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= 0 (181a)
( A P ) ~=
(c) + 2qhw
(a2ezr sinh’ r).
(181b)
For a characteristic measurement time 7 , the fluctuations in the difference between the momenta of the end masses leads to a differential change in the length of the arms of the interferometer by an amount
This is the radiation-pressure error in z. In the limit a2 9 sinh’ r, this simplifies to
2. Photon-Counting Error We assume ideal photodetection, i.e., the photodetectors have a quantum efficiency of unity. Then the photodetection statistics are the same as photon statistics. The photon-counting error can be determined by calculating the fluctuations in the number of output photons. The output fields Ei:\ and Ei”, have the form
where
s=-, 2qwz C
SQUEEZED STATES OF THE RADIATION FIELD
195
is the phase difference due to the different arm lengths of the interferometer. Once again, a 4 2 phase shift between the reflected and transmitted beams is assumed at the beam splitter. The annihilation operators c1 and c2 for the fields EL!,! and EL:/ are
+ i(eid+ 1)u2],
(1 86a)
+ l)al - (eia- 1)u2].
(1 86b)
c1 = +[(eid- l)al c2 = +[i(eib
The photon-number operator the difference measurement is Ad
Ad
= ctc2 - cicl for the total output field in
=
+ ata,)sin 6.
(187)
For an input field I+) given by Eq. (180), the mean number of photons and the fluctuations in the photon number in the output are given by
(A,)
= (a2 - sinh2 r)cos
(Ah,), = (a'e-"
6,
+ sinh2 r)sin2 6.
(188) ( 189)
In the limit a2 B sinh' r, these simplify to (Ad)
AA,
= a'
cos 6,
= ae-'
sin 6 .
Changes in z are detected by looking at changes in (A,). (188), we find that a change Az produces a change
Using Eqs. (185) and
('7)
An, = -a2 - Az sin 6, in (n,,). On using Eq. (192) to transform An, into a corresponding error in z, we obtain the photon-counting error
It is clear from Eqs. (183) and (193) that squeezing the field Ei:) increases the radiation-pressure error and decreases the photon-counting error. 3. Optimum Sensitivity and Optimum Power
The total error in measuring the differential change in the length of the arms of the interferometer due to radiation-pressure and photon-counting errors is AZ = [(Az),Z, (Az)$]~'~. (194)
+
196
KhaIid Zaheer and M . Suhail Zubairy
Minimizing the total error with respect to u2, we have
which is of the order of the standard quantum limit. This minimum is achieved for an optimum value of u2 given by 2
opt
- 2 -21 - aoe
9
(196)
with
Correspondingly, the optimum input power is Popt= Poe-2r,
(198)
where P o , given by Eq. (176), is the optimum power for a standard (r = 0) interferometer. The optimum power can therefore be adjusted by squeezing the vacuum entering from the unused port. It should be mentioned here that in an interferometer that measures changes in phase (2 = & ( z ,+ z2) rather than z), one could squeeze the input laser light to obtain an optimum power as in Eq. (198). Equations (196)-( 198) are only valid for the condition u2 % sinh’ r. If the squeeze parameter is large enough to violate this condition, the terms dropped in Eqs. (1 82) and (189) dominate. Increasing the squeeze factor beyond a value rmaxx $ln(ao) degrades the optimum sensitivity. Given the experimental difficulties in achieving large degrees of squeezing, this is not likely to be a serious concern in practice. A more stringent condition is that the sensitivity depends critically on the quantum efficiency of photodetectors and the linear losses inside the interferometer. Both of these degrade the squeezing and may even destroy it completely. Other limitations are unequal losses in both arms, wave-front distortions, depolarization, and imperfect overlap between the interfering beams. These have been analyzed by Gea-Banacloche and Leuchs (1987,1989). Remedial measures may be taken by appropriately modifying the spectrum of squeezed light. These analyses also show that squeezing the input-laser-intensity fluctuations might, if practicable, be very useful in these “nonideal” systems. In conclusion, in addition to increasing the arm length, frequency, power of the laser, and the storage time to increase the strain-sensitivity, “squeezing the vacuum” is an additional option. A number of prototype gravitational-
SQUEEZED STATES OF THE RADIATION FIELD
197
wave-detector interferometers around the world have reached the photoncounting limit, and the next generation (the real detectors) being planned as of 1990 is expected to be (for some conditions at least) limited by the photoncounting error. Squeezing offers a great opportunity to operate these devices at reduced laser power, which would considerably limit damage to the mirrors, as well as make the required laser powers themselves more readily accessible.
B. ATOMIC DECAY I N SQUEEZED VACUUM The decay of an atom in an excited state is well understood in a model in which the atom is coupled to a reservoir of simple harmonic oscillators. This model of quantum damping, with the reservoir in the vacuum state or in a thermal state, has been treated by many authors using density matrix formalism (Lax, 1966; Louisell, 1973; Weidlich et al., 1967; Sargent et al., 1974) and the Langevin method (Haken, 1970). Collet and Gardiner (1984) derived a master equation for an atomic density matrix when the source term is described by squeezed white noise. This corresponds to a situation in which squeezed vacuum is incident on a two-level atom. We consider a system denoted by S interacting with a reservoir denoted by R. Denoting the combined density matrix by pSR, the reduced density operator for the system is obtained by taking a trace over the reservoir variables, i.e. Ps = TrRIPSRl*
( 199)
In order to obtain an equation of motion for the reduced density operator, one can formally integrate the Liouville equation jAbs
= TrR[V(t), PSR(t)l,
(200)
and iterate it to yield in Born approximation
bS
=
-i/h TrR[V(t), ps(ti) @ PR(ti)] - l/hz TrR
l:
(201)
{ v(t>, Cv(t'), P S ( ~@) PR(ti)l}dt'*
Here V ( t ) is the interaction Hamiltonian and we have made the reasonable assumption that the system and reservoir are statistically independent at the initial time ti, i.e., PSR(ti) = d t i ) @ PR(li)*
(202)
Khalid Zaheer and M. Suhail Zubairy
198
We consider a system of a two-level atom (ps = p A ) interacting with a reservoir that consists of a set of harmonic oscillators described by annihilation (creation) operators bk(bf) and densely distributed frequencies wk. In the interaction picture and the rotating-wave-approximation, the Hamiltonian is
v(t)= h
c Sk(bfo-
+
- e-i(mo-mk)f
k
b ei(mO-mk)r1,
+ k
(203)
where u- = I b ) (a1 in terms of the excited (la)) and ground (1 b ) ) states of the atom. On inserting Eq. (203) in Eq. (201), one obtains
1:
P A = - i 1gk(b~)U-pA(c~)e-(Wo-W)r - lft' k
x
ei(CXO-U3)r-i(m0
-Ok')f'
(bkbkt')
+ (a-a+pA
gkgk'[(a+a-pA
- a-pAa+)
kk'
- a+pAu-)
- i(mo - m)r + i(oo - rOk,)t' (bl bk,) - (2a- p A +)e- [(no-Q')r + i(aO-mk*)t'
+ H.C.
In Eq. (204), the expectation values such as ( b k b t ) depend upon the initial state of the reservoir. For a reservoir in a squeezed state, @,) = ( b l ) = 0,
(205a) (205b) (20%) (205d)
(205e)
The parameters N and M depend upon the state of the reservoir and thus the particular scheme used to generate squeezing. They obey the inequality JM122 N ( N
+ l),
(206)
where the equality is achieved for an ideal squeezed state. For the multimode squeezed vacuum discussed in Sec. II.C, one symmetric about the atomic frequency wo and having a bandwidth that is small compared to coo, N =sinhZr and M
= eie sinh r
cosh r.
(207a) (207b)
With these substitutions in Eq. (204) and following the same procedure as in Wigner-Weisskopf theory of spontaneous emission (which essentially in-
199
SQUEEZED STATES OF THE RADIATION FIELD
volves taking the continuum limit), the master equation is (Gea-Banacloche et al., 1988b)
PA = - -Y cosh' r(cr+o-p,
- 2o-p,a+
+ p,a+a-)
2
- ye-"
sinh r cosh r a+pAa+,
as the decay rate. Here D(o)is the density of modes in free space. For r = 0, Eq. (208) is readily seen to reduce to the well-known result for decay into vacuum. (See, e.g., Sargent et al., 1974.) From Eq. (208), the equations of motion for the expectation values of the dipole quadratures ol,o2 and inversion g3 are < k l > = -ye2r(al),
(210a)
( k 2 ) = -ye-"(az),
(210b)
( k 3 ) = -y(2 sinh2 r
Y + l)(a3) - -, 2
(210c)
where (21la) (211b) and we have chosen the phase 0 = 0. Clearly, the two components of the polarization decay at different rates y1 = y e2r,y 2 = y e - 2 rdepending upon its initial phase relative to the phase (0) of the squeezed vacuum, while the inversion decays rapidly to its equilibrium value at rate y3 = y(2 sinh' r 1). In time scales that are short as compared with y;' but larger than y;' and
+
Y i 1 9
(al) -,0,
(03) -,
- 1/(2 sinh' r
+ I),
(212)
but ( 0 2 ) remains essentially unchanged. This can be detected from the spectrum of fluorescent light. Gardiner (1986) has shown that the spectrum contains a peak corresponding to y1 that can be made as narrow as desired depending upon the squeeze factor r.
200
Khalid Zaheer and M. Suhail Zubairy
C. LINEARAMPLIFIERS Linear amplifiers were discussed in the early days of quantum electronics (Gordon et al., 1963b; Glassgold and Holliday, 1965; Mollow and Glauber, 1967a). With the generation of squeezed states of the radiation field, there has been a renewed interest in this area. This interest arises due to two reasons: first, on the application side, to study the possibilities of amplification of these nonclassical states of light with particular reference to their use in communication and photon-correlation experiments; second, squeezed states themselves may be employed to improve upon the quantum limits of the linear amplifiers. The principal use of amplifiers is in the measurement process; they can bridge the gap between the quantum theory and classical theory by strengthening a quantum signal enough to bring it into the classical domain, thus allowing its measurement without any significant disturbance. Analysis of the linear amplifiers shows that the amplification process is invariably associated to noise that is uncorrelated to the signal and obeys thermal statistics (Friberg and Mandel, 1983). Indeed, an operator-transformation equation describing a linear-amplification process
where G is the amplification factor, requires the second term for consistency so that the commutation relation at the output is satisfied. Here Ft is an operator obeying a boson-commutation relation, which commutes with a,. Defining the Hermitian quadratures for the bosonic signals as X, = f(ae-"
+ ate"),
(2 14)
the variances in the two quadratures are Var(Xr') Var(Xr:,lz)
= GVar(Xk)
+ (G - l)Var(F,),
= GVar(Xk+,/,)
+ (G - l)Var(Fe+(,/zl).
(215a) (215b)
The first term in the preceding equations represents the amplified input noise and the second term represents the "additive noise" due to amplification. Such a gain-dependent noise is a vivid manifestation of the fluctuationdissipation theorem. Based on Caves' (1982) classification, one calls a phase-insensitive amplifier one that amplifies both quadratures by the same factor and also adds equal noise to the two quadratures. Consequently, (a) it is incapable of giving
SQUEEZED STATES OF THE RADIATION FIELD
20 1
squeezed output for an unsqueezed input, and (b) any squeezing in the input signal is destroyed due to the addition of phase-insensitive noise. From Eqs. (215a) and (215b), the maximum gain preserving any squeezing at the output is
which for a highly squeezed input and the additive noise in a vacuum state is 2. A phase-sensitive amplifier, on the other hand, is one that responds differently to the two quadrature phases in the form of unequal gains or unequal noise or both. The added noise obeys a fundamental theorem due to Caves (1982) :
Here we have taken the gain for the two quadratures to be equal. For unequal gains, the theorem takes the general form of an “amplifier-uncertainty principle.” The added noise in an amplifier arises due to internal degrees of freedom. For example, in a nondegenerate parametric amplifier the operator F represents the multimode idler field. Clearly, by Eq. (217), the added noise in one quadrature can be reduced at the expense of a larger added noise to the conjugate quadrature, thus satisfying the amplifier-uncertainty principle. Yurke and Denker (1984) and later Milburn et al. (1987) have proposed the squeezing of the idler field in a nondegenerate parametric amplifier. In this case, it is possible to surpass the “cloning limit” of 2 given by Eq. (216). In the following, we consider two explicit amplifier models involving the use of squeezed states. Even though the two models correspond in principle to different physical systems, they are intimately similar in the sense that both rely on rigging the gain medium. As will be seen in the following, they are mathematically equivalent, and the latter may in fact be regarded as possible physical realization of the former. 1. Rigged Reservoir Scheme
In order to amplify a signal, one needs a reservoir full of energy that it can supply to the signal. But a reservoir at high temperature would simply feed thermal noise into the signal. The model proposed by Glauber (1986)
Khalid Zaheer and M. Suhail Zubairy
202
involves the interaction of a boson mode with a bath of inverted harmonic oscillators through the interaction Hamiltonian
HI= h(a’R’
+ uR).
(218)
Here the coupling takes place through the reservoir variable =
k
where 1, are the coupling constants and bk are the annihilation operators for the boson modes of the bath. Linearity of operation implies a weak coupling in the bath. This amplifier still adds thermal noise to the system since the state of the reservoir resembles a canonical distribution with negative temperature. Dupertuis and Stenholm (1987) first proposed the “rigging” or modification of the reservoir by preparing it in a squeezed vacuum centered around the frequency of the mode that is to be amplified. The boson modes of the bath obey correlation functions (205a)-(205e). The master equation can be obtained following the Langevin formalism developed by Gardiner and Collet (1985) that applies to the cases when the state of the input to, and output from, the reservoir consists of “squeezed white noise.” ap NA - = __ (apa’ at 2
A + a’pa - a’ap - pa’a) - (N + l)(aa’p 2
+ A2 M*(aap - 2apa + paa) + A2 M(a’a’p -
-
- 2a’pa’
- 2a’pa
+ pa’a’),
+ paa’) (220)
where A is an amplification constant. This is the linear-response master equation for an amplifier with squeezed inverted harmonic oscillators. As it stands, this is of course only a mathematical model. Its implementation by means of some physical system remains an open question. We shall presently show an example of a specific physical system that leads to an equation essentially identical to (220). 2. Two-Photon Linear Amplijier A practical linear amplifier involves the coupling of the boson mode of interest to an inverted population of atoms. The linearity of operation restricts it to one-photon processes only. This situation corresponds to a laser with the end mirrors removed being perturbed by a weak external field. The equations of motion have been known for a long time and have even been solved in general terms (Louisell, 1969; Carusotto, 1975; Abraham and Smith, 1977; Rockower et al., 1978).
SQUEEZED STATES OF THE RADIATION FIELD
203
Since amplification is achieved through an emissive process, it is inevitably associated with spontaneous emission. Consider the usual master equation for a laser below saturation and without the cavity losses:
where A is the linear gain coefficient. The spontaneous emission can readily be seen from the equation for the photon number (ti) = A ( ( n )
+ 1).
(222)
The amplifier therefore adds spontaneous photons at a rate A and the output contains a component of "noise photons" with random phases. The two quadratures of the field receive equal amount of noise and the amplifier is phase-insensitive. Scully and Zubairy (1988) proposed a phase-sensitive amplifier in which the gain medium consists of three-level atoms in a cascade configuration, prepared in a coherent superposition of states. The density operator for the atom at time t = 0 from Eq. (66) is = Paala)(bl - Pacla)(cl - ~calc)(al + ~ccIc)(cl-
(223)
The interaction Hamiltonian at resonance and in the rotating-wave approximation is = Ma(la>(bl
+ Ib>(cl) + a'(Ib)(QI + Ic>(bl)I,
(224)
where g is the atom-field coupling constant, taken to be equal for the two transitions. The equation of motion for the reduced density operator pF for the field can be derived using standard methods (Sargent et al., 1974) and in Born-Markov approximation is given as
where A is the linear gain coefficient. Apart from the usual emission and absorption terms, the equation of motion contains anomalous terms that arise due to atomic coherence. In Eq. (220), which has a form similar to (225), such terms result from squeezing of the state of the amplifying reservoir.
204
Khalid Zaheer and M . Suhail Zubairy
3. Phase-Sensitive Amplijication
From Eq. (225), the operator expectation values at the output (time t) can be determined in terms of the expectation values at the input (t = 0). These are (226a)
,= J G ( a ) o , (ata), = G(ata),
+
Paa ~
Paa - P c c
(G - 11,
(226b)
The variances at the output in the two quadratures defined by (214) satisfy Eqs. (215a) and (215b), but now the two quadratures receive unequal noise given by
For pall = pee = lpacl and the phase choice 28 - @ = n, the added noise is zero and the signal-to-noise ratio (SNR) at the output is equal to the SNR at the input. From Eqs. (227) and (228) we see that the additive noise quenching and gain are competing factors. For complete quenching of additive noise, the gain is 1. Defining a parameter E such that paa = (1
+ 6)/Z
pCC= (1 - 6)/2,
Jpacl = (1 - c2)’l2/2,
(229)
it follows that an amplified signal without added noise in the quadrature component X , is obtained in the limits 28 - @ = n,
At
+ 00,
E + 0,
At6 = x (finite).
(230)
Under these conditions Var(F,)
+ Var(Fo+z/2)= 2€1 -
1
Z’
so that Caves’ theorem is satisfied. For this amplifier, it is possible to have sub-Poisson statistics at the output for a particular class of super-Poisson states at the input (Scully and Zubairy,
SQUEEZED STATES OF THE RADIATION FIELD
205
1988; Dupertuis and Stenholm, 1988). For this rather couterintuitive process, one requires the initial field state to satisfy 0 < (2(da), - (d), - (a+2)o) I 1,
(232)
and the amplifier is limited to a certain range of gain. So far no explicit expression for such a field state has been found. Zaheer and Zubairy ( 1988a) discussed a two-level phase-sensitive amplifier in which the gain medium consists of correlated pairs of atoms. They showed that the additive noise is in a squeezed state, the effective squeeze parameter being a function of initial atomic variables. Dupertuis et al. (1987a, b) show that for a strongly squeezed reservoir, squeezing can be retained at the output for large gains. In the attenuator configuration, the system acts as a “noise cleaner.”
D. LASERSWITH
SQUEEZED VACUUM
1. Laser with Injected Squeezed Light In the absence of all sources of noise (such as thermal and mechanical), the laser linewidth is limited by spontaneous emission. A simple pictorial model in this regard envisions it as being due to a random-phase-diffusion process arising due to the addition of spontaneously emitted photons with random phases to the laser field (Loudon, 1983). From Eq. (222) we see that spontaneous emission in a particular mode with n photons appears as the 1 in a factor (n + 1). In turn, the role of vacuum fluctuations in spontaneous emission is well known (Dalibard et al., 1982). It was natural to apply Caves’ idea to active systems such as a laser in which the vacuum fluctuations leak into the cavity through the out-coupling mirror. The analysis by GeaBanacloche (1987) gives an interesting result that is somewhat at variance with the usual picture of phase-noise in a laser. The physical system considered is a ring laser shown in Fig. 15 with one running-wave mode above threshold. An external field is coupled to the intracavity field through an end mirror. Such a coupling is also possible in a standing-wave cavity by using a polarization rotation device and reflection polarizers. The state of the external field, which is assumed to be squeezed vacuum centered around o, the operating frequency of the laser, is given by (cf. Eq. (53))
I Jlo) =
fl exp[ - reieb(o + c)b(o - + re-%t(w + c)bt(w - c)] 10). 6)
fro
(233)
Khalid Zaheer and M. Suhail Zubairy
206
FIG.15. Ring laser arrangement considered to couple squeezed vacuum (dashed line) to the intracavity field. (From Gea-Banacloche, 1987.)
