. The group velocities vg± for the optical pulses are deter mined by 1/vg± = Re(k± ) = [∂k± (ω) /∂ω]|ω=0 . Note that k± (0) = β± + iα± , where β± denote the phase shifts per unit length and α± are the absorption coefficients for the two propagation modes. These parameters are given by β+ ≈ −Δa2 α+ /Γ20 ≈ −Δa2 ξ10 ξ30 /A1 β− ≈ −A1 A2 /(A22 + A23 )

α− ≈ A1 A3 /(A22 + A23 ) with 2

2

A1 = ξ10 |G3 | + ξ30 |G2 | 2

2

A2 = Γ30 |G2 | + Γ10 |G3 | 2

A3 = −Δf |G2 | − Δ1 |G3 | Then, Eq. (8.7) can be recast into ⎧ + − iβ− z−α− z ⎪ G1 (z, t) = G− + ⎪ 1 (z, t) + G1 (z, t) = [G1 (0, t − z/vg )ϕ+ e ⎪ ⎪ ⎪ + iβ+ z−α+ z ⎨ ]/(ϕ+ + ϕ− ) G1 (0, t − z/vg )ϕ− e ⎪ + − iβ− z−α− z ⎪ − Gf (z, t) = G− ⎪ f (z, t) + Gf (z, t) = −ϕ+ ϕ− [G1 (0, t − z/vg )e ⎪ ⎪ ⎩ + iβ+ z−α+ z G1 (0, t − z/vg )e ]/(ϕ+ + ϕ− ) (8.8)

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where ± G± f (z, t) = ±ϕ± G1 (z, t)   z G1 0, t − ± ϕ∓ eiβ± z−α± z vg G± 1 (z, t) = ϕ+ + ϕ− G∗3 ϕ+ = ψ+ (0) ≈ ∗ G2 ξ30 G2 ϕ− = −ψ− (0) ≈ ξ10 G3 1 1 ξ10 ξ30 1 + > + ≈ c A1 c vg

The group velocities of the probe and the NDFWM signal pulses can show subluminal and superluminal behaviors at proper G2 and G3 values. In order to obtain efficient NDFWM processes in this system, the balance condition for the matched pulses (i.e., the probe and generated NDFWM pulses with the same ultraslow group velocity and matching pulse shape during propagation through the atomic medium) needs to be satisfied. The absorption coefficient α− is usually much larger than the absorption coefficient α+ [10, 11]. In this case, one of components in Eq. (8.8) decays much faster than the other. So, after a short characteristic propagation distance, Eq. (8.8) becomes Gf (z, t) ≈ G1 (z, t)ϕ+ ≈ G1 (0, t − z/vg+ )ϕ+ ϕ− eiβ+ z−α+ z /(ϕ+ + ϕ− ) (8.9) These results show that after a characteristic distance the matched forward probe and NDFWM pulses (with the ratio Gf (z, t)/G1 (z, t) ≈ ϕ+ ≈ G∗3 /G∗2 (i.e., the balance condition) to be a constant) propagate with the same ultraslow group velocity (vg+ << c) in this double-ladder system. The NDFWM efficiency for the forward configuration is defined as the ratio of the generated NDFWM field intensity at the exit face of z = l and the probe field intensity at the entrance face of z = 0, i.e.,   ωf ξ10   |Gf (z = l)|2 ξ10 ξ30 |G2 G3 |2 e−2α+ l If (z = l) δ ω1 ξ30 = ηf = ≈ 1 + I1 (z = 0) |G1 (z = 0)|2 Ω1 A21 where I1(f ) = |ε1(f ) |2 =

|G1(f ) |2 2 μ21(f )

The maximum NDFWM efficiency is achieved at 2

2

ξ10 |G3 | ≈ ξ30 |G2 |

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8 Multi-dressed MWM Processes

Then, it is straightforward to obtain the maximum NDFWM efficiency as 2

2

e−Γ20 ξ30 l/|G3 | (1 + δ/Ω 1 ) e−Γ20 ξ10 l/|G2 | (1 + δ/Ω 1 ) ηf ≈ ≈ 4 4 In the double-ladder system [see Figs. 8.1(c) – (d)], if δ  Ω 1 (Ω 2  Ω 1 +Ω 3 ) and e−2α+ l ≈ 1 (α+ ≈ 0), the NDFWM efficiency can reach ηf  50% for practical parameters. Under the balance conditions of Qf /Q1 ≈ ψ+ ≈ G∗3 /G∗2 , D10 = 0 and D30 = 0 are satisfied simultaneously for all ω [see Eq. (8.4)]. Two multiphoton destructive interferences occur between the excitation pathways for the states |1 and |3, as depicted in Figs. 8.1(c) and(d). Equation (8.7) can be reduced into Qf (z, ω) ≈ Q1 (z, ω)ψ+ ≈ −

Q1 (0, ω)ψ+ ψ− eizk+ ψ+ − ψ−

By inserting these results into Eq. (8.4) and carrying out its inverse Fourier transform, the probe field and the NDFWM signal field are G1  −G∗2 ρ20 and Gf  −G∗3 ρ20 , respectively. With these results and Eq. (8.2), it is straightforward to see that at the propagation distance where Eq. (8.9) is valid, ρ10  0 and ρ30  0 are satisfied simultaneously, and, therefore, the probe and the NDFWM fields will propagate in the medium without any further attenuation or amplification. Physically, this interesting phenomenon can be understood as follows. When deep inside the medium (so that G1  −G∗2 ρ20 and Gf  −G∗3 ρ20 are valid), two multi-photon (threephoton and one-photon) destructive interferences on the |1 and |3 states are simultaneously established through: (a) State |3 is excited through two pathω1 ω2 −ω |1 −→ |2 −→3 |3 via phase-matched G1 G2 G∗3 [see Fig. 8.1(c)] ways: |0 −→ ωf and |0 −→ |3 via feedback coupling of the generated Gf ; and (b) state |1 ωf

ω

−ω

3 is coupled through |0 −→ |3 −→ |2 −→2 |1 via Gf G3 G∗2 feedback exciω1 tation [see Fig. 8.1(d)] and |0 −→ |1 via probe beam G1 . These processes lead to simultaneous suppressions of the amplitudes of states |1 and |3 from these multi-photon destructive interferences through two pathways. This is qualitatively different from the standard EIT process where the destructive interference (the induced transparency) is between two one-photon channels. Before the destructive interference becomes effective, the medium is highly dispersive and also absorptive to both the probe and NDFWM fields, as can be seen from Eqs. (8.5) and (8.6) at the weak NDFWM (Gf ≈ 0), nondepleted coupling and pump limit. As the destructive interference rapidly builds up (for increasing Gf ) with increasing distance, both ρ10 and ρ30 are strongly suppressed, and the medium becomes highly transparent to both the probe and NDFWM fields. So, at a certain distance (or optical depth) in the medium, the probe and NDFWM pulses will evolve into a pair of temporally and group-velocity matched pulses that propagate without any

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further distortion [10 – 12]. It is remarkable that the generated NDFWM field can lead to such an efficiently induced transparency effect. From Eqs. (8.8a) and (8.8b), there are two mixed dressed states as ⎧ ⎪ G (z, t) = G1 (z, t) + Gf (z, t) ⎪ ⎪ + ⎪ ⎪ − + + ⎨ = [G− 1 (z, t) + Gf (z, t)] + [G1 (z, t) + Gf (z, t)] (8.10) ⎪ ⎪ (z, t) = G (z, t) − G (z, t) G ⎪ − 1 f ⎪ ⎪ ⎩ − + + = [G− 1 (z, t) − Gf (z, t)] + [G1 (z, t) − Gf (z, t)] First, since the absorption coefficient α− >> α+ is satisfied, the equations become ⎧ + ⎪ G+ (z, t) = G+ ⎪ 1 (z, t) + Gf (z, t) ⎪ ⎪ ⎪ ⎨ = G1 (0, t − z/vg+ )ϕ− eiβ+ z−α+ z (1 + ϕ+ )/(ϕ+ + ϕ− ) (8.11) ⎪ + + ⎪ ⎪ G− (z, t) = G1 (z, t) − Gf (z, t) ⎪ ⎪ ⎩ = G1 (0, t − z/vg+ )ϕ− eiβ+ z−α+ z (1 − ϕ+ )/(ϕ+ + ϕ− ) The maximum NDFWM efficiency is achieved at ξ10 |G3 |2 ≈ ξ30 |G2 |2 (ϕ+ ≈ 1) Equation (8.11) evolve into G+ (z, t) = 2G1 (0, t − z/vg+ )eiβ+ z−α+ z G− (z, t) = 0 Since both the probe G1 (z, t) and NDFWM Gf (z, t) photons have EIT, only one mixed dressed state G+ (z, t), or polariton of the matched probe and NDFWM pulses, survives after propagating through the atomic midium. Second, under different conditions of vg+ = vg− = vg , |α± | << 1, β+ z = 0, β− z = π, Δ2 = 0, Δ3 /Γ30 >> 1, and ϕ+ ≈ ϕ− , solutions for the propagation equations become G1 (z, t) ≈ 0 −2ϕ+ ϕ− G1 (0, t − z/vg ) Gf (z, t) = ϕ+ + ϕ− Then G+ (z, t) = −G− (z, t) = −2ϕ+ ϕ− G1 (0, t − z/vg )/(ϕ+ + ϕ− )

(8.12)

Finally, under yet another set of conditions with vg+ = vg− = vg , |α± | << 1, β+ z = β− z = 0, Δ2 /Γ20 >> 1, and Δ3 = 0, solutions are then G1 (z, t) = G1 (0, t − z/vg ) and Gf (z, t) = 0, satisfying conditions of   t−z G+ (z, t) = G− (z, t) = G1 0, (8.13) vg

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8 Multi-dressed MWM Processes

From the above discussions one can see that the perfect transmissions of the matched probe and NDFWM pulses are established at Δ1 = Δ2 = Δ3 = 0, i.e., all lasers are exactly on the unperturbed resonances. However, one can detune two cw strong lasers from their respective resonances to avoid the multi-photon destructive interferences. In the case when the constructive interference between two modes occurs in Eq. (8.7) or (8.8), the NDFWM conversion efficiency can reach close to 100% when the difference between vg+ and vg− is small, the absorption factor |α± | << 1, the difference of the phase shifts is π (i.e., β+ z = 0, β− z = π (or β+ z = π, β− z = 0)), Δ2 = 0, and Δ3 /Γ30 >> 1. Under these conditions the probe efficiency η1 = |G1 (z = l)|2 /|G1 (z = 0)|2 is close to zero, so all photons for the probe beam are converted to the signal photons through the NDFWM process. This case is actually very similar to the backward scheme to be discussed next. Now, the multi-photon destructive interference of state |3 no longer occurs and the propagating NDFWM field is strongly enhanced by the two-mode constructive interference, while the multi-photon destructive interference of state |1 may occur in this case. On the other hand, NDFWM conversion efficiency is close to zero when the phase shifts are the same, i.e., β+ z = β− z = nπ(n is integer), Δ2 /Γ20 >> 1, and Δ3 = 0. From Eq. (8.6), the field propagation equations for the forward probe and the backward NDFWM scheme [see Fig. 8.1(f)] can be rewritten as ⎧ ∂Q1 ⎪ ⎪ ⎨ ∂z = iθ1 Q1 + iθ2 Qf ⎪ ⎪ ⎩ ∂Qf = iθ3 Qf + iθ4 Q1 ∂z

(8.14)

where ω ξ10 M1 + M c ξ10 G3 G∗2 θ2 = M  ξ30 Mf ω + θ3 = − M c ∗ ξ30 G2 G3 θ4 = − M θ1 =

For given initial conditions of Q1 (z = 0, ω) and Qf (z = l, ω) = 0 at the entrance face in this backward NDFWM scheme, Eq. (8.14) can be solved analytically as ⎧    izk−  izk+   ⎨ Q1 (z, ω) = Q1 (0, ω)(ψ+ e − ψ− e )/(ψ+ − ψ− )     ⎩ Q (z, ω) = Q (0, ω)ψ  ψ  [eizk− +il(k+ −k− )   − eizk+ ]/(ψ+ − ψ− ) f 1 + −

(8.15)

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273

where  − θ1 k±  = ψ± (0) + o(ω) θ2 {(θ1 + θ4 ) ± [(θ1 − θ4 )2 + 4θ2 θ3 ]1/2 } (1)   k± = k± = (0) + k± ω + o(ω 2 ) 2

 = ψ±

After the inverse Fourier transformation, Eqs. (8.15a) and (8.15b) become ⎧ + ⎪ G1 (z, t) = G− ⎪ 1 (z, t) + G1 (z, t) ⎪ ⎪   ⎪ ⎪ = [G1 (0, t − z/vg− )ϕ+ eiβ− z−α− z + ⎪ ⎪ ⎪   ⎪ ⎨ G1 (0, t − z/vg+ )ϕ− eiβ+ z−α+ z ]/(ϕ+ + ϕ− ) ⎪ + ⎪ Gf (z, t) = G− ⎪ f (z, t) + Gf (z, t) ⎪ ⎪     ⎪ ⎪ = −ϕ+ ϕ− [G1 (0, t − z/vg− )eiβ− z−α− z eil(k+ −k− ) − ⎪ ⎪ ⎪   ⎩ G1 (0, t − z/vg+ )eiβ+ z−α+ z ]/(ϕ+ + ϕ− )

(8.16)

where vg+ = vg− ≈ vg = Re(k± ) = (1/c + ξ10 ξ30 /A1 )−1 (1)

  k± (0) = β± + iα±  ϕ+ = ψ+ (0)  ϕ− = −ψ− (0)

As discussed above, the NDFWM efficiency in the backward scheme is typically greater than that in the forward scheme and may even reach 100% under certain conditions. The balance condition may be satisfied throughout the entire medium [12], which is mainly due to the two-mode constructive interference. The backward NDFWM efficiency can be defined as the ratio of the generated NDFWM field intensity at its exit face of z = 0 and the probe field intensity at its entrance face of z = 0. This NDFWM efficiency can be calculated to be ηf = If (z = 0)/I1 (z = 0) = (ωf ξ10 /ω1 ξ30 )|Gf (z = 0)|2 /|G1 (z = 2 l/(2ξ30 Γ30 + ξ10 ξ30 l)] [14]. For a sufficiently high atomic 0)|2 ≈ (1 + δ/Ω1 )[ξ10 density or a long enough propagation distance (or a high optical density), the efficiency is close to 100%.

