Advances in multi-photon processes and spectroscopy. Volume 15

(15)

where Ip is the ionization potential. However, as emphasized by us [19] in the molecular case, collision will occur also with neighbouring ions. The above maximum velocity condition (cot + cp) ="in12 gives the velocity: z=(EMl'ft>)[l + sin(9>)l, i.e., z(max) = 2£'MIco at cp = n!2. Thus the maximum energy in this case is £ (max) = 2El, / co2 =8Up in agreement with numerical simulations for the corresponding distance traveled by the electron in a half cycle cot = n, i.e., z(n/co) = naM [19]. The simple classical model described above where one assumes an electron is ionized with initial zero velocity, the basis of tunnelling ionization models [7-9] allows us to deduce the laser induced dynamics of the electron after ionization. Thus, for A = 1064nm and intensity 7 = 1014W/cm2, one obtains aM =29 a.u. and Up = 9co~ lOeV. Then distances of TUXU = lOOa.u. and energies SUP ~ 80eV are readily achieved by ionized electrons. This implies the necessity of using large grids and high spatial and temporal resolutions in FFT's used in calculating the split-operators, Eq. (11). As we have shown above, the grid size for high intensity simulations are governed by the dimensions of the ponderomotive radius OTM and ponderomotive energy UP , Eq. (14). As mentioned above, the first recollision can occur at phase cot-\.l>n , i.e., in 0.65 cycle of the laser field [11,19,56]. This period corresponds to 2.3 fs at X = 1064 nm and 1.8 fs at 800 nm. Clearly, the laser-induced electron dynamics occur on near-femtosecond time scale and on attosecond time scale for shorter wavelengths [15]. In simulations, we generally set zmax = ± 500 a.u. corresponding to ~ 4000 grid points; in 3D simulations, the cylindrical coordinate pm3X = (x^,ax + y^) is set to be 125 a.u. (500 grid points). The step sizes in z and t are determined by the limits imposed by the uncertainty principle SzSp - SeSt ^ 1 (e.g., for £ = 5 a.u., i.e., p = lO^a.u., 8x < 0.3a.u. and St < 0.2a.u.). For ionization rate calculations, an absorbing potential is used at both z and p boundaries [49]. The ionization rate r(s~l) is obtained from the decrease of the time dependent norm N of the wave function,

158

dNldt = -rN

,

N(t)=

fy/(t)\2dv.

(16)

Calculations of complete ATI and ATD spectra can be readily obtained in 1-D models, i.e., one electron coordinate z and one nuclear coordinate R, thus reducing the numerical solution of the TDSE to a tractable 2-D problem with current computer technology. This allows for using ever larger numerical grids and circumventing the problem of loss of information of high-energy components of the total wavefunction y/(z,R,t) by projecting onto exact electron wavefunctions in the laser field, called Volkov states [4, 57]. This exact non-Born-Oppenheimer 1-D numerical procedure is described in detail for one-colour (single frequency) [45] and for two colour [46] dissociative ionization of H 2 + . Such exact 1-D non-Born-Oppenheimer have allowed us to propose new methods of dynamic imaging of moving nuclear wave packets on near femtosecond time-scales for rapid proton motion [58-59]. 3 Charge Resonance Enhanced Ionization and Quasistatic Models: One-Electron Systems The first attempt to derive simple analytic formulae for atomic multiphoton transitions beyond usual perturbation (Fermi's golden rule) theory was undertaken by Keldysh [7] who showed that a parameter y, today called the Keldysh parameter [34], allows to separate the perturbative multiphoton regime from the nonperturbative tunnelling region. This parameter is obtained from the low frequency limit of the transition probability from an initial bound state to a Volkov state [4, 8] and is given by: r

= yllP/2Up,

(17)

where Ip is the ionization potential and Up the ponderomotive energy defined in Eq. (14). This can also be interpreted as the ratio of the laser frequency co to the tunnelling frequency CO, through a barrier of width l = Ipl E where E is the electric field amplitude corresponding to the peak intensity I-cE2 I%K [7, 60]. Thus, provided the ionization proceeds faster than the laser induced tunnel barrier lifetime or alternatively the tunnelling frequency co, is larger than the laser frequency co, then the tunnelling ionization model should apply for y<\ where E is calculated at the peak of the field. In this tunnelling regime of y <\, we can estimate the intensity at which ionization will occur completely by over-barrier passage for any given ionization potential Ip and charge q, using a simple electrostatic potential V(z) = -q/\z\-Ez. The barrier height of the total electrostatic potential V(z) takes the maximum value

159 at the position zm = jq/E . Equating —Ip to V(zm) gives the electric field Eb or intensity Ib at which complete over-barrier ionization should occur: Eb=I2p/4q,

Ib=I4p/l6q2,

(18)

For an H atom for which Ip = 0.5 a.u., the critical intensity Ib for over-barrier ionization can be readily estimated from the data in Eqs. (l)-(3) as 1.4xlOl4W/cm2. The corresponding Keldysh parameter is rb=l6qa)/(2Ipf\

(19)

which now depends on frequency a>. Numerical comparisons of TDSE solutions for the H atom [18] with tunnelling rates obtained from quasistatic ADK theory [17] show that indeed for E field values below Eb, the tunnelling model gives satisfactory ionization rates but fails above Eb > since tunnelling no longer applies. It was further concluded that for noble gases and for laser wavelengths in the visible and the near infrared, the Keldysh parameter yb is ~ 1 , which means that in this frequency range multiphoton ionization goes over directly into above-barrier ionization and tunnel ionization plays a minor role. Molecules in intense laser fields will undergo dissociative ionization upon multiphoton absorption. This is due to the competition between dissociation giving rise to ATD spectra [20, 27] and ionization giving rise to ATI spectra [45]. In particular for H2+ for which a detailed comparison between experiment and theory has been performed [61-62], the extremely rapid motion of the protons on a timescale of ~ 10 fs allows for dissociated H2+ protons to travel to large distances R > Re and Rc, where j^, is the equilibrium and Rc is the critical distance for Charge Resonance Enhanced Ionization (CREI). Enhanced ionization then occurs at a critical distance Rc> Re, which results in rapid creation of H + H + . Likewise, in the case of multielectron molecules, multiply charged fragments are created [38-41,44,61-66]. Following the procedure described in the previous section, i.e., Eq. (16), we have obtained ionization rates from numerical solutions of the TDSE for H 2 + . In Fig. 1, we show 3-D ionization rates for H2+ at intensity 7 = 1014 W/cm2 and the wavelength X = 800 nm. In Fig. 2, we give 1-D ionization rates at A = 1064nm (YAG) and 10.6 um(C0 2 ). All rates are obtained for fixed internuclear distance R and have been performed in two different gauges, the length (E -r) gauge and a new space-translation representation in which Vc(r) is replaced with Vc(r — a(t)), where a(t) = [ dt'\ dt"E(t") is the ponderomotive radius as given in Eq. (14). The exact wave functions in the two representations are related to each other by a unitary transformation [ 49, 50 ] so that the ionization rates must be independent of these as

160

10

12

R (a.u.)

Figure 1 3-D H* ionization rates (units of lO'V ) as a function of R at A = 800nm : 0 - for circularly polarized light with / = 2 X10M w/cm 2 ; + -linearly polarized light with/ = 1014 w/cm 2 , parallel to the molecular axis R : o - linearly polarized light with i" = 1014 w/cm2 , perpendicular to R.

1

I

a)

2

4 6 8 10 12 Irrtemuelear distance (a.u.)

&H-13 5e+13

1c o 1 c o

4e+13 3e+13 2e+13 1e+13 0

~^1 2

4 6 8 10 12 Internuclear distance (a.u.)

b)

14

14 Figure 2 1-D Hj ionization rates as a function of R at intensity / :10 (a) X = 1064 nm (YAG); (b) X = 10.6 um ( C 0 2 ) .

162

found numerically. This serves as a validation of the numerical procedure. It is to be observed that in the near-infrared region, ionization rates are large in a window 610 a.u. Figure 2b shows a single ionization maximum at R = 7 a.u. for the IR wavelength X (C0 2 ) = 10.6 urn. All three figures show considerable enhanced ionization for 6
163

/-

0.4 - R*4a.u. B=o.o 0 2 -r>2.1x10"'aii. r_.2.9x10'Wu. 0.0 02

-0.6 0.8 10

l°-\

1.4

t\

I

10

:

I _

'-/) -V \

1.2

-r,.i.9xio / r..i.exio W -

/ -

1° +

-0.4

/-

"R=9.5a.u. p . o.o

'4

l

-

r -3.9x10 s a.u iot

L

t/l 1

_ -

•R-14a.u. B-0.0 r.»3.5x10"*a.u

-

L

-15-10-5 0 5 10 1515-10-5 0 5 10 15-15-10-5 0 5 10 15-15-10-5 0 5 10 1515-10-5 0 5 10 15 "R=6a.u. B = 0.1 -r,.9.5x10"'a.il r..8.1x10''ai

10.

^_

-15-10-5 0 5 10 1515-10-5 0 5 10 15-15-10-5 0 5 10 15-15-10-5 0 5 10 1515-10-5 0 5 10 15

> • Z(a.u.)

Figure 3 3-D H2+ energies £_ and £+ of HOMO \a_ and LUMO lcr+ (horizontal lines) and corresponding linewidths r_ 12 and r+12 (in a.u.) in a static field E = 0.053 a.u. (/ = 1014 W/cm 2 j: (a) zero magnetic field/? = 0; (b) magnetic field /? = 0.1 a.u. = 106Gauss. The internuclear distance is fixed at various values of R = 4, 6, 7.5, 9.5, and 14 a.u. The electronic potential along the molecular axis at the nonzero field, - l / | / ? / 2 + z| - l / | i ? / 2 - z | +Ez , is plotted as a function of the electron coordinate z. The 1 <7_ and 1 <7+ are localized in the descending and ascending wells, respectively.

164

magnetic bottle and confines (contracts) the electron density along the internuclear axis [54]. We give the linewidths T of these levels, which provide estimates of ionization rates x = \lT [ 1 a.u. = 24xl0~18s, See Eq. (5)]. One observes from Fig.3 that the LUMO (lcr + ) is situated above the internal barrier in the range 5 < R < 10 a.u. in good agreement with the TDSE ionization rate maxima illustrated in Figs. 1-2. The effect of a magnetic field is to lower the energies of the LUMO, thus resulting in a narrower range of critical distances Rc [54]. The above correlation between exact TDSE (Figs. 1-2) and static (Fig. 3) ionization rates suggest a wide applicability of the quasistatic model not only to atoms but also to molecules to interpret intense laser-molecule interactions. Figure 4 gives a detailed variation of the ionization rates in the static field of intensity / = 1014 W/cm2 for zero magnetic field. Figure 5 gives the corresponding energies of the Stark-shifted HOMO and LUMO, i.e., £_ and £+, as a function of R for the same intensity as in Fig. 4. Vin is the maximum height of the inner (central) barrier of the instantaneous electronic potential and Vout is the maximum of the outer barrier (left barrier when E > 0, as shown in Fig. 3). One sees in Fig. 5 that both barriers are of equal height around R- 6a.u. where an ionization maximum appears in the TDSE calculation (Figs. 1-2) and in the static line widths /+ (Fig. 4). The equality of the two barrier heights causes an additional enhancement at R - 6 a.u. because around this distance the energy difference between the LUMO and the higher one of these barriers is maximized. Finally, one observes the gradual trapping of the LUMO in the ascending well for R>l0a.u., resulting in the disappearance of over-barrier ionization. The above static over-barrier model involving Stark-shifted HOMO and LUMO is devoid of any time dependent dynamics. Thus, various time scales must be considered in order to understand the complete time dependent dynamics. Firstly, the energy separation A(i?) = f,CT„ (R)-£]r7g(R) can be taken to be a measure of the tunnelling time t, = \/A between the two Coulomb wells created by the two protons (Fig. 3) at zero fields. Another time scale is the laser photon cycle time tL = 1 / co. The maximum radiative coupling strength is determined by the maximum amplitude EM and the transition moment//g!/(^?) = < l parallel to the internuclear axis. For one or odd electron symmetric systems this type of transition moment is due to Charge Resonance (CR) or charge transfer phenomena. In the case of H2+, fi^ (R) is ~ R/2 corresponding to half an electron transfer from one end of the molecule to the other [28]. The relation //^ (i?) - R/2 can be easily proven by approximating \ag and \ou as |l<7g>B) = (±a + Z>)/v2 , where a and b are Is atomic orbitals located on the left and right protons, respectively. Then the maximum radiative coupling coR=EMRI2 is the Charge Resonance Rabi frequency with a time scale TR = 2n I coR . Finally, one has to consider the ionization rate r -1 /1, and the

165

ft (a.u.)

Figure 4 3-D H* linewidths T_ and r + of HOMO and LUMO in a static field E 0.053 a.u. ( / = 1014 w/cm 2 ) as a function of R. The case corresponds to Fig.3 with

166

Figure 5 HOMO (£_) and LUMO (£ + ) energies of 3-D H+2 as a function of R. Vin and Vm, are the maxima of the inner and outer barriers (Fig. 3) at p = {x2+y2) =la.u.

167

nuclear motion time scale ^ s l O - l S f s (the vibrational frequency 0)p of H2+ is 2000 cm"', i.e., tp — 15 fs). We now compare these various parameters at 7 = 1014 W/cm2 and A= 1064 nm [in atomic units, EM =0.053 a.u. and co = 0.043 a.u. ( ^ = 3.5 fs) ]. Thus for 6A> 0.02 a.u. Thus, in the enhanced ionization range, the strong radiativecoupling limit holds; coR >A> coL, coP [67]. Clearly, the radiative coupling between the HOMO and LUMO dominates around the ionization maxima 6< J R<10a.u. so that these two electronic states can be considered as the essential states, i.e., the CR states of the system. We develop next a two-level model to describe the time-dependent dynamics for this strongly driven system based on Bloch equations [68] which we used previously to describe HOHG in such a system [30, 67]. The wave function of a two-level system can be written as \¥{t))-Cg{t)\¥g)

+

Cu{t)\¥u).