The effect of squeezed vacuum is incorporated in the quantum-Langevin equation through the Langevin-force operators which is given as (Yamamoto et al., 1986) da dt
-=
1 2
- - [r
- A(N2
- N l ) l a + G,(t) + F,(t)
+ F,(t),
(234)
where A is the linear gain coefficient and N , and N, are the operators for the populations of the upper and lower levels, respectively. Here atomic variables have been adiabatically eliminated and G, is the corresponding force operator. The correlation functions for G, are (Lax and Louisell, 1969) (Gd(t)G,(t')) = AN26(t - t'),
(235a)
- t').
(235b)
(G,(t)Gf(t'))
= A N , d(t
The operators Fa and F , represent the absorption and transmission losses for the cavity field, respectively, and in terms of boson creation and destruction operators (236a) (236b)
SQUEEZED STATES OF THE RADIATION FIELD
207
where ya and y, are the corresponding decay rates. The field described by c(w) is taken to be in a thermal state and that described by b(w) to be in a squeezed-vacuum state. The number of thermal photons at optical frequencies at room temperature is very small and can be neglected. The correlation functions for F , are (FT(t)FI(t')) = y, sinh' r d(t - t'),
(237a)
(F,(t)FT(t')> = y, cash' r d(t - t'),
(237b)
(F,(t)F,(t')) = y, sinh r cosh r e-ie d(t - t'),
(237c)
(F,'(t)Ff(t')) = y, sinh r cosh r eie d(t - t'),
(237d)
and Fa satisfies these correlation functions with r = 0. On solving the quantum-Langevin equations for a and the populations N , and N, in steady state, the diffusion rate is obtained as (Gea-Banacloche, 1987)
- 2y, sinh r
cosh r cos(24 - O)].
In steady state A ( N , , - Nl0) = ya + y , = y, the total loss rate. In the absence of squeezing, r = 0, the diffusion rate is y/2ii. If the external field is in a squeezed-vacuumstate, then for the choice of phase 8 = 24 + mn (m = 0, 1, .. .)
D
= (4n)-'(y
+ ya + y1e-").
(239)
For ya 4 yI, r -,00, the linewidth is reduced by a factor of 4.Thus, vacuum fluctuations appear to account for only one-half of the spontaneous emission in a particular mode. This becomes more obvious when one looks at the noise that the spontaneously emitted photons add to the field. From Eq. (221), we have A ($(AXl)') = A(AX,), + -. 4 gain
Note that Eq. (240) contains both added and amplified noise. This is at variance with the usual picture of spontaneous emission as adding, at a rate A (4 Eq. (222)), randomly oriented one-photon phasors to the preexisting field: that picture would yield
208
Khalid Zaheer and M. Suhail Zubairy
independently of noise in the initial field. Of course, if the initial field is in vacuum or coherent state, all the noise appears to be the “added noise” and we obtain Eq. (241) from Eq. (240). But it is more natural to interpret Eq. (240) as saying that half of the spontaneous emission noise is “amplifiedvacuum fluctuations” (when the initial field is in a coherent state) and half is “added noise.” In any case, Eq. (240) shows the distinct possibility of reducing the phase-diffusion rate by reducing (AX,)’ in some way. Ordinary losses do not change (AX,)’ if the field leaking into the cavity is in a coherent state, but if it is replaced by light with reduced fluctuations in the quadrature X , , then the loss term causes (AX,)’ in the cavity to decrease. Specifically, for ordinary losses of the form
one finds
(:(AXl)’)
=
-y(AX,)’
+ i.Y
(243)
loss
Note that if A = y and Eqs. (240) and (243) are added, the result is precisely Eq. (241), i.e., pure added noise as in the usual geometric picture; only in this case half of it appears to come from the losses and half of it from the gain! It is the loss part, the y/4 in Eq. (243), that can be reduced by injecting squeezed light. Hence, the laser-phase diffusion cannot be attributed entirely to purely random spontaneous emission by the active atoms; rather, half of it comes from amplifying the fluctuations of the field already present. In this sense, we should note that it is really not consistent to speak of spontaneous emission in a mode that is not in a vacuum state (such as the lasing mode in a laser). If there is already a field present, there is a correlation between this field and the “spontaneously” emitted field. If, on the other hand, there is no field originally present, the two points of view are quite compatible: one may think of half of spontaneous emission as amplified vacuum fluctuation and half as added noise. Such a viewpoint agrees well with the interpretation proposed by Dalibard et al. (1982). The remaining A/4 in Eq. (240) can only be eliminated by rigging the active medium, turning it into a phase-sensitive amplifier, which would add no noise in the preferred quadrature. This has been considered by Lu and Bergou (1989) in the context of CEL. It is also the basic idea behind some of the systems discussed in the following section.
SQUEEZED STATES O F THE RADIATION FIELD
209
2. Laser with Squeezed Pump
Pump noise plays an important role in the photon statistics. This is evident from the studies of the micromaser. In a micromaser, if the velocity of the atoms is assumed to be constant, the output has sub-Poissonian statistics. However, when a velocity distribution equivalent to the pump noise is included, the usual Poisson statistics of a laser are obtained. Yamamoto et al. (1986a) have shown that if the amplitude fluctuations in the pump are reduced, the output has reduced number fluctuations compared to an incoherently pumped laser. They have also demonstrated a 7.3% reduction (below the Poissonian level) in photon-number fluctuations in a semiconductor laser pumped with an electron beam having reduced fluctuations. The light generated thus has An smaller than a coherent state and is therefore in a kind of amplitude-squeezed state that has been studied quite thoroughly by Yamamoto and coworkers. (cf: Section 1I.D.) Marte and Walls (1988) have considered a laser with a pump in a squeezedvacuum state. They showed that with such a pump, phase-sensitive gain and a phase-locked steady-state laser field are obtained. In this case, however, the phase fluctuations are not lower than the standard quantum limit.
V. Detection of Squeezed States Direct-photon-count experiments, in which light of photonumber distribution p ( n ) falls directly on a photodetector, provide information about the mean photon number and higher-order moments only. Direct detection of an ideal squeezed state, for example, would yield the photon-number variance given by Eq. (45). Such intensity measurements, therefore, are not particularly sensitive to squeezing but are sensitive to antibunching and sub-or superPoisson statistics that can also occur for nonsqueezed fields. Detection of squeezed states, on the other hand, requires a phase-sensitive scheme that measures the variance of a quadrature of the field. Shapiro et al. (1979) and Yuen and Shapiro (1980) considered the problem of detection of squeezed states of radiation in detail. They showed that homodyne detection measures a single quadrature of the field and heterodyne detection results in the two-quadrature field measurement. Both these schemes involve the interference of the light in a squeezed state with a coherent field and, hence, add a large coherent component to it. The resulting field may always be made sub-Poissonian for a particular choice of the phase
Khalid Zaheer and M . Suhail Zubairy
210
(Mandel, 1982). As would be seen in the following, measurements based on photoemissive process in these schemes are equivalent to field-quadrature measurements. Yurke (1988) has discussed a scattering technique to detect squeezed boson fields. A. HOMODYNEDETECTION
The schematic arrangement for homodyne detection is shown in Fig. 16. The input field is superposed on the field from a local oscillator (LO) at a lossless beam splitter of transmissivity T and reflectivity R. The input and the oscillator modes are described by the annihilation operators a and b, respectively. Then denoting the two out-modes reaching the photodetectors 1 and 2 by c and d, respectively, we have c =i f i a -J m b ,
d
=
- J r T a
+ifib.
Here we have assumed a 7c/2 phase-shift in the reflected mode at the beam splitter. The signals measured by the two detectors are determined by the operators
+ (1 - T)btb + i , / m ( a t b - bta), dtd = (1 - T)ata + Tbtb - i J T 0 ( a t b - b'a). ctc = Ta'a
DETECTOR
-
___ ----_- --__ -__
2
a
----- -- -3
QBALANmD
C
>
DETECTOR
1
______________* ORDINARY
(245) (246)
SQUEEZED STATES OF THE RADIATION FIELD
21 1
The frequency of the LO is equal to the input frequency so that the preceding operators do not have any time-dependence. In the following we discuss the ordinary and balanced homodyne detectors. 1. Ordinary Homodyne Detection
In ordinary homodyne detection, the transmissivity of the beam splitter is close to unity, i.e.,
T % R,
(247)
and only the photocurrent from detector 1 is measured. The LO mode is excited into a large-amplitude coherent state I/?,) with phase q,. From Eq. (245) the signal reaching the detector 1 is obtained as
Here X ( q ) = X, and we have used the definition (214) for the quadrature phase X,. We see that the signal contains the transmitted part of the input photons, reflected LO field, and, most importantly, an interference term between the input field and the LO field. It is precisely this interference term that contains a quadrature of the input field depending upon the phase of the LO. In this detection scheme, a strong LO is used so that
The inequalities (247) and (249) together imply that almost all the input field reaches the photodetector but the fraction of the LO field reaching the detector is still dominant. We can therefore neglect the first term in Eq. (248) and the mean number of photons in mode c is
Equation (250) substantiates the discussion following Eq. (44), i.e., the detection of a squeezed state usually adds a large coherent component to it. As far as detection is concerned, the first term constitutes a known constant value that can be subtracted from the signal and the remaining signal contains the quadrature of the input only.
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Khalid Zaheer and M . Suhail Zubairy
The input and the LO modes are independent, i.e., (ab) = (a)(b). The photon-number fluctuations can then be calculated in a straightforward manner using Eqs. (245) and (248):
(An,)’ = ( 1
-
T)1&1’
In obtaining Eq. (251), we have used the inequality (249) and retained terms of second order in 1/3,1. The signal noise is now seen to contain reflected LO noise (first term) and the transmitted input quadrature noise (second term). When the input is in coherent (or vacuum) state, [AX(’p, + n/2)]’ = 1/4 and the remaining term represents the LO shot-noise. The squeezing condition for the input is
[AX( q1+
i)]’-= f ,
for certain values of the LO phase ‘p, for which either quadrature XI or X, is squeezed. In practice, the input is first blocked to determine the shot-noise level. The input is then allowed to reach the beam splitter and the variance is determined with reference to the shot-noise level. Squeezing therefore manifests itself in sub-Poisson statistics in homodyne detection. Note, however, that the intensity measurements in homodyne detection are quite different from those in direct detection, i.e., (a) intensity fluctuations in this case directly measure the fluctuations in a quadrature of the input and (b), the signal and its variance depend upon the local oscillator phase angle which is an external parameter. 2. Balanced Homodyne Detection In the discussion following Eq. (250), we have assumed a perfectly coherent LO field and the oscillator excess noise has been neglected. The LO shotnoise and the excess noise that enter through the reflectivity of the beam splitter cannot be suppressed in ordinary homodyne detection because T in principle can never be 1. The LO noise can therefore limit the ordinary homodyne detection (Yuen and Chan, 1983). In particular, the detection is not quantum limited, if the transmitted input noise is smaller than the reflected oscillator noise, as may be the case when the input noise is too small or when a semiconductor laser local oscillator is used (Yamamoto and Kimura, 1981).
SQUEEZED STATES OF THE RADIATlON FlELD
213
Yuen and Chan (1983) proposed two-port homodyne detection that balances the output from the two ports of the beam splitter. The fact that the noninterference terms at the two ports have the same sign and the interference terms appear with opposite signs (Eqs. (245) and (246)) can be exploited to completely eliminate the noninterference terms. In this scheme, a 50-50 beam splitter is used and the difference of two photodetector measurements is obtained. The output signal is determined by the operator ncd = d'd - C t C = -i(a+b - b t U ) .
(253)
The measured signal then is
and we see that the LO contribution to the signal has now been eliminated and only the interference between the LO mean field and the input quadrature survives. The variance of the output signal can be found as
Here once again we assume a strong LO. The dominant term now is only due to the interference between the input signal noise and the LO power, and the LO noise is eliminated completely, which makes the strong LO condition rather less stringent in this case. Schumaker (1984) has shown that significant improvement in signal-to-noise ratio at the output is achieved in balanced homodyning as compared to the ordinary homodyne detection. With a further arrangement of beam splitters, simultaneous balanced homodyne measurements on both quadratures of the input can be made (Walker and Carrol, 1986; Walker, 1987). It is easy to see that such a scheme requires at least four beam splitters. The measurements are therefore always plagued by the vacuum fluctuations entering through the unused ports. An alternative scheme proposed by Ou et al. (1987a) employs the same setup as the balanced homodyne detection. But instead of taking difference measurements, it relies on coincidence counting or cross correlation of current from two detectors. They show that positive cross correlations arise due to squeezing in the input signal whereas a classical (or coherent) input would always lead to negative cross correlations. The degree of squeezing can be obtained directly from cross correlation and autocorrelation measurements, through a squeezing parameter (Ou et al., 1987b) whose value ranges between 0 and - 1, a value of - 1 indicating 100% squeezing. The spectral
Khalid Zaheer and M.Suhail Zubairy
214
width can be determined by choosing amplifiers with particular spectral response. Since the LO noise in this case does not cancel out, it may be a serious source of error in squeezing measurements. However, a procedure similar to homodyne detection can be followed, i.e., with the input signal blocked, the noise contribution from terms of the order 1&14 can be determined (such terms cancel out in balanced homodyning) and subtracted. The terms of the order 1f1113 are proportional to ( a ) or ( a ’ ) and do not contribute if these expectation values for the signal are zero. Tan and Tan (1989) proposed a holographic technique based on homodyning. The two outputs can be recorded on a hologram whose transmittance will then depend upon the interference term. In practice, balanced homodyne (or, rather, the conceptually similar balanced heterodyne to be introduced shortly) is the most frequently used detection scheme for squeezed states. 3. Quantum Eficiency of the Detectors
The preceding discussion assumes ideal photodetectors of unit quantum efficiency. In practice, however, the photodetector output is a random-point process and the quantum efficiency of photodetectors is always less than 1. Consequently, the photon-number mean and its variance are different from the photocount mean and variance. Nonideal photodetection, then, is a considerable source of noise in homodyne detection. A photodetector of quantum efficiency q can in principle be modeled by a lossless beam splitter of transmissivity q followed by an ideal photodetector (Yuen and Shapiro, 1978b).The model is shown in Fig. 17 where the mode a, represents the vacuum. The input to the 100% detector is obtained through the transformation a’ = &a
a
a’
+f i a , .
IDEAL DETECTOR
(256)
PHOTOCURRENT
>
FIG.17. A beam-splitter-100%-detectorarrangement to model nonideal photodetector.
SQUEEZED STATES OF THE RADIATION FIELD
215
For the sake of completeness, in the following we first discuss the effect of detector inefficiency in direct detection and then in homodyne detection. a. Direct Detection. Equation (256) can now be used to obtain the photocount mean and variance. Since all the normally ordered expectation values for the vacuum field are zero, we have
(nb) = (at's') = ?(PIa), (An:) = V’(Ana)’
+~
(1 V)(na>-
(257) (258)
Equations (257) and (258) give the measured signal and its variance for an arbitrary state of the input mode a. For a two-photon squeezed state, the photocount distribution for direct detection has been calculated by Walls et al. (1982). Using Eqs. (26) and (45) in the preceding equations, (nb)
(259)
=Vl4’9
+ 21’
(260) sinh’ r cosh’ r.
When the coherent component of the mean photon number is larger than the squeeze component, the last term in Eq. (260) can be neglected. The first factor of 1 in the bracket represents the shot-noise associated with the coherent light. The remaining term is negative when the input field is subPoissonian and positive when it is super-Poissonian. However, when the squeeze component is larger than the coherent component, the last term in Eq. (260) is dominant and the photocount statistic is above the shot-noise level. Note however, that if the quantum efficiency q is small, the squeezing effect becomes harder to observe. It should be mentioned here that we have not considered the effects of finite measurement time on photocount statistics. For single-mode detection, the measurement time is much smaller than the fluctuation time and no further distortion is introduced. For a continuous-mode case, the relative sizes of the spectral width and the detection bandwidth need to be considered. These have been discussed by Collet and Loudon (1987) and Collet et al. (1987) in the context of the output from a parametric amplifier. The single-mode analysis remains valid for the broadband squeezed state when the spectral width is smaller than the detection bandwidth. In the opposite limit, that is, with very narrow bandwidth detection, one looks at “spectral squeezing” or
216
Khalid Zaheer and M . Suhail Zubairy
noise reduction in the selected frequency interval near the central (carrier) frequency (heterodyne case, see below).
b. Homodyne Detection. In order to discuss nonideal photodetectors in homodyne detection, we transform the out-mode operators (cJ Fig. 16) given by Eqs. (244a) and (244b) as ct = &c
d = &d
+ &a,, + &a,.
(261) (262)
The photocount mean and variance for ordinary homodyne detection under the conditions (247) and (249) are (nb> = 2 r l J T o I s , I
(263)
In Eq. (263) we have subtracted a constant contribution to the signal from reflected LO power. The discussion following Eq. (251) is still valid. For balanced homodyne detection
These equations show that the randomness effects due to quantum efficiency of the detectors do not qualitatively change the idealized results, but they show (just as for direct detection) that when q is small the squeezing effect becomes very hard to observe. B. HETERODYNE DETECTION
The schematic arrangement for heterodyne detection is the same as that for homodyne detection (Fig. 16), but the LO frequency is offset from the frequency of the input signal. Let the frequency of the local oscillator and input signal be R and R + 6 , respectively. The output of the photodetector is filtered to select the beat frequency components at the intermediate frequency 6 . The intermediate frequency output results from the beating of the signal at
SQUEEZED STATES OF THE RADIATION FIELD
217
+
frequency R 6 as well as its image at R - L with the oscillator field. In conventional heterodyne detection, the image field is in vacuum state. In the TCS heterodyne-detection scheme of Yuen and Shapiro (1980), the image field is placed in a squeezed state and mixed with the input signal through an image oscillator beam splitter that has transmissivities T, and Ti for signal and image frequencies, respectively (Fig. 18). For T, x 1, Ti x 0, the part of the out-mode Eq. (244a) that will give oscillation at the intermediate frequency L will be c =ifi[a(R
+ t)ei('+')' + a(Q - ~)e~('-')']- J m b e ' "
(267)
where a(R + L ) and a(R - 6 ) are the annihilation operators for signal and image modes, respectively. It is easy to show that the intermediate frequency output directly measures the two-quadrature signal component. Note, however, that the term in square brackets represents the collective destruction operator (49b) for the two-mode squeezed state discussed in Section II,C. Heterodyne detection is therefore most suited to measurements on two-mode squeezed states. The quantum treatment of the problem is the same as that for homodyne detection discussed earlier. The mean signal measures the coupled-mode field quadratures given by Eqs. (50a) and (50b) for the particular choice of the LO phase angle. Milburn (1987) has analysed optical heterodyne detection of an intracavity field within the context of quantum measurement theory.
7 DETECTOR
DETECTOR
a ( ~ - € )
1
-
IF FILTER
LOCAL OSCILLATOR
FIG. 18. Schematic diagram for TCS heterodyne detection scheme.
4
218
Khalid Zaheer and M. Suhail Zubairy
In practice, most sources of squeezing produce a broadband squeezed field (made out of pairs of two-mode squeezed states, that is, something like the output of the first beam splitter in Fig. 18), and what is usually detected is precisely the two-mode squeezing at some intermediate frequency 6 # 0, using this technique. This is done to avoid many sources of noise which usually dominate the detection electronics near 6 = 0. It should be mentioned here that the ionization rates in photodetectors depend upon the square root of the frequency of the mode (Caves and Schumaker, 1985). When the two modes are far apart ( 6 is large), the different ionization rates need to be considered. These tend to diminish the influence of squeezing on the photocurrent. Finally, the coupled-mode field operators are analogous to the single-mode operators and the detection is quite similar to homodyne detection, i.e., it is characterized by photocurrent noise below the shot-noise level.
c. MEASUREDPHOTOCURRENT AND ITS RELATIONSHIPTO SQUEEZING As explained earlier in this section, nonideal photodetectors tend to diminish the effect of squeezing on the measured photocurrent. Apart from this, a number of other loss mechanisms in the experimental setup have a similar effect on the observed noise reduction. In the following, we dicuss the experimentally measured quantities and their relationship with the squeezing parameter. The treatment presented is fairly general and is relevant to most of the experimental setups used today (Wu et al., 1987; Orozco et al., 1987). In balanced homodyne (or heterodyne detection), the photocurrents il(t) and i2(t) from the two detectors are combined (with one of them phaseshifted by 180") to obtain the resultant photocurrent i(t). The photocurrent spectral density, @(v) =
s
(Ai(t)Ai(t
+ z))e-'"
dz,
is measured where Ai(t) = i ( t ) - ( i ( t ) ) . Assuming delta-function response for the photodetector, a strong local oscillator condition and straightforward application of theory of photodetection give (Mandel, 1981; Shapiro, 1985; Kimble and Mandel, 1984)
ww) = (QA +
P,T,V'V:S(R,0)).