8.1.2 Steady-state Analysis After the optical pulses reach the balance condition of G1 (z, t)G∗3 ≈Gf (z, t)G∗2 through propagating in the medium, we can consider the high efficiency and ultraslow propagation behaviors of the probe and NDFWM fields in steady

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8 Multi-dressed MWM Processes

state. By setting ρ˙ 10 = ρ˙ 20 = ρ˙ 30 = 0 in Eq. (8.2), the density matrices can be solved to give  2 ρ10 = i[(iΔa2 + Γ20 )(iΔf + Γ30 )G1 + |G3 | G1 − G∗2 G3 Gf ]eik1 ·r /D ρ30 = i[(iΔ1 + Γ10 )(iΔa2 + Γ20 )Gf + |G2 | Gf − G∗3 G2 G1 ]eikf ·r /D (8.17) 2

where D = (iΔ1 + Γ10 )(iΔa2 + Γ20 )(iΔf + Γ30 ) + |G2 |2 (iΔf + Γ30 ) + |G3 |2 (iΔ1 + Γ10 ) The required phase-matching conditions are kf = k1 + k2 − k3 and k1 = kf + k3 − k2 for NDFWM G1 G2 G∗3 and for the feedback NDFWM Gf G3 G∗2 , respectively. Equation (8.17) show three key contributions in the propagation characteristics of the probe and NDFWM fields, i.e., (1) the linear response term; (2) the cross-Kerr nonlinear term; and (3) the phase-matched coherent NDFWM term, which determine the total linear and nonlinear absorptions and dispersions of the field in the atomic medium. When the generated NDFWM field is weak enough (i.e., Gf << |G1 ||G3 |/|G2 |, which is far away from the balance condition), Eq. (8.17) can be written as i[(iΔa2 + Γ20 )(iΔf + Γ30 )G1 + |G3 |2 G1 ]eik1 ·r D iG∗3 G2 G1 eikf ·r =− D

ρ10 = ρ30

As the NDFWM field gets stronger due to propagating in the medium, the ratio Gf (z, t) G∗ ≈ ϕ+ ≈ 3∗ (balanced condition) G1 (z, t) G2 approaches to a constant. Then, it is straightforward to obtain iG1 eik1 ·r (iΔa2 + Γ20 )(iΔf + Γ30 ) D iGf (iΔ1 + Γ10 )(iΔa2 + Γ20 )eikf ·r = D

ρ10 = ρ30

(i.e., the two nonlinear terms cancel with each other). The opposite signs of the cross-Kerr term and the NDFWM term in Eq. (8.17) show the normal and abnormal nonlinear dispersions, respectively. After a characteristic propagation distance, the nonlinear absorption and dispersion contributions from these two terms exactly cancel with each other. As a consequence, a pair of ultraslow, temporally, as well as group-velocity, the matched probe and NDFWM pulses are generated. The relations ε0 χ1 E1 = N μ10 ρ10 and ε0 χf Ef = N μ30 ρ30 define the susceptibilities of the probe and NDFWM fields, respectively. The group

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275

Fig. 8.2. Probe dispersions versus Δ1 with different G2 intensities, for (a) total probe dispersion; (b) linear part; (c) cross-Kerr part; and (d) NDFWM part of probe dispersion, respectively. Adopted from Ref. [13].

velocities of the probe (χ1 ) and the NDFWM (χf ) pulses are given by vg1 = c/[1 + ω1 (∂χ1 /∂ω1 )Δ1 =0 /2] vgf = c/[1 + ωf (∂χf /∂ωf )Δf =0 /2] [(∂χ1(f ) /∂ω1(f ) )Δ1(f ) =0 = −(∂χ1(f ) /∂Δ1(f ) )Δ1(f ) =0 ] Comparing with the result for the N-type system [9], here we have considered the additional NDFWM contribution in the probe group velocity, and obtained the NDFWM group velocity directly [see Figs. 8.2 – 8.4). Parameters in Fig. 8.2 are δ = 0, Δ2 = Δ3 = 0, Γ10 /π = 5.4 MHz, Γ20 /π = 1.8 MHz, Γ30 /π = 5.9 MHz, Gf /2π = 0.15 MHz, G3 /2π = 5 MHz, and G2 /2π = 7 MHz (solid curve), 10 MHz (dashed curve), 25 MHz (dotted curve), and 30 MHz (dash dotted curve), respectively. Next, we show that, under the appropriate condition, cross-Kerr modulation and NDFWM coupling can precisely balance the nonlinear group velocity dispersion in the ultraslow propagation regime, leading to the formation of the ultraslow matched probe and NDFWM pulses (i.e., vg1 ≈ vgf ). The total real part χr of the susceptibility includes both linear and nonlinear parts (cross-Kerr and NDFWM), i.e., χr = χrl + χrnl (χrnl = χKerr + χF W M ). Figures 8.2 and 8.3 show the competitions between these three dispersion slope contributions for the group velocities of the probe and NDFWM pulses. The NDFWM part always shows normal dispersion (slow light propagation) at resonance [(∂χFWM /∂ω1 )Δ1 =0 > 0], while the cross-Kerr part corresponds to anomalous dispersion (superluminal light propagation) in general ((∂χKerr /∂ω1 )Δ1 =0 < 0). The total dispersions in Fig. 8.2 are dominated by the linear contribution, while the total dispersions of Fig. 8.3 are dominated by the NDFWM part. The parameters in Fig. 8.3 are δ = 0, Δ1 = Δ2 = Δ3 = 0, Γ10 /π = 5.4 MHz, Γ20 /π = 1.8 MHz, Γ30 /π = 5.9 MHz, Gf /2π = 0.15 MHz, G2 /2π = 5 MHz, and G3 /2π = 3 MHz (solid

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8 Multi-dressed MWM Processes

Fig. 8.3. NDFWM field dispersion versus Δf with different G3 powers, for (a) total NDFWM dispersion; (b) linear part; (c) cross-Kerr part; and (d) NDFWM part of NDFWM field dispersion, respectively. Adopted from Ref. [13].

curve), 10 MHz (dashed curve), 25 MHz (dotted curve), and 100 MHz (dash dotted curve), respectively. The subluminal (normal dispersion) and superluminal (anomalous dispersion) characteristics of the probe and NDFWM pulses are shown in Fig. 8.4 with the parameters of δ = 0, G2 /Γ30 = 7, Ω1 /Γ30 = 2.7 × 106 , Δ1 = Δ2 = Δ3 = 0, Γ10 /Γ30 = 0.002, Γ20 /Γ30 = 0.7, and Gf /Γ30 = 0.15. The matched group velocities vg1 and vgf of the probe and NDFWM pulses exist at the conditions of G2 /Γ30 = 7 and G3 /Γ30 = 1.35 (see Fig. 8.4). The probe light prefers G2  G3 to have ultraslow propagation (see Fig. 8.2), while the NDFWM ultraslow light prefers G3  G2 (see Fig. 8.3).

Fig. 8.4. Matched group velocities vg1 (dashed curve) and vgf (solid curve) for probe and NDFWM pulses, respectively, versus G3 /Γ30 . Adopted from Ref. [13].

The induced dual transparency windows are created by the one- and threephoton destructive interferences between two different excitation pathways connecting |0 → |3 and |0 → |1 in this four-level double-ladder system

8.1 Matched Ultraslow Pulse Propagations in Highly-Efficient FWM

277

(Figs. 8.1(c) and (d)). The balance condition G1 (z, t)G∗3 ≈ Gf (z, t)G∗2 is obtained from inspecting the ratios in Eq. (8.9) after a sufficient long propagation distance in the medium. Then, it is straightforward to show that G1 eik1 ·r +G∗2 e−ik2 ·r ρ20  0, G2 eik2 ·r ρ10 +G3 eik3 ·r ρ30  0 and G∗3 e−ik3 ·r ρ20 + Gf eikf ·r  0. When these results are applied to Eq. (8.2) (note ρ00  1), the amplitudes of all three upper atomic states are strongly suppressed to have ρ10  ρ20  ρ30  0. Physically, when the NDFWM field is sufficiently intense an additional feedback excitation channel to the state |1 is opened via ωf ω3 −ω |0 −→ |3 −→ |2 −→2 |1 for the Gf G3 G∗2 NDFWM feedback excitation [see Fig. 8.1(d)]. This excitation is π out of phase with respect to the excitation ω1 |1, resulting in a significant suppression of state provided by G1 via |0 −→ |1, as indicated by G1 eik1 ·r + G∗2 e−ik2 ·r ρ20  0.

Fig. 8.5. Forward NDFWM conversion efficiency ηf = If (ξ10 z/Γ30 = 10)/I1 (z = 0) versus (a) pump Rabi frequency G3 and (b) coupling Rabi frequency G2 , respectively. Adopted from Ref. [13].

By letting G∗3 e−ik3 ·r ρ20  −Gf eikf ·r ρ00 in Eq. (8.2c), we find that at a sufficient depth in the atomic medium (where G∗3 e−ik3 ·r ρ20  −Gf eikf ·r is valid), ∂ρ30 /∂t = −[i(Δ1 + Δ2 − Δ3 ) + Γ30 ]ρ30 . This indicates that the two coupling terms in Eq. (8.2) interfere destructively after this point, and no further excitation can be made to the NDFWM generating the state |3. This means that when the generated NDFWM field is sufficiently intense, the absorption of the generated NDFWM field opens the second excitation pathway to state |3. This excitation is π out of phase with respect to the three-photon NDFWM excitation G1 G2 G∗3 to the same state, resulting in a destructive interference that suppresses further excitation to the state |3. Therefore, the production of the phase-matched NDFWM field saturates. When the NDFWM field becomes sufficiently intense, a small detuning Δf will also lead to a strong absorption of this generated wave via the one-photon process. Such a robust three-photon and one-photon destructive interference inevitably limits the maximum conversion efficiency achievable with the ultraslow-propagation technique. However, even with the three-photon destructive interference shown in the present double-ladder system, the

278

8 Multi-dressed MWM Processes

forward NDFWM conversion efficiency can still reach as high as 50% (see Figs. 8.5 – 8.8). We can detune the two strong cw lasers from their respective resonances in order to avoid these multi-photon destructive interferences. Under certain detuning conditions, the backward NDFWM efficiency can reach 100%, which results from constructive interference between the two propagating modes.

Fig. 8.6. Forward NDFWM efficiency ηf (z, G2 , G3 ) versus (a) G3 and ξ10 z/Γ30 , and (b) G2 and ξ10 z/Γ30 , or (c) G2 and G3 , respectively. Adopted from Ref. [13].

Fig. 8.7. Calculated forward NDFWM intensity s (solid curve) and probe intensity Ip (z)/I1 (z = 0) (dashed curve) versus ξ10 z/Γ30 . Adopted from Ref. [13].

Now, let us consider the steady-state field propagation equations. Assuming no depletions of the coupling (G2 ) and pump (G3 ) fields, the forward probe beam G1 and NDFWM signal Gf , as a function of distance, are  ∂G1 /∂z = iξ10 ρ10 = ξ10 (D2 Gf /D − D1 G1 /D) (8.18) ∂Gf /∂z = iξ30 ρ30 = ξ30 (D2∗ Gp /D − S1 Gf /D) where D1 = (iΔa2 + Γ20 )(iΔf + Γ30 ) + |G3 |2 D2 = G∗2 G3 S1 = (iΔ1 + Γ10 )(iΔa2 + Γ20 ) + |G2 |2

8.1 Matched Ultraslow Pulse Propagations in Highly-Efficient FWM

From these first-order differential equations, we can easily obtain  G 1 + [(S1 + D1 )/D]G1 − [(D22 − S1 D1 )/D2 ]G1 = 0 G f + [(S1 + D1 )/D]Gf − [(D22 − S1 D1 )/D2 ]Gf = 0

279

(8.19)

For a given G1 (z = 0) and with Gf (z = 0) = 0 in the forward NDFWM configuration [see Fig. 8.1(e)], Figs. 8.5–8.8 show the result from the steadystate numerical solutions. For comparison, the field propagation equations for the forward probe and the backward NDFWM [see Fig. 8.1(f)] are ∂G1 /∂z = iξ10 ρ10 = ξ10 (D2 Gf /D − D1 G1 /D) ∂Gf /∂z = −iξ30 ρ30 = −ξ30 (D2∗ Gp /D − S1 Gf /D) The typical initial conditions are for the given G1 (z = 0) and Gf (z = l) = 0 at the entrance face of the medium.

Fig. 8.8. Calculated forward NDFWM intensity If (z)/I1 (z = 0) (solid curve) and probe intensity Ip (z)/I1 (z = 0) (dashed curve) versus ξ10 z/Γ30 , respectively. Adopted from Ref. [13].

The efficiently generated NDFWM field can acquire the same ultraslow group velocity and pulse shape of the probe pulse and the maximum forward NDFWM efficiency can be greater than 50% in the four-level doubleladder system (as shown in Figs. 8.5 – 8.8). From the simulated results (see Fig. 8.9), it is clear that the high NDFWM efficiency prefers the backward scheme [see Fig. 8.1(f)], with smaller coupling and pump frequency detunings, and larger coupling and pump Rabi frequencies. Parameters for Fig. 8.9 are δ = 0, Δ1 /Γ30 = 0, Δ2 /Γ30 = Δ3 /Γ30 = 15, Γ10 /Γ30 = Γ20 /Γ30 = 1, and G2 /Γ30 = G3 /Γ30 = 150. The pulse matching for the probe and NDFWM pulses needs sufficient propagation distance (or optical depth) in the atomic medium (see Figs. 8.6 – 8.9). Specifically, large G2 is good for NDFWM generation, while large G3 is good for the probe field conversion (see Figs. 8.5 and 8.6). The maximal achievable NDFWM efficiency (as shown

280

8 Multi-dressed MWM Processes

Fig. 8.9. (a) Backward NDFWM intensity If (z = 0)/I1 (z = 0) (solid curve) and probe intensity Ip (z = l)/I1 (z = 0) (dashed curve) versus ξ10 l/Γ30 ; (b) backward NDFWM intensity If (z)/I1 (z = 0) and (solid curve) probe intensity Ip (z)/I1 (z = 0) (dashed curve) versus ξ10 z/Γ30 (ξ10 l/Γ30 = 15). Adopted from Ref. [13].

in Figs. 8.5 and 8.6) qualitatively correlates with slower group velocities of the pulses which has also been demonstrated in the four-level double-Λ system [8,9]. Parameters for Fig. 8.5 are δ = Ω 1 , Δ1 = Δ2 = Δ3 = 0, Γ10 /π = 5.4 MHz, Γ20 /π = 1.8 MHz, Γ30 /π = 5.9 MHz, and (a) G2 /2π = 2 MHz (solid curve), 3MHz (dashed curve), and 4MHz (dotted curve); (b) G3 /2π = 2 MHz (solid curve), 3MHz (dashed curve), and 4MHz (dotted curve), respectively. Parameters for Fig. 8.6 are δ = Ω 1 , Δ1 = Δ2 = Δ3 = 0, Γ10 /π = 5.4 MHz, Γ20 /π = 1.8 MHz, Γ30 /π = 5.9 MHz, and (a) G2 /2π = 8 MHz and (b) G3 /2π = 8 MHz, respectively. Due to factors sin(Δ−1 i z) and cos(Δ−1 z), obtained from Eq. (8.9) [10], the spatial period in the probe and i NDFWM propagation curves increases, and NDFWM generation efficiency gets to be extremely low when Δi is increased (see Fig. 8.8). Parameters in Fig. 8.8 are Δ1 /Γ30 = 0 and Γ10 /Γ30 = Γ20 /Γ30 = 1, with (a) δ = Ω 1 , Δ2 /Γ30 = 6.5, Δ3 /Γ30 = 15 and G2 /Γ30 = G3 /Γ30 = 150; (b) δ = 0, Δ2 /Γ30 = 200, Δ3 /Γ30 = 0 and G2 /Γ30 = G3 /Γ30 = 100, respectively. If Δ2 /Γ30 = Δ3 /Γ30 = 15, only a relatively short propagation distance is needed to establish matched pulses, and the spatial oscillation also disappears [see Fig. 8.7(b)]. Parameters in Fig. 8.7 are δ = Ω 1 , Δ1 /Γ30 = 0 and Γ10 /Γ30 = Γ20 /Γ30 = 1, with (a) Δ2 /Γ30 = Δ3 /Γ30 = 0 and G2 /Γ30 = G3 /Γ30 = 5; (b) Δ2 /Γ30 = Δ3 /Γ30 = 15 and G2 /Γ30 = G3 /Γ30 = 150, respectively. This is similar to the case of Δ2 /Γ30 = Δ3 /Γ30 = 0 [12]. Since transparencies for the probe and NDFWM fields are degraded, Figs. 8.7(a) and 8.8(b) with a small G2 (G3 ) or a large frequency detuning show the decay effect. Larger G2 is good for the NDFWM generation, while smaller G2 is good for NDFWM ultraslow propagation. However, larger G3 is good for the probe conversion, while smaller G3 is good for the probe ultraslow propagation. For a specific double-ladder scheme, balance parameters can be chosen to produce highly efficient and ultraslow matched pulse pairs of the probe and NDFWM fields. Actually, when the two pulses are matched,

8.1 Matched Ultraslow Pulse Propagations in Highly-Efficient FWM

281

the group velocities are slowed, but not necessarily minimized, and can generate the highest conversion efficiency. However, if one wants one of fields (either probe or NDFWM) to propagate very slowly, then the other field will propagate faster, which will reduce the conversion efficiency.