(20)

where \y/g) and \y/u) are the wave functions of field-free time-independent two states g and u. The corresponding TDSE takes on the form:

''^H^te+rCOC,,, id

u= + {Y2)Cu+V{t)Cg,

(2D (22)

where A = (su-£g} is the field free transition energy and V(t) = jUguE(t). //^ is the transition dipole moment. Defining the three real functions: x{t) = CgC'u+C„C'g, (23) y{t) = i{C«C]-CgC:),

(24)

z(0 = |c„|2-|Cg|\

(25)

one obtains two important physical parameters : d(t) = Hmx{t) is the induced dipole moment responsible for HOHG and z(t) is the population inversion. One derives easily the corresponding two level Bloch equations by differentiating Eqs. (23)-(25) and using Eqs. (21) and (22) x = -Ay, z=

y = Ax + 2V(t)z,

-2V(t)y.

(26) (27)

168

These equations have the constant of motion, x2 + y2 + z2, i.e., the norm of the wave function, which is set equal to one. This means that negligible ionization is assumed. In the strong field limit which is valid in the enhanced ionization region 6 < R < 10 a.u. as discussed above, V (t)» A. Neglecting A in Eq. (26), one obtains readily the strong field solutions: y(t) = y{0)cos[F(t)]

+ z{0) sin [ > ( , ) ] ,

(28)

z[t) = z ( 0 ) c o s [ F ( / ) ] - y{0)sin[F(t)],

(29)

F(t) = 2J'oV(t')dt'.

(30)

where

The quantity F(t) is therefore the total field area [67]. For a monochromatic field E(t) = EM cos {cot) , then selecting the initial conditions j ( 0 ) = 0 and z ( 0 ) = + l (as shown in [67] this gives only odd harmonics), and using the trigonometric relation [69], cos[Ssm(a)t)]^J0(S)

+ 2YjJu(2kci)t),

(31)

one obtains the expression for the population difference between the l
=J0(2o)R/ o)) + 2^J2k(2o)R/

co)cos(2kcot).

(32)

The Js are Bessel functions and coR = EMR12 is the Rabi frequency. In fact, AxJ0 (2coRl 0)) is the energy difference of the dressed (Floquet) states, i.e., the field free energy separation A is renormalized (dressed) by the time dependent field [70]. Clearly, at zeros of J0, there is a maximum charge asymmetry, i.e., Cg (t) = ±CU [t) which corresponds to localization in atomic electronic states, (lcr, ±\au) = b or a, the atomic orbitals to which the HOMO and LUMO dissociate. Thus, the electron is localized at one well or another [67]. This is known as laser-induced or dynamical electron localization due to tunnelling suppression [71-72]. The second timedependent term in Eq. (32) corresponds to the electron following the field at multiple laser frequencies ka>, which in the long wavelength limit can be viewed as appearance of the upper state lcr+ or lower state \a_, i.e., the Stark-induced linear combinations of the field-free HOMO ( lag) and LUMO ( lau). At large EM>0, lcr and 1<7+ approach the localized atomic orbitals a and b, respectively; for EM <0,

169 vice versa. As an example, we take a case in which the distance R = 7a.u., intensity / = 1014 W/cm2, and X = 1064 nm. Then, the argument of J 0 is 2caRl a> = %.l, which is very close to the third zero (x = 8.6) of J0(x) , thus suggesting dynamical localization. Such electron localization has been confirmed in full 3-D TDSE calculations of the ion H2+ [31]. In general, due to the strong radiative coupling oR being larger than the essential state energy separation A charge asymmetry due to the field induced charge transfer results in laser induced population of the upper state la+ as illustrated in Fig.3. This population inversion can be more quantitatively described in a nonadiabatic description [73] and is discussed in Sees. 7 and 8 where it is shown that the result (32) is an asymptotic, strong field limit. We next derive analytic formulae for the critical distance Rc where enhanced ionization is observed from numerical simulations. As this is due to charge resonance or charge transfer effects induced by the laser field as discussed above, we call this CREI-Charge Resonance Enhanced Ionization. The simplest one-electron system H2+ may be viewed as a prototype of odd-electron molecules and general features of odd-electron molecular ionization can be elucidated by investigating the dynamics of H2+ for which we have performed the first full non-Born-Oppenheimer calculations in 3-D [47] and also in 1-D [45-46]. The electronic dynamics in H2 prior to ionization is dominated by the strong radiative coupling between the essential frontier orbitals, i.e., the HOMO ( l c g ) and LUMO ( \ a u ). The transition moment between these increases linearly with intermiclear distances R as R/2, a spectroscopic observation first made by Mulliken [28], which he called a charge resonance (CR) transition between a bonding and corresponding antibonding molecular orbital. The strong radiative coupling changes the electronic potential surfaces of the bound 2~L* and repulsive 2E* states into new distorted (dressed) potentials as a result of the creation of adiabatic HOMO (1<7_) and LUMO (lcr+) energies, £± (R) - -Ip ± EMR12, where /,, is the ionization potential of H atom and EM is the maximum field amplitude. This quasistatic Stark shifted orbital picture was earlier discussed with respect to intense static field dissociation of molecules, where the concept of barrier-suppression of ground state molecular potentials was inferred as an efficient mechanism of dissociation in intense fields [74-75]. This static picture has been shown to be very useful in describing thresholds for laser-induced photodissociation at high intensities and low frequencies [76-77] and also in controlling dissociative ionization in two colour excitation schemes [46, 78]. As shown in Fig. 5, while £_ is usually well below the barrier heights, £+ is higher than the heights of the inner and outer barriers, Vjn and Vm, , in the range 6
170 Enhanced Ionization (CREI). In this critical range ofR, ionization proceeds rapidly from the upper adiabatic state |lc + ) (when E(t) > 0, |lcr+) - b, i.e., the right atomic orbital of a dissociating H 2 + ). Assuming that at Rc, £+(Rc) is at the top of both barriers maxima Vin and Voul, one obtains simple analytic expressions for Rc first derived for highly charged one-electron ions A22,?+ [32] and further confirmed later [79]. The analytical expression for H2 is Rc-AIIp whereas it is Rc—5IIP for linear H32+ (where R is the total bond length between the two outer protons) [32, 34, 42]. The latter result was also confirmed numerically for highly charged ionsA2?+ where q>2 [33-34, 80]. As we show next for H2+ and H32+ , these results are independent of charge, depend weakly on field strength, are further consistent with the exact TDSE numerical simulations described in the previous section, and confirm experimental results in Coulomb explosion of H2 and other molecules at high intensities [38-41,44, 61-66]. We follow here the original 1 -D derivation [32] for one-electron models of A22,+ systems in the one-electron approximation undergoing Coulomb explosion at high intensities and summarized for 1-D models [42]. Exact Born-Oppenheimer (static nuclei) TDSE simulations of such highly charged ions show ionization maxima for 6
+ z\-q/\R/2-z\+Euz,

(34)

and £+(R) is given by £+(R) = -qIp-q/R

+ EMR/2.

(35)

Ip is the ionization potential of a neutral atom A. In Eq. (35), we use the empirical formula for the ionization potential of electrons from the same shell, i.e., 7ion - qlp for an ion of charge q-\. For the CI atom, this formula is exact within 2-3% for q>2 and 10% for q = 2 [32]. The potential (34) has maxima at z_ (left) and z+(right) for EM >0, i.e., Vin(R) = V(z+,R) and Vml (R) = V(z_,R): z_=-R/2-E^>2,

z^EqR?IZ2,

(36)

171

where

Eq=EMlq.

The approximate solution of the inequalities (33) gives the following condition for Rc, Rt < Rc < R2, where CREI is predicted to occur: R =1.464£-" 2 , R2 =

l-(l-6^/V)"

hIE,-

(37)

For Eq «. I2P16, one obtains Rc-R2='i/Ip. This result used by Seideman et al. to predict onset of electron localization [33] neglects the Stark-shift of the LUMO, which is an essential feature of CREI as discussed above. Defining the optimal condition for CREI by the equation RI=R2 [i.e., Vin(R) and Vm, (R) equal to £+(R)] gives Ec= 0.12912pq ,RC=4.07/Ip.

(38)

2

For fields EM >I q/4, the CREI effect will disappear since the electron in the separated ion of charge q - 1 is itself above the barrier. This was confirmed in Fig. 1 of [32] for the system Cl25+ -»Cl 3+ +C1 3+ , which shows a plateau instead of a CREI peak for 7>2xl0 1 5 W/cm2. Formula (38) shows the independence of Rc from field strengths and charge q. Such independence was originally discovered in Coulomb explosion kinetic energy spectra of ion fragments [41], thus confirming the existence of a critical distance Rc for enhanced ionization. More detailed experiments on Hj [61-62] and comparison with full non-Born-Oppenheimer simulations [45, 62] confirm CREI as the principle mechanism for the production of highly charged ions at critical distances Rc predicted by Eq. (38). The Coulomb explosion measurements must necessarily depend on the pulse length since the molecular ion must reach Rc from its equilibrium Re in order to undergo CREI. Thus using a classical model of evolution of highly charged ions from Re to Rc on a purely Coulombic potential V = q21R, one obtains for the time tc in which dissociating fragment of charge +^move from Re to Rcthe expression given by [32], tc=(/l/2)"2j£(£0-q2R)-"2dR, = (M/2)V2ReV2[x/(x2-\)-0.5\n({x

+ l)/(x-l))]/q,

(39) (40)

where £0 is the initial potential energy q21 Re, |X the molecular reduced mass, and x = [Rc /(Rc -Re)] . Thus, times 17 > tc > 75 fs are obtained from molecules N 2 to

172 I2 for fragments of charge q = +2 to reach Rc as predicted by Eq. (38). Recently, the effect of short pulses on Coulomb explosion of I2 has been observed [81-82] demonstrating indeed deviations from Eq. (38) as the heavy fragments F + do not reach Rc for ultrashort pulses (t he = 30fs). Extending the quasistatic ionization model of H2 to one-electron triatomics such as Hj + involves consideration now of at least three doorway states. Both for the linear [31, 34] and nonlinear H32+ [43] sharp ionization maxima are found to occur at Rc -lOa.u. (as shown in Fig. 6) and 7a.u., respectively, from TDSE BornOppenheimer (static nuclei) calculation. R is the total bond length of the linear molecule. For the nonlinear geometry, an ionization maximum occurs also at a critical bond angle 6C - 87° [43] which can be also explained as over-barrier ionization of the LUMO. Thus, the CREI mechanism is seen to be operative even in nonlinear molecules through the field induced displacement of the three doorway molecular orbitals (MO's), as we show next. In the linear case of H32+ with equidistant atoms, the three MO's are given in the LCAO approximation by [83]: lc7g=r1,2[c

+

2-V2(a + b)],

(41)

lau=2-V2[b-a], 2(Tg=2m[c-2-l'2(a

(42) + bj],

(43)

where a, b, and c are the Is atomic orbitals on the left, right, and central protons, respectively. The transition moment /x = U<7 g |z|lt7\ can readily be shown to be R/2in in contrast to H2+ where ju = R/2. In both cases, such divergent moments are due to the general phenomenon of CR effects [28]. In the presence of a static field of amplitude EM (maximum amplitude of a laser field), one obtains three new field-induced states as shown in Figs. 7 and 8. The energies are approximately equal to £l=-Ip-3/R-\EM\R/2 £2=-Ip-4/R,

;£i=-Ip-3/R

+ \EM\R/2,

(44) (45)

where £2 of the middle level localized in the central proton remains undisplaced with energy corresponding to the atomic ionization potential, - / , perturbed by the Coulomb potential 2{—2IR) due to the two protons adjacent to the central one. Efficient ionization from the state of energy £2, the LUMO for this system, will occur at critical distances Rc where £2(RC) equals the highest of the outermost

173

(a) v

'

H 3 2 *, 3D l = 10 14 W/cm : X = 1064 nm .

Figure 6 Ionization rates of linear H32+ at / = 1014 w/cm2 and A = 1064nm (units of 1 0 ' V ) : (a)3-D;(b)l-D.

174 0.2

H3a*, 3D l = 1014W/em2 R = 12a.u.

d -0.2

I

-0.4 -0.6

0.0000

-6.6

0.3962

1

-1

0.5861

-1.2 -1.4 h -16 0.2

-10 i

0 .

1

'T"

1

1

T

Hj 4 *, 3D i = id14w/ci«s fi = 19a.u.

-0.2

1

10 /

15 1

/ / f

0.1196

e3

0.Z7X1O-, r 3 .

>^K

-0.4 -0.6

1• x -0.6

0.3831

-1.4

2 j

\

e

: W II

-20

^^\0.1?32/

-15

\

o.3?xibia,r2_

/

1 U

-1

-1.2

e

^K

/

- 1 0 - 6

0.64x1 o - i , r t .

.

0

6

10

15

20

2 (a.u.)

Figure 7 Energies £t and linewidths r, (in a.u.) for linear Hj+ at/? = 12 and 19 a.u. in a static field of E = 0.053 a.u. (/ = 1014 w / c m 2 ) . The values on horizontal energy lines are the occupations of the corresponding three states. The energies Fj, V2, and V3 denote the barrier maxima of the electronic potential.

175

barrier, i.e., Vx shown in Fig. 7. The electronic potential V{z,R) can be expressed in terms of the shift x of the electron's position from the left proton (when EM > 0): r(x,R) = -^-—L-~^--EM(x x x + R/2 x + R

+ R/2),

(46)

where x - -[z-(-R/2)] = -z-R/2. Evaluating the height of the outermost barrier of V(x, R) and equating the obtained Vx to £2 gives the simple result [34,42]

X-5/V

(4?)

Since 7P (H) = 0.5a.u., this gives readily the value of the critical distance Rc = lOa.u. in agreement with numerical solutions of the TDSE for 3-D [34, 85] and 1-D H32+ [80, 86]. This result can also be obtained by considering the value of R at which the unperturbed middle level £2 of energy -IP-AIR is degenerate with the field free barriers at z = ±R IA, i.e., setting Ip+l/Rc=2/(Rc/4)

+ l/(Rc/2 + RJ4) = 28/3Rc

(48)

gives RC=16/3IP. Inserting Ip =0.5a.u. gives Rc =lla.u., which is in satisfactory agreement with the major ionization peak of H32+ shown in Fig. 6 [34]. The fieldfree value of Rc, Eq. (48), is again close to Rc obtained from a rigorous calculation with the field Stark displacements included in Eq. (47). As Eq. (38) in the case of H j , R c in Eq. (47) is independent of field strength and agrees with the previous estimate Eq. (48) from a field free model. The coincidence of this last result confirms the field strength independence of Rc, which comes from the near equality of the classical energy shifts of the barrier maxima Vt and K3 in Figs. 7-8 (or Vin and Vmt in Fig. 5) with the charge resonance energy shifts of the quantum levels. Thus both classical and quantum energy shifts cancel each other rendering the values of the CREI distance Rc nearly independent of field strength EM as in H 2 + . Similarly, the independence of Rc with charge q comes from the narrowness of the wells with increasing q, thus leading to potential maxima, for instance, Vin and Vm, of H2+ shown in Fig. 5 [or Vx ,V2, and V3 in H32+ (Figs. 7-8)] equal to the Stark shifts of the electronic levels. We end this section emphasizing that CREI occurs also in nonlinear, i.e., triangular H 3 2+ , with a critical distance Rc -7 a.u. and a critical angle 0C ~ 87° [43]. Thus in nonlinear molecules, CREI is expected to occur also at critical bond distances Rc and critical bond angles 6C. This is a general phenomenon of enhanced ionization via over-barrier ionization induced by Stark-displacements of LUMO's.