(269)
SQUEEZED STATES OF THE RADIATION FIELD
219
Here Qi is the total charge per photopulse from the detector-amplifier combination and the dimensionless frequency R = v/T,, where Tt represent the total losses from the cavity. The efficiency factors appearing in the second term typically depend upon the optics of the experimental setup. For example, pl is the ratio of the measured cavity finesse to the finesse inferred from the transmissivity of the mirror, (1-To)represents the propagation losses due to the optical components between the out-coupling mirror and the homodyne detector, q' characterizes the efficiency of the homodyne detector, and q,,, arises from the imperfect matching of the two beams and is given by the mode overlap of the signal and LO beams over the detector surface. From Eq. (269) we see that in the absence of squeezing, the noise level in i ( t ) is the sum of shot-noise for the two channels that can be readily determined. Equation (269) can be normalized to the shot-noise level as R(SZe) = 1
+ ~J-,~'V;S(R,
el.
(270)
Experimentally, the noise voltage V(R, 0) is measured and squeezing is related to the noise power by the relation
where V, represents the shot-noise level.
VI. Experimental Results Following the detailed theoretical analyses of various sources of squeezing and the prospects of some novel applications, there have been extensive efforts to generate squeezed radiation fields. As of 1989, five groups have reported generation of squeezed states in different systems. The first generation of squeezed states ever reported was by Slusher et al. (1985a) in intracavity backward four-wave mixing (FWM) in an atomic beam. The minimum noise level achieved by them was 7 % below the vacuum limit. Shelby et al. (1986b) observed squeezing in forward FWM in an optical fiber and the maximum squeezing obtained by them was 12.5%. Squeezing in optical fiber arises due to optical Kerr effect (Milburn et al., 1987) and is somewhat different from the parametric interaction involved in intracavity FWM. Maeda et al. performed a series of experiments on forward FWM
220
Khalid Zaheer and M. Suhail Zubairy
using sodium vapor as the nonlinear medium. In single-beam forward FWM, they found optically phase-sensitive noise that was always at or above the vacuum level (Maeda et al., 1985). In forward FWM with direct detection and probe-conjugate correlation measurement, they observed positive correlations, but these correlations did not exceed the excess noise. However, in homodyne detection, they observed a 4% reduction in noise below the vacuum level (Maeda et al., 1987). Optical bistability has been another process through which squeezed states have been generated. Raizen et al. (1987) investigated the dynamics of a collection of two level atoms in a high finesse cavity under a condition when the Rabi frequency is larger than the field, atomic polarization, and inversion decay rates. A dynamic exchange of excitation between the atomic polarization and the cavity modes results in a rather broad band squeezing (f75 MHz) around the central frequency g f i where N is the number of atoms (Orozco et al., 1987). A 30% noise reduction below the vacuum level was observed and, considering the propagation and detection losses, a squeezing of about 53% was inferred. Periera et al. (1988) have obtained squeezing in second-harmonic generation in an optical cavity. They observed a 13% reduction in the photocurrent noise in the fundamental field reflected from the nonlinear cavity. By far, the maximum reported squeezing as of 1989 has been in an optical parametric oscillator (OPO) by Wu et al. (1987), which is 63 % below the vacuum level. Generation of amplitude-squeezedstates (number-phase minimum-uncertainty states) has been reported by Machida et al. (1987) in a semiconductor laser driven by a constant current source. The maximum observed noise reduction was 7.3% below the SQL; considering the effect of detector quantum efficiency, they inferred a 31 % maximum noise reduction and 21 % average noise reduction at the laser output. Heidmann et al. (1987) used a two-mode optical parametric oscillator operating above threshold to generate highly correlated twin-beams. The measured noise in the intensity difference of the two beams was 30% below the shot-noise level. In an improved experiment, Debuisschert et al. (1989) have observed a 69 % noise reduction in the intensity difference. For the sake of brevity, in the following, we discuss only two experiments, namely, intracavity backward FWM by Slusher et al. (1987a) and OPO experiment by Wu et al. (1987). Detailed descriptions of the other experiments can be found in the Journal of the Optical Society of America B, special issue on squeezed states of the electromagnetic field (October 1987). Also see Slusher et al. (1987b) and Yurke et al. (1987).
SQUEEZED STATES OF THE RADIATION FIELD
22 1
A. INTRACAVITY FOUR-WAVE MIXINGIN ATOMICBEAM 1. Experimental Setup The schematic setup of the experiment is given in Fig. 19. A CW ring dye laser is used to pump a sodium beam in a confocal pump cavity (PC) formed by mirrors P M l and PM2. The dye laser is tuned to a frequency 1.5 GHz above the D, resonance of sodium at 589.0 nm. A TEM,, mode from the pump is matched t 0 . a PC mode and the cavity was servo-controlled to remain in resonance. For simplicity, various modulators used for cavity locking are not shown. The atomic beam was 1 cm wide with a density of about 10" cm-3 and an angular divergence of 15". A standing wave builds up in the confocal squeezing cavity (SC) formed by the mirrors SM1 and SM2 which is placed at an angle of 0.86" from the pump beam. The SC was also servo-controlled to remain at a detuning 2v,, = 281 MHz where v,, is the node-spacing frequency. In this setup, then, the standing wave in the PC drives a polarization in the atoms to generate pairs of frequency-shifted photons at f3v,, = +421.5 MHz. The conjugate pairs of photons thus generated in four-wave mixing lead to a squeezed state buildup in the cavity. 2. Detection of Squeezing Light from the out-coupling mirror SM2 was fed into a balanced detector. The LO (3-mW) beam was separated from the pump by beam splitter BS1.
RING
DYE
RSI
\
LASER
FIG.19. Schematic diagram of the experiment for squeezed-state generation by four-wave mixing as described in the text. (After Slusher et al., 1987a.)
Khalid Zaheer and M.Suhail Zubairy
222 -58
-59
1
E
m
3 J W
> -60
W
A
w
-I
-62 ._
I
0
V I
I
TI2
T
I
3~/2
+LO
FIG.20. Noise levels corresponding to the rms photocurrent from the balanced homodyne detector shown as a function of local-oscillator phase &o. With the squeezing cavity blocked, the mean noise level indicated by the dashed line is obtained from the dotted trace. The photodetector amplifier noise at - 10 db is shown relative to the dashed line. The measured noise level increases 1.3 db above and decreases 0.7 db below the vacuum noise level as a function of when the four-wave mixing output from the cavity is matched to the local-oscillator mode. The radio frequency is centered at 594.6 MHz with a bandwidth of 300 KHz. A theoretical model predicts squeezing of & 2 db (solid curve) for an ideal measurement. For the experimental detection efficiency and amplifier noise, this ideal behavior is degraded to the dashed-dotted curve. The pump-frequency detuning for both theory and experiment is A = -400, the nondegenerate detuning is S = 60, and the cooperativity parameter is C = 1O00, corresponding to the model prediction in Fig. 4. The pump intensity is I/I, = 0.056. (From Slusher et al., 1987a.)
SQUEEZED STATES OF THE RADIATION FIELD
223
The LO phase was controlled by a piezoelectrically mounted mirror. Two P-1-N silicon photodiodes of efficiency ql,2 = 0.7 were used in conjunction with low-noise wide-bandwidth amplifiers to amplify the photocurrents to 60 dB. Typical values for q, were 0.75. The noise spectrum was observed on a spectrum analyzer at frequenciesnear the beat frequency between the LO and the cavity-output frequency at f3v,,. The maximum noise reduction below the vacuum level was 0.3 dB, which is nearly 7%. Taking into account the system efficiency, phase jitter, etc., nearly 20 % squeezing was inferred at the cavity output. In a following experiment (Slusher et al., 1987a), the frequency locking of the pump to SC was improved by injecting a locking beam through the highreflectivity mirror SM1. It was shifted from the sodium absorption line by 2.5 GHz to eliminate any absorption or instabilities. Conjugate pairs of photons at frequency shifts of approximately & 595 MHz were generated. Mode-matching efficiency q,,, was improved to 0.8 which, along with an improved detector efficiency or q = 0.8, resulted in a net detection efficiency of 0.65. Figures 20 and 21 show the measured noise levels as a function of the local-oscillator phase angle for different detunings of the pump from the Na resonance. From Fig. 20, the maximum noise reduction is 0.7 dB which TIME (msec)
0
lT/2 +LO
FIG.21. Homodyne-detector noise level as a function of local phase for a detuning A = -280 and side-mode detuning from the pump mode S = 60.(From Slusher et al., 1987a.)
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Khalid Zaheer and M . Suhail Zubairy
corresponds to 15% reduction below the vacuum level. Note that the maximum increase in noise levels is 1.3 dB which is in good agreement with theory (dash-dotted line). However, the maximum reduction is only in marginal agreement with theory (which predicts 1.05 dB reduction). Figure 21 shows the noise level for a reduced detuning A = -280. The maximum observed reduction is 1.5 dB, corresponding to 25 % reduction below the vacuum level. Both the figures, in particular Fig. 21, indicate some extra phase-insensitive noise so that the noise floor about which the noise level oscillates is raised. The origin of the extra noise has not been determined. Spontaneous emission into the SC resonance is a very small component since the SC frequency shift of 595 MHz is much larger than the Doppler shift, which is less than 200 MHz. Linear absorption in the cavity due to some scattered atoms and the phase jitter of the pump have also been ruled out by Slusher et al. (1987a). A possible explanation could be that the nonlinear mixing of the pump, the locking beam (shifted seven cavity modes away from resonance), and the spontaneous emission could generate light near the squeezed-light frequency to degrade the predicted squeezing. B. OPTICAL PARAMETRIC OSCILLATOR Because of the reasons discussed in Section II.B, parametric down conversion has been of particular interest for the generation of squeezed states. While with the nonlinear optical materials available now (Fejer et al., 1985; Luh et al., 1986), it may be feasible to generate squeezing in the amplifier configuration, experiments conducted so far have been in the optical parametric oscillator configuration. Wu et al. (1986) reported more than 50% reduction below the vacuum noise level in an optical parametric oscillator. The same group later observed a 63 % reduction below the vacuum level (Wu et al., 1987). They concluded from their .analysis that the observed noise reduction results from a field that in the absence of linear loss mechanism would be more than ten-fold squeezed. The field emitted through the mirror in the OPO was inferred to be fivefold squeezed and in an ideal squeezed state. As of 1989, this is the largest observed reduction in noise and also the largest inferred squeezing. 1. Experimental Setup
Figure 22 shows the experimental setup of Wu et al. The pump beam at 0.53 pm was generated by a Ba,NaNb,O,, crystal placed inside a Nd: YAG
SQUEEZED STATES OF THE RADIATION FIELD
225
Ba2NaNb 5015
Polarizer 0.53
ccm
1.06 pm Signal
FIG.22. Diagram of principal elements of the apparatus for squeezed state generation in OPO. (From Wu et al., 1987.)
ring-laser cavity. An rms linewidth of 100 KHz was achieved by locking to the transmission peak of a reference cavity. The two orthogonally polarized components (green and infrared for pump and local oscillator, respectively) were separated by a polarizer. The pump power needs a high degree of stabilization because variations in pump power would result in a fluctuating crystal temperature that could drive the system away from the condition of simultaneous resonance. Pump and local oscillator intensities were stabilized with acousto-optic modulators. In order to avoid the complexities involved in the construction of a single-mode cavity, a multimode OPO cavity consisting of two mirrors M and M' was used. The mirror M, through which the pump field enters the cavity, had transmission coefficients of 3.5 % and 0.06% at 0.53 pm (pump) and 1.06 pm (signal), respectively. The mirror M that couples to the output field had high reflectivity for the pump wave and 4.3% (or 7.0 %) transmissivity for the
226
Khalid Zaheer and M.Suhail Zubairy
subharmonic field, The length of the cavity was servo-controlled to lock the pump beam at 0.53 pm to a longitudinal mode. A MgO:LiNbO, crystal, heated to phase-matching temperature of 98°C was used for nonlinear coupling. The crystal was coated with dual band antireflection coatings to achieve transmissions of 96-97% and 98-99% for the pump and signal waves, respectively. In this setup, a number of modes are phase-matched and participate in the down-conversion process. Above threshold, this multiplicity of modes Ieads to instability in the OPO (Zernike and Midwinter, 1973; Byer, 1975). Consequently, the investigation by Wu et al. (1987) was mainly in the subthreshold region. As they point out, however, injected signal at the subharmonic can be used to stabilize the operation above threshold. A twofold search for the degenerate operation is conducted by first scanning the length with the OPO cavity unlocked followed by a search in temperature to tune the birefringence of the crystal so that a longitudinal mode of the cavity at 1.06 pm comes into simultaneous resonance with one at 0.53 pm. The lowest temperature is identified for which the oscillation occurs and the length scan is stopped. The pump power is then reduced below the critical power P, for operation below threshold. The cavity length is then locked and the temperature is varied slowly within the range of degenerate oscillation to bring successive longitudinal modes at 1.06 pm into resonance with the intracavity field at 0.53 pm.
2. Observation of Squeezing
The light exiting from the mirror M was combined with the light from the original laser, which acts as local oscillator (P = 1 mW) in a balanced homodyne detection system. The two photodetectors consisted of InGaAs photodiodes with quantum efficiencies q1 = 0.91 f 0.02 and qz = 0.87 f 0.02. The spectral distribution of fluctuations in the difference photocurrent O(Q, 0) was recorded where R = v/T, is a dimensionless frequency. Figures 23 and 24 show the rms noise voltage V(0) from the balanced detector as a function of the local oscillator phase at fixed frequency R.Various efficiency factors in the experiment were 0.70 > pI > 0.95, To = 0.94, q’ = 0.89, and tf, = 0.95. Consistently reproducible minimum-noise level was R = 0.37, which corresponds to S - -0.90. However, the authors concluded that “nothing fundamental in the parametric process itself or in the materials employed
SQUEEZED STATES O F THE RADIATION FIELD
227
2.0
we) 1.0
0 FIG.23. Dependence of nns voltage V ( 0 ) on local-oscillator phase 0 for the signal from the balanced homodyne detector shown in Fig. 22. With the output of the OPO blocked, the vacuum field entering the detector produces the noise voltage V, labelled by (i) with no sensitivity on 0. With the OPO input present, trace (ii) exhibits phase-sensitive deviation both below and above the vacuum level, with the dips below trace (i) representing a 61 % reduction in noise power relative to the vacuum level. Trace (iii) is the amplifier noise level. Note that for traces (i)-(iii), the ordinate is linear in noise voltage (amplitude). Trace (iv) is actually two curves almost superimposed that give the levels of dc photocurrent (with zero at the bottom of the figure) during the acquisition of traces (i) and (ii). For traces (i)-(iii), v/2n = 1.6 MHz. The sharp feature is generated by the flyback of the piezoelectric ceramic used to scan the local-oscillator phase. The time for the entire sweep is 0.2 sec. (From Wu et al., 1987.)
was found that intrinsically limits the achievable squeezing” (Wu et al., 1987, p. 1466). Furthermore, by considering the enhancement and reduction in fluctuations, it was inferred that the field generated was in a minimum uncertainty state. C. CONCLUSIONS
The experimental results achieved as of 1989 are by no means ideal and we may still be a long way off from the actual applications of the sort discussed in Section IV. Yet at the same time, these hold great promise for the future. In particular, as the noise sources are eliminated, which may be an imperfect
228
Khalid Zaheer and M . Suhail Zubairy
l r
....................................... ........................ ............ I
8,
I
6,+R
I
I
6,*2?l
e
8,
8,t
H
8,+2H
FIG.24. Dependence of noise voltage V ( 0 ) on local-oscillator phase 0 for the signal beam produced by the subthreshold OPO. Operating conditions are as in Fig. 23, with traces (i) and (iv) deleted; the dashed line is the vacuum level obtained by multiple averaging. (From Wu et al., 1987.)
coating of crystals or some avoidable linear losses as for example, in OPO, “squeezed-state generators” may be available in any ordinary laboratory in the near future. In certain cases, however, some extra noise is associated with the nonlinearity that generates squeezing (as, for example, the spontaneous emission noise in four-wave mixing). This obviously limits the maximum achievable squeezing. Search for new materials as well as investigation of new systems is of course imminent. In this sense, the research in squeezed-state generation has just begun. ACKNOWLEDGMENTS
We are grateful to Dr. J. Gea-Banacloche for a critical reading of the manuscript and many useful suggestions. We wish to thank Mr. Shahid Qamar and Mr.Aftab A. Rizvi for their help in the preparation of the manuscript. We also thank Mr. Duri-Iman for his help in typing the manuscript. This work was supported by the Pakistan Science Foundation, by a research grant from the Pakistan Atomic Energy Commission, and the World Laboratory Centre for High Energy Physics and Cosmology, Islamabad.
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ll
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 28
CAVITY QUANTUM ELECTRODYNAMICS E. A . HINDS Physics Departmeni
Yale University New Haven, CT
I. Introduction . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . A. Irreversible Spontaneous Radiation . . . . . . . . . . . B. Observations of Modified Irreversible Spontaneous Radiation . C. Modified Emission from Dressed Atoms . . . . . . . . . D. Reversible Spontaneous Radiation in High-Q Cavities . . . . 111. Shifted Energy Levels . . . . . . . . . . . . . . . . . . A. Perturbation Approximation for Energy-Level Shifts. . . . . B. van der Waals Shift . . . . . . . . . . . . . . . . C. Observations of Perturbative Level Shifts. . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . 11. Modified Radiative-Decay Rates
. . . . . . . . . . .
. . . . . . . . . . .
237 239 239 249 257 261 269 . 269 . 277 . 280 286 286
. . . . .
I. Introduction Charged particles are coupled to the electromagnetic radiation field at a fundamental and inescapable level, and in simple atomic systems this coupling is responsible for some basic phenomena such as the Lamb shift and spontaneous radiative decay. These radiative effects have been studied extensively in cases where the atom is in free space, and remarkable agreement has been found between theory and experiment. One is led to conclude that quantum electrodynamics (QED) provides a reliable description of the coupling between charged particles and electromagnetic fields. In free space an atom is coupled to many modes of the electromagnetic field and spontaneous emission is an irreversible process, well approximated using perturbation theory. Consequently, the spontaneous radiation rate is proportional to the density of electromagnetic modes at the radiation frequency in accordance with Fermi’s golden rule, while the radiative-level shift (Lamb shift), due to emission and reabsorption of virtual photons, is a broadband phenomenon that depends on the whole electromagnetic spectrum. 237 Copyright a 1991 by Academic Press. Inc.