8.1.3 Discussion and Outlook The matched ultraslow propagation of the probe and NDFWM is caused by the back-and-forth population transfer and coherent coupling in this fourlevel double-ladder system. One possible experimental candidate for this proposed double-ladder system is in the Na atoms with energy levels |0 = |3S1/2 , |1 = |3P1/2 , |2 = |7D3/2 , and |3 = |4P3/2 . The respective transitions are |0 → |1 at 590 nm (a weak pulsed probe laser) (Γ−1 1  16.9ns, Γ−1 10  5.7 ps), |1 → |2 at 449 nm (a strong cw or long pulsed coupling laser), |3 → |2 at 1.12 μm (a strong cw or long pulsed pump laser), and |0 → |3 at 330 nm (the generated UV NDFWM short pulse radiation). This experimental system will not be very good for the two-photon Dopplerfree conditions (to generate EITs) in an atomic vapor, since the frequency difference between ω3 and ωf is too big, but it is good for UV light generation [similar to the four-level cascade system as in Fig. 8.1(b)]. Another possible experimental candidate for the proposed double-ladder system is in 87 Rb atoms with energy levels |0 = |5S1/2 , |1 = |5P1/2 , |2 = |5D3/2 , and |3 = |5P3/2 . The respective transitions are |0 → |1 at 795 nm (a weak short-pulse probe laser) (γ10  5.4 MHz, where γij is term due to spontaneous emission (longitudinal relaxation rate) between states |i and |j), |1 → |2 at 762 nm (a strong cw or long-pulse coupling laser) (γ21  0.8 MHz), |3 → |2 at 776 nm (a strong cw or long-pulse pump laser) (γ23  0.97 MHz), and |0 → |3 at 780 nm (the generated NDFWM short-pulse radiation) (γ30  5.9 MHz). This system behaves similar to the four-level double-Λ system (with a pair of transitions flips up) and can easily satisfy the two-photon Dopplerfree configurations for EITs in a hot Rb vapor [3,14]. For comparisons with results in other systems [10 – 12], we adopted near-unity efficiency in plotting Figs. 8.7 – 8.9. However, realistic experimental parameters were used (for 87 Rb atom) in plotting Figs. 8.5 and 8.6. The transverse relaxation rate Γij between states |i and |j can be obtained by Γij = (Γi + Γj )/2 (Γ0 = 0, Γ1 = γ10 , Γ2 = γ21 + γ23 , Γ3 = γ30 ). It is interesting to consider the competition between the NDFWM process (given by G1 G2 G∗3 ) and the feedback NDFWM process (given by Gf G3 G∗2 ) in this four-level double-ladder atomic system with two counter-propagating beam pairs [i.e., the probe beam ω1 and the coupling beam ω2 as one ladder system, and the pump beam ω3 and the generated NDFWM beam ωf as another ladder system in Figs. 8.1(e) and (f)]. The forward NDFWM scheme (with NDFWM efficiency 50%), as shown in Fig. 8.1(e), is good for

282

8 Multi-dressed MWM Processes

matched pulse pair propagation; while the backward NDFWM scheme (with NDFWM efficiency≈100%), as shown in Fig. 8.1(f), is good for the NDFWM generation. Since this four-level double-ladder system has features that combine advantages of the double-Λ [see Fig. 8.1(a)] and the four-level cascade [see Fig. 8.1(b)] configurations, it can be used to optimize the system parameters for different desired objectives. Entangled photon pairs (or correlated bright beams) for the matched probe and NDFWM pulses can be generated in this double-ladder system [13]. The maximum entanglement between the wellmatched probe and NDFWM pulses (that propagate with the same ultraslow group velocity) can only be obtained after a characteristic propagation distance (or certain optical depth) in the atomic medium. This system with the matched probe and NDFWM pulses (with high degree of intensity correlation, or bright squeezing) is an ideal candidate as the quantum correlated photon source for quantum information processing and quantum networking [15]. As have shown in the above discussions, suppressing the linear absorptions of the probe and NDFWM fields (via double EIT windows) facilitates the enhanced NDFWM efficiency in this double-ladder system. After a characteristic propagation distance, the nonlinear absorption and dispersion contributions of the cross-Kerr term and the NDFWM term cancel out with each other. As a consequence, a pair of ultraslow, the temporally (as well as group-velocity) matched probe and NDFWM pulses are generated in the medium. The generations of the phase-matched coherent NDFWM field and its feedback NDFWM process are limited by the two destructive interferences between the three-photon and one-photon excitation channels. The maximally generated NDFWM field is achieved at the balance condition of Gf (z, t)/G1 (z, t) ≈ G∗3 /G∗2 . These multi-photon destructive interferences can be avoided by detuning the two strong cw (coupling and pump) lasers away from their respective resonances. For certain parameters, the backward NDFWM efficiency can reach 100%, caused by the constructive interference between two modes. The balance condition may be satisfied throughout the whole medium in the backward NDFWM scheme. To get the matched pulses (or correlated photon pairs), one will want to have 50% each for the probe and the NDFWM pulses, so the forward scheme with two multi-photon destructive interferences is better for this purpose. There are three configurations to generate the entangled photon pairs in this system: (1) two multi-photon destructive interferences occur in the states |1 and |3; (2) the multi-photon destructive interference of the state |3 no longer occurs and the NDFWM field is strongly enhanced by constructive interference of the two propagating modes; (3) the multi-photon destructive interference of the state |1 no longer occurs and the propagating probe field is strongly enhanced by constructive interference of the two modes. These three modes of generating entangled NDFWM and probe photon pairs can be switched from one to the other by the frequency detuning which might give rise to a potential application in

8.2 Generalized Dressed and Doubly-dressed MWM Processes

283

quantum memory. Although the propagation characteristics in this four-level double-ladder system with pulsed probe and generated NDFWM fields are fundamentally different from the case for energy exchange between the generated FWM and SWM signal fields, as demonstrated in Section 7.2, some features are in common. For example, certain propagation distance (or optical depth) is required to reach equilibrium conditions for the probe and the generated signal fields. Paired beams (or pulses) with the same intensity can be generated during their strong interactions with the coherently-prepared atomic medium. More research works, especially experiments, are needed to fully understand the propagation effects of nonlinearly generated optical signals in multi-level atomic systems.

8.2 Generalized Dressed and Doubly-dressed MWM Processes In studying multi-FWM processes in four-level atomic systems (Chapters 6 – 7), it is often convenient to describe the atom-field interactions in the dressed-state picture, i.e., by considering FWM in a three-level sub-system and the additional strong coupling/pumping field(s) connecting another (additional) transition as the dressing field(s), as demonstrated in Sections 6.1 and 6.4[16] and 7.1 and 7.2 [17]. By taking into account interference between two dressed-state channels for the FWM processes, enhancement or suppression of the total FWM signal intensity can be obtained [17]. The dressedFWM systems can also generate coexisting FWM and SWM processes [18]. In the previously considered dressing situations, the dressing schemes are quite simple with only one dressing field (or two dressing fields with same frequency driving the same transition), which is called the “singly-dressing” scheme. However, doubly-dressed scheme has been applied to the N-type four-level atomic system with the metastable excited state showing sharp dark resonance due to the destructive interference between the secondarilydressed states [19]. Also a triple-peak absorption spectrum was observed in a doubly-dressed four-level N-type atomic system, which exhibits constructive interference due to the decoherence of the Raman coherence [20]. However, the constructive interference has been shown to occur between two FWM excitation paths of the doubly-dressed states in a five-level system [17]. These high-order multi-photon interferences and light-induced atomic coherence are very important in nonlinear wave-mixing processes, and might be used to open and optimize multi-channel nonlinear optical processes in multi-level atomic systems that are otherwise closed due to high absorption [16 – 20]. As the order of the nonlinearity increases, more complex beam geometries are usually required to satisfy the phase-matching conditions. Also, the nonlinear signal decreases by several orders of magnitude with an increase in

284

8 Multi-dressed MWM Processes

each order of the nonlinearity of the interaction [21]. Since higher-order nonlinear optical processes are usually much smaller in an amplitude than lower order ones, the interplays between nonlinear optical processes of different orders, if it exists, are usually very difficult to observe. In recent years, many schemes have been developed to enhance higher-order the nonlinear wavemixing processes. More importantly, with induced the atomic coherence and interference, the higher-order processes (such as SWM) can become comparable or even greater in an amplitude than the lower order wave-mixing processes (such as FWM), as we have described in the previous chapters. In this section we describe a generalized scheme for resonantly dressed (2n − 2) wave mixing [denoted as (2n − 2)WM] and doubly-dressed (2n − 4) wave mixing [denoted as (2n − 4)WM] processes in a (n + 1)-level atomic system. Co-existing FWM, SWM, and eight-wave mixing (EWM) processes will be considered in a close-cycled five-level folded system as one example (for n = 4) of this generalized doubly-dressed (2n − 4)WM system. Investigations of such intermixing and interplays between different types of the nonlinear wave-mixing processes will help us to understand and optimize the generated high-order multi-channel nonlinear optical signals.

8.2.1 Generalized Dressed-(2n–2)WM and Doubly-dressed(2n–4)WM Processes For a close-cycled (n+1)-level cascade system (as shown in Fig. 8.10), where states |i − 1 to |i are coupled by laser field Ei (Ei ) [ωi , ki (ki ), and Rabi frequency Gi (Gi )]. The Rabi frequencies are defined as Gi = εi μij / Gi = εi μij / where μij are the transition dipole moments between level i and level j. Fields  with the same frequency propagate along beam 2 and beam En−2 and En−2 3 with a small angle [see Fig. 8.10(a)]. Fields E2 , E3 to En−3 propagate along the direction of beam 2, while a weak probe field E1 (beam 1) propagates along the opposite direction of beam 2. The simultaneous interactions of atoms with fields E1 , E2 to En−2 will induce the atomic coherence between |0 and |n − 2 through resonant (n − 2)-photon transitions. This (n − 2) , and En−3 to E2 , and as a photon coherence is then probed by fields En−2 (2n−5) result a (2n − 4)WM (ρ10 ) signal of frequency ω1 in beam 4 is generated almost opposite to the direction of beam 3, satisfying the phase-matching  . When two strong dressing fields En−1 condition k2n−4 = k1 + kn−2 − kn−2 and En are used to drive the transitions |n − 2 to |n − 1 and |n − 1 to |n, respectively, as shown in Figs. 8.1 (b), there exist one doubly-dressed   (2n−5) (2n−3) (2n − 4)WM (ρ10 ), one singly-dressed (2n − 2)WM (ρ10 ) and one (2n−1) ) processes, satisfying the same k2n−4 . 2nWM (ρ10

8.2 Generalized Dressed and Doubly-dressed MWM Processes

285

Fig. 8.10. (a) Schematic diagram of phase-conjugate doubly-dressed (2n − 4)WM. (b) energy-level diagram for doubly-dressed (2n − 4)WM in a closed-cycle (n + 1)level cascade system.

To quantitatively understand such phenomenon of interplay between coexisting 2nWM, dressed-(2n − 2)WM and doubly-dressed-(2n − 4)WM processes, we need to use perturbation chain expressions involving all the (2n−1) (2n−3) (2n−5) ρ10 , ρ10 , and ρ10 nonlinear wave-mixing processes for arbitrary field strengths of Ei . The simple (2n − 4)WM via Liouville pathway (n−2) −ωn−2

ωn−2

(0) ω

−ω

(2n−5)

1 (Cn−2 )ρ00 −→ · · · −→ ρn−2,0 −→ · · · −→2 ρ10

gives (2n−5)

=

ρ10

i(−1)n+1 Ga Gn−2 (Gn−2 )∗ eik2n−4 ·r d21 d22 · · · d2n−3 dn−2

where Ga = G1 |G2 |2 · · · |Gn−3 |2 di = Γi0 + i(Δ1 + Δ2 + · · · + Δi ) with Δi = Ω i − ωi . Γi0 is the transverse relaxation rate between states |i and |0. Similarly, we can easily obtain (2n−5)

(2n−3)

=−

(2n−1)

=

ρ10 ρ10

|Gn−1 |2 ρ10 dn−2 dn−1

(2n−5)

ρ10

|Gn−1 |2 |Gn |2 dn−2 d2n−1 dn

via perturbation chains ωn−1

(0) ω

(n−1) −ωn−1

−ω

(2n−3)

1 · · · −→ ρn−1,0 −→ · · · −→2 ρ10 (Cn−1 )ρ00 −→

(0) ω

ω

(n) −ω

−ω

(2n−1)

1 n (Cn )ρ00 −→ · · · −→ ρn,0 −→n · · · −→2 ρ10

The non-dressed generalized 2nWM with phase-conjugate geometry has also been considered in an (n+1)-level system [22]. When both fields En−1 and En are turned on, there exist three physical mechanisms of interest. First, the (2n− 4)WM process will be dressed by two

286

8 Multi-dressed MWM Processes

strong fields En−1 and En and a perturbative approach for such interaction can be described by following coupled equations: ∂ρn0 /∂t = −dn ρn0 + iGn eikn ·r ρ(n−1)0

(8.20)

∂ρ(n−1)0 /∂t = −dn−1 ρ(n−1)0 + iGn−1 eikn−1 ·r ρ(n−2)0 + iG∗n e−ikn ·r ρn0

(8.21)

∂ρ(n−2)0 /∂t = −dn−2 ρ(n−2)0 + iGn−2 eikn−2 ·r ρ(n−3)0 + iG∗n−1 e−ikn−1 ·r ρ(n−1)0

(8.22)

In the steady state, Eqs. (8.20) – (8.22) can be solved together with perturbation chain (Cn−2 ) to give the doubly-dressed-(2n − 4)WM ρ 10

(2n−5)

=

i(−1)n+1 Ga Gn−2 (Gn−2 )∗ eik2n−4 ·r (dn−1 dn + |Gn |2 ) {d21 d22 · · · d2n−3 [dn−2 (dn−1 dn + |Gn |2 ) + dn |Gn−1 |2 ]} 2

2

Under the condition of |Gn | << Γn0 Γ(n−1)0 and |Gn−1 | << Γ(n−1)0 Γ(n−2)0 , (2n−5) can be expanded to be ρ 10   ⎧ |Gn−1 |2 |Gn−1 |2 |Gn |2 1 (2n−5) ⎪  (2n−5) ⎪ = dn−2 ρ10 − 2 + 2 ⎪ ⎪ ρ 10 dn−2 dn−2 dn−1 dn−2 d2n−1 dn ⎪ ⎨ (8.23) (2n−5) (2n−5) (2n−5) (2n−3) (2n−1) (2n−1) ⎪ = ρ10 + ρ10 + ρ10 = ρ 10 + ρ10 ρ 10 ⎪ ⎪ ⎪ ⎪ ⎩ (2n−3) (2n−5) = ρ10 + ρ 10 This expansion shows that, the doubly-dressed-(2n − 4)WM process converts to a coherent superposition of signals from (2n − 4)WM, (2n − 2)WM (2n−5) (2n−3) (2n−1) and 2nWM (ρ10 + ρ10 + ρ10 ), or dressed-(2n − 4)WM and 2nWM (2n−1) (2n−5)  (2n−5) + ρ10 ), or (2n − 4)WM and dressed-(2n − 2)WM (ρ10 + (ρ 10  (2n−3) ) in the weak dressing field limit. ρ 10 (2n−3) term in Eq. (8.23) results from the (2n − 2)WM Second, the ρ 10 process dressed by the strong field En and a perturbative approach for such interactions can be described by the following coupled equations: ⎧ ikn ·r ⎪ ρ(n−1)0 ⎪ ⎨ ∂ρn0 /∂t = −dn ρn0 + iGn e (8.24) ∂ρ(n−1)0 /∂t = −dn−1 ρ(n−1)0 + iGn−1 eikn−1 ·r ρ(n−2)0 + ⎪ ⎪ ⎩ ∗ −ikn ·r ρ iG e n0

n

In the steady state, Eq. (8.24) can be solved together with the perturbation chain (Cn−1 ) to give ρ 10

(2n−3)

=−

(2n−5)

|Gn−1 |2 dn ρ10 [dn−2 (dn dn−1 + |Gn |2 )]

8.2 Generalized Dressed and Doubly-dressed MWM Processes

Under the condition of |Gn |2 << Γn0 Γ(n−1)0 , ρ 10

(2n−3)

(2n−3) ρ 10

(2n−3) ρ10

287

can be expanded fur-

(2n) ρ10 ,

ther to be ≈ + and the dressed-(2n − 2)WM process converts to a coherent superposition of signals from (2n − 2)WM and 2nWM. (2n−5) term in Eq. (8.23) results from (2n − 4)WM process Third, the ρ 10 dressed by the strong field En−1 . Similarly, we can obtain ρ 10

(2n−5)

(2n−5)

=

dn−2 ρ10 dn−1 (dn−1 dn−2 + |Gn−1 |2 )

via the En−1 coupled equations and the perturbation chain (Cn−2 ). Under (2n−5) |Gn−1 |2 << Γ(n−1)0 Γ(n−2)0 , ρ 10 can also be expanded to be ρ 10

(2n−5)

(2n−5)

≈ ρ10

(2n−3)

+ ρ10

and the dressed-(2n − 4)WM process converts to a coherent superposition of signals from (2n − 4)WM and (2n − 2)WM.