176

.1.6 L: 6

1 8

1 10

1 12

1 14

1 16

1 IB

1 20

1 22

24

R (a.u.)

Figure 8 Energy levels £t and barrier maxima V^, V2, and V3 plotted against R. The notations are the same as in Fig. 7.

177

4 Two-Electron Systems The first detailed experimental study of ATI for molecules was performed in H2 by Verschuur et al. [84] where it was discovered that the photoelectrons obtained in intense fields showed anomalous vibrational structure of H2+ later interpreted as due to bond softening [26] via LIMP'S dominated by laser induced avoided crossings between dressed molecular potentials [20, 21-23, 27]. 3-D ionization rates for H2 and H3+ in intense fields were first attempted using time-dependent Hartree-Fock methods and finite-element techniques [85]. However, since such methods fail at large R due to configuration interaction (CI), exact 1 -D calculations were pursued in the long wavelength regime (A = 1064nm) where it is expected that 1-D models should be adequate and that CR effects should dominate. Thus, maxima in the ionization rate with respect to R have been again found for 1-D two-electron models such as H2, H3+ [19, 34, 85], H42+ and H53+ [34]. These maxima have recently been interpreted as due to strong radiative interactions between essential doorway states resulting in charge transfer at the peak of the laser pulse. Ionic states thus created as ground states in intense fields play the role of main doorway states to ionization [3536, 42]. The electron transfer and resulting enhanced ionization were confirmed by recent wave packet calculations of a 3-D model of H2 [86]. The results of static field calculations of H2 also agree with the interpretation of two-electron enhanced ionization [34, 87-88]. The existence of similar critical Rc 's as in one or odd-electron systems indicates that enhanced ionization is a universal phenomenon. The mechanism of enhanced ionization of even-electron molecules is however expected to differ nevertheless from that of one or odd-electron systems due to electron correlation. As an example, we have found that the HOHG atomic plateau which continues up to the maximum IP+3A7UP given by Eq. (15) can extend to IP+\2UP in two-electron extended molecular systems [19]. The two-electron diatomic prototype is H2 for which we have discussed the mechanism of enhanced ionization in [35, 86]. In this case, ionization is enhanced when the ionic state H+H~ is most efficiently created from the covalent ground state HH in the level dynamics prior to ionization. The importance of ionic states in the spectroscopy of symmetric molecules was emphasized by Mulliken [28] and recently such a state in 0 2 (0"0 + ) has been confirmed experimentally [89]. In H2, ab initio calculations have proven the existence of H+H" states with multiple crossings with valence (Rydberg) states at R = 6a.u.[90]. An analytic expression for the ionic and covalent crossing condition was obtained for H2 in terms of three doorway states [42, 86] which agrees well with the numerical results [34-35].

178

2\ 4

,

,

,

,

,

,

1

6

8

10

12

14

16

18

Internuclear distance R (a.u.)

Figure 9 Ionization rates (units of l O ' V ) for linear symmetric 1-D H* at A = 800 nm : (a) / = 3.2xl0' 4 w/cm 2 ; (b) / = 1014 w/cm 2 ; (c) / = 1013 w/cm 2 . In (a), the result for the static field of the same peak intensity 7 = 3.2xl014 W/cm2 is presented as well as the ac field case: (i) ac field; (ii) dc field.

179 The two-electron triatomic system, H3+, is the simplest model to understand the electron dynamics of extended systems. We already have reported calculations of enhanced ionization of both symmetric and non-symmetric 1-D H3+ [34]. We examine here in detail the mechanism of enhanced ionization (see Fig. 9) of linear symmetric H3+ by employing the adiabatic-fleld state method previously used for H2+ (Sec. 3) and for H2 [35]. Simple Stark-shifted MO's will be again shown to give a clear interpretation of the numerical TDSE results and of the origin of the R dependence of the ionization rate. Figure 9 illustrates the ionization rate for fixed nuclei (Born-Oppenheimer) for H3+ at three different intensities. A maximum occurs around Rc =9 a.u., close to Rc =10 a.u. for H32+ (Fig. 6). A static field calculation for the same static electric field amplitude as the maximum of the laser field essentially reproduces the same Rc. See the static field case (ii) in Fig. 9(a). We shall develop next a quasistatic model based on radiatively coupled doorway states to predict the occurrence of such critical distance for enhanced ionization as resulting of the crossing of the ground covalent state HH+H with the ionic state H+H+FT or H"H + H + , similar to H2 [34-35,42]. At large R the lowest three states of symmetric H3+ become degenerate corresponding to the fact that H3+ dissociates into H + HH, HH+H and HHH+. The electronic transition moments ju between states i and j are defined as Mij =(^(z 1 ,z 2 )|(z,+z 2 )|^.(z 1 ,z 2 )).

(49)

The first transition moment //,2 corresponding to the transition XlZg —¥ S'X„+ has a linear R dependence at large R (up to R ~ 4 a.u.) as well as the second moment //23 for BlZ„ —> Ex£g (up to R ~ 10 a.u.), confirming the CR character of these states [28], The linear R dependence of these moments can be estimated from the LCAOMO method [34] which give {in=R/2 and jU2i=R/23'2 for the states described next. In order to incorporate the ionic character one now needs to incorporate six doorway states for H3+ whereas for H2 only three states were essential [35]. Using the lowest three MO's of H32+ in Eqs. (41)-(43), we obtain the lowest six states of H3+: fl(l,2) = UT,(l)l
(50)

%(l,2)

(51)

= [l(7g(l)lcTli(2) + lC7 g (2)l ( T u (l)]/V2,

ft(l,2) = [l/2 ^ ( l , 2 ) = lff„(l)l < r.(2),

,

(52) (53)

180
= [lau(\)2<7g(2)

+

lau(2)2c7g(l)]/^2,

%(l,2) = 2<7g(l)2c7g(2).

(54) (55)

If \<7g and 12, and (p4 become the three essential states of H2. For H3 , the transition moments as discussed above are //,2 = //24 = -ju45 = -fiSi = R12 ; //23 = - / ^ = R12'1, where R is the total bond length of the molecule; other matrix elements are zero. The state /' and j are thus coupled with each other by the radiative matrix element EM(i\z\ j) at maximum field strength £ M , i.e., the peaks of the laser field. Assuming that these radiative matrix elements far exceed the zero-field energy separation between the six electronic states (^, -(p6) , we obtain the following adiabatic states {p.} and energies {£}} in the field EM by diagonalizing the 6x6 Hamiltonian matrix: for E„>0, £,=-\EM\R, £2=-\EM\R/2, £3=0,

^,=a(l)a(2), yr2=[a(l)c(2)

(56) + a(2)c(l)]/>/2,

j/3=c(l)c(2),

£3' = 0, y/i=[a(\)b(2) £A=\EM\RI2, e5=\EM\R,

(57) (58)

+ a{2)b{\)\l42

i/ 4 =[6(l)c(2) + A(2)c(l)]/>/2 > ¥5=b{\)b{2),

(59) (60) (61)

where a, b, and c are the Is atomic orbitals on the left, right, and central protons, respectively. For EM < 0 , a and b in Eqs. (56)-(61) should be interchanged with each other. The two electron states y/2 , \f/\ , and y/4 in a long distance case of R = 14 a.u. are illustrated in Fig. 10 as states (a), (b) and (c), respectively. The states \ffx , i//2 , and y/5 are clearly the ionic states H^H^Hj, H^H^H* and H^H^H* . Equation (56) shows that the lowest y/{ state is ionic and the energy £x agrees with the electrostatic energy of a charge transferred through the distance R by the field E

M-

As is illustrated by the two-electron distribution ^(z^Zj,?) in Fig.l 1 obtained from exact TDSE numerical solutions, the ionization dynamics via laser-induced charge transfer or charge resonance [36] is determined by three electronic configurations H^H^H^, H^H^H*, and H^H^Hj for EM > 0. Which doorway states (configurations) to ionization mainly appear depend on the characteristic parameters of laser-molecule interaction. The electronic response to the ac field can be classified

181

-15 -10 -5 0 5 z,(a.u.)

10 15

-15 -10 -5 0 5 z, (a.u.)

10 15

-15 -10 -5 0 5 z, (a.u.)

10 15

Figure 10 Two-electron probability densities of the three adiabatic field states of linear symmetric H* at E(t) =0.1 a.u.; (a) y/2 of Eq. (57); (b) y/[ of Eq. (59); (c) ^ of Eq. (60). z, and z2 are the coordinates of electrons 1 and 2. The internuclear distance is fixed at R = 14 a.u. The corresponding electronic spatial configurations are schematically illustrated in the right panels, where the closed circles denote the positions of two electrons and the open circles denote the protons. The dipole interaction with the field is represented by a slanted bar (at this moment, the field is positive).

182

30 20 Si

10

CVJ

N

0 •

-10 • -10

0

10

20

30

10 ~

5

si N"

o -5 -10 -10

-5

0

5

10

15

z, (a.u.)

Figure 11 Two-electron probability densities for a pulse ofA = 800nm and the peak intensity / = 3.5xlO14 w/cm 2 (EM=0Aa.u.) . Two example are shown: (a) a diabatic case where R is fixed at a large value of R = 14 a.u.; (b) an adiabatic case where R = 8 a.u. In (a), a snapshot at t =3.875 cycles is presented. At this moment, E{t)=0.54a.u. In (b), a snapshot is taken at t= 3.75 cycles at which E{t)= -0.72 a.u. The ionic site z{=z2=R/2 of the lowest adiabatic state y/\ is denoted by the symbol I, while the covalent site z, = -z 2 = R12 of y/[ is denoted by C. The site M (z, = R12 and z2 = 0) corresponds to the second lowest adiabatic state y/2. In order to obtain the wave functions of adiabatic states for E(t) < 0 , a and b in Eqs. (56)-(61) should be interchanged with each other. In (b), three types of ionization occur: from the ionic site I denoted by the solid line (4); from the covalent site C denoted by the line (1); from the site M of the second lowest adiabatic state yr2, denoted by the line (2).

183

into three different categories [67]: (i) in the adiabatic regime of R < 9 a.u., H~H*Hj (i.e., ionic \ffx) appears for EM > 0; (ii) in the diabatic regime of R>9a.u., HoH*H4 (covalent y/'z); (iii) in the mixed regime of R ~ 9a.u., H^HCH4 (y/4). The adiabatic energy of each electronic configuration can be estimated from electrostatic principles assuming negligible overlap between atomic orbitals on neighbouring sites: £l{R) = -Ip-Aa-6/R-\EM\R,

(62)

fS{R) = -2Ip-4/R,

(63)

£4{R) = -2Ip-3/R + \Eu\R/2,

(64)

where Aa is the ionisation potential of H" (0.06 a.u.) and Ip is that of H (0.67 a.u.) in 1-D. The adiabatic regime of R 0 o r H+H^H~ for EM < 0 [the ionic site I in Fig. 11 (b)] with the covalent state HaH*H4 (the covalent site C). Thus equating £\{R) and E[{R), we obtain an expression similar to the case of H 2 . The maximum field strength at which the two energies can be equalized is given by Emax = (lp - Aa) /8 = 0.045 a.u. At this strength, the energy gap between the two adiabatic states is maximized, indicating adiabatic electron transfer from the covalent state y/'^ to the ionic state ^, as a doorway state to ionization. At Emax , the two states cross each other in energy at the internuclear distance Rc K = (/, - Aa )/2£max = 4/(/„ - A J .

(65)

This gives Rc=6-1 a.u. by using Ip-Aa = 0.6 a.u., which is in agreement with peaks around R ~ 8 a.u. in Fig. 9 or the first high intensity peak in Fig. 9(c). It should be pointed out that in Fig. 11(b) the probability of ionization from the second lowest adiabatic state y/2 (site M) is fairly large. The state is regarded as a transition state from the covalent y/'} to ionic state y/x . The ionization process via the intermediate state y/2 is not taken into account in the derivation of Eq. (65).

184

A snapshot of the wave packet in the diabatic regime (ii) is shown in Fig. 11(a) where if is large (14 a.u.). It is clearly demonstrated that ionization proceeds from the covalent configuration denoted by the symbol C in Fig. 11(a). There exist two routes to ionization. As shown in Fig. 10(b), one electron is localized in the lowest well and the other one is in the uppermost well. In the ionization process denoted by the solid line in Fig. 11(a), the electron in the lowest well is ionized; in the ionization process denoted by the broken line, the electron ejected from the uppermost well penetrates the middle and lowest wells. The mixed regime around Rc=9 a.u.corresponds to laser-induced transitions mainly between the three field-free electronic states XXZ^, BxZt and E]Ug which are degenerate at large R as they dissociate symmetrically to the three degenerate asymptoties H+HH , HH+H and HHH+. As in the case of H2+ , these three asymptotic configurations appear in the adiabatic states illustrated in Fig. 10. We therefore adopt again the quasistatic above-barrier method described in the previous section for the one-electron systems H2+ and H32+. We write the energy of the adiabatic state y/4 in the mixed regime (iii) as £4(R) = £up{R) + £mid{R), £up{R) = -Ip-\/R

(66)

+ \EM\R/2,

£mid(R) = -Ip-2/R,

(67)

where one electron (z, = R12) is on the outer right proton H*, i.e., in the uppermost well (yielding £up) and the other one (z2 =0) is on the central atom Hc (yielding £mid). The total energy (66) is equal to Eq. (64). Thus the ionizing upper electron (zi > 0), i.e., leaving from the LUMO, only interacts with the bare proton Ho+ and the electrostatic field EM since the central proton Hc is shielded by the other electron (z2 =0). The potential of the upper electron with z = z,> 0 at distance z from FL is then V(z) = -\ V ; (R/2) + z

\ + EMz. (R/2)-z

(68)

We note the similarity between the H2+ CREI model potential and the present H3+ model, i.e., between Eqs. (34) and (68). Defining z = (R14) + y and searching for the maximum barrier Vm from the condition (dV/dz) = 0 gives readily y-EMRi /128-R/9. Setting next Vm(y) = £up(R) gives the equation for Rc Ip^EMRj2

+ U/4Rc.