All righls 01 reproduction in any form reserved. ISBN 0-12-003828-5
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E. A . H i n h
In reality, of course, the atom is never isolated in free space, but usually all other objects are far away and their presence can be neglected. If, on the other hand, conducting or dielectric material is intentionally placed nearby, it is possible to perturb the electromagnetic field in such a way that the radiative properties of an atom are substantially modified. This fact was first recognized by Purcell(l946) and the early development of the theory was carried out by Bloembergen and Pound (1954), Power (1965), Morawitz (1969), Barton (1970), and Stehle (1970). The earliest experimental demonstrations of modified radiation rates were reported by Feher et al. (1958) (at microwave frequencies) and by Drexhage et al. (1968) (in the visible). Inside a cavity or waveguide, the spectrum of the electromagnetic field modes is strongly modified for wavelengths that are comparable with the physical dimensions. Thus, it is possible by a suitable choice of cavity geometry to influence the fluctuations of the quantized radiation field and, hence, to change both the Lamb shift and spontaneous decay rate. In the extreme case in which the atom can couple only to a single;sharp mode of the cavity, the system of atom-plus-cavity becomes a beautiful example of coupled quantum oscillators whose quantum beat is the coherent exchange of a photon back and forth between the atom and the cavity. This type of system lends itself naturally to a study of the connection between the dynamics of quantized and nonquantized theories, which was first discussed in depth by Jaynes and Cummings (1963) and later, for example, by Milonni (1976) and Boyer (1980). In the laboratory the study of atoms in cavities is a relatively young field in which much of the effort is still aimed at elucidating the basic physical principles at work and demonstrating the elementary modifications of energies and rates of an atom in a cavity. However, it is already clear that our ability to adjust the electromagnetic spectrum of the “vacuum” is leading us into a remarkable new domain of quantum physics in which atoms can “decay” from the ground state to an excited state, “radiative corrections” can be far larger than fine structure, and electromagnetic fields can be prepared with an exact number of photons. This realm of physics is now known as “cavity quantum electrodynamics.’ The remainder of this chapter consists of two principal parts. The first, Section 11, is devoted mostly to modifications of the spontaneous radiation rate in a cavity, while the second, Section 111, concerns the shifts of energy levels. While our focus is primarily on experimental work, each section begins with an outline of a theoretical framework in which the experiments may be understood. Within these theoretical parts we attempt, with the minimum of
CAVITY QUANTUM ELECTRODYNAMICS
239
mathematics, to develop an intuitive physical model of the processes involved. For those who are interested in the finer details we have provided references to the original literature. The experiments described are those that demonstrate the basic physical ideas concerning radiation rates and level shifts in a cavity. The nature of this topic, however, is so fundamental that it connects with many different aspects of atomic physics and quantum optics, such as the quantum theory of measurement (Scully and Walther, 1989), squeezed states (Lewenstein and Mossberg, 1988b), spectroscopy of trapped ions and electrons (Brown and Gabrielse, 1986), and micromasers (Meschede et al., 1985; Brune et al., 1987). No attempt has been made at a complete exploration of all the possible applications of cavity QED, but it is hoped that such discussion as appears here will provide a useful starting point for those interested in learning more. In addition to this chapter, the interested reader might wish to consult some of the excellent reviews of this field that appeared in the 1980s (Haroche, 1984; Haroche and Raimond, 1985; Walther, 1988; Haroche and Kleppner, 1989).
11. Modified Radiative-Decay Rates A. IRREVERSIBLE SPONTANEOUS RADIATION
When an excited atom emits a photon through spontaneous decay in free space, the process can be treated by perturbation theory and the decay rate computed using Fermi’s golden rule (Sakurai, 1982):
Here the spontaneous transition is between states le) and Ig) of the atom at frequency w, and the sum is over photon modes of wavevector k,polarization E , and frequency wk. The state 10) denotes the electromagnetic vacuum. Of course the spontaneous radiation is not really monochromatic; we are making the assumption here that the atomic line is much too narrow for there to be any variation in the spectral density of the vacuum. The operator H, is the interaction Hamiltonian responsible for the transition. For present purposes it may be written in the dipole approximation as e
HI - --A-Q. m
We employ the Coulomb gauge throughout this chapter.
E. A. Hinds
240
The &function ensures that q = ma, in which case the matrix elements of HIare identical to those of the electric-dipole interaction - D,(r) E(r) where D,(r) is the electric-dipole operator for the atom and E(r) is the electric-field operator evaluated at the site of the atom. To a good approximation we can view the atom as a two-level system when calculating the radiation rate (Jaynes and Cummings, 1963); it is then convenient to write the dipole operator in the form D, = ( D + - D-)d. (3)
-
D + and D- are the raising and lowering operators for the two-level atom, and d is the electric-dipole matrix element. When the atom is in free space the result is
In a confined space, the electromagnetic modes of the vacuum are altered and the number of modes per unit frequency interval can vary quite rapidly with frequency. Nevertheless, the preceding perturbative approach can still be used to calculate the transition rate provided the natural width of the atomic line remains narrow on the scale of these variations in the vacuum spectrum. In the following subsections we discuss in this approximation three different examples of confined space. They are (a) the gap between two infinite parallel plane mirrors, (b) the region in front of a single infinite plane mirror, and (c) the space inside a single-mode resonant cavity. 1. Atom between Parallel Plates
Figure 1 shows the coordinates we use to discuss this system. Only one of the two mirrors is shown in the drawing. The position vector is r with the origin at the surface of one mirror [Fig. l(a)]. The components of r normal and parallel to the mirrors are z and p. The modes of the electromagnetic field between the mirors have wavevectors k with components k, and K. The modes can,be divided into two types as shown in Figs. l(b) and l(c). We call these E-modes (E, = 0) and M-modes ( B , = 0) by analogy with the TE and TM modes of a waveguide (Jackson, 1975).The field operators for E- and Mmodes can be written as (Barton, 1970)
+
sin(k,z)R x ,fei(Kp-wkr)afh.c.
AM(k) =
J”(r
(54
k cos(k,z),f - i 2 sin(k,z)R ei(Kp-wkr)af + h.c. (5b) E ~ o ~k V k
24 1
CAVITY QUANTUM ELECTRODYNAMICS
FIG.1. This figure defines the notation and coordinate system. (a) The position vector of an atom and its cylindrical components. (b) Electric field E, wavevector k, and its cylindrical components for an E-mode of the field. (c) Magnetic field B, wavevector k, and its cylindrical components for an M-mode of the field.
in which V is an arbitrarily large normalization volume, a t and a r are the annihilation operators for the E- and M-modes having wavevector k, and h.c. means the Hermitian conjugate. These operators are normalized so that the energy of the field is hw,(n + f) when there are n photons present in the mode. In the special case when k , = 0, the E-mode is absent and the normalization of the M-mode in Eq. (5b) is wrong by a factor of
&:
(64
AE(k) = 0
Returning to the general case, the boundary condition imposed by the perfectly conducting mirror surfaces, i.e., that A;= 0 at the walls, constrains the possible values of k,:
where L is the gap between the mirrors and n is zero or any positive integer. The total electric field is given by
E(r, t ) =
k
a ~
at
[AE(k)
+ AM(k)].
242
E. A . H i n h
After substituting this field into Fermi's golden rule (Eq. 1) and integrating over all possible values of K, one finds (Stehle, 1970; Barton, 1970, 1987a; Philpott, 1973) that a dipole parallel to the mirrors radiates at a rate
where To is the free-space rate, 1 is the free-space wavelength of the radiation, and (2L/1)is the largest integer number of half-wavelengths that can fit in the gap. Similarly, a dipole normal to the mirrors has a radiation rate.
[
1-
(3'1 YLZ)} -
COS~-
(10)
where the first term comes from the n = 0 modes. Figure 2 shows the variation of Tp/T,and rz/ro with gap width L when the atom is at the center of the gap. The most striking feature is the complete suppression of spontaneous emission from a dipole parallel to the mirrors when the gap is narrower than 1/2 [curve (a)]. In terms of the mode expansion this is quite natural, for at wavelengths longer than 2L, only the n = 0 modes exist and these are polarized along z. Consequently, there is no long-wavelength fluctuation of the vacuum field parallel to the mirrors. An alternative, classical point of view is that the wave impedance in the parallel-
GAP
WIDTH / WAVELENGTH
FIG.2. Radiative-decay rate of an atom at the center of an ideal parallel-plate waveguide relative to the rate in free space. The abscissa is the spacing between the plates relative to the wavelength of the radiation in free space. (a) The rate for a dipole parallel to the plates. (b) The rate for a dipole normal to the plates.
CAVITY QUANTUM ELECTRODYNAMICS
243
plate waveguide becomes purely imaginary at frequencies below the c/2L cutoff value. As a result, atomic radiation that has p-polarization is unable to propagate in the confined space and can exist only as an evanescent field surrounding the atom. At the same time, the emission rate for the z polarization [curve (b)] is enhanced relative to the free-space rate and increases inversely,with L as the gap approaches zero. This is because the energy density in the n = 0 vacuum modes is inversely proportional to the volume between the plates. (See Eq. 6.) When the gap width increases above 1/2, the decay rate is alternately enhanced and suppressed (Fig. 2) and, as is to be expected, approaches the free-space rate To as successively higher values of n contribute to the vacuum field at the atomic-transition frequency. The contribution of each mode to the radiation rate also depends generally upon the position of the atom, through the spatial distribution of the field in that mode. For example, when the atom is at the center of the gap, the even-n modes contribute nothing to f, because the center is a node of the electric field. This explains why r,,in Fig. 2 jumps at odd half-integer values of L/3, but not at integer values. Although it is assumed in Fig. 2 that the atom is at the center of the gap, the main features-including inhibition of r pand enhancement of r,, when the gap is small-remain true for atoms at any position because the field in the n = 0 modes of the vacuum is uniform across the gap. Thus, we see that the spontaneous emission rate of an atom between two mirrors depends on three factors: the size of the gap, the position of the atom, and the polarization of the dipole. These reflect the dependence of the coupling between the atom and the vacuum upon mode density, spatial distribution, and polarization of the radiation field.
2. Atom in Front of a Single Mirror The emission rates for an atom in front of a single mirror can readily be found by taking one of the mirrors to infinity ( L + 00 in Eqs. 9 and 10) while maintaining a fixed distance z from the atom to the other mirror. In this limit, the sum over n tends to an integral over k and the two decay rates become
2kz
{' 3
1
r , = 3 r o {3
cos2kz (2kz)'
-} m]'
cos2kz sin2kz + ( 2 k ~ ) and (2kz)' ~
r = - r -2_ - _sin2kz ___
sin2kz
+
244
E. A . Hinds
DISTANCE / WAVELENGTH FIG.3. Radiative-decayrate of an atom in front of an ideal plane mirror relative to the rate in free space. The abscissa is the distance from the mirror relative to the wavelength of the radiation in free space. (a) The rate for a dipole parallel to the mirror. (b) The rate for a dipole normal to the mirror.
Here as usual k = 2n/L and z is the distance from the mirror to the atom. These rates are plotted in Fig. 3. Very close to the mirror, where 2nz 4 A, the radiation rate r pfor a parallel dipole is suppressed [curve (a)], while r, is enhanced [curve (b)]. In this case the modification of the decay rate originates from the spatial distribution of field, which is constrained by the proximity of the conducting boundary, rather than by the spectral density of the modes. As the distance from the mirror becomes large compared with the radiation wavelength, the decay becomes isotropic and the rates both tend to the .free-space value To, as obviously they must. 3. Atom in a Resonator of Moderate Q
In this section we consider the decay of an atom in a resonator whose quality factor, Q,is high but in which the variation of the vacuum spectrum is still small over the linewidth of the atomic transition. Suppose now that the radiating atom is placed inside a resonant cavity, tuned to have a single mode of oscillation at frequency o close to the atomic-transition frequency. The electric-field operator for the standing wave mode is E(r) = v :;/ -[f(r).ta
+ h.c.1
CAVITY QUANTUM ELECTRODYNAMICS
245
where f (r) expresses the spatial variation of amplitude and is equal to unity where the field is largest. 2 is the polarization of the field. V is an effective mode volume defined by
s
I f(r)l’ d3r = V .
For the fundamental Gaussian mode T E M , , of an open confocal resonator of length L, this volume has the value IL2/8.Thus, the square of the coupling matrix element for a transition polarized in the same direction as the field is
Here 0 and 1 indicate the vacuum and one-photon states of the cavity mode, while e and g refer to the upper and lower states of the atom, respectively. When Q is much greater than unity, but still finite, the lifetime for dissipation of a photon in the cavity walls is Q / o and the corresponding spectral distribution is the Lorentzian of width o/Q,
whose peak spectral density is 2Q/nw0. Suppose the photon is absorbed by the cavity walls in a time that is short compared with the one-photon Rabi oscillation period; then there is no chance that it will re-excite the atom and spontaneous emission remains an irreversible, dissipative process, just as it is in free space. Equivalently, we see that in this situation, the width of the cavity-frequency distribution is large in comparison with the natural width of the transition. Thus, it is appropriate once again to use Fermi’s golden rule. When the peak of the cavity resonance coincides with the atomic-transition frequency, the spontaneous rate of radiation into the cavity modes is easily found (Eqs. 1, 15, and 16) to be
In addition, the atom may radiate photons out of the open sides of the cavity. Let us assume that the atom is placed at the center of the cavity, that the dipole is polarized at right angles to the cavity axis, and that the total solid angle An subtended by the cavity mirrors at the position of the atom is
E. A . Hinds
246
much less than 411.In this case, the spontaneous emission rate out the side of the cavity is ro[l - (3AR/8n)] and the total rate is
r = r , l - - +3AR 8n
[
3Q A3 - If(r)l 472 ( V )
’1.
In the nodes of the field, where f(r) = 0, the partial spontaneous emission rate into the cavity solid angle is zero. However, the total rate does not change substantially unless the solid angle is a large fraction of 411. Clearly a large-cavity solid angle is more suitable for the suppression of radiative decay. On the other hand, significant enhancements are possible, even in an open cavity of limited solid angle, in the strong field regions (f(r) = 1). In that case, the radiation rate is substantially enhanced provided QA3/V % 411. Hence, in a low-order microwave cavity mode where A3/V can be of order unity, we see that the free-space rate can be increased by a factor of order Q and large enhancements of the decay rate should be possible in practice. Of course, if Q becomes too large, the storage time of the photon in the cavity can be longer than the period of the one-photon Rabi oscillations. In that case the decay ceases to be irreversible and the excitation energy can oscillate back and forth between the atom and the cavity. This situation is considered further in Section 1I.D. 4. Classical Image Model for Modifed Radiation Rates
While we have previously viewed the alteration of radiative rates due to the presence of a cavity in the context of perturbative QED, identical results can also be derived from purely classical arguments (Morawitz, 1969; Milonni and Knight, 1973). In these we consider the power radiated by a dipole antenna in the presence of the cavity walls. We illustrate this point first using the simplest example: an antenna in front of a single, plane conductor. In free space, the electric field at position r due to an oscillating electric dipole deio‘ is
k3e‘” E(r, t ) = 411&0
x d) x
f(L) +
[3P(P.d) - d ] (k# (l-
and the power delivered from the dipole into this field is dZk30 Po=-. 12n&O
L (kr)’))F(19)
241
CAVITY QUANTUM ELECTRODYNAMICS
When the dipole is placed at a distance z from a plane conducting surface, the field at the site of the dipole is modified by the presence of the reflected wave whose components follow from Eq. (19):
[
d,k3ei@ Ercf,.2 = 211.50
- - + - eimr f3]
+
where stands for 2kz. The change in radiated power due to the presence of the mirror is just that power, Im{d Er,,,}w/2, that is extracted from the dipole by virtue of its interaction with the reflected field. Hence, the parallel and perpendicular dipoles radiate as follows: a
Since the classical decay rate is proportional to the radiated power, these results are identical to the quantum-mechanical ones (Eqs. 11 and 12). In this model the reflected field can be regarded as the radiation from an image dipole d = d, - d, located at a distance z behind the mirror. From this point of view the radiative properties of a dipole placed very close to the surface of a mirror (Fig. 3 ) can be understood quite simply. The vanishing of the radiation from a parallel dipole d , is due to the cancellation between the physical dipole and its inverted image. Similarly, the doubling of the radiation rate from a dipole d , normal to the mirror is due to the doubling of the radiation field when the image dipole is included. This produces four times the power density in half the volume of free space, corresponding to twice the free-space decay rate. The agreement concerning the average radiation rate between this classical image model and the full QED treatment is related to the fact that only a single pair of atomic levels is involved. By contrast, it will be necessary to consider the whole spectrum of atomic excitations when in Section I11 we consider the cavity-induced level shifts. In that case we shall encounter phenomena that have no classical counterparts. In the case where the mirror is not a perfect conductor, the reflected fields must be multiplied by an additional reflection coefficient of the form {e", which has the effect in Eq. (21) of replacing the cos and sin 4 by { cos(+ + 6) and { sin(+ + 6). Figure 4 shows the decay rates r p[curve (a)]
+
248
E. A . Hinds
DISTANCE
/ WAVELENGTH
FIG.4. Radiative-decay rate of an atom in front of an imperfectly reflecting plane mirror relative to the rate in free space. The amplitude-reflectioncoefficient is taken to be 0.96ei/100.The abscissa is the distance from the mirror relative to the wavelength of the radiation in free space. (a) rp,the rate for a dipole parallel to the mirror. (b) rz,the rate for a dipole normal to the mirror.
and Tz[curve (b)] for an atom in front of a very slightly lossy mirror for which 5 = 0.96 and 6 = 0.01. Both rates diverge at small distances from the mirror, in complete contrast with the rates shown for the perfect mirror in Fig. 3. This divergence is associated with the near field of the dipole. In free space the component of the field out of phase with the dipole tends to a finite value close to the dipole. Near a perfect mirror, this is the field whose reflection cancels or doubles the decay rate. By contrast, the field in free space that is in phase with the dipole diverges as l/r3. The reflection of this field from a perfect mirror is prevented from modifying the decay rate (at least in this first-order perturbation model) only by virtue of its phase. When the field is reflected with a phase shift from a mirror of finite conductivity, this is no longer the case and the decay rate diverges close to the mirror for both polarizations. The simple physical explanation of this effect is that the excitation energy of the atom is dissipated through direct Joule heating of the surface by the oscillating Coulomb field of the dipole. In the same way, an atom in a parallel-plate waveguide can be viewed as interacting with an array of image dipoles corresponding to the multiple reflections of the field between the mirrors. This model duplicates the results shown in Fig. 2 when the mirrors are perfectly conducting. However, when the conductivity is finite, the discontinuities of Fig. 2 become smooth as shown in Fig. 5, where we have once again taken the amplitude-reflection
CAVITY QUANTUM ELECTRODYNAMICS
l-xii [ I
w
u
249
3-
--
U
a
Y
W W ( I
LL
2--
__
\
a 1-U
z
H
--
W
GAP WIDTH / WAVELENGTH FIG.5. Radiative-decay rate of an atom at the center of an imperfectly reflecting parallelplate waveguide relative to the rate in free space. The amplitude-reflectioncoefficient is again taken to be 0.96ei/’00.The abscissa is the spacing between the plates relative to the wavelength of the radiation in free space. (a) The rate for a dipole parallel to the plates. (b) The rate for a dipole normal to the plates.
coefficient to be 0.96ei”00.In terms of the mode expansion, this reflects the fact that the modes no longer have to satisfy k, = nn/L but can exist with nonvanishing parallel fields at the surface. Equivalently, the smoothing can be understood as a spectral broadening due to the finite lifetime of a photon in the cavity. In particular, r pno longer goes to zero abruptly when the gap is less than A/2 but instead approaches zero as the wing of a Lorentzian. When the gap is sufficientlysmall, the decay rate for both polarizations diverges as in the case of the single mirror.
B. OBSERVATIONS OF MODIFIED IRREVERSIBLE SPONTANEOUS RADIATION I . Radiation near a Single Plane Mirror The cavity modification of spontaneous decay rates was observed in an early series of experiments described by Drexhage (1974), who studied the fluorescence from a thin layer of optically excited organic-dye molecules that were separated from a metal surface by a dielectric layer of known thickness. These experiments showed variations in both lifetime and the angular distribution of the fluorescence,due to the spatial variation and anisotropy of the vacuum field. In addition they showed at short range the effects of direct coupling between the excited molecules and surface excitations of the mirror.