8.2.2 Interplays Among Coexisting FWM, SWM, and EWM Processes One case (n = 4) of the generalized doubly-dressed-(2n − 4)WM system described above can be employed, as an example, to study the intermixing and interplays between FWM, SWM and EWM processes (see Table 8.1). The laser beams are aligned spatially in the pattern, as shown in Fig. 8.11(a), with seven beams (E1 , E2 , E2 , E3 , E3 , E4 , E4 ) propagating through the atomic medium with small angles between them in a square-box pattern. For a close-cycled folded five-level system, Figs. 8.11(b) – (f) generally correspond to cases of blocking beams E2 and E3 (EWM) [see Fig. 8.11(b)], E2 and E4 (dressed-SWM) [see Fig. 8.11(c)], or E3 and E4 (doubly-dressed-FWM) [see (3) Fig. 8.11(e)], respectively. However, the doubly-dressed FWM (ρ 10 ) process [see Figs. 8.11(e) and (f)] converts to a coherent superposition of signals from (5) (3) (5) (7) FWM (ρ10 ), SWM (ρ10 ), and EWM (ρ10 ). The dressed-SWM (ρ 10 ) process [see Figs. 8.11(c) and (d)] converts to a coherent superposition of signals from FWM and SWM in the weak dressing field limit. Under the conditions of |G3 |2 << Γ30 Γ20 and |G4 |2 << Γ10 Γ40 , Eq. (8.23) reduces to ρ 10 =

−Ga (d3 d4 + |G4 |2 )

(3)

≈

d21 [d2 (d3 d4 + |G4 |2 ) + |G3 | d4 ] 2

2

2

−Ga Ga |G3 | −Ga |G3 | |G4 |2 + 2 2  + 2 d1 d2 d1 d2 d3 d21 d22 d 23 d4

= ρ10 + ρ10 + ρ10 = ρ 10 + ρ10 = ρ10 + ρ 10 (3)

(5)

(7)

(3)

(7)

(3)

(5)

288

8 Multi-dressed MWM Processes

Fig. 8.11. (a) Three-dimensional beam geometry(b) five-level atomic system for EWM process; (c) System for dressed SWM process and (d) corresponding dressedstate picture; (e) system for doubly-dressed FWM process and (f) corresponding dressed-state picture.

where, ρ10 = −Ga /d21 d2 , ρ10 = Ga |G3 |2 /(d21 d22 d3 ), ρ10 = −Ga |G3 |2 |G4 |2 / (3) (5) 2 (d21 d22 d 3 d4 ), ρ 10 = −Ga [1 − |G3 |2 /(d3 d2 )]/(d21 d2 ), and ρ 10 = Ga |G3 |2 [1 − 2  2 2    |G4 | /(d3 d4 )]/(d1 d2 d3 ) with kF = k1 +k2 −k2 , d3 = Γ30 +iΔa3 , Δa3 = Δa2 −Δ3 , Δa2 = Δ1 + Δ,2 and Ga = iG1 G2 (G2 )∗ eikF ·r . (3)

(5)

(7)

Table 8.1. Phase-matching conditions and perturbation chains of EWM, dressedSWM and doubly-dressed-FWM in a close-cycled five-level system. Doubly dressed (3) FWM ρ 10 kF = k1 + k2 − k2

(5)

None None

(7)

Dressed SWM ρ 10

EWM ρ10

kS = k1 + k2 − k2 + k3 − k3 kS = k1 + k2 − k2 + k3 − k3 None

kE = k1 + k2 − k2 + k3 − k3 + k4 − k4 kE = k1 + k2 − k2 + k3 − k3 + k4 − k4 kE = k1 + k2 − k2 + k3 − k3 + k4 − k4

Perturbation chains (3)

(0)

ω

(1)

ω

(2) −ω

(3)

(5)

(0)

ω

(1)

ω

(2) −ω

(3)

ω

(4) −ω

(5)

(7)

(0)

ω

(1)

ω

(2) −ω

(3)

ω

(4) −ω

(5)

1 2 ρ10 : ρ00 −→ ρ10 −→ ρ20 −→2 ρ10

1 2 3 ρ10 : ρ00 −→ ρ10 −→ ρ20 −→3 ρ30 −→ ρ20 −→2 ρ10

ω

(6) −ω

(7)

1 2 4 3 ρ10 : ρ00 −→ ρ10 −→ ρ20 −→3 ρ30 −→ ρ40 −→4 ρ30 −→ ρ20 −→2 ρ10

(3)

(0)

ω

(1)

ω

−ω

(2)

(3)

1 2 ρ 10 : ρ00 −→ ρ10 −→ ρ(G4 ±G3 ±)0 −→2 ρ10

(5)

(0)

ω

(1)

ω

(2) −ω

(3)

ω

(4) −ω

(5)

1 2 3 ρ 10 : ρ00 −→ ρ10 −→ ρ20 −→3 ρ(G4 ±)0 −→ ρ20 −→2 ρ10

8.2 Generalized Dressed and Doubly-dressed MWM Processes

289

Let us look at the dressed-SWM spectrum versus Δa3 /Γ20 or Δ4 /Γ30 . Figure 8.12(a) shows that as the dressed field G4 is increased a dip appears at the line center first, then the spectrum splits into two separate peaks. Parameters are Γ30 /Γ20 = 1, Γ10 /Γ20 = 0.5, Δ4 /Γ20 = 6, and G4 /Γ20 = 0 (solid curve), G4 /Γ20 = 2 (dash curve), G4 /Γ20 = 10 (dot curve), G4 /Γ20 = 20 (dash dot curve). This is a typical Autler-Townes (AT) splitting [The left and right peaks of Fig. 8.12(a) correspond to the |+ and |− levels created by the dressed field G4 in Fig. 8.11(d), respectively]. Two peaks are located asymmetrically due to Δ3 = 0. Figures 8.12(b) and (c) present the suppression and enhancement of the dressed SWM signal intensity. The SWM signal intensity with no dressing field is normalized to 1. At the exact three-photon resonance Δa3 = 0, we see that the SWM signal intensity is suppressed when the frequency of the dressing field is scanned across the resonance (Δ4 = 0). The presence of the weak dressing field can either suppress or enhance the SWM signal when Δa3 = 0 [see Fig. 8.12(b)]. Parameters in Fig. 8.12(b) are Γ20 /Γ30 = 1, Γ10 /Γ30 = 0.5, G4 /Γ30 = 0.5, and Δa3 /Γ30 = 0 (solid curve), Δa3 /Γ30 = −0.5 (dash curve), Δa3 /Γ30 = −2 (dot curve), Δa3 /Γ30 = −6 (dash dot curve). Such suppression and enhancement mainly result from the absorption and dispersion of SWM and EWM signals and their interference. When G4 /Γ30 = 50 the SWM signal is strongly enhanced by a factor of 620 in the presence of the dressing field when Δa3 /Γ30 = −50 [dash curve in Fig. 8.12(c)], which is mainly due to the three-photon (|0 → |1 → |2 → |+) resonance. Parameters are Γ20 /Γ30 = 1, Γ10 /Γ30 = 0.5, G4 /Γ30 = 50, and Δa3 /Γ30 = −30 (solid curve), Δa3 /Γ30 = −50 (dash curve), Δa3 /Γ30 = −70 (dot curve), Δa3 /Γ30 = −100 (dash dot curve). In general, the constructive and destructive interferences between the |+ and |− SWM channels (see Table 8.1) result in the enhancement and suppression of SWM signal, respectively. However, such enhancement mainly originates from the dispersion of dressed-SWM in the weak dressing field limit [1]. Next, we consider the doubly-dressed-FWM spectrum versus Δa2 /Γ20 or Δ3 /Γ30 [see Figs. 8.11(e) and (f)]. A dressing field G3 (G3 > G4 ) creates dressed atomic states |+ and |− from the unperturbed states |2 and |3 [see solid curve in Fig. 8.13(a)] (the separation between the two peaks is ΔAT1 ≈ 2{G3 [G23 +2Γ30 (Γ20 +Γ30 )]1/2 −Γ230 }1/2 ). Parameters in Fig. 8.13(a) are Γ30 = Γ20 = Γ10 , G3 /Γ20 = 13, G4 /Γ20 = 5, Δ3 /Γ20 = 0, and Δ4 /Γ20 = 200 (solid curve), Δ4 /Γ20 = 10.5 (dash curve), Δ4 /Γ20 = −10.5 (dot curve). When the dressing field G4 is tuned close to one of the primarily-dressed states |+ (or |−), basically the dressing field only couples the dressed state |+ (or |−) to the state |4 and leaves the other dressed state |− (or |+) unperturbed [28]. Thus, when Δ4 = ΔAT1 /2 there exist the secondarily-dressed states |++ and |+– around the primarily-dressed state |+, the FWM signal amplitude for the two doubly-dressed states will approximately be half of the amplitude for the singly-dressed state |− [see dash curve in Fig. 8.13(a)]; when Δ4 = −ΔAT1 /2 the secondarily-dressed states |–+ and |– – (around the primarilydressed state |−) can be generated (dot curve in [see Fig. 8.13(a)]. That is

290

8 Multi-dressed MWM Processes

Fig. 8.12. (a) Dressed-SWM signal intensity versus Δa3 /Γ20 (the maximum is normalized to 1)(b) dressed-SWM signal intensity (normalized by no dressing (5) (5) field (G4 = 0) case, i.e., ρ 10 /ρ10 ) versus Δ4 /Γ30 ; (c) dressed-SWM signal intensity (normalized by no dressing field case) versus Δ4 /Γ30 . Adopted from Ref. [23]. (0)

ω

(1)

ω

(2)

−ω

(3)

1 2 to say, the two FWM Liouville pathways, ρ00 −→ ρ10 −→ ρ(++)0 −→2 ρ10

(0)

ω

(1)

ω

(2)

−ω

(3)

1 2 and ρ00 −→ ρ10 −→ ρ(+−)0 −→2 ρ10 , interfere constructively, leading to an enhanced FWM signal. Due to the decoherence of the Raman coherence ρ30 , the doubly-dressed four-level system also exhibits a constructive interference [20]. By contrast, the doubly-dressed system with a metastable excited state shows the sharp dark resonance due to the destructive interference between the secondarily-dressed states [19]. In Figs. 8.13(b) and (c) the doubly-dressed FWM signal intensity with no (3) (3) dressing field (G3 = G4 = 0) is normalized to 1 (i.e., ρ 10 /ρ10 ). There are two groups of the suppression and enhancement curves due to the primarilydressed states created by the field G4 (G4 > G3 ). When Δa2 = Δ4 = 0 the splitting ΔAT2 is approximately proportional to ΔAT2 ≈ 2{G4 [G24 + 2Γ40 (Γ30 + Γ40 )]1/2 − Γ240 }1/2 . Based on the same secondarily-dressed state |+ created by the field G3 , the two photon (|0 → |1 → |+) resonant FWM signals (corresponding to each group of curves) are enhanced dramatically [see Figs. 8.13(b) and (c)]. Specifically, at exact two-photon resonance Δa2 = 0 we see that the FWM signal intensity is suppressed when the frequency of the dressing field G3 is scanned across Δ3 = ±ΔAT2 /2. The presence of the weak dressing field G3 (G3 < G4 ) can either suppress or enhance the

8.2 Generalized Dressed and Doubly-dressed MWM Processes

291

Fig. 8.13. (a) Doubly-dressed FWM signal intensity versus Δa2 /Γ20 (b) doublydressed FWM signal intensity (normalized by two dressing field G3 = G4 = 0 case, (3) (3) i.e., ρ 10 /ρ10 ) versus Δ3 /Γ30 ; (c) doubly-dressed FWM signal intensity (normalized by no dressing field case) versus Δ3 /Γ30 . Adopted from Ref. [23].

FWM signal when Δa2 = 0 [see Fig. 8.13(b)]. Parameters in Fig. 8.13 (b) are Γ20 /Γ30 = 1, Γ10 /Γ30 = 0.5, G3 /Γ30 = 0.5, G4 /Γ30 = 50, Δ1 = Δ4 = 0, and Δ2 /Γ30 = 0 (solid curve), Δ2 /Γ30 = −0.5 (dash curve), Δ2 /Γ30 = −2 (dot curve), Δ2 /Γ30 = −6 (dash-dot curve). With two strong dressing fields G3 = G4 = 50Γ30 the FWM signal with dual peaks is enhanced by a factor of 440 when Δa2 + Δ4 = −50Γ30 . The dual enhanced FWM channels have been opened simultaneously [see Fig. 8.13(c)] by the two strong dressing fields, which provide the energy for such large enhancement. Parameters in Fig. 8.13(c) are Γ20 /Γ30 = 1, Γ10 /Γ30 = 0.5, G3 /Γ30 = G4 /Γ30 = 50, Δ1 = Δ4 = 0, and Δ2 /Γ30 = −30 (solid curve), Δ2 /Γ30 = −50 (dash curve), Δ2 /Γ30 = −70 (dot curve), Δ2 /Γ30 = −100 (dash-dot curve). The coexistence of these three nonlinear wave-mixing processes in this five-level system can be used to evaluate the high-order nonlinear susceptibility χ(7) by beating the FWM and EWM, or SWM and EWM signals. Since (3) (7) (5) (7) |ρ10 | >> |ρ10 | and |ρ10 | >> |ρ10 | is generally true and the FWM, SWM and EWM signals are diffracted in the same direction with the same frequency, the real and imaginary parts of χ(7) can be measured by homodyne detection with the FWM (or SWM) signal as the strong local oscillator. Multi-wave mixing possesses features of excellent spatial signal resolution, free choice of interaction volume and simple optical alignment. Moreover,

292

8 Multi-dressed MWM Processes

phase matching can be achieved for a very wide frequency range from many hundreds to thousands of cm−1 . Specifically, in doubly-dressed-(2n − 4)WM, the coherence length is given by Lc = 2c/[n0 (ωn−2 /ω1 )|ωn−2 − ω1 |θ2 ], with θ being the angle between beam 2 and beam 3 [see Fig. 8.10(a)], where n0 is the refractive index. For a typical experiment, θ is very small (< 0.5◦ ) so that Lc is larger than the interaction length L, as has been demonstrated in Refs. [18, 24]. Thus, the phase mismatch due to such small angles between laser beams can be neglected. Moreover, the angle θ can be adjusted for individual experiments to optimize the tradeoff between the better phase matching and the larger interaction volume or the better spatial resolution in Figs. 8.10(a) and 8.11(a). In this section, we described a generalized treatment for generating highorder (up to 2n) nonlinear wave-mixing processes in a close-cycled (n + 1)level atomic system. By applying dressed and doubly-dressed laser beams in such (n + 1)-level cascade system, various high-order nonlinear wave-mixing processes can be significantly enhanced and coexisting multi-order nonlinear wave-mixing processes can be generated. A five-level folded atomic system has been used, as an example for this general (n + 1)-level system (with n = 4), to illustrate the coexisting FWM, SWM, and EWM processes, and the great enhancement (as well as suppression) of the generated FWM and SWM signals during to the dressing fields at different parametric conditions. Understanding the higher-order multi-channel nonlinear optical processes can help in optimizing these nonlinear optical processes, which have potential applications in achieving better nonlinear optical materials and opto electronic devices.

8.3 Interacting MWM Processes in a Five-level System with Doubly-dressing Fields In Section 8.3, a five-level atomic system was used to show the doubly-dressed FWM with these two dressing fields connecting to each other, as shown in Fig. 8.11(e). The FWM process is generated from a simple three-level laddertype sub-system, and these two dressing fields are applied (sequentially) to the upper excited state of the ladder-type sub-system. FWM, SWM, and EWM processes are shown to coexist in such five-level close-cycled atomic system with the simple double-dressing scheme. However, depending on the arrangements of the multiple laser beams, there are several ways to generate multi-wave mixing processes and to dress the multi-level atomic systems. As we have demonstrated in Chapters 6 and 7, the third-order (FWM) and fifth-order (SWM) nonlinear processes can coexist in the open-cycled (such as V-type, Y-type and inverted Y-type) atomic systems. By applying more laser beams to dress these open-cycled three- and four-level systems with an additional energy level, more complicated interaction configurations and

8.3 Interacting MWM Processes in a Five-level System with · · ·

293

wave-mixing processes can be achieved. When the five-level atomic system, as shown in Fig. 8.14(a), is considered, different singly- and doubly-dressing schemes can be realized by selectively blocking different laser beams. In some cases and under certain conditions, coexisting FWM, SWM, and EWM processes can be generated in those dressed five-level systems. Investigations on the interactions of doubly-dressed states and the corresponding effects in multi-level atomic systems have attracted great attentions in recent years. The interaction of double-dark states (in a nested-cascade scheme of doubly-dressing) and splitting of dark states (with the secondarilydressed states) in a four-level atomic system were studied theoretically in an EIT system [19]. The triple-peak absorption spectrum was observed in the Ntype cold-atomic system, which shows the existence of the secondarily-dressed states in the nested-cascade scheme [20]. Similar result was obtained in the inverted-Y system [20, 25]. Also, the doubly-dressed FWM (DDFWM) in the nested-cascade, close-cycled atomic system was reported [23]. Two other kinds of DDFWM processes (i.e., in parallel- and nested-cascade schemes) were considered in an open five-level atomic system [26]. In this section, we present detail studies of three kinds of doubly-dressing schemes for DDFWM and show similarities and differences among these different dressing schemes in the open five-level atomic system shown in Fig. 8.14. The experimental study of the mutual-dressing processes existed between two competing, dressed-FWM channels in a four-level Y-type atomic system has been presented in Section 7.1 [17]. The dressing fields perturb both FWM processes and modify the total signal amplitude when these two FWM signals are tuned together in frequency. Constructive or destructive interference, as controlled by the phase difference between the two dressed-FWM processes, exists in such system. Such interference is also considered in two-level and three-level atomic systems as the coupling-field detuning is adjusted [27]. However, the contributions from the mutual dressing effects can be an order of magnitude larger than the interference effect in the four-level Y system, as discussed in Section 7.1 [17]. In the open five-level atomic system (see Fig. 8.14) to be discussed below, several features are different and advantageous [28] over the previously studied multi-wave mixing processes. First, there coexist DDFWM (with three different dressing schemes), singly-dressed SWM (DSWM), and EWM in this open five-level atomic system. So, this is a good system for studying the interplays and interactions between nonlinear optical processes of different orders. Second, three DDFWM processes (in nested-cascade, parallel-cascade, and sequential-cascade schemes) and their relationships can be considered in detail. Third, the AT splitting and suppression/enhancement of FWM spectra in this system can be well interpreted by the dressed-state diagrams and by the competitions between dispersion and absorption of the dressed MWM. Fourth, by controlling the DSWM or DDFWM signal, mutual-dressing processes and the constructive/destructive interference existing in this open fivelevel system are considered.