(69)

185

The maximum field strength that satisfies Eq. (69) is Em!iX -ill5. The internuclear distance for enhanced ionization is found to be i?c — 9 a.u. at Emax, which is in agreement with the dominant ionization maximum shown in Fig. 9. We conclude that the quasistatic model of over-barrier ionization for molecules exposed to intense laser fields as originally proposed by Codling and Frasinski et al. [65-66] needs to be modified to take into account the important Stark or equivalently Charge Resonance [20, 28] energy shifts of the LUMO. By examining the heights of barriers for tunnelling and the field-induced energy shifts of adiabatic states, we explained qualitatively the main enhanced ionization critical distance Rc — 4/Ip - 8 a.u. for H2+, Rc = 51 Ip =10 a.u. for H23+, and Rc=9a.u. for H3+ calculated from appropriate TDSE's. In this last case, enhanced ionization at Rc - 9 a.u. occurs from the laser-induced excitation of the configuration H^H^Hj with the electron situated on the uppermost quasistatic well at H4 contributing mainly to the ionization by over-barrier passage. We emphasize once again the similarity of this ionization regime to H2+ where the electron trapped in the upper Stark-shifted LUMO state <J+ (see Fig. 3) contributes mainly to the ionization. This is the fundamental difference between atoms where only one well exists and no LUMO. Thus in the molecular case, at Rc the electron must cross the whole molecule in order to ionize via CREI. Electron repulsion at lower laser intensities as in Fig. 9 (a) and (b) suppresses the creation of the ionic state, H^H^Hj, i.e., electron correlation limits charge transfer. The ionic state begins to contribute to the ionization dynamics only at higher intensities I> 1015W/cm2 [as in Fig. 9 (c)] due to the high energy of these states. The critical distance Rc for this ionic mechanism, Eq. (65), is obtained under the condition that this ionic state and the ground covalent state cross each other at the maximum field strength for crossing, £max. There are at least two mechanisms of enhanced ionization, that is, CREI and ionic mechanism. In the ionic mechanism, the ionization step proceeds mainly through the lowest adiabatic state (localized ionic state such as H + H") created by field-induced transfer of an electron; in the CREI mechanism, it proceeds through the upper LUMO or an upper adiabatic states. Upper LUMO's are populated by HOMO-LUMO crossings at near-zero fields, as we show next; ionic states are populated by crossings between covalent and ionic states at nonzero fields, as discussed above. We have introduced in Sec. 3 a strong-field excitation two-level model, Eqs. (20)-(32), which we have used previously to explain molecular HOHG [19, 67] and CREI in H2+ [67] around the critical distance Rc. This strong field regime where coR> A is called the diabatic regime [see Eq. (32)]. The more precise definitions of adiabatic and diabatic regimes will be given in Sees. 5 and 7. In the next section, we extend this theory to describe the full nonadiabatic dynamics of

186

electron-nuclear-photon dynamics in intense laser fields [91] based on analytical solutions of the two-level model in strong fields [73]. 5 Adiabatic State Formalism 5.1 Time-dependent adiabatic states As has been shown in previous sections, bond stretching in a low-frequency intense laser field enhances the rate of tunnel ionization (known as enhanced ionization). The experimentally observed kinetic energies of ionized fragments of a molecule are large (> a few eV) and are consistent with Coulomb explosions of multiply charged cations at a specific internuclear distance Rc which is much longer than the equilibrium internuclear distance Rg. Other numerical simulations also indicate that the tunnel ionization rates around Rc exceed those near Re and those of dissociative fragments. In what follows, we present numerical simulations and analyses of abovementioned nonperturbative phenomena in intense fields in terms of strictly defined field-following adiabatic states. The analyses are devoted to identifying doorway states to ionization [86, 91, 92]. To analyze laser-induced electronic and nuclear dynamics in intense fields, we first define time-dependent " field-following " adiabatic states { \n) } as eigenfunctions of the "instantaneous" electronic Hamiltonian HQ(t). The H0(t) is the sum of the Born-Oppenheimer (B-O) electronic Hamiltonian Hel at zero field strengths and the interaction with light, VE{t) [92, 93]: H0(t) = Hel +VE(t). (70) To obtain {!«)}, we diagonalize H0(t) by using bound eigenstates of Hel as a basis set [94]. The time t and the internuclear distance R are treated as adiabatic parameters. 13

2

In an intense field ( 7 > 1 0 W/cm ), an adiabatic state or adiabatic states are populated by laser-induced intramolecular electronic motion. Tunnel ionization to Volkov states (quantum states of a free electron in a laser field) proceeds through such an adiabatic state (adiabatic states). Intramolecular electronic motion also affects nuclear motion; e.g., after one-electron ionization from H 2 , the bond distance of the resultant H2+ stretches on the lowest adiabatic potential surface [26, 62, 95]. Field-induced nonadiabatic transitions through avoided crossing points in time and internuclear coordinate space govern the electronic and nuclear dynamics in intense fields. As explained below, the dynamics of bound electrons and the subsequent ionization process, as well as laser-induced nuclear motion, can be understood by analyzing the time-dependent populations of adiabatic states.

187

We take a one-electron molecule H2+ as an example. In the intense and lowfrequency regime, H2+ is aligned by a linearly polarized laser field, parallel to the polarization direction. Then, VE(t) in the dipole approximation is given by jeE(t) = zE(t) as in Eq. (7) [74]. Here, z is the electronic coordinate parallel to the molecular axis, and E(t) is the linearly polarized laser electric field. The electronic dynamics of H2+ prior to tunnel ionization is determined by the large radiative dipole coupling of zE(t) between the lowest two B-0 electronic states, ls<7g and ls<7u, respectively (abbreviated as |g) and |u)). The dipole transition moment between them, parallel to the molecular axis, increases as RI2, where R is the internuclear distance. This large transition moment is characteristic of a charge resonance transition between a bonding and a corresponding antibonding molecular orbital, which was originally pointed out by Mulliken [28]. We diagonalize the instantaneous electronic Hamiltonian including the radiative interaction, H0(t)-Hel +VE{t), by using the radiatively coupled |g) and |u) states as basis functions. The eigenvalues of Hel for ls(7g and \s(Ju, are denoted by E (R) and £U(R) , respectively. The resulting eigenfiinctions of H0(t) are given by [91, 94] |l) = cos0|g)-sin<9|w)

(71a)

|2) = cos0|w) + sin0|g)

(71b)

~2(g|z|u)ff(/r 6 = — arctan 2 . Aeag(R) _

(72)

where

with the B-0 energy separation A£ug(R) A£ug(R) = £u(R)-£g(R).

(73)

The phases of the two wave functions |g) and |u) are chosen so that (g|z|u) is positive. The corresponding eigenvalues of H0(t) are given by £2,(R,t)=Ueg(R) 2 ~±[£,(R)

+

£a(R)±jA£2ag+4\(g\z\u)E(t)\

+ £u(R)±R\E(t)\]

forlargetf.

(74)

The strong radiative coupling of the charge resonance transition yields the "fieldfollowing" time-dependent adiabatic surfaces, £x{R,t) and £2(R,t) , which are adiabatically connected with lscrg and ls(7 u , respectively. The adiabatic energies £x 2(R,0 at E(t)=0.053 a.u. are shown in Fig. 12 as well as £„ and £u. If £ is

188

assumed to be equal to the field envelope / ( / ) , the field strength E in atomic units corresponds to the intensity/as/= 3.5x10 E W/cm . The instantaneous electrostatic potential for the electron, V0(t), has two wells around the nuclei, i.e., z=±R/2. The dipole interaction energy for an electron is E(t)R/2 at the right nucleus and —E(t)R/2 at the left nucleus. As E(t) increases from zero, the potential well formed around the right nucleus ascends and the well formed around the left nucleus descends. Therefore, the ascending and descending wells yield the adiabatic energies £2 and £], respectively; |2) and |l) are localized near the ascending and descending and wells, respectively (irrespective of the sign of the electric field). In addition to the barrier between the two wells (inner barrier), when \E{t)\^Q, there exists a barrier of finite width outside the descending well (outer barrier). The barrier heights of V0(t) evaluated along the molecular axis are also plotted in Fig. 12. It is clearly demonstrated that while £x is usually below both of the barrier heights, £2 can be higher than the barrier heights in a range around ^.=6 a.u. Except the atomic limit where R is much larger than the equilibrium internuclear distance, the upper adiabatic state |2) is easier to ionize than is |l). In Fig. 13, the rate of ionization from |2) in a DC field, ri, is plotted as a function of R. The field strength is fixed at a constant E(t)= 0.053 a.u. In the numerical calculation, we use cylindrical coordinates z and p, where p is the coordinate perpendicular to z. Then, the z-component of the electronic angular momentum is conserved: only z and p are necessary to describe the electronic state. We solve the time-dependent Schrodinger equation for H2+, with nuclei frozen at various values of R, by using a grid point method for Coulomb systems (the dual transformation method) [52,91]. To eliminate the outgoing ionizing flux and to calculate the ionization probability, we set absorbing boundaries for p and z: the ionization rate is obtained by fitting the norm within the space encircled by the boundary planes to an exponential decay form. Roughly speaking, owing to barrier suppressions favorable for tunnel ionization from |2), 7"2 is large in a wide range around R^,. However, a couple of peaks are observed. The maximum ionization rate from |2) is found at R ~9 a.u. For this field strength, at R ~9 a.u., £2{t) is however only a little above the inner barrier, as shown in Fig. 12. The rapid ionization for the instantaneous potential is attributed to the following facts: at R~9 a.u., |2) is resonant with an adiabatic state in the descending well that is higher than |l); the outer barrier is thin and is much lower than £2 . Another peak due to resonance is also found at R~6 a.u. but it is lower than that at R ~9 a.u. The rate of ionization from |2) seems to be more sensitive to the outer barrier than the inner barrier. In Fig. 13, we also show ionization rates r in an alternating intense field under the condition that the initial state is the ground state Is (J . The laser electric field is

189 -0.2-1— l -0.3-

7 -0-4" 3g5 -0.5
Inner barrier height 1SO c

. - - * *

Outer barrier .height

— i

1-

4 6 8 Internuclear Distance R (a.u.)

10

Figure 12. Potential energies of the lowest two field-following adiabatic states, £i and £2, of H2+ (denoted by solid lines) and the heights of the inner and outer barriers for tunnel ionization at £ ( 0 = 0.053 a.u. i.e., 7 = 10 W / c m (dotted lines). The broken lines denote the Born-Oppenheimer potential surfaces of lscr and lsou . The internuclear repulsion l/R is included in the energies.

190

i i i i | i i i i | i i i i | i i i "'I | i i i i | i i

5

6

7

8

9

U

10

Internuclear Distance R (a.u.)

Figure 13. Ionization rates of H2+ as functions of the internuclear distance R (1 a.u. in ionization rate = 4.1x10 /s). The open squares denote ionization rates r2 of the upper adiabatic state \l) at a constant field strength E{i) = 0.0533a.u. (which corresponds to 7 = 10' 4 W/ cm ); the open circles denote ionization rates 7" in an alternating intense field of A=1064 nm and 1= 1014 W / cm2 under the condition that the initial state is the ground state IscT, ( the field envelope has a five-cycle linear ramp). The scales for r2 and r are marked on the right and left ordinates, respectively.

191 assumed to take the form E(t) = f(t)cos(cot), where a =0.0428 a.u. (Z = 1064 nm) and the envelope / ( / ) is linearly ramped with t so that after five optical cycles f(t) attains its peak value / 0 =0.0533 a.u. ( / = 1014 W/cm ). As clearly demonstrated in Fig. 13, the ionization rate r is correlated with 7^2 (e.g. both have peaks at R=6 and 9 a.u.), which indicates that ionization proceeds via the |2) created from the initial state lscr The mechanism of this correlation is revealed in the following sections. 5.2 Coupled equations for the two-adiabatic-state model The initial state of Hj created from one-electron tunnel ionization of H2 is the lower adiabatic state |1) (equal to ls(Xg at zero field strengths). Using the two-adiabaticstate model, we next show how the upper adiabatic state 12) is populated from |l). The electronic and nuclear dynamics of H2+ within the model is expressed as \¥) = &{R)\\) + X2{Rp),

(75)

where ^ ( - K ) and Xi^^ a r e m e nuclear wave functions associated with |1) and |2). We insert Eq. (75) into the time-dependent Schrodinger equation for the total Hamiltonian 1 d2 • + H0(t), (76) H(t) = 2{mJ2)dRwhere mp is the mass of a proton. The following coupled equations are obtained [91]: 1 d2 de_X Zl{R)-A(R,t)Z2(R), (77a) -£l(R,t)+w p dR m] dR)

j;xW-

VtxM)-i

mp dR

mp

de_ dR

Z2(R) + A(R,t)zl{R),

(77b)

where the coupling that induces transitions between the t w o states (laser-induced nonadiabatic transitions) is . , _ , dd 2 / dd d d2e (78) A(R,t)=— —-— mpdR2 dt mvdR dR Here the small terms {g\d2ldR2\g)/mp and (u\d2/dR':|w)/mp appearing in the diagonal and off-diagonal elements in Eq. (77) are ignored. T h e term ddjdt is the nonadiabatic coupling due to the change in the electric field, and the other coupling terms in A(R,t) are due to the joint effect o f the electric field and the nuclear motion.