250
E. A . Hinds
2. Enhanced Radiation in a Single-Mode Cavity In 1983, Goy et al. made the first demonstration on isolated atoms that atomic lifetimes can be altered in a cavity. Their apparatus, shown schematically in Fig. 6, consisted of a sodium atomic beam, laser light to excite the atoms, a cavity, and a detector of Rydberg atoms. The cavity was a nearly confocal resonator formed by two spherical superconducting niobium mirrors of 20 mm diameter spaced 25 mm apart and cooled to 7 K. This was able to oscillate in a TEM,, mode at 340 GHz and could be tuned to either of the transitions 23s + 22P3,, or 23s -P 22P1,, in sodium. The mode volume V was 70 mm3. Although the Q of this cavity was high ( - lo6), the damping time was still short compared with the Rabi frequency, allowing the decay to be described by the theory outlined in Section II.A.3, which is based on Fermi’s golden rule. Pulsed laser light was focused into the cavity to excite the sodium atoms to the 23s state. The beam then passed through the cavity and into a detector where an electric field ionized the excited atoms and an electron multiplier detected them. After a laser pulse, the detector field strength was swept through the ionization thresholds of the 23s and 22P levels to produce two time-resolved signals corresponding to the populations of these levels. The experimental results are shown in Fig. 7. When the cavity was not resonant with any atomic transition the spontaneous decay was negligible (free-space decay rate To= 150 s-’) and the atoms were detected in the 23s state. This is shown by the dotted lines for three different runs in Figs. 7(a), (b), and (c). However, when the cavity was tuned to the frequency of the transition 23s -P 22P3,,, the atoms decayed to the 22P term as shown by the solid lines in Figs. 7(a) and (c). For curve (a) the average number of atoms in
Laser/
Electron
Multiplier FIG.6. The main features of the apparatus used by Goy et al. (1983). This arrangement of atomic beam, exciting laser, cavity, and field-ionizationdetector is typical of all the cavity QED experiments on Rydberg states of atoms.
CAVITY QUANTUM ELECTRODYNAMICS
25 1
Ab :-..
I1
... ...
Ac ...ti.-
.-
.... :.-:-. i.-
12
...
I 255
1 22P
& TIME
FIG. 7. First evidence of enhanced radiation from isolated atoms in a cavity (Goy et al., 1983). Dotted line: cavity not resonant with radiation, atoms remain in the 23s state. Solid line: cavity resonant with radiation, atoms transferred to 23P state. (a) 23s + 22P,,, transition with 3.5 atoms in the cavity on average. (a) 23.5 + 22P,,, transition with 2 atoms in the cavity on average. (c) 23s + 22P,,, transition with 1.3 atoms in the cavity on average.
the cavity was 3.5. This was reduced in curve (c) to an average of 1.3 atoms, which is close to the single-atom case assumed in the theory of Section 1I.A. The experiment was also performed on the other fine-structure transition 23s -,22P,,, with the similar result shown in Fig. 7(b). Analysis of these enhanced decay signals demonstrated that the very slow free-space partialradiation rate for the transition 23s -+ 22P could be enhanced by a factor of about 500. This was the first quantitative confirmation of the theory for radiation by a single atom coupled to a single mode of the field. The solid angle subtended a t the center of the cavity by the mirrors was of order 2n. Hence, when the cavity was not resonant with the atoms, there was presumably some 50% suppression of the free-space emission rate, with a corresponding effect on the number of atoms in the 22P state. Unfortunately, the free-space rate was so slow that this inhibition could not be detected. In 1981 Kleppner had pointed out that a more dramatic result could be found by placing an atom in a waveguide whose cutoff frequency is higher than the frequency of the radiation; in this case the density of all modes is zero and spontaneous decay would be completely suppressed. The space between two plane, parallel conducting mirrors is a simple realization of such a structure as we have discussed previously.
252
E. A . Hinds
3. Suppressed Radiation in a Parallel-Plate Waveguide
This waveguide effect was first observed at MIT by Hulet et al. (1985). They used an atomic beam of cesium atoms that had been prepared in the “circular” state ( n = 22, 1 = 21, m = 21) in which the wavefunction of the excited electron is a thin torus of radius n2ao.This state can decay only to the next circular state ( n = 21, I = 20, m = 20) and it does so by radiating a photon of wavelength 0.45 mm with the electric field in the plane of the torus. The excited atoms flew through a parallel-plate waveguide whose length was 12.7 cm, corresponding to approximately 0.5 natural lifetimes. At first, the width of the gap (230 pm) was slightly larger than half the photon wavelength, and spontaneous emission was found to occur as expected. As in the preceding experiment, detection was by field ionization. Next, the atomic levels were shifted by an applied electric field in order to tune the transition frequency below the waveguide cutoff. Figure 8 shows the number of n = 22
/2L FIG.8. Observation by Hulet et al. (1985) of inhibited spontaneous emission from n = 22 circular Rydberg states in a parallel-plate waveguide. The graph shows the number of atoms surviving the flight through the waveguide as the radiation wavelength was varied using an electric field. The sharp increase in the number of survivors near I = 2L is due to inhibition of the spontaneous decay. The loss of signal at long wavelength is due to ionization in the electric field.
CAVITY QUANTUM ELECTRODYNAMICS
253
atoms that were detected versus the spontaneous-emission wavelength in units of twice the gap width. The large increase in signal at 1/2L = 1 was due to the inhibition of spontaneous decay, while the eventual disappearance of the signal was due to field ionization of the n = 22 level in the electric field used to tune the transition frequency. The measurements implied that the spontaneous lifetime was increased in the gap by at least a factor of 20. With the use of a very much smaller parallel-plate waveguide, Jhe et al. (1987) were able to demonstrate very strong inhibition of a near-infrared (3.48-micron) Cs transition 5D,,, + 6P312. In this case the waveguide consisted of two gold-coated mirrors arranged face to face with a spacing of 1.1 microns between them. Figure 9 is a schematic view of the apparatus used in the experiment and Fig. 10 shows the relevant energy levels of the Cs atom.
FIG.9. Experimental setup of Jhe et al. (1987). Inset: scanning-electron-microscope picture of the exit from the mirror gap.
E. A . Hinds
254
FIELD IONlZATKm
P
......... ......... ....... .....
F.5 F':4 F'z3 F's2
6p3,2
FIG.10. (a) Cesium energy levels and transitions relevant in the experiment of Jhe (1987). (b) Close-up showing the hyperline structure of the 5D,/,and 6P,/,states.
et al.
In a preliminary measurement, the ground-state atomic beam passed through the waveguide and was then laser-excited (position A) to the 7P,/, level from which 13% of the atoms cascaded into the 5Ds/z manifold. In this way, three hyperfine sublevels of 5Ds/z were populated: F = 4, 5, 6 . These atoms were detected by exciting a second laser transition to the 26F state which was subsequently field-ionized in front of a channeltron. Figure 1l(a) shows the spectrum of the three 5Ds/z hyperfine levels obtained by scanning the wavelength of the detection laser. In order to observe the inhibition of spontaneous decay, the first laser was then moved to position B (Fig. 9) immediately in front of the waveguide entrance. The 5D,,, atoms produced here were obliged to spent 13 natural lifetimes passing through the (8 mm) channel before reaching the detection laser. Figure 1l(b) shows the spectrum that was measured. Some 10% of the F = 6 atoms had survived the flight whereas the F = 4 and F = 5 atoms had almost entirely decayed from the beam. The survivors were the mF = & 6 atoms whose decay was inhibited in the waveguide because they could only radiate an electric field parallel to the mirror surfaces. From the measured spectra it was determined that their radiative lifetime had been increased by a factor of at least 25. This enhancement was close to the best that was theoretically possible in view of the finite conductivity of the gold mirrors at the frequency of the transition.
CAVITY QUANTUM ELECTRODYNAMICS 1
2
255
3
i CPS
!50
I 00
50
6
0
I00 2GO3’00
” (MHzJ
FIG.1 1 . Spectra of the 5D,,, + 26F transition taken from Jhe et al. (1987). (a) Recorded with laser in position A of Fig. 9. The hyperfine lines 1, 2, and 3 are defined in Fig. 10. (b) Recorded with laser in position B. The presence of hyperfine line 3 is evidence for suppression of spontaneous decay from the level 5D,,,, F = 6, mF = 6.
The magnetic sublevels of an excited atomic state decay with different polarizations of the transition moment and therefore couple to different components of the vacuum field. In free space, the vacuum is isotropic, so the decay rate of an atom is independent of its polarization. However, in a cavity this is not generally the case. Indeed, Fig. 5 shows how the decay rates can be radically different for the two polarizations of the transition dipole in a parallel-plate waveguide. In the experiment described previously, Jhe et al. (1987) were able to demonstrate this anisotropy of the vacuum by varying the magnetic-field axis along which the 5D,,,atoms were polarized, as shown in Fig. 12. When the angular momentum was polarized exactly normal to the mirrors, nearly all the atoms survived because they were coupled only to the suppressed mode in the waveguide, but as the polarization axis was rotated, the decay rate increased, reaching a maximum at 90 degrees that was greater than the free-space value. As a result of this anistropy it should be possible to polarize an excited-state population in such a waveguide, simply by allowing it to decay. A further demonstration of modified fluorescence, this time in the visible, was reported De Martini et al. (1987). In their experiment, a pulsed laser was used to excite organic-dye molecules flowing between two closely spaced mirrors. The transient broadband fluorescence was then filtered and the intensity near the wavelength 633 nm recorded. When the mirror gap was
E. A . Hinds
256 CPS 20
10
0
FIG.12. Excited-state transmission through the parallel-plate structure as a function of the angle between the magnetic field and the normal to the mirrors (Jhe et al., 1987). This demonstrates the anisotropy of the cavity vacuum.
close to half this wavelength they found a shorter, higher-intensity fluorescence pulse, while on closing the gap to less than half the wavelength the pulse became weaker and longer. 4. Observations in Large Cavities
So far we have discussed experiments to enhance and inhibit spontaneous decay using low-order cavities. Now we turn to observations involving cavities whose dimensions are much larger than the radiation wavelength. Gabrielse and Dehmelt (1985) noticed while studying single electrons in a Penning trap that the excited states of the cyclotron motion, which decay by electric-dipole radiation at the cyclotron frequency, were surviving for longer than the natural lifetime at certain values of the magnetic field. This was understood to be due to the formation by the electrodes of the trap of a crude cavity whose standing-wave field had an electric node at the center, the region occupied by the electron. When the cyclotron frequency was tuned to this resonance (164 GHz), the coupling of the electron to the vacuum was reduced below the free-space strength. The change in decay rate was a consequence of confining the electron to a region that was small compared with the wavelength and where the vacuum field was suppressed. In this respect the experiment reminds us of the case of an atom close to a plane mirror. A different example of altered decay rates in a macroscopic cavity is the work of Heinzen et al. (1987), who measured the visible fluorescence at 556 nm from a beam of ytterbium atoms that had been weakly excited near
CAVITY QUANTUM ELECTRODYNAMICS
257
the center of an optical resonator. This cavity was formed from two spherical mirrors spaced 5 cm apart, having a useful aperture of 1-2 mm and a finesse of approximately 70. In the experiment the atoms were spread over a region much larger than the wavelength of the light so that the main effect was simply due to the mode density, which is high at the resonance frequencies of the cavity and low in between. According to Eq. (18) the enhancement of the radiation rate at the frequency of a cavity resonance depends upon the factor QA3/V,which tends naturally to be larger in the microwave domain than it is in the optical. However, since the optical cavity in this experiment was confocal, a large number of Gaussian radial modes could be excited simultaneously by the atom, in contrast with the single TEM,,-mode cavity of Goy et al. discussed previously. The number of radial modes that could contribute to the enhanced radiation rate was limited by the diameter of the mirrors, for higher radial modes extend further in the radial direction and are damped more strongly by diffraction losses. An equivalent point of view is to regard the cavity field as a single mode whose waist size is the diffraction-limited focal spot of the cavity. From this perspective the advantage of confocality appears as a much reduced mode volume compared with that of the fundamental Gaussian TEM,, mode. In any case, the vacuum field at the center of the cavity was high enough to permit the observation of cavity effects in the visible. The modified decay rate was observed in two different ways. First, the fluorescence intensity in the cavity was measured by detecting the light coupled out through one of the mirrors. When the cavity length was scanned, the spontaneous rate into the mode was seen to be alternately enhanced and suppressed (Fig. 13) as the cavity passed in and out of resonance with the atoms. The figure also shows the free-space fluorescence rate into the cavity solid angle, which. was measured by blocking one of the mirrors. This experiment showed an enhancement by 19 and a suppression by 42 of the partial radiation rate into the cavity. Their second observation was of the fluorescence out of the side of the cavity. When the radiation into the cavity mode was enhanced, this intensity decreased because the two decay modes were in competition. The observed intensity variation was small, however, (-2%) because the cavity solid angle was quite small.
c. MODIFIEDEMISSIONFROM DRESSED ATOMS Up to this point we have considered the spontaneous decay of atoms coupled only to the vacuum. The lasers responsible for excitation of the
E. A . Hinds
258
0
CAVITY TUNING (MHZ)
4000
FIG.13. Spontaneous photon count rate obtained by Heinzen et al. (1987) with and without resonant cavity.
atoms have played no significant role in their subsequent radiation. It is also interesting, however (Lewenstein and Mossberg, 1988a, b), to consider the effects of a cavity on an atom that is strongly coupled to a laser field. In this case it is physically realistic to adopt a hierarchy of perturbations in which the atom is first coupled to the radiation mode containing the photons of the exciting laser, i.e., the atom is “dressed” (Haroche, 1971; Cohen-Tannoudji and Reynaud, 1977; Dalibard and Cohen-Tannoudji, 1985), and then the dressed atom is allowed to decay through its coupling to the vacuum states of all the remaining modes. 1. An Outline of the Dressed-Atom Picture
We begin by recalling the main features of the dressed-atom picture for an atom interacting with a single mode of the electromagnetic field. The state In) describes a field which has n photons in the mode and whose energy is hw(n + 1/2). We take the atom to be a two-level system whose unperturbed eigenstates are Ig) and le) with eigenvalues 0 and ha,. This is a reasonable approximation since we assume furthermore that the radiation frequency o is not far from the transition frequency o,.To begin, we neglect the electricdipole interaction between the atom and the field, in which case the eigenstates are the product states 19, n ) and le, n). The state of lowest energy is )g, 0), above which lie a series of doublets 19, l), le, 0), etc. as shown on the left of Fig. 14. The doublet splitting is just the detuning 6 = o - o,of the radiation from resonance. If we now allow the atom to interact with the field,
CAVITY QUANTUM ELECTRODYNAMICS
259
ENERGY
7 . 1
wi 9l-
0
e
0
9
0
--' - +
W
7
2
~
FIG.14. Energy levels of an atom coupled to a field of n photons. The ladder of states on the left shows the energies of the atom + field system when the two are not coupled. The energy levels on the right are those of the coupled atom-field system or dressed atom. The upper-right portion shows the four spontaneous transitions that give rise to the Mollow fluorescence triplet in a strong field.
each state 19, n + 1) is coupled with its nearly degenerate partner le, n), the matrix element being (e, nld.Elg, n
+ 1)
=
d/g
h R o w . 2
=-
(23)
This defines the Rabi frequency R o w in the field E, and the vacuum Rabi frequency R,. In the rotating-wave approximation, which we are assuming to be valid, there are no other couplings, so the Hamiltonian consists of two-by-two blocks whose eigenvalues relative to the ground state are
E*(n)
= (n
+ 1)hw - hS2 +- 2hR -
where
R
= J(n
+ l)n; + s2.
The eigenstates are the dressed states:
-
E. A . Hinds
260
The right-hand side of Fig. 14 shows the energy levels of the dressed atom which are split by R rather than by 6, the laser detuning. In general this splitting depends on the number of photons in the mode, and differs from one doublet to the next. However, when n is large, the splittings of adjacent doublets are essentially equal as in the upper part of Fig. 14. The stimulated absorption and emission processes are now part of the internal Hamiltonian of the dressed atom and the dressed eigenfunctions incorporate the associated radiative level shifts. Spontaneous emission, on the other hand, occurs at the next level of perturbations when we couple the dressed atom to all the other (empty) modes of the radiation field. The spectrum includes therefore three different frequencies, as indicated in the top-right portion of Fig. 14, which form the well-known fluorescence .triplet (Mollow, 1969; Kimble and Mandel, 1976). The central line is generated by the two transitions I n ) -+ I n - 1) in which a photon is scattered out of the laser field with no other change. The rate for each of these follows at once from the eigenfunctions:
+,
+,
r(+
-+
+) = r(-
-+
-1
=
R2 - 6 2 4R2 To ~
where To is the free-space rate. Symmetrically placed at fR on either side are lines due to the transitions I n ) -+ I T , n - 1) which involve changes of the internal state of the bare atom. These can be regarded as sidebands of the fluorescence that are generated by the Rabi flopping of the bare atom between its ground and excited states. The transition rates are
+,
Under steady conditions, these decay rates control the probabilities P( + ) and P( -) for a dressed atom to be in the I n ) and I -, n) states:
+,
Since this ratio cannot exceed unity, the dressed atom is more often in states I +, n) than I -, n ) and hence the bare atom in equilibrium with the field is more often in its ground state than excited. In short, a laser field cannot produce an equilibrium population inversion of the atom.
CAVITY QUANTUM ELECTRODYNAMICS
26 1
1. Cavity-Mod$ed Radiation of a Dressed Atom Consider now the dressed laser-atom system placed in an empty cavity; the laser mode is not one of the cavity modes. If the spectral density of the cavity vacuum varies sufficiently rapidly with frequency, it should be possible to control separately the three decay rates in the Mollow triplet and hence to modify the steady state of the system. This has been demonstrated by Zhu et al. (1988). A particularly dramatic result can be obtained if the rate I-( - + ) for the Mollow sideband I -, n ) -+ I +, n - 1 ) is strongly inhibited by the cavity while r( -+ -) is not, for then the dressed atom can be predominantly in states 1 - , n) and correspondingly the bare atom is more often excited than not. In this way it is possible (although this has not been achieved in the laboratory yet) for the cavity to create a steady atomicpopulation inversion-which is impossible to achieve when the vacuum is that of free space. It is worth restating that this result is due simply to the alteration of the vacuum mode structure by a cavity that is virtually empty. -+
+
D. REVERSIBLE SPONTANEOUS RADIATIONIN HIGH-QCAVITIES We turn now to the case of a single atom coupled to a cavity of very high-Q (Jaynes and Cummings, 1963; Cummings, 1965; Yo0 and Eberly, 1985; Knight 1986). Under this heading we consider first the regime of transient oscillations and then the continuously oscillating micromaser. 1. Transient Rabi Oscillations
The damping rate for the field in a high-Q cavity is ycaY= w/(2Q) while the Rabi frequency for the atom-field coupling is R,. If the damping rate is less than the Rabi frequency, the photon can remain for long enough in the cavity to be reabsorbed and drive the atom back up to the excited state. Under these conditions one is not surprised to find (Haroche and Raimond, 1985) that the probability for the atom to be excited at time t has damped oscillations:
Whereas Rabi oscillation is very familiar when it is an external laser or microwave field that drives the transition, in the case considered here there is by contrast no applied field; the cavity contains no photons when the atom is excited (le, 0)) and just one when it is in the ground state (Is,1)). These two
E. A . Hinds states are coupled, of course, by the electric-dipole interaction and the eigenstates of the coupled atom-cavity system are precisely the dressed-atom states I f ,0) discussed in the previous section. When the cavity is tuned to the transition, i.e., 6 = 0, the energy difference between these states is ha,. This can be viewed as the dynamic Stark splitting due to the vacuum field or “vacuum-field Rabi splitting” (Sanchez-Mondragon et al., 1983). If we place an excited atom suddenly in the empty cavity, the initial state le, 0) is a superposition of the normal modes I &, 0) and the subsequent coherent exchange of the photon between atom and cavity is a quantum beat. The frequency of the beat measures the dynamic Stark splitting of the levels. The first experimental study of such “self-induced” Rabi oscillations was performed by Kaluzny et al. (1983) using the high-lying transition 36S1,, -+ 35P1,, of sodium in an atomic beam apparatus of the type illustrated in Fig. 6. Rydberg states were chosen because the large dipole matrix element (d n2) gave a strong coupling to the vacuum, while the low (microwave) transition frequency allowed the cavity-damping time to be long. Even so, with a cavity Q of lo5 the damping rate was approximately 20 times higher than the single-atom Rabi frequency R, so that the oscillations described previously could not occur. In order to obtain a fast enough Rabi frequency, several thousand atoms were excited together in the cavity. We digress for a moment to discuss the behavior to be expected when many atoms are excited in the cavity. An ensemble of N identical excited atoms in a cavity behaves as a single collective system in which the individual atoms are coupled by their common interaction with the cavity mode (Tavis and Cummings, 1969; Bonifacio and Preparata, 1970; Scharf, 1970). The system can be treated as a generalization of the single dressed atom in which there are now not two but N + 1 degenerate basis states coupled by the electric-dipole interaction. They are the states (N excited atoms, no photons), (N - 1 excited atoms, 1 photon) ...(no excited atoms, N photons). The N 1 eigenstates of this manifold are analogous to the two dressed states I k,0) of the single atom. If N excited atoms are introduced suddenly into a cavity, the initial state is a coherent superposition of these dressed states and the subsequent evolution of excited-state probability, P,(t), should exhibit beats at all the Bohr frequencies of the dressed manifold. In general the beat pattern is complicated, but as N becomes sufficiently large, a dominant beat frequency emerges (Scharf, 1970) and the oscillations are quasi-periodic with frequency f2,fi.This is a natural scaling law, for the dipole interaction is proportional to the amplitude of the field and hence to the square root of the number of photons in the cavity.