294

8 Multi-dressed MWM Processes

8.3.1 Three Doubly-Dressing Schemes Let us consider the open five-level system as shown in Fig. 8.14(a). In the three-level ladder-type (|0 − |1 − |2) sub-system, the beam E1 (ω1 ,k1 and Rabi frequency G1 ) probes the lower transition |0 to |1 while two coupling beams, E2 (ω2 ,k2 and G2 ) and E2 (ω2 ,k2 and G2 ), drive the transition |1 to |2. One FWM signal field Ef1 (with ω1 , kf1 = k1 + k2 − k2 ) is generated (0) ω1 (1) ω2 (2) −ω (3) via the FWM perturbation chain (f1) ρ00 −→ ρ10 −→ ρ20 −→2 ρ10 and we can obtain (3) ρf1 = ρ10 = −iGA exp(ikf1 · r)/d21 d2 (8.25) where GA = G1 G2 (G2 )∗ , d1 = Γ10 +iΔ1 , and d2 = Γ20 +i(Δ1 + Δ2 ) with frequency detuning Δi = Ω i − ωi . Γij is transverse relaxation rate between states |i and |j and Ω i is atomic resonance frequency.

Fig. 8.14. (a) Energy-level diagram of an open five-level system for EWM(b1) nested-cascade DDFWM and (b2)–(b4) the dressed-state pictures; (c1) paralleland sequential-cascade DDFWM and (c2) dressed-state picture for parallel-cascade DDFWM. (d1)&(d3) two DSWM and the (d2)&(d4) dressed-state pictures.

Similarly, in the three-level V-type (|3 − |0 − |1) sub-system E3 (ω3 ,k3 , G3 ) and E3 (ω3 ,k3 ,G3 ) drive the transition |0 to |3. The fields E1 , E3 , and E3 generate another FWM signal field Ef2 (ω1 ,kf2 = k1 + k3 − k3 ) via (0) ω1 (1) −ω (2) ω3 (3) another FWM perturbation chain (f2) ρ00 −→ ρ10 −→3 ρ13 −→ ρ10 , which gives (3) ρf2 = ρ10 = −iGB exp(ikf2 · r)/d21 d3 (8.26) ∗

where GB = G1 G3 G 3 and d3 = Γ13 + i(Δ1 − Δ3 ).

8.3 Interacting MWM Processes in a Five-level System with · · ·

295

At the same time, there are coexisting SWM processes in different fourlevel sub-systems and EWM processes in the five-level system, as discussed in the previous section [23, 26]. The appropriate perturbation chains are quite helpful in investigating the interactions in the DDFWM processes. In this five-level open atomic system, two strong coupling fields are applied to dress the FWM processes with three different doubly-dressing (i.e., nested-cascade, parallel-cascade and sequential-cascade) schemes [29, 30]. For the nested-cascade DDFWM, the outer dressing field E4 (ω4 ,k4 ,G4 ) drives the transition from |3 to |4 and the inner dressing field E3 dresses the level |0 [see Fig. 8.14(b1)] via a segment (sub-chain) of the EWM per(0) ω1 (1) ω2 (2) −ω (3) ω4 (4) −ω (5) ω3 ρ10 −→ ρ20 −→3 ρ23 −→ ρ24 −→4 ρ23 −→ turbation chain (e1) ρ00 −→ (6) −ω (7) (3) ω4 (4) −ω (5) ρ20 −→2 ρ10 . In detail, the sub-chain ρ23 −→ ρ24 −→4 ρ23 (related to (2) −ω (3) the dressing field E4 ) is nested between the sub-chains ρ20 −→3 ρ23 and (5) ω3 (6) ρ23 −→ ρ20 (due to the dressing field E3 ). Hence, such doubly-dressing configuration is denoted as the nested-cascade scheme. By virtue of the perturbation chain (e1), we can modify the FWM chain (f1) as the DDFWM −ω (0) ω1 (1) ω2 (2) (3) chain (Df1) ρ00 −→ ρ10 −→ ρ2(G4±G3±) −→2 ρ10 . Here, the subscript “0” (2)

of ρ20 in (f1) is replaced by “G4 ± G3 ±” in (Df1), which indicates that two dressing fields dress the level |0 and both influence the atomic coherence between states |0 and |2 . (1) According to the chain (Df1), we can obtain equations ρ10 =iG1 exp(ik1 · (0) (3) (2) ∗ r)ρ00 /d1 and ρ10 =iG 2 exp(−ik2 · r)ρ2(G4 ±G3 ±) /d1 for the dressed-FWM processes. The perturbation approach for such dressing cases can be well deduced by the following coupled equations, as described in Section 1.3: (2)

(2)

(1)

∂ρ20 /∂t = −d2 ρ20 + iG2 exp(ik2 · r)ρ10 − iG3 exp(ik3 · r)ρ23 ∂ρ23 /∂t = −d4 ρ23 − iG∗3 exp(−ik3 · r)ρ20 − iG∗4 exp(−ik4 · r)ρ24 (2)

∂ρ24 /∂t = −d5 ρ24 − iG4 exp(ik4 · r)ρ23 Here, d4 = Γ23 + i(Δ1 + Δ2 − Δ3 ) and d5 = Γ24 + i(Δ1 + Δ2 − Δ3 + Δ4 ). (2) In the steady state, ∂ρ20 /∂t = ∂ρ23 /∂t = ∂ρ24 /∂t = 0, we can solve the equations to get (2)

(2)

ρ2(G4 ±G3 ±) = ρ20 =

(1)

iG2 exp(ik2 · r)ρ10 G23 d2 + d4 + G24 /d5

(0)

Under the condition of ρ00 ≈ 1 (the probe field is much weaker than the other fields), we have (3)

ρDf1 = ρ10 =

−iGA exp(ikf1 · r)   |G3 |2 d21 d2 + d4 + |G4 |2 /d5

(8.27)

296

8 Multi-dressed MWM Processes

In fact, without these dressing fields (G3 = G4 = 0), Eq. (8.27) can be converted into Eq. (8.25). Item “d2 ” representing the “ω1 + ω2 ” two-photon process in ρf 1 is modified by the intensity |G3 |2 , while |G3 |2 is modified by the intensity |G4 |2 . Hence, Eq. (8.27) shows that the two dressing fields are entangled with each other in such a nested-cascade scheme. Moreover, in the weak dressing-field limit (i.e., |G3 |2 << Γ20 Γ23 and |G4 |2 << Γ23 Γ24 ), Eq. (8.27) can be expanded to be [31] ρDf1 = ρf1 + ρs1 + ρe1

(8.28)

Here iGA |G3 |2 exp(iks1 · r) (with ks1 = kf1 + k3 − k3 ) (d21 d22 d4 ) −iGA |G3 |2 |G4 |2 exp(ike1 · r) (with ke1 = kf1 + k3 − k3 + k4 − k4 ) = (d21 d22 d24 d5 )

ρs1 = ρe1

represent the SWM and EWM processes, respectively. Equation (8.28) shows that the nested-cascade DDFWM is a coherent superposition of the signals from FWM, SWM, and EWM under the weak dressing-field condition. For the parallel-cascade DDFWM, as shown in Fig. 8.14(c1), the two dressing fields E2 and E4 dress the energy levels |1 and |3, respectively, (0) ω1 (1) ω2 (2) −ω (3) −ω ρ10 −→ ρ20 −→2 ρ10 −→3 via the sub-chains of the EWM chain (e2) ρ00 −→ (4) ω4 (5) −ω4 (6) ω3 (7) ρ13 −→ ρ14 −→ ρ13 −→ ρ10 . Two sub-chains of the dressing fields (1) ω2 (2) −ω (3) (4) ω4 (5) −ω (6) “ρ10 −→ ρ20 −→2 ρ10 ” (for E2 ) and “ρ13 −→ ρ14 −→4 ρ13 ” (for E4 ) lie parallel within the chain (e2) and such doubly-dressing scheme is denoted as the parallel-cascade scheme. The FWM chain (f2) is modified by the −ω ω3 (0) ω1 (1) (2) (3) ρG2±0 −→3 ρ1G4± −→ ρ10 . Here the subDDFWM chain (Df2) as ρ00 −→ (1)

(2)

scripts “1” of ρ10 and “3” of ρ13 in (f2) are replaced by “G2 ±” and “G4 ±” in (Df2), respectively. These two dressing fields now dress two different energy (1) (2) levels and influence different coherences ρ10 and ρ13 for the FWM processes. From the perturbation chain (Df2), the equation for the modified FWM (3) (2) is ρ10 = −iG3 exp(ik3 · r)ρ1G4± /d1 and the coupled equations are ∂ρ10 /∂t = −d1 ρ10 + iG1 exp(ik1 · r)ρ00 + iG∗2 exp(−ik2 · r)ρ20 (1)

(1)

(0)

(1)

∂ρ20 /∂t = −d2 ρ20 + iG2 exp(ik2 · r)ρ10 ∗

∂ρ13 /∂t = −d3 ρ13 − iG 3 exp(−ik3 · k)ρ10 − iG∗4 exp(−ik4 · r)ρ14 (2)

(2)

(1)

(2)

∂ρ14 /∂t = −d6 ρ14 − iG4 exp(ik4 · r)ρ13

Here, d6 = Γ14 + i(Δ1 − Δ3 + Δ4 ). Under the steady-state condition, (1) (2) ∂ρ10 /∂t = ∂ρ20 /∂t = ∂ρ13 /∂t = ∂ρ14 /∂t = 0. The steady-state solutions (1) (1) (0) (2) (2) are ρG2±0 = ρ10 = iG1 exp(ik1 · r)ρ00 /(d1 + |G2 |2 /d2 ) and ρ1G4± = ρ13 =

8.3 Interacting MWM Processes in a Five-level System with · · · ∗

297

−iG 3 exp(−ik3 · r)ρ10 /(d3 + |G4 |2 /d6 ). With ρ00 ≈ 1, the above equations can be solved to give (1)

(0)

(3)

ρDf2 = ρ10 =

−iGB exp(ikf2 · r) d1 (d3 + |G4 |2 /d6 ) (d1 + |G2 |2 /d2 )

(8.29)

Comparing with Eq. (8.26), the items d3 and d1 in Eq. (8.29) are modified by the intensities of the dressing fields |G2 |2 and |G4 |2 , respectively. In the weak dressing-field limit (|G2 |2 << Γ10 Γ20 and |G4 |2 << Γ13 Γ14 ), Eq. (8.29) can be expanded to be ρDf2 = ρf2 + ρs2 + ρs3 + ρe2

(8.30)

Here i|G2 |2 GB exp(iks2 · r) (with ks2 = kf2 + k2 − k2 ) (d21 d2 d3 ) i|G4 |2 GB exp(iks3 · r) = (with ks3 = kf2 + k4 − k4 ) (d21 d23 d6 ) i|G2 |2 |G4 |2 GB exp(ike2 · r) (with ke2 = kf2 + k2 − k2 + k4 − k4 ) =− (d31 d2 d23 d6 )

ρs2 = ρs3 ρe2

are the expressions for the SWM and EWM processes, respectively. This means that under the weak dressing-field condition the parallel-cascade DDFWM can be considered as a coherent superposition of the signals from one FWM, two SWM, and one EWM processes. For the sequential-cascade DDFWM, the energy-level diagram is the same as the parallel-cascade DDFWM [see Fig. 8.14(c1)]. However, these two dressing fields E2 and E4 dress ρf2 via the sub-chains of a different EWM per(0) ω1 (1) −ω (2) ω2 (3) −ω (4) ω4 (5) −ω turbation chain (e3) ρ00 −→ ρ10 −→3 ρ13 −→ ρ23 −→2 ρ13 −→ ρ14 −→4 (6) ω3 (7) (2) ω2 ρ10 . Note that the two sub-chains of the dressing fields ρ13 −→ ρ13 −→ (3) −ω2 (4) (4) ω4 (5) −ω4 (6) ρ23 −→ ρ13 and ρ13 −→ ρ14 −→ ρ13 join together sequentially. Such doubly-dressing scheme is denoted as the sequential-cascade scheme. According to the chain (e3), the FWM chain (f2) is modified by the DDFWM ω3 (0) ω1 (1) −ω (2) (3) ρ10 −→3 ρG2±G4± −→ ρ10 . Here subscripts “1” and “3” to be (Df3) ρ00 −→ (2)

of ρ13 in (f2) are replaced by “G2 ±” and “G4 ±” in (Df3), respectively. This means that these two dressing fields dress different energy levels, but they influence the same coherence induced between states |1 and |3. Similarly, (1) (0) from the chain (Df3) we have the equations ρ10 = iG1 exp(ik1 · r)ρ00 /d1 and (3) (2) ρ10 = −iG3 exp(ik3 · r)ρG2±G4± /d1 , and the coupled equations ∂ρ13 /∂t = −d3 ρ13 + iG∗2 exp(−ik2 · r)ρ23 − (2)

(2)

∗

iG 3 exp(−ik3 · r)ρ10 − iG∗4 exp(−ik4 · r)ρ14 (1)

(2)

∂ρ23 /∂t = −d4 ρ23 + iG2 exp(ik2 · r)ρ13

(2)

∂ρ14 /∂t = −d6 ρ14 − iG4 exp(ik4 · r)ρ13

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8 Multi-dressed MWM Processes (2)

In the steady state, ∂ρ13 /∂t = ∂ρ23 /∂t = ∂ρ14 /∂t = 0 and we obtain ∗

ρG2 ±G4 ± = ρ13 = −iG 3 exp(−ik3 · r)ρ10 /(d3 + |G2 |2 /d4 + |G4 |2 /d6 ) (2)

(2)

(1)

(0)

which, under the condition of ρ00 ≈ 1, gives (3)

ρDf3 = ρ10 =

−iGB exp(ikf2 · r) d21 (d3 + |G2 |2 /d4 + |G4 |2 /d6 )

(8.31)

Comparing with Eq. (8.26), the item “d3 ” in Eq. (8.31) is modified by the intensities G2 and G4 together. In weak dressing-field limit (|G2 |2 << Γ13 Γ23 and |G4 |2 << Γ13 Γ14 ), Eq. (8.31) can be expanded to be ρDf3 = ρf2 + ρs3 + ρs4

(8.32)

which shows that the sequential-cascade DDFWM can be considered as a coherent superposition of the signals from one FWM and two SWM processes. Here, ρs4 = i|G2 |2 GB exp(iks2 · r)/(d21 d23 d4 ) represents the SWM process. These three different doubly-dressing configurations show different interactions between the two dressing fields and the FWM channels. In the following, we discuss and present major differences and similarities among these different doubly-dressing schemes. First, in the energy-level diagrams, the two dressing fields in the nestedcascade scheme connect three neighboring levels in the sub-system |0 − |3 − |4 and the outer dressing field is based on the inner dressing field while the inner one dresses the state |0 [see Fig. 8.14(b1)]. However, the dressing fields in the parallel-cascade and sequential-cascade schemes dress two different states independently [see Fig. 8.14(c1)]. Second, the perturbation chains for different DDFWM processes also represent differences between the three doubly-dressing schemes. In the nestedcascade scheme, the two dressing fields are entangled tightly with each other (2) −ω (3) ω4 (4) −ω (5) ω3 (6) ρ24 −→4 ρ23 −→ ρ20 in the according to the sub-chain ρ20 −→3 ρ23 −→ chain (e1). However, for the parallel-cascade scheme, the two dressing pro(1) ω2 (2) −ω (3) (4) ω4 (5) −ω (6) cesses “ρ10 −→ ρ20 −→2 ρ10 ” and “ρ13 −→ ρ14 −→4 ρ13 ” are separated in the chain (e2) (not connected). For the sequential-cascade scheme, although (2) ω2 (3) −ω (4) (4) ω4 (5) −ω (6) “ρ13 −→ ρ23 −→2 ρ13 ” and “ρ13 −→ ρ14 −→4 ρ13 ” lie independently in (4) the perturbation chain (e3), they are conjoined by ρ13 in the chain. This means that the interaction between the two dressing fields in the nestedcascade scheme is the strongest and that in the parallel-cascade scheme is the weakest. The sequential-cascade scheme is an intermediate case between them. Third, according to the DDFWM expressions, for the nested-cascade DDFWM in Eq. (8.27) if |G3 |2 = 0, |G4 |2 will have no effect on the result. However, the intensities of the dressing fields are independent for both parallel- and sequential-cascade DDFWM cases as can be seen in Eqs. (8.29)

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299

and (31). On the other hand, one term of FWM is modified by the dressing fields for the nested- and sequential-cascade schemes while two terms are modified for the parallel-cascade scheme. In fact, forms of the nestedand parallel-cascade DDFWM expressions can be converted into the form of the sequential-cascade DDFWM under certain conditions, which shows that the sequential-cascade scheme is an intermediate case between the other two schemes. In the limit of small G4 or large Δ4 , Eq. (8.27) can be written as ρDf1 ≈

−iGA exp(ikf1 · r) d21 (d2 + |G3 |2 /d4 + |G4 |2 /D(Δ4 ))

(8.33)

where D(Δ4 ) = −d24 d5 /|G3 |2 . Thus, the form of Eq. (8.33) is in the same form as Eq. (8.31). Also, according to Eq. (8.30) (for the parallel-cascade DDFWM), if |G2 |2 and |G4 |2 are small enough, the EWM term ρe2 can be ignored and we have (8.34) ρDf2 ≈ ρf2 + ρs2 + ρs3 which is similar to Eq. (8.32) for the sequential-cascade DDFWM. Hence, weaker the dressing fields are, more alike the parallel- and sequential-cascade DDFWM processes will become.