192

It is expected from Eqs. (77) and (78) that nonadiabatic transitions occur if the main coupling term dOjdt is large in comparison with the gap between the two adiabatic potential surfaces, £2 - £\ • The term dd/dt is expressed as

d6ldtM2\U)dE«\COs28f. 1

A£ag(R)

(79)

dt

Since E(t) is a sinusoidal function of time, | dE{t)ldt | and (cos2#) becomes largest when the field E(t) changes its sign, i.e., t„ = nn/a> (n is an integer) for E(t) = smcot. The gap £2 - £\ becomes A£ ug , i.e., smallest when the two adiabatic potential surfaces come closest to each other, i.e., again, when the field E(t) changes its sign. Thus, a nonadiabatic transition between the two adiabatic states is expected to occur within a very short period rtr before and after the field E{t) changes its sign [96]. In this case, a nonadiabatic transition between |l) and |2) corresponds to suppression of electron transfer between the two wells, i.e., two nuclei. We will later show the condition for temporally localized noadiabatic transition. It should be noted that (g\z\u)/A£ug(R) increases as R increases [(g|z|w) is an increasing function of R and A£ug(R) is a decreasing function of/?]. Larger internuclear distances are thus favorable for nonadiabatic transition. If the nuclear motion is slow in comparison with the time scale of rlr, the nonadiabatic transition probability per level crossing at R can be defined and it is given by exp(-2 nS), where S = A£ug(R)2/s{g\z\u)f(t)o).

(80)

The meaning of 8 will be discussed later. The probability exp(-2;rJ) can be increased as the intensity, frequency, and R are increased. Thachuk et al. [97] have developed a semiclassical formalism for treating timedependent Hamiltonians ( nuclei are propagated classically on the surfaces) and applied it to the dissociation of diatomic ions. They have derived the nonadiabatic couplings dd/dt and vdOl dR, where v is the relative nuclear velocity. These two terms correspond to the first and second terms in Eq. (78). For homonuclear ions, dd/dt is much larger than vddldR , except when A£ug » ( g | z | w ) x (pulse envelope). 6 Adiabatic State Population Analysis 6.1 Numerical simulation of ionization of a dissociating H2* The above qualitative analysis of the coupled equations is consistent with the actual numerical simulations of nuclear dynamics and ionization in the intense- and low-

193

frequency regime. In the case of H2+, it is possible beyond the two-state model. The exact electronic wave packet 0(p,z,R;t) can be calculated [47, 91]. The components ;fr(i?) and z2($ a r e m u s obtained by projecting the exact electronic and nuclear wave packet onto |l) and |2). The mechanism of enhanced ionization has been directly proved using the populations of adiabatic states. We show that tunnel ionization proceeds in the region R>2Re through the |2) state populated from |l), where R,, is the equilibrium internuclear distance. Unlike the case of Fig. 13, here, the role of bond stretching is investigated by treating the internuclear distance quantummechanically. The molecular axis is again assumed to be fixed, parallel to the polarization direction, by the field E(t). For the following numerical calculation, the grid ends in z are ± 30 a.u. and the grid ends in pare 0 and 30 a.u. We assume that H2+ interacts with the following laser pulse: E(t) = f(t)sm(wt),

(81)

where a =0.06 a.u. (A = 760 run) and the envelope f{t) is linearly ramped with t so that after two optical cycles / ( / ) attains its peak value f0 = 0.12 a.u. (/ = 5.0xl0 14 W/cm2). The total pulse duration is 20 cycles (40 fs). Ff2 is prepared by one-electron ionization from H 2 . The resultant state of H2+ is |l). Since the creation of an ionizing electronic wave packet through tunneling begins and ends within a very short period, i.e., a half-optical cycle (characteristic of tunnel ionization), the vibrational wave packet of H2 created is nearly equal to the ground vibrational state of H2 (vertical transition). We thus assume that the initial total wave function of H2+ at t = 0 has the following form [62, 95]:

0{p,z,R;t = V) = X«^(.R)V),

(82)

where %vjb=a{R) is the lowest vibrational state of H2 in the ground electronic state X ILg . Here, we simply replace |l) at t-0 with the Lserg B-O electronic wavefunction of H2+ , yf\sa (p,z;R). This replacement does not affect the following conclusions. The dynamics of the nuclear motion is summarized as follows. The H2+ wave packet starts from the equilibrium internuclear distance ^,=1.5 a.u. of H 2 . Since Rg is longer for H2+ than for H 2 , the nuclear component of the H2+ wave packet moves toward the outer turning point R ~ 4 a.u. on £ . Without an external field, the main component of the packet reflects at the outer turning point, coming back toward the inner turning point. On the other hand, under the condition of an intense pulse, dissociation takes place. When the instantaneous field strength \E(t)\ is large, the potentials £]{R,t) and £2(R,t) given by Eq. (74) are distorted as shown in Fig. 12.

194 The nuclear wave packet is moved toward larger internuclear distances by the fieldinduced barrier suppression in £x(R,t) (bond softening due to a laser field). We map the calculated &(p,z,R;t) onto |1) and |2) to obtain the projected vibrational components %l(R,t) = (l\0) and %2(R,t) = (2\0) [92, 93]. The overall populations /;(*)= j|;ft(/?,*)|2<#* a n d ^ ( 0 = ^Zii^OfdR are shown in Fig. 14 together with the total norm in the grid space enclosed with the absorbing boundaries, PsM(t). In the early stage, in which the internucear distance is small (R<3 a.u.), the response to the field is adiabatic: the main component of the wave packet is still electronically |l). As the packet reaches internuclear distances around R-3.5 a.u., the adiabatic parameter 8 becomes as small as 0.2; exp(—2nd) = 0.23 . Nonadiabatic transitions between |1) and |2) occur when the field E(t) changes its sign, i.e., when the two adiabatic potential surfaces come closest to each other: around ? =210 a.u. (=47u/a>), a part of the population in |l) is transferred to |2), as shown in Fig.14. The increase in P^it) at a crossing point coincides with the decrease in Px (t). During the next half cycle after a nonadiabatic transition (as the field strength approaches a local maximum), a reduction in the population of |2), Pi(t), is clearly observed, whereas P\{t) changes very little. According to nonadiabatic transition theory, without ionization, Px(t) and Pitt) are constant between adjacent crossing points. The reduction in P^it) denoted by open circles is hence due to ionization. This is also confirmed by the correlation in reduction between PgM(t) and Pi(t) as indicated by the line with arrows at both ends. After the incident pulse has fully decayed, the relation P](t) + P2(t) = PgM(t) holds. We hence conclude that the state 12) is the main doorway state to tunnel ionization. Enhancement of ionization in H^ around R > 2Re is due to the joint effect of the following three factors: (i) the upper adiabatic state |2) is easier to ionize than is |l); (ii) nonadiabatic transitions from |l) to 12) occur at large R; (iii) the ionization probability of |2) has peaks around certain internuclear distances R>2Re. The above mechanism of ionization is called charge resonance-enhanced ionization because nonadiabatic transitions occur mainly between the two adiabatic states |l) and |2) arising from a charge resonance pair such as lscr, and lsC7a. The analysis using the populations of the two adiabatic states is validated by the fact that in the high-intensity and low-frequency regime only these states are mainly populated before ionization [92]. We have diagonalized H0(t) by using the lowest six a -states. The total population of the resultant six adiabatic states is nearly equal to the sum of the populations of |l) and |2). The lifetimes of intermediate states other than |l) and |2) are shorter than a half optical cycle. Using the right- and left-centered localized bases,

195

1.0-

'"w

> 0.8c o 12 Q. O Q_

,

P, \

P

fSi 0.6

* 5

0.4-

v\-

0.20.0-

r—

r

I

ft N 1

200

grid

\

\ \

^

I

400

1

600

Time (a.u.)

Figure 14. Overall populations P{(i) and P2 (t) on the lowest two adiabatic states, 11) and 12), of H2+ . The populations P, (?) and P2 (t) integrated over R are denoted by a dotted line and a solid line, respectively. To eliminate the outgoing ionizing flux, we set absorbing boundaries for the electronic coordinates p and z. Ionization corresponds to the reduction of the norm of the wave function remaining in the grid space. The total norm in the grid space enclosed with the absorbing boundaries, Pgnd(t), is denoted by a broken line. Level crossing points for H2+ , where E{i) = 0, are indicated by vertical dotted lines. Around t =210 a.u.(—4TT / ffl\ the wave packet approaches i?~3.5 a.u.; then, nonadiabatic transitions between 11) and 12) begin. Nonadiabatic transitions occur around level crossing points t = nn\ (O for n >4. The reduction in P2 (?) denoted by open circles is due to ionization. The upper adiabatic state 12) is regarded as the main doorway state to tunnel ionization. A reduction in P2 (t) corresponds to the subsequent reduction in Pgrid (?) as indicated by the line with arrows at both ends.

196

l L )=(ls>-l M »A/2'

(83b)

we can also describe the dynamics as |^>= /! r R (^)|R> + j L ( J R)|L),

(84)

where the vibrational parts ZR and %h are related with Z\

an

^ Xi

as

ZR(R) = [(COS 6 - sin 0)Zi (R) + (cos 6 + sin ff)z2 (R)]/yf2,

(85a)

ZL (R) = [(cos 6 + sinff)Xi(R) ~ (cos 6 - sin 0)Xl (R)]/J2.

(85b)

The populations in the right and left wells associated with two nuclei, PR and PL , are expressed as follows [91] PAR) = \zA^2

=^2[^+^)\MR)\2

+^d+^\Z2(R)\2

+ Re[z1\R)z2(R)cos28],

PdR) = \zd^

-^2^fj\x(R)\2

(86a)

+^^^\z2(R)\2

-Re[z1'(R)Z2(R)™s28].

(86b)

The third term in both equations represents the interference between |1> and |2>. During a half cycle between two adjacent crossing points, the populations of |l) and |2) are nearly constant. From Eq. (77), we notice that Zj(R) i s expressed, in the period, as the product of the dynamical phase factor expj -i\ Ej{R,t') + {ddldR)1 jmv \dt'\ and the modulus. The interference term thus takes the form of Zx{R)Zi{R)^%26

-

cos.*20 \z'{R)Z2{R)I

e x p | - / J ' [ f 2 ( * / ) - £ , (*/)]<*']>

( 87 )

which oscillates with the period of the phase factor, 2nj\e1 [R,t)-£{ (R,t)~\ • In intense fields, the period ~2K ^Keli+A\l^g\z\u)E[t)\2 ~7tj\ig\z\u)E{t)\ can be much shorter than a half cycle, which results in ultrafast interwell electron transfer (whose period is neither intramolecular one 2#7A£ nor 2nl co) [91]. Field-induced ultrafast interwell transition is pronounced when both |l) and |2) are equally

197 populated and cos 28 is large. In the diabatic regime, \8\ is close to K/A except in the vicinities of crossing points; consequently, the interference term in Eq. (86) becomes negligible. This type of interference disappears when Z\(R) a n d %2^) do not overlap with each other in .ft-space (e.g. if the speeds of Z\(R) and %2(R) are extremely different from each other). 6.2 Adiabatic doorway states of H2 The tunnel ionization of H2 in an intense laser field (7=10 Wcm~ and A=760 nm) has been examined with accurate evaluation of two-electron dynamics by the dual transformation method [48, 86]. The molecular axis is assumed to be parallel to the polarization direction of the applied laser field. We have estimated the ionization probabilities at different values of/? to reveal the mechanism of enhanced ionization in a two-electron molecular system. An ionic component characterized by the electronic structure H H~ or H~H is created near the descending well owing to laser-induced electron transfer from the ascending well. As R increases, while the population of H~H decreases, a pure ionic state H~H+ becomes more unstable in an intense field because of the less attractive force of the distant nucleus. As a result, ionization is enhanced at the critical distance Rc — 4-6 a.u. Although H2 in the ground vibrational state does not expand to R^ [98], the peak of the ionization probability around Rv indicates that ionization would be strongly enhanced when H2 is vibrationally excited. Ionization proceeds via the formation of a localized ionic component in the descending well, in contrast to the H2+ case where the electron is ejected most easily from the ascending well. We have estimated that the change in R from 1.6 a.u. to 4 a.u. would enhance the ionization rate by a factor of 10-20. In the small R (<4 a.u.) region, the main doorway states to ionization are identical to the localized ionic states H H~ and ETH . Around R = 6 a.u., the ionization from the covalent state competes with that from the created localized ionic state. As R increases further, the electron density transferred between the nuclei is suppressed: the main character becomes covalent. Although the rate of direct ionization from a covalent state is much smaller than that from an ionic state, the ionization at large R (>8 a.u.) mainly proceeds from the remaining covalent component. We have also investigated the intramolecular electronic dynamics that governs the ionization process by analyzing the populations of field-following adiabatic states defined as eigenfunctions of the instantaneous electronic Hamiltonian. In a highintensity and low-frequency regime, only a limited number of adiabatic states participate in the intramolecular electronic dynamics, i.e., dynamics of bound

198 electrons. The effective instantaneous Hamiltonian for H2 is constructed from three main electronic states, X, B I a + , and EF [99] (At large R, the last two electronic states are replaced with the ungerade and gerade ionic states.)- We thus have three time-dependent adiabatic states |l) , |2), and |3). By solving the time-dependent Schrodinger equation for the 3 x 3 effective Hamiltonian, we have found that the difference in electronic and ionization dynamics between the small R case and the large R case originates in the character of the level crossing of the lowest two adiabatic states. As the field strength increases, the lowering second lowest adiabatic state |2) comes closer to the lowest adiabatic state |l) starting from the X state. The transition period in which a nonadiabatic transition between |l) and |2) is completed is much smaller than the quarter optical cycle 7tj2co. Thus, a nonadiabatic transition is localized around the time tc when the field strength E{t) reaches the value required for a crossing, E{tc). In the case of R < 4 a.u., the energy gap at the avoided crossing between |l) and |2) is as large as that at zero field strengths. As a result, nonadiabtic transitions to upper adiabatic states hardly occur. When E(t) is larger than E(tc), |1) is ionic, while |2) is covalent. Therefore, ionization occurs from state |l) characterized by a localized ionic state H H~ or H~H directly to Volkov states. In the case of large R (>4 a.u.), nonadiabatic transitions occur from |l) to |2) when these two states cross each other. For R>6 a.u., ionization proceeds mainly through state 12), which is a covalent character-dominated state when E(t)> E{tc). The three-state problem can be reduced to a two-state problem by prediagonalizing the 2 x 2 matrix constructed in terms of the upper two states B and EF. On the basis of the two-state model, an analytical expression of the field strength required for the crossing of |1) and |2) is derived; moreover, the probability of a nonadiabatic transition between |1) and |2) is expressed by the Landau-Zener formula. The results based on these simple formulas agree with those in the three-state treatment. The characteristic features of electronic dynamics of H2 in an intense laser field leads to a simple electrostatic view that each atom in a molecule is charged by fieldinduced electron transfer and ionization proceeds via the most unstable (most negatively charged) atomic site. The success of the adiabatic state analysis for H2 means that the dynamics of bound electrons and the subsequent ionization process can be clarified in terms of a small number of "field-following" adiabatic states. The properties of adiabatic states of multi-electron molecules can be calculated by ab initio MO methods. While the time-dependent adiabatic potentials calculated by an MO method are used to evaluate the nuclear dynamics (such as bond stretching) till the next ionization process, the charge distributions on individual atomic sites are used to estimate the ionization probability. We have already applied this approach to