-
+
CAVITY QUANTUM ELECTRODYNAMICS
263
In the experiment of Kaluzny et al. (1983), several thousand atoms were simultaneously excited in order to achieve a collective Rabi frequency no@ greater than the cavity damping rate. After the pulsed-laser excitation, the atoms were allowed to interact with the cavity for a preset time after which an electric field was used to Stark shift the levels, decoupling them from the cavity and “freezing” the population distribution. Next, the beam passed into a detector that measured the state populations by means of the usual selective field-ionization method (Section II.B.2). Figure 15 shows recordings of the no@ oscillations in the 36s population for several values of N. Superimposed on the experimental curves are dotted lines indicating the calculated time development that was found by solving the optical Bloch equations. In order to obtain good agreement with the observations it was necessary to include the effect of blackbody radiation in the cavity which triggered the start of the Rabi oscillations by stimulating the emission of the first few photons. However, the average number of thermal photons in the cavity was very much less than the number of atoms, so the subsequent evolution of the ensemble was virtually the same as that in a cavity at absolute zero. Raizen et al. (1989) carried out a closely related experiment in the optical domain. The aim of their experiment was to measure by direct spectroscopy
0
(15
1
TIME (p5)
0
0.5 1 TIME ( ~ 5 )
FIG.15. Collective oscillationsin the state of an ensemble of N Rydberg atoms in a resonant cavity (Kaluzny et al., 1983). Initially all the atoms are in the upper state. In the first trace, the rate Ro,,h is less than the cavity damping rate and the oscillations are overdamped.At higher N the oscillations are evident. Dotted lines indicate theory.
264
E. A . Hinds
the energy-level splitting of the lowest doublet I &, 0) (Fig. 14). When there are N identical atoms in the cavity, the atomic ground state is
(31)
IG) = 1g1)1g2)*.. IgN-1)IgN)
and the first atomic excited state is
where 0: is the raising operator for the ith atom. In this basis, the lowest dressed states are 1 I+, 0) = CIG, 1) IE, 0)l (33)
*
Jz
~
and since the dipole matrix element between I G) and I E ) is d f i , the first doublet splitting of the atom-cavity system is no,/%. In the experiment, a high-finesse ( 20,000) optical resonator was formed by two spherical mirrors of radius 1 m, placed one on each side of a beam of ground-state sodium atoms and separated by 2-3 mm. The cavity was tuned to oscillate in the T E M , , mode at a frequency close to the “two-level” ( F = 2, mF = 2) + ( F = 3, mF = 3) component of the 589-nm sodium resonance line. The coupled atom-cavity system was then probed by laser spectroscopy using a weak laser beam close to the same frequency. The laser light was coupled into the cavity mode through one of the mirrors and the fraction emerging from the other mirror was collected for analysis. Figure 16(a) shows the single narrow transmission peak of the cavity when it was tuned to the atomic transition but devoid of atoms. By contrast, when the cavity contained 300 sodium atoms, the transmission pattern split into two lines and became broader, as shown in Fig. 16(b), indicating that the and having a atom-cavity system had two normal modes split by decay rate equal to the average of the cavity and atomic-polarization decay rates (Haroche and Raimond, 1985). A similar pattern is shown in Fig. 16(c) which was taken with N 40. When the number of atoms in the cavity was reduced to unity, the Rabi frequency became small compared to the damping rate and hence the coupled levels were broadened but not split (Lamb, 1952; Haroche and Raimond, 1985). The solid lines in Fig. 16 show the exact solutions of the optical Bloch equations which agree excellently with the measurements. Of course, thermal effects were negligible at this wavelength. Since the first microwave observations of Goy et al. in 1983, the techniques for making high-Q cavities have improved and it has become possible at
-
-
Qofi
-
CAVITY QUANTUM ELECTRODYNAMICS
0
-30 -15
0
15
30
265
"(MHz)
FIG. 16. Data of Raizen et al. (1989). (a) Spectrum of optical cavity alone. (b) Lowest doublet in the spectrum of -300 atoms coupled collectively to the cavity. (c) As in (b) but with only -40 atoms.
microwave frequencies to study the basic single-atom system. In this case, since the Rabi oscillation involves only one photon, it is obviously important to consider the effects of thermal radiation. Suppose the cavity were at a finite temperature T so that the field included some blackbody photons. The probability of having n thermal photons in the cavity is distributed in accordance with Bose-Einstein statistics,
where Ti is the mean number of thermal photons, (eho'lrT- l ) - ' (Loudon, 1986). Correspondingly there is a statistical distribution of the system over the dressed-atom doublets. Since the doublet splittings n o wvary with n, the quantum beat following the sudden introduction of an excited atom into a resonant high-Q cavity would be a mixture of these frequencies, each weighted according to the probability p(n) (Cummings, 1965) 1 " P,(t) = 2 p(n)[l
1
n=O
+ cos(Jn+lRot)l.
(35)
266
E. A. Hinds
We have assumed here that the cavity damping is negligible on the time scale of interest. (A more elaborate analytic expression incorporating weak cavity damping is given by Haroche, 1984). Whereas the thermal distribution of photon number has a simple temperature-dependence, the beat pattern P,(t) is a complicated and sensitive function of temperature (Cummings, 1965; von Foerster, 1975; Knight and Radmore, 1982a). This time-dependence of the upper-level population was studied experimentally by Rempe et al. (1987). The apparatus involved a beam of rubidium atoms in the highly excited state 63P,/, and a superconducting niobium cavity, cooled to 2.5 K and tuned to the 63P,/, + 6lD,,, transition at 21.5 GHz. The average number of thermal photons in the cavity was two. Atoms of a particular velocity, selected by a set of spinning slotted wheels, were permitted one at a time to enter the cavity where they interacted with the resonant mode for a known period before emerging. A detector then determined whether the final state was 63P,/, or 6lD,/,. In order to trace out the time evolution of the Rabi oscillations in the cavity, the velocity of the beam was varied. Figure 17 shows the experimental results obtained using a low atomic-beam flux for which the average number of maser photons in the cavity was only 0.5. The solid line indicates the prediction of the theory outlined previously and its shape is determined mainly by the thermal photons in the cavity. It is clearly in agreement with the measurements. The atomic flux was then increased so that the average number of maser photons in the cavity grew to 3 and, correspondingly, the frequency of the Rabi nutation increased. In this case the experiment showed oscillations at short times that soon collapsed to P,(t) = and then revived at a later time. This collapse and revival can be understood as a beat between the various
4
FIG.17. Oscillations in the state of a single Rydberg atom subjected to the thermal field of a cavity at 2.5 K. The measurements were made by Rempe et al. (1987).
CAVITY QUANTUM ELECTRODYNAMICS
267
superposed Rabi frequencies (due to the distribution of photon number in the cavity) which at first interfered constructively, but later became dephased so that P,(t) approached the average value of +. However, the number of significant oscillation frequencies was limited (because of the small number of photons in the cavity) so that eventually the interference was again constructive and the pattern repeated (Cummings, 1965; Eberly et al., 1980; Knight and Radmore, 1982b). By contrast, a classical thermal field (i.e., an exponential distribution of intensity) would have produced a completely different time evolution of the population in which there would have been no revivals of the inversion. We note in closing this section that the single-frequency oscillation expected at zero temperature has yet to be observed. 2. Micromasers In the experiment previously described, the rate at which atoms entered the cavity was made less than or comparable with the cavity-damping time w/Q in order that the number of photons in the cavity would be small. If, on the other hand, the atoms arrive more frequently, it becomes possible for each atom to interact with the photon that was left behind by its predecessor. Under these conditions, the field in the cavity builds up through a series of stimulated photon emissions until it reaches a steady state. Such a microwave oscillator is a maser, but if the Q is sufficiently high, it is an unusual one in which the number of photons is small and the number of atoms is frequently zero! Consequently, this system has been named a micromaser. The operation of a micromaser was first realized by Meschede et al. (1985) on the rubidium Rydberg transition 63P,/, -+ 61D3,, using a superconducting niobium cavity of exceedingly high Q-as large as 8 x lo8 at 2 K. They were able to sustain the oscillations of the maser with an average of only two coherent photons in the mode and an atom in the cavity only 4% of the time. At this low level, the maser field was comparable with the blackbody field in the mode. More recently Brune et al. (1987) have succeeded in operating a micromaser based on the two-photon transition, 40s-39s in rubidium. In this maser the mode structure of the high-Q cavity serves a dual role. First, the cavity frequency was tuned to half the 40s-39s interval, approximately 68 GHz, strongly enhancing the atom-field coupling for the emission of two equal-frequency photons through intermediate P-states. This coupling was also helped by the presence of the 39P3/, state, which was only 39 MHz away
E. A . Hinds
from resonance. The second function of the cavity was to provide extremely low mode density for one-photon transitions to the real 39P,,, level. Thus, the cavity both enhanced the two-photon rate and suppressed the onephoton transition that would normally proceed at a vastly greater rate. One interesting aspect of the two-photon maser is the role of quantum noise in triggering the start of oscillation (Davidovich et al., 1987). When the field in the cavity is very small, the gain of this maser is a quadratic function of the photon number N whereas the loss varies only linearly. Hence, the gain of the two-photon maser always falls below the loss as N approaches zero. This means that N must exceed some minimum number before the maser can oscillate continuously-classically the two-photon maser does not start. This is quite different from the case of a one-photon oscillator where the gain at low levels is linear in N . In fact, the quantum-field fluctuations are responsible for starting the maser in a way that is analogous to the activation of a chemical reaction by thermal fluctuations. The triggering time depends upon the details of the “barrier” and times as long as tens of seconds have been observed in the laboratory (Raimond et al., 1989). 3. Quantum Measurements Using Micromasers
The input to a micromaser is a beam of Rydberg atoms, all of which are prepared in a specific excited level. After passing through the cavity, each atom can be interrogated using electric-field ionization by a detector such as that shown in Fig. 6. If the atom is still in the original excited state, it did not leave a photon in the cavity, but if it is found in the lower level, it did. Thus, the state of the atom and the state of the cavity field are entangled in such a way that a measurement of the former reduces the wavefunction of the whole atom-field system and influences our knowledge concerning the latter. Specifically, the state of the cavity field makes a quantum jump upward each time an atom is detected in the lower excited state (Krause et al., 1987). Since the atoms can be detected with very high efficiency and their state determined unambiguously, it is possible to know exactly how many photons have been deposited in the cavity. If the cavity losses are negligible, the result is a Fock state in which there is a definite number of photons. When realistic losses and efficiencies are taken into account, the variance of photon number is no longer zero but can still be much less than the Poisson variance associated with a coherent field (Filipowicz et al., 1986). This sub-Poissonian character of the maser field has been demonstrated by Rempe and Walther (1989).
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269
111. Shifted Energy Levels A. PERTURBATION APPROXIMATION FOR ENERGY-LEVEL SHIFTS
So far we have focused on changes in the radiation rate of an atom, changes that are due to the component of field in phase quadrature with the dipole. In general, however, a cavity will also modify the component of field in phase with the dipole and this affects the atomic energy levels. Some of the earliest discussions of this phenomenon are given by Lennard-Jones ( 1 932), Casimir and Polder (1948), and Power (1965). In order to discuss these radiative level shifts, it is convenient to adopt the Coulomb gauge and the dipole approximation. (See, for example, CohenTannoudji et al., 1989.) We split the Hamiltonian into three parts, H , , Hfield, and HI, of which the first describes the atom alone: -2
H,
Y
=-
2m
+ U(r,)
where ra is the position coordinate of the electron within the atom. The second part refers to the transverse radiation field alone,
E?,,Jr)d3r
e2 e +A2(r) - A(r). p, 2m m -
(38)
is the Hamiltonian for the interaction between the atom and the field. The coordinate r indicates the position of the atom in space. In HI, the first term represents the energy of the instantaneous Coulomb field associated with the electron charge, while the second term can be understood as kinetic energy of the electron motion induced by fluctuations of the vacuum. The third term is the dipole interaction that we used in Section 11. Since the first two of these operators cannot drive electric-dipole transitions in the atom, they were ignored in our earlier discussion of the radiation rates, but now all three terms in HI are important. We take the eigenfunctions l j ) of H, together with the one-photon field states Ik) and the vacuum state 10) as our starting point for a perturbation expansion in the coupling interaction H I . Consider an atom in state la) in the vacuum. The radiative-energy-level shift to order e2 is a combination of
E. A . H i n h
270
first- and second-order perturbations. The first two terms in H, give first-order level shifts that we write as
Ainsl =
j ~ Y ? , ~ ~ ( r and )d~r
(39)
while the last term contributes a second-order shift, l(a, 01
A2=CC j
k.a
wa
~ ( kr).plj, , k) -
4-
(41)
Here k,E, and wk represent, as before, the wavevector, polarization, and frequency of the photon. The sum over j covers all the states of the atom, including levels that lie above state I a) as well as those below. Thus, the level shifts explore the whole spectrum of atomic oscillations, unlike the radiation rate (Section 1I.A) which involves only the initial level and the lower-lying final level of the transition. It is well known that in free space the preceding expressions for Ainsl,Al, and A, are infinite. The energy Ainsl of the instantaneous Coulomb field diverges because of the singularity in the field at the site of the electron. The other terms, A l and A,, both involve sums over photon modes of A2(k) that diverge at high frequency because the mode density is proportional to w z ,but in any case this treatment is inappropriate at high frequency because the dipole approximation fails and because relativity becomes important. If we were interested in calculating the total radiative-level shift of the atom, these difficulties would be of central importance and we would need to discuss the renormalization of the electron mass and charge. (See, for example, Sakurai 1982.) For present purposes, however, we are concerned only with the difference between the energy in a cavity and the energy in free space. This quantity is finite for the following simple reasons. First, the same singularity of the free-space Coulomb field is also present inside the cavity since the field due to induced charges in the cavity walls is finite. Second, same highfrequency divergence of A2(k) is found inside the cavity because the mode density tends to the free-space value for wavelengths that are sufficiently small compared with the cavity dimensions. Hence, the change in energy when an atom is placed in a cavity can, after all, be evaluated using straightforward perturbation theory in which the electron has the physical free-space mass and charge. The subtleties of renormalization in cavity QED
CAVITY QUANTUM ELECTRODYNAMICS
27 1
are discussed more fully by Power (1965), Dalibard et al. (1982), and Barton (1987a).
I . Atom and Plane Mirror: A Model Problem A good paradigm for the study of cavity QED level shifts is the simple problem of an atom in front of a plane, perfectly conducting mirror surface. This problem has been studied by numerous authors (including Morawitz, 1969; Barton, 1974; Chance et al., 1975; Spruch and Kelsey, 1978; Babiker and Barton, 1976; Lutken and Ravndal, 1983; Wylie and Sipe, 1984, 1985; and Meschede et al., 1990). Figure 1 illustrates the geometry of the problem and Eqs. (5a) and (5b)give the field operators for the E- and M-modes. Since there is only one mirror in the present case, k, of Eqs. ( 5 ) is a continuous variable. The difference between the instantaneous field energies, with and without the presence of the plane-conducting mirror, is just the interaction energy of the atomic charges with their electrical images in the mirror. When the atom is neutral, the leading term in a multipole expansion of this energy difference is the London-van der Waals dipole-dipole interaction (Lennard-Jones, 1932)
where d , and d , are the components of the instantaneous electric-dipole moment of the atom parallel and normal to the mirror surface. Of course, this is a good approximation only when z, the distance to the mirror, is much greater than the radius of the atom. Similarly, the mirror boundary condition also leads to a modification of the transverse fields, which causes A 1 and A2 to differ in the presence of the mirror from their free-space values. The change in A, is (e.g., Barton, 1974)
The expression for energy shift A2 is so cumbersome to write down (Barton, 1974) that for the moment we merely give it a name The total level shift, 6, is then the sum of three parts, 6 = binst+ 6, + 6,. The individual contributions to 6 are, of course, specific to the Coulomb
272
E. A . Hinds
gauge and do not correspond to three physically distinct interactions. Nevertheless it seems natural to remark that on the one hand Sinsthas the appearance of an atom reacting to its own field, while S1, on the other, looks like a response of the atom to external fluctuations of the vacuum field. This observation suggests that it might indeed be possible to divide the total level shift into two parts: one due entirely to self-reaction and the other to vacuumfluctuation (Welton, 1948; Ackerhalt et al., 1973; Milonni et al., 1973; Senitzky, 1973; Milonni and Smith, 1975; Milonni, 1976, 1982; Fain, 1982; Barut and Dowling, 1987). The idea has been clarified by Dalibard et al. (1982, 1984). The principal difficulty is knowing how SZ should be divided. These authors showed that if the two parts are required to be separately Hermitian, as they must be to have separate physical meaning, there is a unique way to make the separation. They went on to analyze the radiative properties of an atom in free space. The evaluation of d2 in separate vacuum-fluctuation and self-reaction parts has been extended to the case of an atom and a plane mirror by Meschede et al. (1990). For the purposes of this chapter, it is enough to give their final results. The shift associated with self-reaction (Sins,plus the appropriate part of 6,) is
where kaj = (E, - Ej)/hc and $aj = 2kajz. It is convenient to abbreviate the summand above as taj,for want of a better symbol. Then the level shift associated with vacuum fluctuation (6, plus the rest of 6,) can be written as
+
where f l + l and g191 stand for Jg [sinx/(x I9l)ldx and jg [cos x/(x 191)]dx, respectively. These rather impenetrable general expressions have interesting limits. When the atom is much closer to the mirror than one wavelength, the phase shift 4ajis small. Naturally, there are always high-frequency dipole oscillations for which 9ajis not small, but when the atom is close enough to the surface, these have negligible effect. In that case Meschede et al. (1990)
+
CAVITY QUANTUM ELECTRODYNAMICS
273
find that the total level shift is dominated by the self-reaction contribution and is simply the London-van der Waals interaction, which varies as l/z3. (See Eq. (42). Thus, we can regard the atom, when it is close to the mirror, as a fluctuating dipole interacting instantaneously with its own electrical image. By contrast, at large distances from the mirror, where 4aj% 1, both hself (Eq. 45) and the first term in a, (Eq. 46) contribute significantly to the level shift, which now has the form
Since cjaj= 2kajz,this is proportional on average to l/z. Note also that only the transverse dipole d , is involved. This result, describing the level shift far from a mirror, also has a clear physical interpretation. Each level j lying below a (positive kaj) contributes to the total shift an amount equal to the energy of a dipole dp interacting with the retarded reflection of its own classical radiation field, as given by the real part of Eq. (21a). For levels lying above a there is no contribution to the shift; a reasonable result since energy conservation prevents those dipoles from radiating into the far field. For an excited level having only one significant decay branch, this result can be neatly recast in terms of the free-space radiation rate To :
where the last factor determines the square of the component of electricdipole moment parallel to the mirror surface. An interesting related problem is that of an electron executing cyclotron motion in a magnetic field near a mirror. This is important in connection with the measurement's of the electron g-factor in a Penning trap (Brown and Gabrielse, 1986), where the cyclotron frequency is compared with the spinprecession frequency. As we have mentioned already in Section II.B.4, the radiative decay of the electron cyclotron motion provided one of the earliest examples of modified spontaneous radiation in a cavity, but here our interest is in the frequency shifts induced by the cavity. Boulware et al. (1985) have shown that the shift of the cyclotron frequency o,at large distances from the mirror is dominated by the classical interaction of the electric dipole with its own reflected far field, just as we have found for the transition frequencies of an atom. Specifically, the shift of the cyclotron frequency is
274
E. A . Hinds
Again, the last factor determines the square of the dipole moment parallel to the mirror surface, 0 being the angle between the magnetic field and the normal to the mirror. In the experiments to measure g-2, the configuration of the trap electrodes is, of course, more complicated than a plane mirror but this model example provides an order-of-magnitude estimate for the cyclotron frequency shift due to the trap, which amounts to several parts in This is close to the level of experimental accuracy. The shift of the spinprecession frequency, on the other hand, is smaller by a factor h o l m 2 x according to Boulware et al. (1985) and is therefore not significant. When the electron or atom is in its ground state, this mechanism is entirely absent since all the levels j lie higher than a. In this instance, the leading contribution to the atomic level shift comes entirely from the second and third terms of a,, given in Eq. (46) (Meschede et al., 1990) and turns out to be proportional to l/z4. When this shift is written in terms of the static electric scalar polarizability a,,,,, it takes on the form of the famous Casimir-Polder interaction (Casimir and Polder, 1948)
where as,a,is defined as -2 C j I(aldlj)12/(3hoaj). This energy shift is most naturally understood as a change in the Stark shift produced by the vacuum-i.e., a change in the Bethe (1947) contribution to the Lamb shiftresulting from the modified vacuum-field distribution. At a given frequency, the vacuum field is modified strongly by the mirror only when the wavelength is long compared with the distance to the surface. (See, for example, Fig. 3.) In the long-range limit that we are considering here, the affected part of the vacuum spectrum is of longer wavelength than all the transition wavelengths and is effectively static. Hence the appearance of the static electric polarizability in Eq. (50) is quite natural. An excellent review of Casimir forces in this and other systems is given in Physics Today by Spruch (1986). 2. Level Shifts in a Parallel-Plate Waveguide The parallel-plate problem has also been analyzed quite fully (Barton, I970,1981,1982,1987a, b; Stehle, 1970; Philpott, 1973; Milonni and Knight, 1973; Chance et al., 1974; Lutken and Ravndal, 1985). Formally, it is the same as that of the single plane mirror except that in the case of two plates k, is restricted to the values nn/L because the tangential field must go to zero on
CAVITY QUANTUM ELECTRODYNAMICS
275
both boundaries. The general result for an atom arbitrarily placed in a waveguide of mirror spacing L is quite complicated and the reader is referred to the work of Barton (1974) for the details. Here we consider only the limiting cases when the gap is either narrow or wide. If the width of the gap is small compared with the important transition wavelengths, the level shift is dominated by the instantaneous London-van der Waals interaction, just as it is in the case of an atom close to a single mirror. This result is true regardless of the position of the atom within the waveguide. In this case the potential involves a sum over all the image dipoles :
where a and b are the distances from each mirror to the atom and L = a + b is the gap width. While the first sum, due to images formed by an odd number of reflections, depends upon the position of the atom, the second, due to even reflections, does not. The level shift given in Eq. (51) diverges negatively at each mirror surface and has a maximum in the center of the gap given by
6 inst .