8.3.2 Aulter-Townes Splitting, Suppression, and Enhancement One possible experimental candidate for proposed five-level system is in 85 Rb atoms with states |0 = |5S1/2  (F=2), |1 = |5P1/2 , |2 = |5D3/2 , |3 = |5P3/2  and |4 = |5S1/2  (F=3). The respective transitions are |0 → |1 at 795 nm (γ10 ≈ 5.4 MHz, where γij gives decay due to spontaneous emission (longitudinal relaxation rate) between states |i and |j), |1 → |2 at 762 nm (γ21 ≈ 0.98 MHz), |0 → |3at 780 nm (γ30 ≈ 5.9 MHz), |3 → |4at 780 nm (γ34 ≈ 5.9 MHz), |4 → |1at 795 nm (γ14 ≈ 5.4 MHz) and |3 → |2 at 776 nm (γ23 ≈ 0.8 MHz). The transverse relaxation rate Γij between states |i and |j is given by Γij = (Γi + Γj )/2 (Γ0 = γ40 ,Γ1 = γ10 + γ14 ,Γ2 = γ21 + γ23 ,Γ3 = γ30 + γ34 and Γ4 = γ40 ). The Rabi frequency G1 is assumed to be small while G2 , G2 , G3 , G3 , G4 , and G4 can be of arbitrary magnitudes [28]. In the following, we will plot the normalized DDFWM signal intensities based on the analytic expressions given above. Let us first look at the spectra of the nested-cascade DDFWM. Figure 8.15 presents signal intensity versus the pump field detuning Δ2 corresponding to the primarily-dressed states and the secondarily-dressed states [see Fig. 8.14(b2)]. Parameters are Γ20 /Γ10 = 0.8, Γ23 /Γ10 = 1.2, Γ24 /Γ10 = 0.8 Δ1 = Δ3 = 0, G3 /Γ10 = 20, G4 /Γ10 = 2, and Δ4 /Γ10 = 2000 (solid curve), Δ4 /Γ10 = −20 (dashed curve), G4 /Γ10 = 5, Δ4 /Γ10 = −20 (dot-dashed curve). According to Eq. (8.27) (in the solid curve), the inner dressing field E3 splits the “ω1 + ω2 ” two-photon resonant peak into two peaks located at Δ2 = ±ΔG3 /2 ≈ ±G3 under the condition |G3 |2 >> Γ20 Γ23 , which is the

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8 Multi-dressed MWM Processes

primary AT splitting. Here ΔG3 is the separation induced by the dressing field E3 . The outer dressing field E4 with a large detuning almost has no impact in this case. According to dressed-state analysis, the right and left peaks represent the transitions from the two primarily-dressed states |G3 + and |G3 −, separated from the ground state |0 to the exciting state |2. More importantly, when the outer dressing field E4 dresses the primarilydressed state |G3 + (Δ4 = −G3 ) and becomes stronger, the dashed and dotted curves show the secondary AT splitting of the right peak which represents that |G3 + is separated into the secondary dressed states |G3 +|G4 ± [see Fig. 8.14(b2)].

Fig. 8.15. Nested-cascade DDFWM signal intensity versus Δ2 /Γ10 . The maximum of the intensity is normalized to be 1. Adopted from Ref. [28].

Next, we investigate the spectra versus the inner dressing field detuning Δ3 while fixing the pump field frequency detuning. Figure 8.16 shows the impact of the two dressing fields on the FWM signal intensity. Here, the FWM signal intensity with no dressing fields (G3 = G4 = 0) is normalized to 1. The intensity above or below “1” means enhancement or suppression of the FWM signal. Figure 8.16(a) shows that there is a suppressed dip at the line center first. As G4 is increased, this dip becomes shallow and then splits into two dips. Parameters are Δ1 = Δ2 = Δ4 = 0, G3 /Γ10 = 0.5, Γ20 /Γ10 = 0.8, Γ23 /Γ10 = 1.2, Γ24 /Γ10 = 0.8, and G4 /Γ10 = 0 (solid curve), G4 /Γ10 = 0.7 (dashed curve), G4 /Γ10 = 1.5 (dotted curve), G4 /Γ10 = 7.5 (dot-dashed curve). In the limit of G4 >> Γ23 ,Γ24 , we have ΔG4 ≈ 2G4 . Here, ΔG4 is the separation induced by the dressing field E4 . It means that the inner dressing field E3 suppresses the resonant FWM signal directly and creates one suppressed dip while the outer dressing field E4 splits such suppressed dip to create a pair of FWM channels. As G4 becomes large, These two channels are separated quite apart as shown by the solid curves of Fig. 8.16(b). Parameters are Γ20 /Γ10 = 0.8, Γ23 /Γ10 = 1.2, Γ24 /Γ10 = 0.8; and (b1) Δ2 = Δ4 = 0, G3 /Γ10 = 3, Δ1 = 0, with G4 /Γ10 = 25 (solid curve) or G4 /Γ10 = 0 (dashed curve); (b2) Δ2 = Δ4 = 0, G3 /Γ10 = 3, Δ1 /Γ10 = 2, with G4 /Γ10 = 25 (solid curve) or G4 /Γ10 = 0 (dashed curve); (b3) Δ2 = Δ4 = 0, G3 /Γ10 = 3, Δ1 = −2,

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301

Fig. 8.16. (a) and (b) Nested-cascade DDFWM signal intensity versus Δ3 /Γ10 ; (c) Values of Re2 ρDf1 scaled by a factor 50, Im2 ρDf1 , Im2 ρf1 (dashed curve), Imρf1 Imρs3 (dotted curve) and Imρf1 Imρe5 (dot-dashed curve) versus Δ3 /Γ10 . The FWM signal intensity with no dressing fields is normalized to be 1. Adopted from Ref. [28].

with G4 /Γ10 = 25 (solid curve) or G4 /Γ10 = 0 (dashed curve). Through such pair of channels the inner dressing field E3 can doubly suppress the resonant FWM signal [the solid curve in Fig. 8.16(b1)] and doubly suppress or enhance the off-resonant FWM signals [the solid curves in Figs. 8.16(b2) and 8.16(b3)], respectively. However, without the outer dressing field E4 , there only exists one suppressed dip [the dashed curves in Figs. 8.16(b1), 8.16(b2) and 8.16(b3)] and one enhanced peak [the dashed curves in Figs. 8.16(b2) and 8.16(b3)]. The outer dressing field E4 splits the peaks and dips in the dashed curves into the dual-enhanced peaks and dual-suppressed dips in the solid curves, respectively. Note that the spectra in Fig. 8.16(b1) have the absorptive shape while those of Figs. 8.16(b2) and (b3) have the dispersive shape due to the dominant contribution from ImρDf1 in resonant case (Δ1 = Δ2 = 0) or ReρDf1 in non-resonant case (Δ1 + Δ2 = 0). These dressed-state analysis agrees well with Fig. 8.16(b). If G4 = 0, only the dressing field E3 drives the transition from |3 to |0 and creates the dressed states |G3 ± from |0. Thus the “ω1 +ω2 ” two-photon resonant or offresonant transition of FWM ρf1 gets off-resonant or more off-resonant, which results in a suppression of the FWM signal intensity as shown by the dips in the dashed curve of Fig. 8.16(b). Hence, we can deduce the suppresseddip condition ω1 + ω2 − ω3 = Ω 1 + Ω 2 − Ω 3 (i.e., Δ3 = Δ1 + Δ2 = 0 or

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8 Multi-dressed MWM Processes

±2). On the other hand, the dressed-state |G3 + can make the transition from off-resonance to resonance, and the FWM signal intensity is enhanced as shown by the peak of the dashed curve in Fig. 8.16(b2) at ω1 + ω2 − ω3 ≈ (Ω 1 − ΔG3 /2) + Ω 2 − Ω 3 (i.e., Δ3 ≈ Δ1 + Δ2 − ΔG3 /2 ≈ −3.8). Similarly, the enhanced peak of the dashed curve in Fig. 8.16(b3) is induced by the other dressed-state |G3 − if ω1 + ω2 − ω3 ≈ (Ω 1 + ΔG3 /2) + Ω 2 − Ω 3 (i.e., Δ3 ≈ Δ1 + Δ2 + ΔG3 /2 ≈ 3.8). Here, we assume that |G3 ± are created symmetrically. As G4 is quite large, the outer dressing field E4 dresses the level |3 and creates the primarily-dressed states |G4 ±, as shown in Fig. 8.14(b3). The inner dressing field E3 (driving the transitions from |G4 ± to the ground level |0) creates the secondarily-dressed states|G4 +|G3 ± [see Fig. 8.14(b4)] or |G4 −|G3 ± from |0 to make the two-photon resonant and off-resonant transitions of ρf1 be off-resonance or even more off-resonance [see Fig. 8.14(b4)], which induces two suppressed dips in the solid curve of Fig. 8.16(b) under the condition of ω1 + ω2 − ω3 + ω4 ≈ Ω 1 + Ω 2 − (Ω 3 ± ΔG4 /2) + Ω 4 [i.e., Δ3 ≈ Δ1 + Δ2 + Δ4 ± ΔG4 /2 = 0 ± 25 for Fig. 8.16(b1), or 2 ± 25 for Fig. 8.16(b2), or −2 ± 25 for Fig. 8.16(b3)]. Here, we assume the primarilydressed states |G4 ± are created symmetrically. However, the secondarilydressed states |G4 ±|G3 + can cause resonant excitations corresponding to the two enhanced peaks of the solid curve in Fig. 8.16 (b2) if ω1 +ω2 −ω3 +ω4 ≈ (Ω 1 − ΔG3 /2) + Ω 2 − (Ω 3 ± ΔG4 /2) + Ω 4 , i.e., Δ3 ≈ Δ1 + Δ2 − ΔG3 /2 + Δ4 ± ΔG4 /2 ≈ −1.5 ± 25. Similarly, dual-enhanced peaks in Fig. 8.16(b3) represent two-photon resonant transitions from |G4 ±|G3 − to |2, respectively, if ω1 + ω2 − ω3 + ω4 ≈ (Ω 1 + ΔG3 /2) + Ω 2 − (Ω 3 ± ΔG4 /2) + Ω 4 , i.e., Δ3 ≈ Δ1 + Δ2 + ΔG3 /2 + Δ4 ± ΔG4 /2 ≈ 1.5 ± 25. In short, the dualenhancement is resulted from the resonant pump fields while the resonance of the inner dressing field results in the dual-suppression. When G4 is small, two dips cannot be separated completely, as shown by dash line in Fig. 8.16(a). Hence, it is better to consider such suppressionspectrum as the result of competition between dispersion [(ReρDf1 )2 ] and absorption [(ImρDf1 )2 ] terms in the weak-dressing-field limit [2,3]. Figure 8.16(c) shows that, with resonant conditions (Δ1 = Δ2 = 0), DDFWM spectrum is mainly induced by absorption term while the dispersion term is negligible [32]. Parameters in Fig. 8.16(c) are G3 /Γ10 = 0.5, G4 /Γ10 = 0.7, Δ1 = Δ2 = Δ4 = 0, Γ20 /Γ10 = 0.8, Γ23 /Γ10 = 1.2, and Γ24 /Γ10 = 0.8. According to Eq. (8.28), we have (ImρDf1 )2 = (Imρf1 + Imρs1 + Imρe1 )2

(8.35)

In Eq. (8.35), the terms (Imρf1 )2 , |Imρf1 Imρs1 | and Imρf1 Imρe1 are larger than other terms by orders of magnitude. Hence we can consider only these three dominant terms. In Fig. 8.16(c) (Imρf1 )2 /|ρf1 |2 equals to 1, Imρf1 Imρs1 has the negative value and Imρf1 Imρe1 has the positive value at the line center. Therefore, under weak dressing-field condition, suppression or enhancement of the FWM signal intensity is determined by destructive or constructive

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303

interferences among various MWM processes. Next, we consider the parallel-cascade DDFWM spectra. Figures 8.17(a) and (b) show that there exist two pairs of AT splitting peaks induced by G2 and G4 , respectively. Parameters are Γ14 /Γ13 = 0.8, Γ20 /Γ13 = 0.5, Γ10 /Γ13 = 0.5, and (a) Δ2 = Δ3 = Δ4 = 0, with G2 /Γ13 = 2, G4 /Γ13 = 15 (solid curve), or G2 /Γ13 = 5, G4 /Γ13 = 15 (dashed curve), or G2 /Γ13 = 5, G4 /Γ13 = 20 (dot-dashed curve); (b) Δ3 = 0, G2 /Γ13 = 25, G4 /Γ13 = 10, Δ2 = Δ4 = 0 (solid curve), Δ4 = 0, Δ2 /Γ13 = −5 (dot-dashed curve), Δ2 = 0, Δ4 /Γ13 = 5 (dashed curve). The positions of these peaks are determined by intensities and dressing field frequency detunings. According to Eq. (8.29), in the limit of |G2 |2 >> Γ10 Γ20 and |G4 |2 >> Γ13 Γ14 , we have ΔG2 = 2G2 and ΔG4 = 2G4 . Here, ΔG2 and ΔG4 are the separations induced by the two dressing fields. In fact, these two pairs of AT splitting peaks represent the transitions |G4 ± → |0 and |0 → |G2 ±, as shown in Fig. 8.14(c2). Note that the secondarily-dressed states do not exist in the parallel-cascade DDFWM.

Fig. 8.17. Parallel-cascade DDFWM signal intensity versus Δ1 /Γ13 . The maximum of the intensity is normalized to be 1. Adopted from Ref. [28].