199

a multi-electron polyatomic molecule, CO2, in an intense field [100, 101] and revealed that in the C022+ stage the two C-0 bonds can be symmetrically (conceitedly) stretched while accompanied by a large-amplitude bending motion. This approach is simple but it has wide applicability in predicting the electronic and nuclear dynamics of polyatomic molecules in intense laser fields. 7 Transfer Matrix Formalism We here treat R as a time-dependent parameter R(t) to solve the coupled equations (77). The nuclear motion can be estimated by semiclassical formalisms. Thachuk et al. [97] have proposed a criterion as to how classical trajectories should be hopped between the £x and £2 time-dependent surfaces. The conservation principle to apply during a hop depends upon its physical origin. The nonadiabatic coupling ddl dt mainly induces energy exchange between the electron and the field. When ddl dt is dominant, nuclear momentum conservation is appropriate. On the other hand, when vddl dt is dominant, energy exchange occurs between the electron and nuclei: total energy conservation is appropriate. The two limiting cases can be smoothly bridged with a physically justified conservation scheme. Since the interference between the two components of |l) and |2) vanish when Z\(R) a n d Zi(^) do n o t overlap with each other, we focus on the case where the difference in nuclear motion between 11) and 12) is negligible. First, we derive a characteristic duration time of a nonadiabatic transition. Near the nth crossing point tn, the gap between the two adiabatic states is given by 2J = A£ug(Rn),

(88)

where Rn = R{t =tn). The coupling strength near the crossing point, {ddldt )t=t in Eq. (77) or (79), is (J0/Jt)l=tn=(g\z\u}R=Rn[ciE(t)/dt]t=t

l2J = Aco/4J,

(89)

where A is a slowly varying function A = 2(g\z\u)R=Rnf(t„)

.

(90)

We have assumed that the internuclear distance hardly changes for the time duration Ttr of a nonadiabatic transition (the Zener transition time given below); i.e., (g|z|w) and A£ug(R) change only a little in the range between Rn±vrtr. The transition process is adiabatic or diabatic according to whether the inverse of the scaled nonadiabatic coupling (ddldt)t=,nlA£ug(R„) is larger or smaller than unity. The quantity A£ag(R„)l(ddldt)t=tn is on the order of

200

8 = J2/Aco.

(91)

Although the Zener transition time ztr is defined by detailed analyses of nonadiabatic transitions in two-level systems [96], we can derive it from the following simple discussion. For 8 > 1 (adiabatic case), the time duration rlr that the system exists in the transition region is given by the temporal width of dOI dt around tn, i.e., JI ACQ , because the coupling dOldt is always smaller than the gap £2 - £\; for <5 < 1 (diabatic case), Ttr is given by the period of effective coupling where 901dt is larger than £2-£\, which leads to Ttrtr < 1/VAm. The condition for \h isolated transition is determined from r,„ « nlco. Conditions are tabulated below.

Adiabatic case

Diabatic case

Adiabatic parameter 8 = J 1 ACQ

8>\

8<\

Transition time xtr

J IACO

\I^~AM

(an overestimated value) Condition for isolated transition

AIJ>\

•JAICO>\

Combining the conditions for 8 and isolated transition, we obtain more specific conditions for isolated transition; Q)< J
dt

zM = -i

d_ dt

ZL(R)

=-

1 d1 + £R(R,t) wp dR2

ZR(R)-i^4^^(R)>

1 d2 + £L(R,*) mp dR2

ZL(R)-i^VXR(R),

(92b)

d2 + (gNw>£(0, mp dR2

(93a)

(92a)

where 1 £R(R,t)=-[£AR)+£u(R)]-—

1

201

£L(R,t)=\[£„(R)+£u(R)]-—(L\^-\L)-(g\z\u)E(t). 2 B mp dR2

(93b)

In the above, we have neglect the small additional coupling terms (L|cr I dR |R)/w p and ( R l ^ / e ^ l O / W p . Since ( R | ^ / c ^ 2 | R ) = {Ud2ldR2\V) , the periodic energy difference A£RL(R,t) = £R(R,t)-£L(R,t) is given by 2{g\iu)E{t). If Ttr « KIco is satisfied (isolated transition), A£RL(Rn,t) can be linearized, in the range between t„ ± Ttr around the «th crossing point t„, as A£RL(Rn,t)«2(gUu)R=RndE(t)/dt\l=ln(t-t„) = Ao)(t-tn),

(94)

where Aco is interpreted as the velocity of the change in the energy difference A£ RL (R n ,t). Here, the field is assume to change from negative to positive at the nth crossing point. In the diabatic representation, the coupling A£u„(Rn)/2-J is independent of time. In the case of rtr « n I a>, the present curve crossing problem is represented by a linear potential model in the time domain with a constant coupling: the exact solution can be given by the Landau-Zener formula [102]. The transition amplitudes ( ^ R ^ L ) J u s t before t-tn-rtr and (Z'R,ZL) Just after t = tn+ ttr is connected, aside from a common irrelevant phase factor, by the transfer matrix Md [73]

Md=( JJL , ^ e ~ * ) t

(95)

where q is the asymptotic Zener transition probability and <j) is the so-called Stokes phase q = e~1"\

(96)

= ;r/4 + argr(l-;<S)+£(ln(5-l).

(97)

The phase

0) = n'I'4 and 0(<S—>°°) = 0. The above Md is obtained for the case where E(t) changes from negative to positive at the crossing point, i.e., |R) crosses |L> from the lower-energy side. In the opposite case, M^ should be replaced with its transpose Md. The transfer matrix in the diabatic representation, Md, can be converted to the adiabatic one Ma which connects (X\->Xi) J u s t before t„ - rtr and {x'\,X'i) J u s t after

202

q e"1

-&

M„

(98)

^ The outgoing state at the «th crossing and the incoming state at the (n+\)th crossing are connected by the propagator (dynamical phase factor) which is in the adiabatic representation expressed as -ia.

G":

0

e

(99)

-ia:•2)

V 0

where the phase a" arises from the terms [...] in Eq. (77), i.e., from the dynamics on they'th adiabatic potential anj =

\,tn+lej{R{t'),t')dt'.

(100)

If the state just after the wth crossing is given by %" -{xl^x'i) > m e subsequent dynamics of the amplitudes, e.g., just after the («+2)th crossing, obeys the following operation

M tt+2 / - i n+\ jk/wn+\

g~in _ , n

(101)

where M^ denotes the transfer matrix for the specific «th crossing event. The general extension to multiple crossings is straightforward. When nuclear motion is taken into account, A and J must be treated as functions of Rn. In the case where the applied periodic field has a constant envelope and the internuclear distance is fixed, an elementary arithmetic of 2x2 matrices as in Eq. (101) yields very simple and physically intuitive formulas. The probabilities of |1) and 12) after the «th crossing, P" and P%, under the condition that the system exists in 11) before the first crossing are expressed as i f =|^ 1 "| 2 =l-g'{sin[«(^ + ^ / 2 ) ] / c o s ^ ] 2 ,

(102a)

Pi = b f =9{sin[n(?7 + *72)]/cos77}2

(102b)

and

where sin7]- VI - q sin(a+ ).

(103)

Here, a is the substantial dynamical phase, namely, the half of the difference between a2 and ax CC: :{a2-ax)l2=

^'l<0[E2{t')-Ex{t')\l2

dt'

203

= _[* "\J4J2+ A2 sin2 at / 2 dt'.

(104)

The maximum of the sequence { P2" }, P™x , is given by q I cos2 rj = q/[l-(\-q)sin2(a+) = 0 , P™* = q (destructive hopping case, localization case). In this case, the condition ri=kn with an integer k holds ; hence, the sequence P2 simply takes the values of 0 (for even n) and q (for odd «): the time-averaged population \P2} = q/2. The transition paths to |2), |l) —>|2) —>|2) and |l) —>|l) —»|2) , interfere completely destructively every two crossings. The condition has a more general meaning. Whenever sin(a+^) = 0 holds, the same populations are fully recovered every two crossings; i.e., {P" ,P2")- (P"+2 ,P2n+2). If both components always exist, i.e., P" * 0 and P2 * 0 for any n, using the transfer matrix formalism, we find that \P2) in the destructive hopping case takes a value in the following range: q/2 + (l-q)P2n±^iq(\-q)P2"(\-P2n)

.

(105)

For sin(«+ at the crossing point. Since tunnel ionization proceeds via the upper adiabatic state 12), the ionization probability increases as the average population of |2) increases. If the system exists in |l) before ionization starts, ionization is enhanced when \P2j is maximized, i.e., in the constructive hopping case (when ionization occurs, (^2/ ' s normalized by setting the sum P" +P2 to be unity). This condition can be fulfilled even in the adiabatic case. If the state just before ionization is |2), the destructive hopping case is favorable for enhanced ionization: then, the hopping to |l) is suppressed. In the diabatic regime, however, the difference between the two hopping cases is very small; \P2")~l/2 in any hopping case. For example, in the case of large R, the value of q is very close to unity. Then, \P2J in the destructive hopping case is ~ql2 from Eq. (105), which is nearly equal to 1/2 of the constructive hopping case. For instance, in the case of Fig.13 (A = 1064 nm and I = 10 14 W/cm 2 ), the situation at R-l a.u. is

204

nearly a destructive hopping case, where sin(0 + a) ~0.1. We then obtain \P2/-0.55 which is in fact close to 1/2. The difference :=0.05 mainly comes from the interference term in Eq. (105), ylq(l-q)P^(l-P2") ( i n t h i s case, g=0.99). In the case of Fig. 13, the average [P?) is nearly independent of R. The tendency of the change in the AC ionization rate r is therefore governed by the change in the DC ionization rate of \2),r2. The ratio of r2 to r, however, peaks at R~6 and 9 a.u. For an instantaneous value of £(/) = 0.0533 a.u., F2 becomes larger around R=6 and 9 a.u owing to resonance with an adiabatic state in the descending well: the role of the ionization from |2) at the peak of the field strength is exaggerated. In the diabatic case, \6\~7tlA at t = tn+Tlr . Thus, except in the isolated transition region, the adiabatic states are identical with the localized states; in the range tn+rlr |L) and 12) —> | R> for E(t„+Ttr)>0 and 11) —>|R) and 12) —»-jL) for E(tn+rlr)<0. See Eq. (71). Even in the adiabatic case, the condition for an isolated transition, AIJ»\, means that the adiabatic states become identical with the localized states around t — (t„+tn+l)/2. Around this moment, the field strength takes a local maximum value, for which |#|= ;r/4 is satisfied. Whenever the transfer matrix formalism is valid, except in the isolated transition region, an adiabatic state can be connected to a localized state by one-toone mapping. Thus, we obtain the population of a localized state ( |R) or |L)) after the wth crossing, P", under the condition that the system starts from the other localized state P2m~l =\-q{cos[(2m~l)7]]/cosT]}2,

(106a)

and P2m = q{s'm(2mT]) /cos T]}2.

(106b)

For parameter values satisfying r\=kn with an integer k, the sequence Pn simply takes two values, namely, 0 (for even ri) and 1 - q (for odd ri): The transition paths to the other localized state interfere completely destructively every two crossings. This is nothing but the destructive hopping case. In the diabatic limit 8«\, the following replacements can be made in Eq. (106): q—¥\, \-q —>2xS , 0—>;r/4, and a-¥ Al co. We find P"=sin 2 [«V2~^?sin04/ft>+;r/4)].

(107)

The rate of tunnelling or transition between the two localized states is reduced from the zero-field value J by the factor 426)17tAsm(AI (0+ x/4). The transition is completely suppressed when the total acquired phase Ala>+ nlA is kn, i.e., in the destructive hopping case. Equation (107) is valid in the diabatic isolated transition

205

case [ max(<w, J) < A ]. In the limit of AI co »1, we can use sin[( AI co) + {n 14)] = \lnAI 2nco J0(A/Q)), where J0 is the zeroth order Bessel function. We thus have P" = sin2 [(naJ/a>)J0(AIco)] .

(108)

The condition for localization in the strong-field limit of AI co»1 is that J0(A/ co)= 0 [103], which is equivalent to the condition J0i2coR I co) = 0 stated in Sec. 3 because AI co = 2coRl co. 8

High-Frequency Limit

We next derive a formula that is valid in the rapid oscillation limit 0)» J and discuss the suppression of transitions between the two localized states [73]. It is convenient to use the coupled equations in a rotating frame as

d

—X*(RA dt'

= -U™9

xdR,t),

(109a)

E(t')dA^{R,t),

(109b)

2i(g\z\u) \ E{t')dt' -2i(g\z\u)^

where ZR(R,t) = exp{ / j'eR(R',t')dt'}

(110a)

ZR(R,t),

ZL(R,t) = exp{ ij'€L(R',t')dt'jzdR,t)

(110b)

•

In Eq. (109), the interwell transition rate / is modified by the exponential phase factors exp[+2('(g|z|w) \ E(t')dt'] . When the condition co»J is satisfied, the exponential phase factors change as fast as the time scale of llco while the state vector changes at most on order of 1 / / . Then, the effective interwell transition rate is determined by cycle-averages of the exponential phase factors. When the pulse envelope / ( / ) , i.e., A, is a slowly carrying function of time in comparison with the oscillation period 2KI co, the cycle-average over one period is given by m

r'\^\±2i(g\z\u)(

E(t")dt"' dt'

In CO

2x

• 2i(g\z\u) fit') cos cot'

rl+itla

exp

• J0[2(g\z\u)fit)/co]

dt'

CO

Jt-jtlco

,

(111)

206

where the pulse is assumed to be turned on adiabatically r E(Odt'=-fit)co&(0t.