r(3) (3d; 4ne0 4L3
=---
+ Sd;),
xr
where 1(3), the Riemann zeta function, symbolizes n - 3 and is approximately equal to 1.20. At the other extreme, if the plate separation is much wider than the wavelengths of the atomic transitions, we can distinguish two regimes of atomic position. Close to either of the mirrors the level shift approaches the single-mirror London-van der Waals shift. O n the other hand, far away from both mirrors, the dominant effects are the classical far-field interaction (proportional to 1/L) for excited states and the Casimir shift (proportional to l/L4) for ground states, just as in the case of the single mirror. At the center of the gap, the excited-state, far-field interaction is
This can also be obtained directly from the one-mirror result simply by summing over the interactions with all the image dipoles formed by multiple reflection.
276
E. A. Hinds
Special consideration needs to be given to the case L = n1/2, when the gap is resonant with the radiation. Since we assusme perfect mirrors, there are infinitely many reflections. In that case the coupling between the atom and the cavity is strong and first-order perturbation theory fails catastrophically unless the atom happens to be positioned exactly at a node of the standing wave. This behavior is evident in Eq. (53), which diverges logarithmically when L is an odd number of half wavelengths, but is completely insensitive to the even resonances because they have nodes at the center of the waveguide where the atom is located. Of course the divergence is not physical; it vanishes as soon as cavity losses are taken into account, which may be done either by summing a suitably attenuated series of image dipoles (as we did in Section II.A.4) or by giving the damped photon modes an appropriate spectral distribution (as in Section II.A.3). Even so, our perturbative approach breaks down unless the level shift is small compared to the width of the resonance, for the essential point about this approximation is that the unshifted frequency ckaj is used in Eq. (53) to compute the shift. It is possible to obtain a more accurate result by iteration (Barton, 1987a), but when the shift is large, the coupled atom-cavity system is described more naturally by the dressed-atom picture as outlined in Section II.C.l. Finally, on the subject of level shifts in a wide partillel-plate waveguide, we turn to ground-state atoms that experience only the l/L4 Casimir shift. At the center of the waveguide this shift is given by (Barton, 1987a)
6Casimir . . = - - - - - - z3hC 4n&,90L4
(
la,,,,
+
+),
(54)
where astatis the static electric-scalar polarizability defined following Eq. (50) and fl is the tensor polarizability -2 (I(aldlj)12-31(ald,lj)12)/(hoaj).
cj
3. Level Shifts in a Resonator of Moderate Q
The level shifts in a cavity of moderate Q are not fundamentally different from those already discussed. Specifically, there is still a van der Waals shift when the atom is close to one of the walls, a Casimir shift for ground-state atoms far from all walls, and a classical radiative shift for excited atoms far from all walls. As usual, moderate Q means here that the bandwidth of the cavity resonances is greater than the vacuum Rabi frequency so that a perturbative treatment is appropriate. Detailed calculations have been carried out to find the level shifts of an atom near the center of an open optical resonator (Heinzen and Feld, 1987;
277
CAVITY QUANTUM ELECTRODYNAMICS
Heinzen, 1988) using the far-field aproximation appropriate for excited states in a large cavity. In the same approximation, the shifts of the eleclroncyclotron frequency have also been computed for an electron at the center of closed cylindrical (Brown et al., 1985) and spherical (Brown et al., 1986) cavities. The full theory for an atom in a spherical cavity, including van der Waals and Casimir terms, has been developed by Jhe et al. (1990). B.
VAN DER
WAALSSHIFT
1. Rydberg States
The van der Waals interaction potential for an atom and one mirror (Eq. 42) can be conveniently written as
where Cb2)indicates (3 cos2 0 - 1)/2 and er, is the magnitude of the electric dipole operator for the atom. Similarly, the potential (Eq. 52) for an atom at the center of a parallel-plate waveguide of width L is
For an atom in a Rydberg state with effective principal quantum number n, and orbital angular momentum quantum numbers 1, m, the relevant matrix elements are
n: [5n; (n*, llrz[n*,1 ) = 2
( I , mlCb2)J1,m)
= -
+ 1 - 31(l + 1)]&
and
(57)
[3m2 - I ( / + I)] (21 3)(21 - 1) ‘
+
The van der Waals shifts of Rydberg atoms can be very large because the rms dipole moment scales approximately as n t . Figure 18 shows the shifts of the nS levels of sodium at the center of a parallel-plate waveguide 600nm wide; for n = 15, the shift is approximately 500 MHz. The situation is more complicated in levels having nonzero orbital angular momentum. For example, Fig. 19 shows the effect of the van der Waals interaction on the Dstates of sodium. Below n = 8, nD multiplets are dominated by fine structure
E. A . Hinds
278
L u W
c m >
6
8 10 12 14 p r i n c i p a l quantum number n
FIG.18. Calculated van der Waals shifts of the nS levels of sodium at the center of a parallelplate waveguide 600 nm wide.
(J = 3/2 and J = 5/2) and the two fine-structure levels are slightly split by the van der Waals interaction according to the values of lmJl. On the other hand, above n = 13 we see a van der Waals triplet, best described by lrnL1,that is weakly split by the spin-orbit interaction. The van der Waals shifts of the Dstates are even larger than those of the S-states. 2. Hydrogen The case of hydrogen requires special treatment because the levels of a given n are almost degenerate with respect to 1. This situation has been 500
-
=-m. -500
-
L W
W c
-1000
I
6
I
;
I
I
I
I
;
I
I
8 10 12 14 p r i n c i p a l quantum number n
FIG.19. Calculated van der Waals shifts of the nD levels of sodium at the center of a parallel-plate waveguide 600 nm wide. The doublet structure atdow n is simply the fine structure (J = 3/2 and J = 5/2). The triplet structure at high n is due to the van der Waals interaction.
CAVITY QUANTUM ELECTRODYNAMICS
279
investigated theoretically by Alhassid et al. (1987), who have studied the level shifts due to the general class of perturbations where y and B are constants. This form includes interactions both with a and with a parallel-plate waveguide (/I in the range single mirror (B = to J8/3 depending upon the position of the atom). In addition it encompasses the diamagnetic interaction (B = 0) of the atom with an applied magnetic field (Gay, 1986). Alhassid et al. restricted their treatment to the case of “weak” V where states of different principal quantum number n are not appreciably mixed, a good approximation to any practical realization of the van der Waals interaction. In this approximation the Runge-Lenz vector A, normally a conserved quantity in the Coulomb potential of hydrogen (Landau and Lifshitz, 1977), varies adiabatically under the influence of the perturbation K It transpires that the variation of A is restricted for all values of B, by the existence of a new conserved quantity:
fi)
fi
A
= (4
- B2)A2 + 5(Bz - 1)Af.
(60)
This invariant is a generalization of the result A = 4A2 - 5Af found previously by Solov’ev (1982) and Herrick (1982) for the diamagnetic case p = 0. Alhassid et al. also obtained a general expression for the energy-level shifts:
+
- (B2 - l)m2 + (B2 + l)n2 + B2 31. (61) 2 Figure 20(a) shows how the energy levels (in units of yn4) of the manifold n = 10, m = 0 change as fi is varied from 0 to 2. In the case of the van der Waals interaction with a single mirror (B = y = -e2/64xe,z3), which is indicated by the dotted line, we see that the six most shifted levels appear as three doublets while the remaining four levels are singlets. Three boundaries are evident in the spectrum and these are reproduced in Fig. 20(b) with labels A, B, and C. The doublets lie between A and B while the singlets are in the region between B and C. An explanation for this behavior can be found in the motion of the Runge-Lenz vector. According to Eq. (60), the tip of the Runge-Lenz vector precesses in space on the surface of the ellipsoid
6
=
yn2 -[A
a,
2A2 + 5Af
= A,
(62)
which varies in size from one end of the manifold to the other as shown in Fig. 20(b). At the same time, the magnitude of the Runge-Lenz vector must be less than n, since L2 + A 2 = n2 - 1 is an invariant within the manifold (Englefield, 1972). The sphere of radius n is also shown in the figure. At the top end
E, A . Hinds
280 A€
0
FIG.20. Energy levels of the manifold n = 10, rn = 0 in hydrogen under the perturbation given in Eq. 59. The dotted line marks the van der Waals case. (a) Energy in units ofyn4 versus 8. (b) Boundary curves A, B, and C for the spectrum above and the constraints on the Runge-Lenz vector in the case of the van der Waals interaction.
of the manifold, we see that the Runge-Lenz vector must be either parallel or antiparallel to the z-axis, corresponding to a degenerate pair of levels. More generally, in the region of the spectrum between A and B, A is too large for the Runge-Lenz vector to lie parallel to the mirror surface (in the x y plane), and therefore A is confined to two regions, one on each side of this plane, that are not connected classically. This is the region of nearly degenerate pairs of levels. At the boundary B, the quantity A becomes small enough that the ellipsoid lies fully inside the sphere and doublets vanish.
c. OBSERVATIONS OF
PERTURBATIVE
LEVELSHIFTS
The cavity shifts previously described theoretically have been partly, but not yet fully, explored in the laboratory. The long-range 1/L shift of a visible transition has been observed in an optical resonator, as we shall describe in
CAVITY QUANTUM ELECTRODYNAMICS
28 1
Section 1II.C.1, while the short-range van der Waals shifts of ground-state and Rydberg atoms have been studied through measurements of atomicbeam deflections due to the gradients of these shifts (Section III.C.2). As of early 1990, there have been no direct spectroscopic observations of the van der Waals shift in a cavity, nor has the Casimir shift of any atomic level been demonstrated in the laboratory.
I . Far-Field Shift The far-field 1/L shift of excited levels has been studied experimentally by Heinzen and Feld (1987) using the 61So-61P1 resonance transition (553 nm) of barium atoms near the center of a concentric optical resonator. This cavity was 5cm in length and the width of its resonances was approximately 150 MHz-much greater than both the width of the atomic transition (-20 MHz) and the cavity shift being measured ( - 1 MHz). The barium atoms traveled in a collimated atomic beam at right angles to the optical axis of the cavity. A weak laser beam was used to excite the lSo-lPl transition in atoms at the middle of the cavity so that the central frequency and width of the atomic line could be determined from the spontaneous decay fluorescence. The laser was scanned through the atomic transition for a variety of cavity lengths. In this way it was possible to plot the width and shift of the atomic transition as a function of cavity tuning as shown in Fig. 21. The 30. N
=5
28.
I 26.
:24. I-
3
22.
CAV I T Y TUN I NG (328 MHZ/O I V) FIG.21. Width (a) and shift (b) of the 6LS,-61P, transition of barium in a resonant cavity versus cavity tuning. As the cavity is tuned through the atomic transition frequency, the width goes through a maximum and the shift varies dispersively (Heinzen and Feld, 1987).
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E. A . Hinds
upper trace shows the width and demonstrates that the rate of spontaneous decay is enhanced when the cavity is tuned to the atomic transition, as we have discussed in Section II.B.4. The lower curve shows the atomic transition frequency being “pushed away” from the cavity resonance as a result of the radiative coupling between the atom and the cavity. On each graph the solid line represents the result of a far-field calculation (Heinzen, 1988) and the good agreement indicates that the atom does indeed behave as a classical antenna under these circumstances. We note that this experiment is closely related to the work of Raizen et al. (1989) (outlined in Section 1I.D.1) in which optical spectroscopy revealed the vacuum Rabi splitting and modified linewidth of atoms strongly coupled to an optical resonator. The essential difference is one of coupling strength. 2. van der Waals Deflection The first demonstration of a van der Waals attraction between free atoms and a metal was performed by Raskin and Kusch (1969) using a beam of Cs in the ground state. The trajectory of the atoms was at grazing incidence to a gold-coated cylindrical surface. Close to the surface of the cylinder, the atoms experienced a force, due to the gradient of the van der Waals energy, that was attractive toward the metal and deflected them into the geometric shadow of the cylinder. Since the radius of the cylinder (10 cm) was very large compared with the relevant atom-surface distances ( z = 50- 100 nm), the surface was essentially a plane mirror. With the assumption of a van der Waals interaction potential, - k / z 3 , Raskin and Kusch were able to compute the beam intensity in the shadow of the cylinder as a function of the deflection angle. The measured-intensity profile was then compared with the calculations in order to obtain a value for k. For a perfectly reflecting surface the expected value of k depends only upon the mean square dipole moment of the atom as indicated in Eq. (42), but in this experiment the spectrum of the dipole fluctuations involves high frequencies (principally the resonance line of Cs) at which the gold surface is not a perfect conductor. A reduction in k of order 20% is required to account for the physical properties of the gold surface at optical frequency (Bardeen, 1940; Lifschitz and Pitaevskii, 1961 ; Mavroyannis, 1963). The experimental result, k = 1.4: ;::(ea,)*, provided qualitative confirmation of the theory, k = (2.1 - 2.4)(eaJ2. Experiments on molecules (Raskin and Kusch, 1969; Shih et al., 1974; Shih, 1974) demonstrated that they too experience a deflecting force, but the
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difficulties in ab initio theory for these systems made it difficult to infer fundamental information from the measurements. More recent experiments (Bardon and Audiffren, 1983, 1984) involving the deflection of tantalum and iridium atoms by a tungsten surface provide further quantitative information concerning the van der Waals deflection but are similarly hard to interpret at the most fundamental level. An improved version of the original Cs experiment was carried out by Shih (1974) and extended to Rb and K (Shih and Parsegian, 1975). From these measurements it was claimed for each species that the potential I - " was consistent with the expected n = 3 and not with n = 2 or 4. However, the measured coupling strength k was systematically lower than the theoretical estimate by a factor of order two, although the experimental uncertainty was only 5510%. Mehl and Schaich (1980) noticed that quantum mechanical diffraction should play a role in the analysis of these experiments because of the small range of impact parameters and transverse momenta involved. They carried out quantum mechanical-scattering calculation and discovered, somewhat surprisingly, that the full theory gave the same results as the classical analysis. The discrepancy between experiment and theory for these ground-state measurements has yet to be resolved. In the preceding experiments, the fluctuating dipole moments were of order eao and the deflections were unobservably small except for the few atoms that came within 50-100 nm of the surface. More recently, Anderson et al. (1988) have used highly excited Cs atoms, in which the rms dipole moment is proportional to n;, both to vary the van der Waals coupling strength k in a systematic way and to increase it by three to four orders of magnitude! In that experiment an atomic beam of Cs was excited to one of the high-lying states nF using cw lasers in a two-step sequence. (Their excitation scheme is shown in Fig. 10 for the particular case of 26F.) The excited atoms then passed into a parallel-plate waveguide, 8 mm long and adjustable from 2 to 10 pm in width, formed by two gold mirrors. Here the atoms were attracted toward the mirrors by the van der Waals force, and those that were deflected sufficiently to strike one of the mirrors stuck to the surface. Thus, the fraction of atoms emerging from the far end of the waveguide was determined by the coupling strength k of the interaction. The transmission through the waveguide was measured for atoms in a variety of excited states in order to find the value of n at which the transmission was reduced to 1%. This principal quantum number was called the maximum value, n,. The width of the gap was then altered and the corresponding n, determined for each new width.
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For a channel of width L and length Z the authors argued that n i a T L S Z - * ,where T is the temperature of the atomic beam source. A numerical simulation confirmed this scaling law and determined the constant of proportionality. Figure 22 shows the computed values of nm versus the width of the channel together with experimental values measured in the Rydberg states of Cs and one point obtained from the 14s state of Na. (The gap width for the Na point is scaled to account for different values of Z and T in the experiment). In the range n, = 14-27, the absolute agreement of theory with experiment is good, which implies that the measured coupling strength k for Rydberg atoms is consistent with the simple van der Waals interaction. On the other hand, for higher principal quantum numbers the atoms were deflected by stray electric-field gradients (Stark effect a n:) rather than the and the effect of this is evident in the point .given van der Waals force (a for n = 32. Figure 22 also shows a data point derived from the ground-state measurements of Raskin and Kusch described previously. Whereas the experiments on ground-state atoms seem to disagree with theory, the measurements on Rydberg atoms do not. This may well be a consequence of the very different frequencies involved in the dipole fluctuation spectrum. In the case of Rydberg atoms the dipole fluctuations are predominantly at microwave frequencies where the gold mirrors are well described as perfect reflectors. In that case the computation of the van der Waals force does not involve any surface physics. The ground-state atoms, on the other hand, fluctuate most strongly at optical frequencies, corresponding to the resonance transitions, where the mirrors are not perfect. In that case it may be necessary to characterize the surface involved in the experiment in some detail in order to obtain agreement between theory and experiment.
nt)
FIG.22. Maximum principal quantum number n, transmitted through a parallel-plate waveguide versus the gap width. The solid line is theoretical. Solid circles are the data from Rydberg states of cesium; the open circle is from the ground state. The cross corresponds to a measurement in sodium. The deviation from theory at large n, is due to stray electric fields.