We consider the parallel-cascade DDFWM spectra versus the dressing field frequency detunings Δ2 and Δ4 , as shown in Fig. 8.18. Parameters are Γ14 /Γ13 = 0.8, Γ20 /Γ13 = 0.5, Γ10 /Γ13 = 0.5, and (a) Δ1 = Δ3 = 0, G2 /Γ13 = G4 /Γ13 = 0.5; (b) Δ1 = Δ3 = 0, G2 /Γ13 = G4 /Γ13 = 5; (c) Δ3 = 0, Δ1 /Γ13 = −5, G2 /Γ13 = G4 /Γ13 = 0.5; (d) Δ3 = 0, Δ1 /Γ13 = −5, G2 /Γ13 = G4 /Γ13 = 5. The FWM signal intensity without any dressing fields is normalized to be 1. Figure 8.18(a) shows the symmetrical suppression spectrum of the resonant FWM signal intensity (Δ1 = Δ3 = 0). As the dressing fields become stronger, the suppressed dips get wider and deeper, as shown in Fig. 8.18(b). Figure 8.18(c) (which has dispersive shapes on each sides due to contribution of ReρDf2 ) presents either suppression or enhancement of the off-resonant FWM (Δ1 /Γ13 = −5) with weak dressing fields. Due to constructive superposition, the FWM signal intensity in the region (Δ2 /Γ13 < 5,Δ4 < 4) is the strongest. When the dressing fields get stronger, a significant enhancement of the FWM signal intensity (by a factor of about 200) can be obtained in Fig. 8.18(d). According to dressed-

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8 Multi-dressed MWM Processes

state analysis and Eq. (8.31), the suppressed-dip conditions in Fig. 8.18 are ω1 + ω2 = Ω 1 + Ω 2 (i.e.,Δ2 = −Δ1 ) and ω1 − ω3 + ω4 = Ω 1 − Ω 3 + Ω 4 (i.e., Δ4 = Δ3 − Δ1 ), respectively, while the enhanced-peak conditions are ω1 + ω2 ≈ (Ω 1 + ΔG2 /2) + Ω 2 (i.e., Δ2 ≈ −Δ1 − ΔG2 /2) and ω1 − ω3 + ω4 ≈ Ω 1 − (Ω 3 + ΔG4 /2) + Ω 4 (i.e., Δ4 ≈ Δ3 − Δ1 − ΔG4 /2), respectively.

Fig. 8.18. Parallel-cascade DDFWM signal intensity versus Δ2 /Γ13 and Δ4 /Γ13 . The FWM signal intensity with no dressing fields is normalized to be 1. Adopted from Ref. [28].

In the sequential-cascade DDFWM spectrum, Figure 8.19 shows the DDFWM signal intensity versus pump field detuning Δ3 . Parameters are Γ23 /Γ13 = 0.5, Γ10 /Γ13 = 0.5, Γ14 /Γ13 = 0.8, and (a) Δ1 = Δ2 = Δ4 = 0, with G2 /Γ13 = 0, G4 /Γ13 = 0 (solid curve), or G2 /Γ13 = 5, G4 /Γ13 = 0 (dashed curve), or G2 /Γ13 = 5, G4 /Γ13 = 14 (dot-dashed curve); (b) Δ1 = Δ4 = 0, G4 /Γ13 = 10, G2 /Γ13 = 3, with Δ2 /Γ13 = 1000 (solid curve), Δ2 /Γ13 = 10 (dashed curve), Δ2 /Γ13 = −10 (dot-dashed curve). In Fig. 8.19(a), according to Eq. (8.31), the central peak of the solid curve is the “ω1 −ω3 ” two-photon resonant peak of FWM. Dash and dot-dash curves show that the two resonant dressing fields (Δ2 = Δ4 = 0) create one pair of AT splitting peaks together. Let Γ = Γ = Γ, in the limit of G22 + G24 >> Γ, 23 14  2 2 we have ΔG2&G4 ≈ 2 G2 + G4 . Here, ΔG2&G4 is the separation induced by the two dressing fields together. From the dressed-state picture, there exist the dual-dressed states |(G2 &G4 )± which make “ω1 − ω3 ” two-photon process resonant. Figure 8.19(b) depicts the primary and the secondary AT splittings by the two dressing fields, which are the same as in the nested-cascade DDFWM

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305

Fig. 8.19. Sequential-cascade DDFWM signal intensity versus Δ3 /Γ13 . The maximum of the intensity is normalized to be 1. Adopted from Ref. [28].

case (see Fig. 8.15). Symmetrical peaks in the solid curve of Fig. 8.19(b) are a pair of the primary AT splitting peaks induced by the dressing field E4 with separation ΔG4 ≈ 2G4 . As Δ2 ≈ ±ΔG4 /2, there exists the secondary AT splitting in the dashed or the dot-dashed curves induced by the dressing field E2 . Similarly, the dressing field E2 with Δ2 = 0 can induce the primarilydressed states while the dressing field E4 with Δ4 ≈ ±ΔG2 /2 induces the secondarily-dressed states. Then we consider the sequential-cascade DDFWM spectrum versus the dressing field detunings Δ2 and Δ4 . The parameters in Fig. 8.20 are Γ23 /Γ13 = 0.5, Γ10 /Γ13 = 0.5, Γ14 /Γ13 = 0.8, and (a) Δ1 = Δ3 = 0, G2 /Γ13 = G4 /Γ13 = 0.5; (b) Δ1 = Δ3 = 0, G2 /Γ13 = G4 /Γ13 = 5; (c) Δ1 = 0, Δ3 /Γ13 = 5, G2 /Γ13 = G4 /Γ13 = 0.5; and (d) Δ1 = 0, Δ3 /Γ13 = 5, G2 /Γ13 = G4 /Γ13 = 5. Figure 8.20(a) gives the symmetrically suppressed spectrum at on-resonant conditions (Δ1 = Δ3 = 0). As G2 and G4 are increased, the suppressed dips get not only wider and deeper but also asymmetrical, as shown in Fig. 8.20(b). Figure 8.20(c) shows either suppression or enhancement of the off-resonant FWM (Δ3 /Γ13 = 5) with weak dressing fields. Due to the constructive superposition, the FWM signal intensity in the region (Δ2 /Γ13 < 5,Δ4 /Γ13 < 4) is the strongest. However, under the condition of strong dressing fields, enhancement (by a factor of about 15) is obtained and there are no significant enhancement peaks in Fig. 8.20(d). Next, we analyze the spectra with strong dressing fields in detail. Figure 8.21(a) with Δ4 /Γ13 = ±200 is the cross section of Fig. 8.20(b). Parameters are Δ4 /Γ13 = −200 (solid curve), Δ4 /Γ13 = −20 (dashed curve), Δ4 /Γ13 = −20 (dotted curve), and Δ4 /Γ13 = 200 (dot-dashed curve). The solid and dot-dashed curves with large detunings (Δ4 /Γ13 = ±200) are almost symmetrical. However, in the case of small detuning Δ4 /Γ13 = −20 (Δ4 /Γ13 = 20), the dashed (dotted) curve has a peak on the right (left) side. According to Eq. (8.31), DDFWM signal intensity is proportional to  −1 (Γ214 |G2 |2 + Γ223 |G4 |2 )2 + (|G2 |2 Δ4 + |G4 |2 Δ2 )2 2 (8.36) Γ13 + (Γ214 + Δ24 )(Γ223 + Δ22 ) When |G2 |2 Δ4 + |G4 |2 Δ2 = 0, peaks in the dashed and dotted curves

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8 Multi-dressed MWM Processes

Fig. 8.20. Sequential-cascade DDFWM signal intensity versus Δ2 /Γ13 and Δ4 /Γ13 . The FWM signal intensity with no dressing fields is normalized to be 1. Adopted from Ref. [28].

can be obtained. Hence, asymmetry in Fig. 8.21(a) is induced by interaction between the two dressing fields. Such interaction weakens suppressed effect of the single dressing field. Figure 8.21(b) versus Δ2 is the cross section of Fig. 8.20(d). Parameters are Δ4 /Γ13 = −100 (solid curve), Δ4 /Γ13 = 100 (dot-dot-dashed curve), Δ4 /Γ13 = 1.5 (dotted curve), Δ4 /Γ13 = 0.2 (dashed curve), and Δ4 /Γ13 = 5 (dot-dashed curve, scaled by a factor of 100). Under the condition of large detunings (Δ4 /Γ13 = ±100) of the dressing field E4 , the solid and dot-dot-dashed curves in Fig. 8.21(b) have almost the similar profiles in which the suppression or enhancement of FWM signal intensity is mainly induced by dressing field E2 . With dressed-state analysis and Eq. (8.31), the suppressed dip and enhanced peak are at ω1 + ω2 − ω3 = Ω 1 + Ω 2 − Ω 3 (i.e., Δ2 = Δ3 − Δ1 ) and ω1 + ω2 − ω3 ≈ (Ω 1 + ΔG2 /2) + Ω 2 − Ω 3 (i.e., Δ2 ≈ Δ3 − Δ1 − ΔG2 /2), respectively. FWM signal intensity is enhanced by a factor of about 13. According to these two curves, one can hardly observe the constructive superposition under the strong dressing-field condition in the sequential-cascade scheme in Fig. 8.20(d). As Δ4 is changed closer to Δ4 /Γ13 = (Δ3 − Δ1 )/Γ13 = 5, profiles of the dash, dot and dot-dash curves fluctuate violently which show that interactions between the two dressing fields are strong and complex. Both the dash and dot-dash curves have the dip profile but difference in height between them is quite large (the dot-dash curve is scaled by a factor of 100). The dot curve has a different profile from that of the solid and the dot-dot-dashed curves (the enhanced peak is on the

8.3 Interacting MWM Processes in a Five-level System with · · ·

307

right of the suppressed peak).

Fig. 8.21. (a) Cross section versus Δ2 /Γ13 of Fig. 8.20(b); (b) Cross section versus Δ2 /Γ13 of Fig. 8.20(d). Adopted from Ref. [28].

Interaction between the two dressing fields in the nested-cascade scheme is the strongest and that of the parallel-cascade scheme is the weakest. This conclusion is verified further in the simulated spectra. First, in the AT splitting spectrum, the dressing fields of the nestedcascade scheme are entangled tightly with each other and have strong interaction. In Fig. 8.15 only the inner dressing field E3 can create the primary AT splitting, based on which the outer dressing field E4 can create the secondary AT splitting. On the other hand, for the parallel-cascade scheme, the dressing fields have a weaker interaction and they can directly create two independent AT splittings (see Fig. 8.17). However, for the sequential-cascade scheme the dressing fields can also directly create AT splitting but they have a strong interaction to create the primary and secondary AT splittings, as shown in Fig. 8.18 (b). Second, in the suppression or enhancement spectrum, both dressing fields E2 and E4 can influence the FWM signal intensity directly (see Figs. 8.18 and 8.20) for the parallel- and sequential-cascade schemes while for the nestedcascade scheme the inner dressing field E3 suppresses or enhances FWM signal intensity directly through a pair of channels created by the outer dressing field E4 . In the case of weak dressing fields, the interaction of the sequential-cascade scheme is negligible, and Figs. 8.20(a) and 8.20(c) are similar to Figs. 8.18(a) and 8.18(c) of the parallel-cascade DDFWM. However, with strong dressing fields the interaction gets stronger, and Figs. 8.20(b) and 8.20(d) become quite different from Figs. 8.18(b) and 8.18(d).

8.3.3 Competition between Two Coexisting Dressed MWM We have investigated the nested-, parallel- and sequential-cascade DDFWM processes in a five-level system interacting with seven laser fields. All the DDFWM processes, as well as DSWM processes, can coexist by carefully arranging the weak probe field E1 and the other six coupling fields E2 , E2 ,

308

8 Multi-dressed MWM Processes

E3 , E3 , E4 , E4 [18]. The interactions (including mutual-dressing processes and constructive (or destructive) interference [17]) between two different DDFWM (DSWM) processes are discussed. In this subsection, we first study the interactions between two DSWM processes and then further investigate the interactions of two DDFWM processes in more complicated situations. As shown in Fig. 8.14(d1), in the |3 − |0 − |1 − |2 sub-system the SWM process uses one photon each from E1 , E2 , E2 and two photons from E3 . The strong coupling field E4 dresses such SWM process via the perturbation ω3 (0) ω1 (1) ω2 (2) −ω (3) −ω (4) (5) ρ10 −→ ρ20 −→2 ρ10 −→3 ρ1G4± −→ ρ10 and we have chain (DS1) ρ00 −→ (5)

ρDs1 = ρ10 =

2

iGA |G3 | exp(iks1 · r) d31 d2 (d3 + |G4 |2 /d6 )

(8.37)

Figure 8.22 presents either suppression or enhancement of the SWM signal intensity versus the dressing field detuning Δ4 . Parameters are Γ20 /Γ10 = 0.8, Γ13 /Γ10 = 1, Γ14 /Γ10 = 0.8, Δ1 = Δ2 = 0, G4 /Γ10 = 5, and Δ3 = 0 (solid curve), Δ3 /Γ10 = 1 (dashed curve), Δ3 /Γ24 = 15 (dotted curve), and Δ3 /Γ10 = 40 (dot-dashed curve). The SWM signal intensity with no dressing field is normalized to be 1 in Fig. 8.22. According to Eq. (8.37) and dressedstate analysis, the dressing field E4 creates the dressed states |G4 ± from |3 to impacting on the “ω1 − ω3 ” two-photon transition of SWM process, as shown in Fig. 8.14(d2). Hence, we can deduce that the suppressed dip and the enhanced peak are at ω1 − ω3 + ω4 = Ω 1 − Ω 3 + Ω 4 (i.e., Δ4 = Δ3 − Δ1 ) and ω1 −ω3 +ω4 ≈ Ω 1 −(Ω 3 −ΔG4 /2)+Ω 4 (i.e. Δ4 ≈ Δ3 −Δ1 −ΔG4 /2 as shown in Fig. 8.14(d2)), respectively. Also, the profile in Fig. 8.22 can be considered as the result of a competition between dispersion and absorption parts of DSWM [32]. At the on-resonant condition (Δ1 = Δ2 = Δ3 = 0), the value of absorption-part (Im = (ImρDs1 )2 ) is much larger than that of the dispersionpart (Re = (ReρDs1 )2 ), as shown in the left inset plot of Fig. 8.22. Thus, the DSWM spectrum has the absorption-profile (the solid curve in Fig. 8.22). However, for the off-resonant SWM, the value of absorption part is decreased while that of the dispersion part is increased simultaneously as shown in the right inset plot. As a result, with large detuning (Δ3 /Γ10 = 40), the profile of DSWM spectrum is dominated by the dispersion part and changes to a dispersion-like profile. Similarly, as shown in Fig. 8.14(d3), other SWM process (using one photon each from E1 , E3 , E3 and two photons from E4 ) is dressed by the strong cou−ω (0) ω1 (1) (2) ω4 pling field E2 via the perturbation chain (DS2) ρ00 −→ ρG2±0 −→3 ρ13 −→ (3) −ω

(4) ω

(5)

3 ρ14 −→4 ρ13 −→ ρ10 , which gives

(5)

ρDs2 = ρ10 =

2

iGB |G4 | exp(iks3 · r) 2

d1 d23 d6 (d1 + |G2 | /d2 )

(8.38)

According to Eq. (8.38) and the dressed-state picture in Fig. 8.14 (d4), ρDs2 is greatly suppressed at ω1 + ω2 = Ω 1 + Ω 2 (i.e., Δ1 + Δ2 = 0), as shown in

8.3 Interacting MWM Processes in a Five-level System with · · ·

309

Fig. 8.22. DSWM signal intensity versus Δ4 /Γ10 . The inset plots show comparisons of Im = Im2 ρDs1 and Re = Re2 ρDs1 . The SWM signal intensity with no dressing field is normalized to be 1. Adopted from Ref. [28].

Fig. 8.14 (d4), or enhanced at ω1 + ω2 ≈ (Ω 1 ± ΔG2 /2) + Ω 2 (i.e. Δ1 + Δ2 ± ΔG2 /2 ≈ 0). Note that these two DSWM processes (ρDs1 and ρDs2 ) have an interesting relationship, i.e. the pump field E2 of ρDs1 is the dressing field of ρDs2 , while the pump field E4 of ρDs2 is the dressing field of ρDs1 . So these two DSWM processes dress each other. Since these two DSWM signal fields copropagate in the same direction, total signal will be proportional to the mod square of ρsum (Δ), where ρsum (Δ) = ρDs1 + ρDs2 with Δ = Δ2 − Δ4 . Figure 8.23 presents evolutions of the total signal intensity versus the probe field detuning Δ1 for different Δ4 values. Parameters are G2 = 50 MHz, G3 = 100 MHz, and G4 = 50 MHz, with Δ3 = 0 and Δ2 = −150 MHz. The moving peak (shifting from left to right) represents the three-photon resonant signal (satisfying Δ1 − Δ3 + Δ4 = 0) of ρDs2 and the fixed peak along the dotted line represents the two-photon resonant signal (Δ1 + Δ2 = 0) of ρDs1 . As the strong coupling field detuning Δ4 is changed, the moving signal of ρDs2 is first greatly suppressed in Fig. 8.23(d) and then enhanced in Fig. 8.23(f) while the fixed signal of ρDs1 is enhanced in Fig. 8.23(b) and greatly suppressed in Figs. 8.23(c) and 8.23(d). In fact, variations of the signal intensities of ρDs1 and ρDs2 are induced through mutual-dressing processes and constructive or destructive interference as to be described in the following. We first consider the mutual-dressing processes. As Δ4 is changed, according to the dressed-state analysis, the moving peak (satisfying Δ1 = Δ3 − Δ4 ) of ρDs2 with the dressing field E2 first satisfies the suppression condition Δ1 = −Δ2 (i.e., Δ4 = Δ2 + Δ3 = −150 MHz) and then the enhancement condition Δ1 ≈ −Δ2 + ΔG2 /2 (i.e., Δ4 ≈ Δ2 + Δ3 − ΔG2 /2 ≈ −170 MHz) while the fixed peak (satisfying Δ1 = −Δ2 ) of ρDs1 with the dressing field E4 first satisfies the enhancement condition (Δ1 ≈ Δ3 − Δ4 + ΔG4 /2, i.e., Δ4 ≈ Δ2 + Δ3 + ΔG4 /2 ≈ −130 MHz) and then the suppression condition (Δ1 = Δ3 − Δ4 , i.e., Δ4 = Δ2 + Δ3 = −150 MHz). Figure 8.24 shows the fixed signal intensity of ρDs1 versus Δ4 . Here, the SWM signal intensity with no dressing field (G4 = 0) is normalized to be 1. The enhanced peak

310

8 Multi-dressed MWM Processes

Fig. 8.23. The total signal intensity constituted by two DSWM ρDs1 and ρDs2 versus Δ1 for the different Δ4 values: (a) Δ4 = −120 MHz, (b) −130 MHz, (c) −145 MHz, (d) −150 MHz, (e) −155 MHz, (f) −170 MHz, (g) −195 MHz. Adopted from Ref. [28].

and suppressed dip are located at Δ4 = −130 MHz and Δ4 = −150 MHz, respectively, which fit well with the evolution curve of Figs. 8.23.