J

~

(H2)

co

In the high-frequency limit, the tunnelling rate is reduced from the zero-field value J by the factor J 0 [2(g|z| u)f(t)/co], as in the strong field case. Inserting Eq. ( I l l ) into Eq. (109), under the condition that it starts from |l_) at t = 0, we obtain

j L (?)= c o s { j r j o[ 2 (^i z i M )^') / ^^'}' XR(t) = -ism{j^J,{2(g\z\u)f{t')lai\dt^

^ii3a> .

(113b)

If / ( / ) has a constant envelope as f(t) = f0, JQ[---] m Eq. ( I l l ) should be multiplied by an additional phase (for the sinusoidal form of sin cot). By choosing proper phase factors for the localized states, we can eliminate the factor from the final formulas as ZL(t)=cos[JtJ0(2(g\z\u)f0/a))], ZRU) = -ism[JtJ0(2{g\z\u)f0/co)]

(114a) .

(114b)

Tunnelling is completely suppressed if J(£2(g\z\u)f0/co) = J0(A/oo) = 0 is satisfied. Upon replacing tl{nl co) with n, i.e.,. the number of crossings, we find that The square of Eq. (114b) is nothing but Eq. (108) which is valid in the asymptotic region of max(ft>, J)«A . The two equations should coincide with each other because the regions where Eq. (114b) and (108) are valid, i.e., J«co and max(
,

(115a) .

(115b)

If Jd^g\z\u)folco) = 0, complete localization is resumed at t=Tp irrespective of the pulse length T. Complete localization is also resumed if JTp[Jo((g\z\u)fo I a>)] = kn ; complete transition to |R) occurs if

207

JTp[J0((g\4u)fo10))] = (k + \I2)JI . The present result is an example that shows a possibility of controlling molecular systems by using pulses of realistic shapes other than pulses of constant amplitude. At larger internuclear distances, the transition between the two wells is suppressed. In the high-frequency limit, the transition rate for interwell tunnelling is given by the well known form Jx J 0[2(g\z\u)f(t)/ 0)] . Interwell tunnelling is further suppressed with increasing field strength. There also exist specific conditions for the Bessel function to be zero. For the zeros of the Bessel function, interwell transition is inhibited. The coherent destruction of tunnelling is due to interference between the two adiabatic (or diabatic) components at periodic crossing points tn = nxl a)(n=\,2,...). 9 Conclusion We have summarized in this review the current status of our knowledge of the behaviour of molecules in intense laser fields. This is a new area of research, which will continue to expand due to the quickly evolving accessible laser technology [1-2]. Needless to say, we have entered already the attosecond era in time scale [ 16] with the foreseeable application to controlling electron dynamics in molecules [104]. Similarly in intensity, we have gone beyond Exawatts and are approaching the Zettawatt limit (see table I) with foreseeable applications to nuclear fusion induced by Coulomb explosion of molecular clusters [105]. Similarities of atomic and molecular processes have been emphasized through the unifying concepts of quasistatic models, which predict tunnelling ionization, barrier suppression mechanism of enhanced dissociation and ionization at high intensities. However, one fundamental difference exists between atomic and molecular multiphoton processes at high intensities: the existence of doorway states in molecular systems due to charge resonance effects predicted as early as by Mulliken in 1939 [28]. Thus as shown in [67], there are various regimes of radiative couplings as a function of internuclear distances and the multiphoton transitions in each regime can be adequately described by a nonadiabatic theory (Sees. 5 - 8). The most interesting regime is the nonadiabatic regime where the excitation energy between the LUMO and HOMO in one (odd) electron systems approaches the laser frequency CO and/or Rabi frequency a>R [Eq. (32)]. This leads to Charge Resonance-Enhanced Ionization (CREI) at critical distances and structures of a molecule where molecular ionization rates exceed that of the dissociative fragments by orders of magnitude. In the case of even-electron systems, the creation of ionic states by laser-induced crossing of these states with the ground state at critical

208

distances also leads to enhanced ionization. It is in these nonadiabatic regimes that field-induced population of excited states, be it LUMO's or ionic states, leads to enhanced ionization where an electron must cross the whole molecule in order to exit the molecule through tunnelling ionization. The combination of the atomic quasistatic tunnelling ionization model and the laser-induced excitation, be it resonant or nonresonant with transitions between molecular essential (doorway) states as described in the nonadiabatic model, offers a consistent simple framework to understand the highly nonlinear nonperturbative response of molecules to intense laser pulses. We reemphasize this fundamental difference between molecules and atoms, i.e., in the former there exist essential or doorway states for enhanced ionization at critical radiative couplings in molecules. These nonperturbative radiative couplings depend on nuclear configurations such as internuclear distances. It is under these conditions that new Laser-Induced Molecular Potentials (LIMP's) are created [20-27]. It has been suggested recently that such doorway states affect ionization, fragmentation, and energetics in large molecules where charge resonance effects dominate [106]. Numerical simulations on the simple systems H2+ and H2 have shown furthermore that it is in this enhanced ionization regime that laser control of ionization is most sensitive to laser parameters such as phase [78, 107]. One outstanding issue is the connection between excitations of the doorway states and plasmons, i.e., the multielectron collective excitations, which appear at critical electron densities. Recent numerical simulations of ionization and Coulomb explosion of large clusters emphasize the persistence of the CREI mechanism at a critical initial configuration as a precursor to the facile ionization and explosion of these large systems [108]. Earlier suggestions that plasmon resonances are the essential doorway states for multielectron ionization [109] require a unifying model, allowing us to link the CREI regime and the collective regime of plasmons. Thus many body effects in strong laser fields, specific to molecules and clusters, remain an issue. Laser-induced electron collisions with neighbours in molecules have been shown to enhance and extend HOHG plateaus [34] and lead to a distinct possibility of cluster heating with long-range electron-atom collisions [109]. The enhanced ionization mechanism discovered in our numerical simulations as described in the preceding sections shows that freed or hopping electrons are created in the high field regime and must traverse a whole molecule or cluster in order to ionize. Such effects have been shown to be very sensitive to electron correlation [86] and have not yet been considered in multielectron excitations of large systems. The extension of the simple models developed in the previous sections for small molecules to large molecules and clusters remains an open problem.

209 The emergence of intense ultrashort pulses (7>10 14 W/cm 2 and r / ^ !e <10fe) is ushering a new era of imaging and diagnostic of molecular dynamics with these new tools. We already have shown through non-Born-Oppenheimer simulations on the simple Hj system that Laser Coulomb Explosion Imaging (LCEI) can be a viable new method of measuring the time evolution of nuclear wave packets [58-59]. As discussed in Sec. 2, intense low frequency fields have a new interesting effect, i.e., such fields induce recollision of the electrons with the ionic core. This suggests the possibility of using such "recolliding" electrons to probe short time molecular dynamics via Laser-Induced Electron Diffraction (LIED) [110]. The first experiment based on this idea has been recently achieved [111] demonstrating the use of sub-laser-cycle electron pulses for probing molecular dynamics and thus opening a new area of science, measurement of molecular dynamics in a new time regime, sub-femto or attosecond time regime. Finally, the spectroscopy of gases interacting with short intense laser pulses [112] offers a new method of detecting small concentrations due to the intense signals generated by such pulses [112]. These are new exciting applications of the new laser technology. Acknowledgments We thank various colleagues, P. B. Corkum, Y. Fujimura, Y. Yamanouchi, S. Chelkowski, Y. Kayanuma, Y. Ohtsuki and I. Kawata for valuable discussions concerning interaction between intense laser fields and molecules. We also thank the Japan Society for Promotion of Science for supporting collaborative projects leading to this review. This work was supported in part by grants-in-aid for scientific research from the Ministry of Education, Culture, Sports, Science, and Technology, Japan (12640484 and 14540463). References [1] [2] [3] [4] [5] [6] [7] [8]

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PART THREE Ultrafast Dynamics and non-Markovian Processes in Four-Photon Spectroscopy

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217

Ultrafast Dynamics and non-Markovian Processes in Four-Photon Spectroscopy B.D.Fainberg Raymond and Beverly Sackler Faculty of Exact Sciences, School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel and Physics Department, Center for Technological Education Holon, 52 Golomb St., Holon 58102, Israel

Contents 1

Introduction

2 Hamiltonian of chromofore molecule in solvent and basic

221

218

m e t h o d s of t h e resonance four-photon spectroscopy

226

3

Calculation of nonlinear polarization

233

4

Stochastic models in transient R F P S

238

4.1

Non-Markovian relaxation effects in two-pulse RFPS with Gaussian random modulation of optical transition frequency

4.2

4.3

5

Transient four-photon spectroscopy of near or overlapping resonances in the presence of spectral exchange

246

Non-Markovian relaxation effects in three-pulse RFPS . . . .

252

Non-Markovian theory of steady-state R F P S 5.1

238

255

Introduction and the cubic susceptibility in the case of Gaussian-Markovian random modulation of an electronic transition

5.2

Model for frequency modulation of electronic transition of complex molecule in solution

5.3

255

257

Cubic susceptibility for detunings larger than reciprocal correlation time

262

5.3.1

Classical system of LF motions

262

5.3.2

Quantum system of LFOA vibrations

265

219

6

Four-photon spectroscopy of superconductors

271

7

Ultrafast spectroscopy w i t h pulses longer t h a n reciprocal bandwidth of t h e absorption spectrum 7.1

Theory of transient RFPS with pulses long compared with reversible electronic dephasing

279

7.1.1

Condon contributions to cubic susceptibility

282

7.1.2

Nonlinear polarization in a Condon case for non-

7.1.3 7.2

276

overlapping pump and probe pulses

286

Non-Condon terms

289

Nonlinear polarization and spectroscopy of vibronic transitions in the field of intense ultrashort pulses 7.2.1

291

Classical nature of the LF vibration system and the exponential correlation function

7.2.2

295

General case. Quantum nature of the LF vibration system

8

299

Experimental s t u d y of ultrafast solvation dynamics

300

8.1

Introduction

302

8.2

Calculation of HOKE signal of i?800 in water and D20

...

303

220

9

8.3

Method of data analysis

309

8.4

Discussion

314

Prospect: Spectroscopy of nonlinear solvation 9.1

318

Four-time correlation functions related to definite electronic states

9.2

321

Simulation of transient four-photon spectroscopy signals for nonlinear solvation

9.3

324

Spectral moments of the non-equilibrium absorption and luminescence of a molecule in solution

331

9.4

Broad and featureless electronic molecular spectra

334

9.5

Time resolved luminescence spectroscopy

336

9.6

Time resolved hole-burning study of nonlinear solvation

9.7

Stochastic approach to transient spectroscopy of nonlinear

9.8

. . . 337

solvation dynamics

339

Summary

344

10 Acknowledgments

350

A Appendix

350

221

1

Introduction

Recent investigations with tuning lasers and lasers for ultrashort (up to a few femtoseconds) light pulses have led to the revision of simple models for the light-condensed matter interaction. The first interpretations of the corresponding experiments with dye solutions were based on a consideration of a model of two-level (or three-level) atoms characterized by certain constants of the energy T\ and phase Ti relaxation and also by the spread of the transition frequencies modeling the inhomogeneous broadening in the system [1-4]. Phenomenological terms taking into account the processes of cross-relaxation T3 [1, 5-7] or vibrational relaxation [8] were also introduced, with the goal of constructing a more realistic model in the case of condensed media. In these studies by comparing the experimental data to the theory, times T2 and T3 were estimated to be of the order of hundred femtoseconds. However, such extremely fast relaxation phenomena, whose duration is of the order of the correlation time r c of the thermal reservoir, can no longer be analyzed by the conventional theory based on the relaxation time description. In this ultrashort time region one should take into account effects of memory

222 in the relaxation (non-Markovian effects) [9, 10]. Another reason for the revision of the simple models is the impossibility to divide a relaxation between closed states into an energetic and a phase one [11-13]. At the beginning of the eighties a non-Markovian theory of the transient and steady-state resonance four-photon spectroscopy (RFPS) began to develop in this respect [14-21]. A non-Markovian character of the optical transition broadening can be illustrated by a simple example of an oscillator whose natural frequency ui' is randomly modulated [9]. The resonance absorption spectrum of such an oscillator at the frequency u> is given by 1 r°° F(u - w0) = — / exp[-i(w -

uQ)t\f{t)dt

Z7T J - o o

where to0 is the time average of UJ',

f(t) = (eXp[i£u1(t')dt')) is the relaxation function of oscillator, uii (t) — UJ (t) — LU0. The resonance intensity distribution F (LJ — UJ0) is an observed quantity and it is broadened around the center w0 by the random modulation LJ\ (<).

223

Consider the shape of this resonance spectrum and its relation to the nature of the modulation. Let us suppose that the frequency modulation is characterized by a probability distribution -P(^'i). If we suppose that P (ui) has only one peak at u>i = 0, the stochastic process u\ (t) can be described in terms of two characteristic parameters. (1) Amplitude of modulation

a: a2 = J uifP (to) dijj\ = (u>f).

(2) Correlation time of modulation, r c . The correlation function of modulation is defined by 5 ( r ) = 4 r ( W l ( i ) W l ( t + r)>. Then the correlation time TC is given by rc = J"0°° S (t) dt, i.e. (wj (t) wi (i + r)) ~ 0 when r ^> r c ; r c thus measures the speed of modulation. It can be demonstrated [9] that two typical situations are distinguished by the relative magnitude of a and TC: (a) Slow

(b) Fast

modulation. a-Tc»l,

(1)

a • rc < 1.

(2)

modulation.