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3. Spectroscopic Measurement of van der Waals Shift In order to obtain more detailed and precise information on the van der Waals interaction, it would be extremely helpful to be able to place the atoms at a fixed, known distance from a mirror and to measure by direct laser spectroscopy the displacement of the energy levels. At first sight any attempt to do this seems bound to be thwarted by the van der Waals force itself, which accelerates the atom toward the mirror. One possible solution to the problem, however, is to balance the van der Waals force in the gap between two mirrors against the optical force produced by a standing light field inside the gap. This technique was demonstrated in an experiment by Anderson et al. (1989) in which the optical dipole force constrained a beam of atomic sodium to travel along the middle of a mirror gap only 600 nm wide over a length of 8 mm. The trapping light was a standing wave with nodal planes at the center of the gap and at each mirror surface and whose frequency was tuned 45 MHz to the blue of the sodium D2 resonance line. Of course, the atoms were perturbed by the light field as well as the van der Waals interaction; indeed, the optical channeling was the result of overwhelming the ground-state van der Waals shift by an optical Stark shift of order 1 MHz. In the Rydberg levels, however, the van der Waals shift is enhanced by the factor n4 and completely overwhelms the light shift. This being the case, it should be possible to perform a new kind of spectroscopy in which atoms, accurately localized by an optical channel within the waveguide, are excited by laser light into a Rydberg state. The resonant frequency for this excitation should differ from that of an atom in free space mainly because of the van der Waals shift of the upper level. The transition would be simple to detect for, once excited, an atom would be strongly accelerated by the van der Waals force to the nearest mirror surface on which it would stick. Figure 23 shows the expected 14s excitation spectrum for sodium atoms trapped in such optical channels; the three spectra correspond to gaps of width 600,900, and 1200 nm. An experiment to observe and measure these very shifts is under way in our laboratory at Yale as of 1990. In principle, it should also be possible to use the van der Waals shift as a method of probing the spatial distribution of atoms close to a surface using, for example, the 900nm mirror gap shown in Fig. 23(b). Here there is a strong gradient of the van der Waals potential across each optical channel and the lineshape of the excitation spectrum should therefore exhibit structure revealing the distribution of the atoms in the channel. When the transverse momentum of the atoms is sufficiently low, the quantum nature of
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I
FREMNCY SHIFT I MHz 1
FIG.23. Excitation spectra calculated for the 14s-state sodium held in a l-dimensional optical trap. (a) A 600-nm gap with atoms trapped at the center should produce a single shifted line. (b) In a 900-nm gap the atoms are trapped away from the center and the line should be broadened but less shifted. (c) Atoms trapped in a 1200-nm gap exhibit two lines of different width.
their oscillations in the channel should be important. In this case, the vibrational ground state (u = 0) the 14s excitation spectrum should exhibit a single peak typically 60 MHz wide, corresponding to a distribution in space of 35 nm, whereas, the u = 1 state, on the other hand, should exhibit two peaks separated by 60 MHz. At the time of writing, this type of spectroscopy is also being developed in our laboratory. It could provide a sensitive new method to probe the de Broglie wave nature of whole atoms and to study the quantum limits of confinement.
ACKNOWLEDGMENTS
The author is indebted to the National Science Foundation for supporting his research in this field.
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Index
A Adiabatic theorem, 108-111 Alignment and orientation, 18, 19 Amplitude fluctuations, 171 Anisotrophy, I , 19, 26, 27 Antenna, dipole electric field, 244 power radiated, 245
resonant cavity, 276, 277 Cave's theorem, 201, 204 Cavity high-Q, 261 low-Q, 239, 269 QED, 237 Cavity quantum electrodynamics, 89-94 cavity-modified spontaneous emission, 90-92 dynamic effects in, 92 Channeling, optical, 285 Classical trajectory, see SCA Coherent state, 147, 151 two-photon, 144, 148, 150, 190 Collapses and revivals, 83-87 Coordinates, 3, 14 Correlated emission laser, 184, 208 Cross sections and SCA, 8 and T-matrix, 7, 21
B Bell inequalities. 119-124 for photon polarization correlations, 121-124 Berry's phase, 108-116 optical experiments, 113-115 Blackbody radiation. 263. 265 Bohm Gedunkenexperiment, 118-121 Born approximation distorted wave, 17, 18 plane wave first, 15-18. 19-26, 49-56 second, 29-48, 56-62
D C
de Broglie wave, 283, 286 Decay, see Radiative-decay, modified Density matrix, 19 Distorted wave Born approximation (DWBA), 17, 18
Casimir interaction experiment, 281 parallel plates, 276 plane mirror, 274 291
292
INDEX
Double collision mechanisms for electron capture, 40, 41 for ionization. 30-33. 48 Dressed atom radiative decay, 260 cavity modification. 257, 261 theory, 258
E Eikonal approximation and SCA, 5-14 and IA, 38 Einstein-Podolsky-Rosen paradox, 116-119 completeness, 117 reality, 117 Electron capture to continuum states, 45-48 from excited states, 44 to excited states, 44, 45 to ground states, 39-43 Energy-level, shifted between parallel plates, 274 experiment, 283, 285 in front of plane mirror, 271 experiment, 282 in high-Q cavity Rabi splitting, 262, 264, 282 image model, 271, 275 perturbation approximation, 269 in resonator, 276 experiment, 281 Excitation, 15-19
F Far-field level shift parallel, plates, 275 plane mirror, 273 resonator, 277, 281 reflected, 273 Fluorescence cavity-modified from atoms, 257, 282 organic dye, 249, 255 Mollow triplet, 260 Fock states, 79, 80, 147 Fokker-Planck equation, 185 Four-wave mixing, 177, 178, 181, 219, 220, 221 Free-electron laser, 161
G Gravity-wave detection, 144
H Hamiltonian, 78, 79 Heisenberg-Langevin equation, 175 Heterodyne detection, 100-102, 166, 210, 213, 216 balanced, 212 Hydrogen invariant quantities, 279 level shift, 278
I
Impact parameter approximation, see SCA Impulse approximation (IA). 34-39 Interferometery, 191 high precision, 144 Internuclear potential, 9, 38 Ionization “capture”, 24-26 direct, 19-24 electron angular distribution, 26-29
J Jaynes-Cummings model. 78-89, 162, 164, 165 chaos in, 89 collapses and revivals in, 83-87 multiphoton, 162
L Lamb shift, 237, 274 Level shift, see Energy-level, shifted Linear amplifier, 190, 200 two-photon, 202 Liouville equation, 197 Local oscillator, 210
M Mach-Zehnder interferometer, 144 Micromaser, 267 quantum measurement, 268 Modes, electromagnetic cutoff of, 241, 449 parallel plate, 238 resonator, 242 TEM,,, 243, 250,255,262 Mollow, see Fluorescence Momentum transfer, 8, 17, 35 Multiphoton absorption process, 161
INDEX
N Non-resonant transfer and excitation (NTE), 62, 63, 65. 66, 69
0 One-atom maser, 87-90 Optical bistability, 220 two-photon, 161 Optical Bloch equation, 166 Optical parametric oscillator (OPO), 175, 186, 224, 226
P Pancharatnam's phase, 113-115 Parametric amplifier, 168, 215 degenerate, 169 nondegenerate, 201 Perturbation approximation level shift. 269. 271, 275 radiation rate, 239, 246 Phase fluctuations, 172, 173 Phase-sensitive amplifier, 201 Photon-counting error, 192, 194, 195 Photon polarization correlations, 121-127 Plane wave Born approximation (PWBA), 20, 27 Poincard sphere. 113-114 P-representation, 146, 150 Pump fluctuations, 170, 174
Q
Q-representation, 150, 151 Quadrature variances, 149 Quantum jumps, 94-97 Quantum recurrence, 83-86 theorem, 83 Quasars, 136
R Rabi frequency, 80, 245, 250, 259, 267, 276 collective, 262 definition, 259 vacuum, 80. 259 oscillation, 238, 246 blackbody, 265 transient, 251 splitting, 262, 264, 282 Radiation pressure error, 192. 193
293
Radiative decay, modified anisotrophy, 255 dressed atom, 257 Fermi's golden rule, 239 in high-Q cavity, 261 micromaser, 267 Rabi oscillation, 261 Rabi splitting, 264 image model, 246 irreversible, spontaneous, 239 between parallel plates, 240 experiment, 252 in front of plane mirror, 242 experiment, 249 in resonator, 244 experiment, 250, 256 Red shift, 136 Renormalization, 270 Resonance fluorescence, 166 Resonant transfer and excitation (RTE), 62, 63, 65, 66, 69 Rotating-wave approximation (RWA). 79 Runge-Lenz vector, 217 Rydberg atom circular states, 252 detection of, 250, 252, 253 van der Waals shift of, 277
S Second-harmonic generation, 161 Semiclassical approximation (SCA) one-electron amplitudes, 57 two-electron amplitudes, 58, 59, 63-67 Source correlations and optical spectra, 127-137 frequency shifts due to, 132-134 Spectrum of light and propogation. 127-131 scattered by turbulent medium, 134-136 Spontaneous emission, 178, 183, 207 cavity-modified, 90-92 cooperative, 93 modification by phase-conjugating mirror, 93, 94 Spontaneous radiation, see Radiative decay Squeezed light, 97-108 applications, 106-108 detection, 100-102
294
INDEX
generation, 102-104 photon statistics, 104-106 Squeezed states, 143, 144, 145, 147, 148, 159, 190, 201, 214 amplitude, 157, 161, 220 atomic, 158 ideal, 150, 156, 157 multiatom, 159 multimode field, 161 two-mode, 217 Squeeze operator, 148 atomic, 159 Squeezing, 221, 226 higher-order, 156, 157 multimode, 155 single-quasimode, 189 spectral, 215 two-mode, 182 State multipoles, 19, 26, 27 Sub-Poisson statistics, 152, 204, 209 Superluminal communication, impossibility Of, 124-126 Super-Poisson statistics, 152, 154, 209, 215 Surface excitation, 249 reflectivity, 247, 282, 284
T Thermal field, 81-85 Time-evolution operator, 163 Time reversal, 59 Transition matrix and impact parameter, 7 definition, 4, 5, 7 Translation factors, 11-15 Two-level atom, 78-86
V Vacuum altered modes of, 238, 240, 243, 251, 256, 257, 261 anisotropy, 249, 255 fluctuation, 238, 269 level shifts due to, 272. 274 versus self-reaction, 272 modes, see Modes, electromagnetic Rabi frequency, 261, 276 Rabi splitting, 262, 282 van der Waals interaction, 271 deflection by, 282 in hydrogen, 278 in Rydberg state, 277 spectroscopic measurement, 285
Contents of Previous Volumes
Volume I Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. T . Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K . Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J . P. Toennies High-Intensity and High-Energy Molecular Beams, J . B. Anderson, R. P. Andres. and J . B. Fenn
Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W . D. Dauison Thermal Diffusion in Gases, E. A. Mason, R. J . Munn, and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W. R. S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . de Heer
Mass Spectrometry of Free Radicals, S . N . Foner
Volume 3 The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy of Stored Ions I : Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H . c. wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas CrystalSurface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood
Volume 4 H. S. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H . G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A. Fraser
CONTENTS OF PREVIOUS VOLUMES
Classical Theory of Atomic Scattering, A. Burgess and I . C . Percival Born Expansions, A. R. Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P . G. Burke Relativistic Inner Shell Ionization, C . B. 0. Mohr Recent Measurements on Charge Transfer, J . B. Hasted Measurements of Electron Excitation Functions, D. W . 0. Heddle and R. G. W . Keesing Some New Experimental Methods in Collision Physics, R . F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M . J . Seaton Collisions in the Ionosphere, A . Dalgarno The Direct Study of Ionization in Space, R. L. F . Boyd
Volume 6 Dissociative Recombination, J . N . Bardsley and M . A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. s. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E. A. Mason and T . R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R . Bates and A. E . Kingston
Volume 7 Volume 5 Flowing Afterglow Measurements of lonNeutral Reactions, E. E. Ferguson, F. C . Fehsenfeld, and A . L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R . J. S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s?Ppq, C. D. H. Chisholm, A. Dalgarno, and F . R. Jnnes Relativistic 2-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle
Physics of the Hydrogen Master, C. Audoin, J . P. Schermann, and P. Grivet Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C. Browne Localized Molecular Orbitals, Harel Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt Diabatic States of Molecules-QuasiStationary Electronic States, Thomas F. OMalley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H . S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J . Greenfield
CONTENTS OF PREVIOUS VOLUMES
Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C . Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y . Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I . Schoen The Auger Effect, E. H. S. Burhop and W . N . Asaad
Volume 9 Correlation in Excited States of Atoms, A. W . Weiss The Calculation of Electron-Atom Excitation Cross Sections, M . R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of LowEnergy Electron- Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy
Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr. and Serge Feneuille The First Born Approximation, K . L. Bell and A. E. Kingston Photoelectron Spectroscopy, W . C. Price Dye Lasers in Atomic Spectroscopy, W . Lange, J . Luther, and A. Steudel
Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley T . Huntress, Jr.
Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I . C. Percival and D. Richards Electron Impact Excitation of Positive Ions, M . J. Seaton The R-Matrix Theory of Atomic Process, P. G. Burke and W . D. Rohb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M . Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M . F . Golde and B. A. Thrush
Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R. K . Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M . Broyer, G. Gouedard, J. C. Lehmann, and J. ViguP Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid
CONTENTS OF PREVIOUS VOLUMES
Volume 13
Volume 15
Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M . Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with LaserExcited Atoms in Crossed Beams, I . V . Hertel and W . Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R . K . Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W. B. Somerville
Negative Ions, H . S . W . Massey Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R . F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J . W . Humberston Experimental Aspects of Positron Collisions in Gases, T . C . Grifjith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer Collisions at Low Energies, J . B. Hasted Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H . Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H . B. Gilbody Inner-Shell Ionization, E. H . S. Burhop Excitation of Atoms by Electron Impact, D. W . 0. Heddle Coherence and Correlation in Atomic Collisions, H . Kleinpoppen Theory of Low Energy Electron-Molecule Collisions. P. G. Burke
Volume 14
Resonances in Electron Atom and Molecule Scattering, D. E . Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J . Jamieson, and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and lan E. McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J . Mohr Semiclassical Effects in Heavy-Particle Collisions, M . S . Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V . Bobashev Rydberg Atoms, S. A. Edelstein and T . F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K . Dupree
Volume 16
Atomic Hartree-Fock Theory, M . Cohen and R. P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Duren Sources of Polarized Electrons, R. J . Celotta and D. T . Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain
CONTENTS OF PREVIOUS VOLUMES
Spectroscopy of Laser-Produced Plasmas, M . H . K e y and R . J . Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L . Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N . Fortson and L. Wilets
Volume 17 Collective Effects in Photoionization of Atoms, M . Ya. Amusia Nonadiabatic Charge Transfer, D. S. F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M . F. H. Schuurmans, Q.H. F. Vrehen, D. Polder, and H . M . Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M . G. Payne, C. H . Chen, G. S. Hurst, and G. W . Foltz Inner-Shell Vacancy Production in IonAtom Collisions, C . D. Lin and Patrick Richard Atomic Processes in the Sun, P. L. Dufton and A . E. Kingston
Volume 18 Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J . Morellec. D. Normand, and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A . S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R . Junker
Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N . Andersen and S. E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W . Norcross and L. A. Collins Quantum Electrodynamic Effects in Few-Electron Atomic Systems, G. W . F. Drake
Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H . Bransden and R . K . Janev Interactions of Simple Ion-Atom Systems, J . T . Park High-Resolution Spectroscopy o f Stored Ions, D. J . Wineland, W a y n e M . Itano, and R . S. V a n Dyck, Jr. Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K . Blum and H . Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. Jen? The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N. A. Cherepkov
Volume 20 Ion-Ion Recombination in an Ambient Gas, D. R . Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, T. D. Mark and A . W . Castleman, Jr.
CONTENTS OF PREVIOUS VOLUMES
Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W . E. Meyerhojand J.-F. Chemin Numerical Calculations on ElectronChristopher Impact Ionization, Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A . Armstrong O n the Problem of Extreme UV and XRay Lasers, I. I. Sobel'man and A . V . Vinogradov Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J . M . Raimond Rydberg Atoms: High-Resolution Spectroscopy and Radiation InteractionRydberg Molecules, J . A. C. Callas, G. Leuchs, H. Walther, and H. Figger
Volume 21 Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O'Brien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M . R. C. McDowell and M . Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M . More
Doubly Excited States, Including New Classification Schemes, C. D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H . B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K . Dolder and B. Peart Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion-Atom Collisions, R. Anholt and Harvey Could Continued-Fraction Methods in Atomic Physics, S. Swain
Volume 23 Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M . Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Y a n Transition Arrays in the Spectra of Ionized Atoms, J. Bauche, C . BaucheArnoult, and M . Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F. J . Wuilleumier, D . L. Ederer, and J . L. PicquP
Volume 24
The Selected Ion Flow Tube (SIFT): Studies of Ion-Neutral Reactions, D . Volume 22 Smith and N. G. Adams Positronium - Its Formation and Inter- Near-Threshold Electron-Molecule action with Simple Systems, J . W . Scattering, Michael A . Morrison Humberston Angular Correlation in Multiphoton Experimental Aspects of Positron and Ionization of Atoms, S. J . Smith and G. Positronium Physics, T . C. GrifJith Leuchs
CONTENTS OF PREVIOUS VOLUMES
Optical Pumping and Spin Exchange in Gas Cells, R . J . Knize, 2. W u , and W . Happer Correlations in Electron-Atom Scattering, A. Crowe
Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A . Williams Dipole Polarizability Measurements, Thomas M . Miller and Benjamin Bederson Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scattering in He-He and He' -He Collisions at KeV Energies, R. F. Stebbings Atomic Excitation in Dense Plasmas, Jon C . Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M . Sando and Shih-I Chu Model-Potential Methods, C. Laughlin and G. A. Victor Z-Expansion Methods, M . Cohen Schwinger Variational Methods, Deborah K a y Watson Fine-Structure Transitions in ProtonIon Collisions, R . H . G. Reid Electron Impact Excitation, R. J. W. Henry and A . E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher
The Numerical Solution of the Equations of Molecular Scattering, A. C. Allison High Energy Charge Transfer, B. H . Bransden and D. P. Dewangan Relativistic Random-Phase Approximation, W. R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W. F . Drake and S. P. Goldman Dissociation Dynamics of Polyatomic Molecules, T. Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F . van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H. Black
Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B. L. Moiseiwitsch The Low-Energy, Heavy Particle Collisions A Close-Coupling Treatment, Mineo Kimura and Neal F . Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, Francoise Masnou-Sweeuws, and Annick Giusti-Suzor O n the /IDecay of '"Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonyhau Chen, Leonard Rosenberg, and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J . Maya and R. Lagushenko ~
CONTENTS OF PREVIOUS VOLUMES
Volume 27
Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in Electron-Atom Collisions, Joachim Kessler
Electron-Atom
Scattering,
I.
E.
I.
E.
McCarthy and E. Weigold
Electron-Atom
Ionization,
McCarthy and E. Weigold
Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V. I. Lengyel and M . I. Haysak
Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule Volume 28 The Theory of Fast Ion-Atom Collisions, J. S. Briggs and J. H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W. Milonni and Surendra Singh
Squeezed States of the Radiation Field, Khalid Zaheer and M . Suhail Zubairy Cavity Quantum Electrodynamics, E. A. Hinds