Fig. 8.24. The DSWM signal intensity of ρDs1 versus Δ4 for Δ1 = 150 MHz, Δ2 = −150 MHz, Δ3 = 0 and G4 = 50 MHz. The SWM signal intensity with no dressing field is normalized to be 1. Adopted from Ref. [28].

Next, we study destructive or constructive interference in this system. Based on the analysis of mutual-dressing processes, when two peaks overlap with each other, the total signal intensity shows a significant suppression as shown in Fig. 8.23(d). However, maximal suppression of the fixed signal occurs at about Δ1 = 150 MHz in Fig. 8.23(c). This is due to destructive interference between ρDs1 and ρDs2 . In fact, constructive and destructive interferences can be converted into each other as the two peaks are tuned to overlap or separate in Fig. 8.23. Figure 8.25(a) shows the intensity values of |ρDs1 |2 , |ρDs2 |2 , |ρDs1 |2 + |ρDs2 |2 , |ρsum |2 , and |ρsum |2 − |ρDs1 |2 − |ρDs2 |2 at Δ1 = − (Δ2 + Δ4 ) /2 (the position is between the two peaks in Fig. 8.23) versus the detuning difference Δ between Δ2 and Δ4 , respectively. Parameters are Δ2 = −150 MHz, Δ3 = 0, Δ1 = − (Δ2 + Δ4 ) /2, G2 = 50 MHz, G3 = 100 MHz and G4 =

8.3 Interacting MWM Processes in a Five-level System with · · ·

311

Fig. 8.25. (a) Total signal intensities of |ρsum |2 (dot-dot-dashed curve), |ρDs1 |2 (dashed curve), |ρDs2 |2 (dotted curve), |ρDs1 |2 + |ρDs2 |2 (dot-dashed curve) and interference item 2|ρDs1 ||ρDs2 | cos θ (solid curve) versus Δ. Here Δ = Δ2 − Δ4 , θ = θDs1 − θDs2 . The maximum intensity is normalized to 1; (b) θDs1 (dashed curve), θDs2 (dotted curve) and θ (solid curve) versus Δ. Adopted from Ref. [28].

50 MHz. |ρDs1 |2 and |ρDs2 |2 have the maximal values at Δ = −32 MHz (dashed curve) and 10 MHz (dotted curve), respectively. Hence, there exist two peaks at around Δ = −32 MHz and 10 MHz in the dot-dashed curve that represents the value of the sum |ρDs1 |2 + |ρDs2 |2 . However, with interference terms, the total signal intensity |ρsum |2 (dot-dot-dashed curve) has two peaks and one deep hole at Δ = −24 MHz, 1.8 MHz and –4 MHz, respectively. It means that the interference has a significant impact on the total signal intensity. The solid curve shows the value of the interference term which is equal to |ρsum |2 − |ρDs1 |2 − |ρDs2 |2 . Here, the value below or above zero means destructive or constructive interference. There are two constructive peaks and two destructive holes, whose amplitudes are comparable to that of both DSWM signals. Actually the variation of phase difference between the two DSWM processes changes the constructive interference into destructive one, and vice versa. More specifically, by letting ρDs1 = |ρDs1 | exp(iθI ) and ρDs2 = |ρDs2 | exp(iθII ), we can write |ρsum |2 −(|ρDs1 |2 +|ρDs2 |2 ) = 2 |ρDs1 | |ρDs2 | cos θ, with θ = θI − θII . Figure 8.25(b) shows the phases θI (the dash curve) and θII (the dot curve), as well as the phase-difference θ (the solid curve), versus Δ. As θI and θII are changed, θ alters between 2π. and −π, and the interference switches from constructive, to destructive, to partly constructive, and finally to partly destructive (as given in Table 8.2). Table 8.2. Evolution of θ, constructive and destructive interference versus Δ. Adopted from Ref. [28]. Δ [–50,–38.6] (–38.6,–10.7] (–10.7,–1] (–1,5] (5,50] θ [7π/5, 3π/2] [3π/2, 8π/5] [π/2, 3π/2] [−π/2, π/2] [−π/2, −π] interference destruction construction destruction construction destruction

Since interference effect between the two DDFWM processes is similar to that of the two DSWM ones (but the contribution can be one order of magnitude smaller than the mutual-dressing effect), in the following we will only consider the mutual-dressing processes between the nested- and parallel-

312

8 Multi-dressed MWM Processes

cascade DDFWM processes. Figure 8.26 shows evolutions of the total signal intensity between the nested-cascade DDFWM ρDf4 in the system [see Fig. 8.14(b1)] and the parallel-cascade DDFWM ρDf5 in the system [see Fig. 8.14 (c1)] versus the probe field detuning Δ1 for different Δ3 values. Parameters in Fig. 8.26 are G2 = 60 MHz, G3 = 30 MHz, G4 = 20 MHz, Δ2 = −120 MHz, Δ4 = 30 MHz. Here −iGA exp(ikf1 · r)   |G3 |2 d1 d2 d1 + d3 + |G4 |2 /d6 −iGB exp(ikf2 · r) = d3 (d1 + |G2 |2 /d2 ) (d7 + |G4 |2 /d8 )

ρDf4 =

(8.39)

ρDf5

(8.40)

Note that the pump field E3 in ρDf5 serves as the dressing field of ρDf4 , while the pump field E2 in ρDf4 is the dressing field of ρDf5 . So, these two DDFWM processes dress each other in this system. From Fig. 8.26(a) to Fig. 8.26(c), right peak is the two-photon resonant (Δ1 + Δ2 = 0) signal of ρDf4 and left peak is the two-photon resonant (Δ1 − Δ3 = 0) signal of ρDf5 . As Δ3 is increased, the moving peak of ρDf5 is suppressed while the fixed peak of ρDf4 is enhanced. In Fig. 8.26(d) as Δ3 is increased continuously, the change in the peak of ρDf5 is too small to be seen and the peak of ρDf4 is also suppressed. From Fig. 8.26(e) to Fig. 8.26(g), the peak of ρDf5 moves to right and is enhanced at Δ3 = 145 MHz, while the fixed peak of ρDf4 is enhanced again at Δ3 = 155 MHz. Finally, in Figs. 8.26(h) and 8.26(i), the fixed peak of ρDf4 is suppressed again at Δ3 = 165 MHz, while the moving peak of ρDf5 gets smaller owing to large detuning Δ3 . In Fig. 8.26, the fixed peak of ρDf4 is doubly enhanced in Figs. 8.26(c) and 8.26(g) and doubly suppressed in Figs. 8.26(d) and 8.26(h). According to dressed-state analysis of the nested-cascade scheme, the dual-suppression (or dual-enhancement) results from the double-resonance of the dressing field E3 (or the probe field E1 ). As the peak of ρDf5 moves to right (Δ3 is changed), the fixed peak of ρDf4 can satisfy the dual-suppression condition of Δ3 ≈ Δ1 +Δ4 ±ΔG4 /2 ≈ 110 MHz or 165 MHz and the dual-enhancement condition of Δ3 ≈ Δ1 + Δ4 ± ΔG4 /2 − ΔG3 /2 ≈ 100 MHz or 155 MHz. Figure 8.27 plots the signal intensity of ρDf4 versus Δ3 . The FWM signal intensity with no dressing fields is normalized to be 1. The pair of suppressed or enhanced channels in Fig. 8.27 agrees well with evolutions shown in Fig. 8.26. The two enhanced peaks are located at Δ3 = 100 MHz and Δ3 = 155 MHz, respectively, while the two suppressed dips are located at Δ3 = 110 MHz and Δ3 = 165 MHz, respectively. Similarly, according to dressed-state analysis of the parallel-cascade scheme, the moving peak (located at Δ1 = Δ3 ) of ρDf5 first satisfies the suppression condition Δ1 = −Δ2 (i.e., Δ3 = −Δ2 = 120 MHz) and then the enhancement condition Δ1 ≈ −Δ2 + ΔG2 /2 (i.e., Δ3 ≈ −Δ2 + ΔG2 /2 ≈ 145 MHz) .

8.3 Interacting MWM Processes in a Five-level System with · · ·

313

Fig. 8.26. The total signal intensity constituted by two DDFWM ρDf4 and ρDf5 versus Δ1 for the different Δ3 values: (a) Δ3 = 45 MHz, (b) 55 MHz, (c) 100 MHz, (d) 110 MHz, (e) 140 MHz, (f) 145 MHz, (g) 155 MHz, (h) 165 MHz, (i) 185 MHz. Adopted from Ref. [28].

Fig. 8.27. DDFWM signal intensity of ρDf1 versus Δ3 for Δ1 = 120 MHz, Δ2 = −120 MHz, Δ4 = 30 MHz, G2 = 60 MHz, G3 = 30 MHz, G4 = 20 MHz. The FWM signal intensity with no dressing field is normalized to be 1. Adopted from Ref. [28].

8.3.4 Conclusion and Outlook It is important to understand competitions and strong interactions (mutualdressing processes and constructive/destructive interferences) between two coexisting MWM processes. Resonances of the dressing field and that of the

314

8 Multi-dressed MWM Processes

pump (probe) fields result in suppressions or enhancements of MWM signal intensities. Constructive/destructive interferences also have significant influences on the total signal intensity. Through adjusting frequency detunings of the strong coupling/dressing fields, two dressed MWM signals can be tuned together or separated, which can modify (suppress or enhance) each other and affect the total signal intensity. Investigations of such interaction processes will help us to understand and optimize the generated multi-channel nonlinear optical signals. In this chapter we have presented theoretical studies of the DDFWM processes in an open five-level atomic system with different multi-dressed schemes. Three doubly-dressing configurations, i.e., the nested-cascade, sequential-cascade and parallel-cascade schemes, have been discussed in detail. By carefully comparing the results of these doubly-dressing schemes, we find that the interactions between the two dressing fields are the strongest in the nested-cascade scheme, but are the weakest in the parallel-cascade scheme. The sequential-cascade scheme is an intermediate case between those two above cases and shares certain common features of the other two doubly-dressing schemes under the weak dressing-field limit. There are also coexisting DSWM and EWM processes in this five-level open system. Competitions and interactions between coexisting nested-cascade DDFWM and parallel-cascade DDFWM or two DSWM processes have been analyzed in detail, which show two interesting physical processes, i.e. mutual-dressing processes and constructive/destructive interferences. Using the dressed-state analysis, the suppressions and enhancements of the MWM signal intensities and the total DDFWM signal intensity are shown to be due to the resonances of the dressing/pumping fields, as well as the constructive/destructive interferences between different channels. Investigations of those different doubly-dressing schemes and interactions between various MWM processes in multi-level atomic systems can help us to understand the underlying physical mechanisms and to effectively optimize the generated multi-channel nonlinear optical signals. Controlling these high-order coexisting MWM processes can have important applications in designing novel nonlinear optical materials and optoelectronic devices in multi-state systems. For example, certain speciallydesigned all-optical switches (or logic gates) or efficient multi-qubit phase gates for quantum information processing can be envisioned, which require coexisting different-order (such as third-order and fifth-order) nonlinearities. Another direction of potential applications for the media with coexisting highorder nonlinearities is in the propagations of high intensity optical pulses. The different nonlinear coefficients with opposite signs, such as competing cubic-quintic nonlinearities, can lead to strong stabilization of high-intensity pulse propagation, forming stable two-dimensional solitons under certain conditions, and pulse shaping. Also, the competing cubic-quintic nonlinearities have been proposed for realizing liquid light condensate in a multi-state system.

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Index

absorption-dispersion 118 adiabatic 264, 268 amplitude-stabilized 156 analogous 56 anti-Stokes 145 arbitrary 56 Autler-Townes 289 auto-correlation 147, 189 autocorrelation 66, 84 back-and-forth 264, 268 bare-state 108, 118 bichromophoric 88, 98 Boltzmann 114 Bragg 31 broadband 19, 25, 29, 36, 39, 51 Brownian-motion 145 chromophore 84, 100 close-cycled 208, 232 closed-loop 265 cold-atomic 293 Color-locking 107, 129 continuous-wave 5 counter-propagating 191 cross-correlation 151, 198 cross-Kerr 122 cross-phase 264 cubic-quintic 211 cubic-quintic-type 4 density-matrix 107, 111 dephasing 115

difference-frequency 1 diffracted 210 dimensionless 20, 46 dipolar 45 dipole-moment 21 distortions 30 Doppler-broadened 18, 47 Doppler-free 63, 92 double-frequency 147, 204 double-peak 225 doubly-dressed 261, 283 dual-enhancement 302 dual-suppression 302 electro-optic 4 ergodic 67 exponential 35, 38, 53 field-correlation 19, 60 finite-bandwidth 30 fluorescence 216, 221 frequency-domain 170, 203 frequency-locked 253 gas-phase 114, 133 Gaussian-amplitude 113 group-velocity 265, 282 homogeneous 25, 51 homogeneously-broadened 107 hybrid 66, 103 hyperfine 216, 231

318

Index

intensity-dependent 3 interaction-induced 173 intermediate 298 intra-atomic 114 isotropic 2 ladder-type 107, 108 light-induced 14, 15 liquid-like 233 Lorentzian 73, 95 Markovian 44 mismatch 292 mod-square 238 molecular-reorientation 6 molecular-reorientational 144, 172, 196 Monochromatic 20, 38 monotonically 222 multi-channel 283, 314 multi-colored 1 multi-dimensional 211 Multi-dressed 1 multi-photon 1 multimode 21, 33, 49 multiple-EIT 211 Narrow-band 22, 47 near-resonant 11 non-degenerate 6 non-depleted 267 non-steady-state 138 non-trivial 114, 138 nonadiabatic 64 off-resonant 301, 305 on-resonant 301 one-photon 66 open-cycled 208, 236 opto-electronic 243 orthogonal 253 oscillatory 85 parallel-cascade 293, 297

partial-coherence 145 peak-to-background 152 perpendicular 35, 55 phase-conjugation 18 phase-dependent 254, 255 phase-diffusion 64, 97 phase-jump 30 phase-matched 108, 122 phase-sensitive 106, 118 photodiode 27 Photon-echo 19, 27, 39 primarily-dressed 289, 300 Quanta-Ray 27 quasi-cw 64 quasi-two-level 18 radiation-matter 66, 85 Raman-active 6 Raman-induced 145 Rayleigh-active 177 Rayleigh-induced 204 refractive 3, 4 reorientational 144 rotating-wave 110 second-harmonic 2 secondarily-dressed 283 self-defocusing 4 self-diffraction 73, 99 self-focusing 4 sequential-cascade 293, 297 singly-dressed 284 sinusoidal 27 solid-state 2 spatial- temporal 255 spatial-temporal 1 splitting 100 steady-state 243 straightforward 120, 129 sublevels 100 subluminal 269 subpulse 100

Index

sum-frequency 1, 63, 65 superluminal 269 superposition 286, 287 susceptibilities 106, 117, 137 synchronization 171 terahertz 66, 103 three-dimensional 163 time-delay 126 time-dependent 264 tradeoff 187 transform-limited 113

triple-peak 283 two-color 172 ultraslow-propagation 277 uncollimated 102 upper-branch 236 vibrations 30 Wiener-Khintchine 114 Z-scan 131

319