In case (a) rc is large compared to 1/a. Then it turns out that F (u> — u>0) =

224

P(u> —u>o)- Thus the intensity distribution reflects directly the distribution of the modulation. The width of the intensity distribution curve will be about a and the response is dynamic and coherent. This case corresponds to an inhomogeneous broadening of the transition under consideration. Thus one can say that the inhomogeneous broadening of the optical transition corresponds to an extreme case of the non-Markovian relaxation, when the "memory" in a system is completely conserved. In case (b), TC becomes small and any modulation UJ\ hardly lasts for any significant time (~ 1/wi), so that the fluctuation is smoothed out and the resonance line becomes sharp around the center. In this limit a • r c —> 0 the line have a Lorentzian form

in which the half-width 7 = a 2 • r c . This case

corresponds to a homogeneous broadening of an optical transition when the relaxation is Markovian with a very short "memory" determined by r c . As will be seen in later sections, the broadening of electronic transitions in dye solutions used in four-photon experiments corresponds to the case of slow modulation (a) (or the intermediate one) rather than to case (6). Therefore, we need to use a non-Markovian theory for the description of corresponding experiments. The literature on non-Markovian effects in nonlinear spectroscopy is cur-

225

rently quite voluminous, but the framework of the review article does not enable us to cover all the issues concerning the subject. Therefore we will confine the review mainly to our recent theoretical results related to an experiment. We do not discuss such interesting problems as the non-Markovian theory of the Resonance Raman Scattering [22-29], spectroscopy of dimers [30, 31], etc. At present, there are a number of review articles [32-34] and a monograph [31] which cover questions which are not discussed here. A large contribution to the theory of four-photon spectroscopy has been made by Mukamel and coauthors (see [31] and references here). Experimentally, the RFPS has been developed in the works of Yajima's, Wiersma's, Shank's and Fleming's groups, Vohringer and Scherer and others (see references here). The outline of this chapter is as follows. In Sec.2 we describe the Hamiltonian of a chromofore molecule in a solvent and basic methods of RFPS. In Sec.3 we present the corresponding theory. In Sec.4 we consider nonMarkovian relaxation effects in transient RFPS by the use of stochastic models. A non-Markovian theory of steady-state RFPS is presented in Sec.5. Sec.6 is devoted to the real time four-photon spectroscopy of superconductors. In Sec.7 we present a theory for transient RFPS with pulses long com-

226

pared with the electronic dephasing and its generalization for strong light fields not satisfying the four-photon approximation. In Sec.8 we describe our experimental results obtained by the heterodyne optical Kerr effect (HOKE) spectroscopy on ultrafast solvation dynamics study of rhodamine 800 (i?800) and DTTCB

in water and D2O. In Sec.9 we shall discuss a prospect of spec-

troscopy with pulses longer than the reciprocal bandwidth of the absorption spectrum: nonlinear solvation study. In the Appendix we carry out auxiliary calculations.

2

Hamiltonian of chromofore molecule in solvent and basic methods of t h e resonance four-photon spectroscopy

Let us consider a molecule with two electronic states n — 1 and 2 in a solvent described by the Hamiltonian

Ho = J2 I") [#» - *'Hi + wn{Q)} H

(3)

n=\

where E2 > Ei,En n,Wn(Q)

and 2jn are the energy and the inverse lifetime of state

is the adiabatic Hamiltonian of reservoir R (the vibrational sub-

227 systems of a molecule and a solvent interacting with the two-level electron system under consideration in state n). The molecule is affected by electromagnetic radiation of three beams 1 3 E ( r , t) = E+(r, t) + E " ( r , t) = - ^ {£m(t) exp[i(k m r - umt)] + c.c.}

(4)

^ m=l

Since we are interested in both the intramolecular and the solvent-solute intermolecular relaxation, we will single out the solvent contribution to Wn(Q): Wn(Q)

= WnM + Wns where Wns is the sum of the Hamiltonian governing

the nuclear degrees of freedom of the solvent in the absence of the solute, and the part which describes interactions between the solute and the nuclear degrees of freedom of the solvent; WnM is the Hamiltonian representing the nuclear degrees of freedom of the solute molecule. A signal in any method of nonlinear spectroscopy can be expressed by the nonlinear polarization PNL.

We will consider both the steady-state and

the transient methods of the RFPS. The steady-state RFPS methods [35, 1, 36, 5, 2, 37, 3, 8, 6, 21, 38-50] are based on an analysis of the frequency dependence of the cubic polarization P ' 3 ' of the medium studied at the signal frequency u>s = u'm + LJ'^ — um.

The

case Lo'm = UJ'^ = wx = const, u>m — u>2 — var corresponds to spectroscopy

228

based on a resonance mixing of the Rayleigh type [35, 1, 36, 5, 21, 39-50]. The optical scheme describing the principles of the method is shown in Fig.l. Two laser beams with different frequencies u,'i ^ u>2 and wave vectors ki

3 =2K,-K 2

Figure 1: The scheme describing the principles of spectroscopy based on a resonance mixing of the Rayleigh type.

and k 2 produce a nonstationary intensity distribution in the medium under investigation. This intensity exhibits a wavelike modulation with a grating vector q = kj—k 2 and a frequency ft — u>i— ui^. The wavelike modulated light intensity changes the optical properties of the material in the interference region, resulting in moving grating structure. If this wavelike modulation is slow in comparison with the relaxation time, rrei of the optical properties of a material (r re ;
229

0. > T~J, the optical properties of a material do not follow the intensity modulation, and the grating structure becomes less contrast, and therefore the grating amplitude decreases. The grating effectiveness measured by the self-diffraction of waves u\ drops for this case. In the moving grating method the relaxation velocity of the optical properties of a material is compared with the motion velocity of the grating (which is proportional to the frequency detuning). When applied to a spectroscopy of inhomogeneously broadened transitions, the self-diffraction signal will change like ~ fl~2 for T 2 -1 S> \Cl\ S> Xi_1 and like ~ fi'4 for \Cl\ > T j - 1 , ^ - 1 [35]. Now let us consider transient methods of RFPS. In a three-pulse timedependent four-wave-mixing experiment (Fig.2a) [51-64], pump pulses propagate in the directions k% and k2 and induce grating in a medium.

The

dependence of the grating efficiency on the delay time r between pulses ki and k 2 is recorded using the scattering of the probing pulse k 3 , delayed by a fixed time T with respect to pulse k 2 . The resonance transient grating spectroscopy (see Fig.2b) is the particular case of a three-pulse time-dependent four-wave-mixing when r = 0 and T is variable [65-67]. For k 3 = k 2 (T = 0) we obtain the spatial parametric effect [4, 68-70] and the two pulse photon

230

kg+lc^k!

k

! - J c2 2T ** t3

b)

Wki

^ - ^ ^ 3

Figure 2: Geometry for three-pulse time-delayed four-wave-mixing experiments: (a) grating on the basis of a polarization (r variable, T = const); (b) population grating (r = 0, T variable).

231

echo case [71-74, 59, 75]. In these experiments, the signal power Is in the direction k s = k 3 ± (k 2 — k t ) at time t, is proportional to the square of the modulus of the corresponding positive frequency component of the cubic polarization P ' 3 ) + : Is(t) ~ | P ' 3 ' + ( r , t) \ . In pulsed experiments, the dependence of the signal energy Js is usually measured on the delay time of the probe pulse relative to the pump ones: J , ~ /°° |P< 3 )+(r,*)| 2 di

(5)

J — oo

When the stimulated photon echo is time gated, for instance, by mixing the echo signal with an ultrashort gating pulse in a nonlinear crystal [76, 77], the signal is proportional to the echo profile at time tg of the arrival of the gating pulse: Jgated(tg,T,T)

~ | P{£P+E{tg,T,T)

| .

Other methods of transient RFPS are: the transmission pump-probe experiment [78-80], the heterodyne optical Kerr effect (HOKE) spectroscopy [81-84], and time resolved hole-burning experiments [78, 85-90]. In the transmission "pump-probe" experiment [78-80], a second pulse (whose duration is the same as the pump pulse) probes the sample transmission AT at a delay r . This dependence A T ( r ) is given by [91] oo

/

-oo

S;r{t-r)VNL+(t)dt

(6)

232

where Spr and VNL+(t)

are the amplitudes of the positive frequency compo-

nent of the probe field and the nonlinear polarization, respectively. In resonance HOKE spectroscopy [81, 83, 84], a linearly polarized pump pulse at frequency w induces anisotropy in an isotropic sample. After the passage of the pump pulse through the sample, a linearly polarized probe pulse at IT/4 rad from the pump field polarization, is incident on the sample. A polarization analyzer is placed after the sample oriented at approximately ~/2 (but not exactly) with respect to the probe pulse polarization. A small portion of the probe pulse that is not related to the induced anisotropy plays the role of a local oscillator (LO) with a controlled magnitude and phase. The HOKE signal can be written in the form: oo

/

£lo(t - r ) exp{irl>)VNL+(t)dt

(7)

-co

where if; is the phase of the LO. If i\> = 0, the resonance HOKE spectroscopy provides information similar to that of the transmission pump-probe spectroscopy (see Eq.(6)). If i\) — 7r/2, the resonance HOKE spectroscopy provides information about the real part of the nonlinear susceptibility (the change in the index of refraction). In the time-resolved hole-burning experiment [78, 85, 86, 92-97], the sam-

233

pie is excited with a ~ 100 fs pump pulse, and the absorption spectrum is measured with a 10 fs probe pulse that is delayed relative to the pump pulse by a variable r. In another variant of such an experiment, a delayed pump pulse, broadened up to a continuum, can play the role of a probe pulse. The difference in the absorption spectrum at w' + u is determined by [91, 86]

^Q(W')

~ -Im[VNL{J)iepr(u')\

(8)

where oo

/

VNL+{t)exv(iLo't)dt

(9)

-oo

is the Fourier transform of the nonlinear polarization, and CO

/

£pr(t - r) exp(iu't)dt

(10)

-oo

is the Fourier transform of the probe field amplitude.

3

Calculation of nonlinear polarization

The electromagnetic field (4) induces an optical polarization in the medium P ( r , t) which can be expanded in powers of E(r, t) [98]. For cubic polarization of the system under investigation, we obtain: P( 3 >(r, t) = p( 3 >+(r, t) + c.c. = iV J L 4 (rr R (D 1 2 ^ 3 1 ) (i)) + c.c.)or

(11)

234

where N is the density of particles in the system; L is the Lorentz correction factor of the local field; D is the dipole moment operator of a solute molecule; (.. .) o r denotes averaging over the different orientations of solute molecules; p' 3 ' is the density matrix of the system calculated in the third approximation with respect to E(r,£). The equation for the density matrix of the system can be written in the form /> = - * ( I o + £i)p

(12)

where Lo and L\ are the Liouville operators denned by the relationships LQp = h~l[Ho,p]

and Lip = &-1[—D • E(r, £),/?].

Using the interaction

representation (int) by means of the transformation pmt = exp(iLot)p £mt _

eXp(iLot)LiexTp(—iLot),

and

solving the resultant equations by perturba-

tion theory with respect to L™' in the third order, and using the resonance approximation, we find p^z\ The diagram representation of p^ can be found in monographs [98, 31]. The a-th component (a, b,c,d = x, y, z) of the amplitude of the positive frequency component of the cubic polarization V^+(t)

describing the gen-

eration of a signal with a wave vector ks = k m / + k m » — k m and a frequency

235 o.'s (P( 3 > + (r,i) = VW+{t)exp[i{k3r-<jjst)})

x

Yl

H

is given by the formula [99]

dTidT2dT3exp{-[i(u21

- ua) + J]TI

mm'm" bed

-[i{iom - um>) + T^]T2 - ~fT3}{exp[i(u;21 - wm)T3]Fla6crf(r1, r 2 , r 3 ) x £ m ' C ( * - Ti - T 2 ) £ ' ^ ( i ( i - Ti - T2 - T 3 )

+ e x p [ - l ( w 2 l - W r o ')r 3 ]F2a6cd(Tl, T 2 , T 3 )

x£m/c(< - TI - r2 - T3)£*md(t - n - T2)}£m»b{t - n )

(13)

where T\ = (27 2 ) - 1 = ( 2 7 ) - 1 is the lifetime of the excited state 2, w2i = ^'ei — (W2 — Wi)/h

is the frequency of the Franck-Condon transition 1 —> 2

(see the definition of W2 and Wi in Sec.2), we; = (£ 2 — Ei)/h

is the frequency

of purely electronic transition with corrections from the electronic degrees of freedom of the solvent [99, 86]. The summation in Eq.(13) is carried out over all fields that satisfy the condition k s = km< + k m » — k m . The functions F\,2abcd{Ti-,'r2,Tz) are sums of four-time correlations functions corresponding to the four photon character of light-matter interaction:

Flabcd(n,

1~2, T 3 ) = Kdcab{0,

^ 3 , n + T2 + T 3 , T2 + T 3 )

+A^6ac(0, T2 + r 3 , Ti+r2

+ r 3 , r 3 ),

(14)

236 F2abcd{n,T2, T 3 ) = K*dba(0, r 3 , r 2 + r 3 , r : + r 2 + r 3 ) + ^ M ( O . T I + T2 + r 3 , r 2 + r 3 , r 3 ),

(15)

where

A^O,^,*,,^) = xDc12exp(iW2(t3

({DltexpiiWih/tyD^expiiWifo-tJ/h) - i2)/^)^1exp(-?W1f3/a)))or

(16)

are the tensor generalizations of the four-time correlation functions A'(0, t\, t2, i 3 ) which were introduced in four-photon spectroscopy by Mukamel [16, 100]. Here (...) = T r # ( . . . pR) denotes the operation of taking a trace over the reservoir variables, pn = exp[—Wi/(kT)]/Trnexp[—

Wi/(kT)]

is the density

matrix of the reservoir in the state 1, W2 = W2 — (W2 — W\) is the adiabatic Hamiltonian in the excited state without the reservoir addition to the frequency of the Franck-Condon transition (the term (W2 — Wi)). It follows from Eqs. (13),(14),(15), (16) that the nuclear response of any four-photon spectroscopy signal, generally speaking, depends on the polarizations of the excited beams because of the tensor character of the values Fl,2abcd a n d

Kabcd{0,ti,t2,t3).

We can represent the latter quantity as a product of the Condon

(FC)

237 and non-Condon (NC) contributions [101, 99]:

Kabcd(0,h,t2,t3)

= KFC(0,h,t2,t3)

The Condon factors KFC(0,ti,t2,t3)

• K^d(0,h,t2,t3).

do not depend on the polarization

states of exciting beams, however the non-Condon ones Kavb^d(0,ti,t2,t3) pend on their polarizations.

(17)

de-

The origin of the non-Condon terms stems

from the dependence of the dipole moment of the electronic transition on the nuclear coordinates D i 2 ( Q ) .

Such a dependence is explained by the

Herzberg-Teller (HT) effect i.e., mixing different electronic molecular states by nuclear motions.

When D 1 2 does not depend on the nuclear coordi-

nates (the Condon approximation), the non-Condon terms are constants: K?£(0,tut3,t3)

= {Dl2Db21D\2Dd2l)or

~ D 4 where D = \Dl2\.

Let us introduce the quantity u = W2 — W\ which determines the strength of the bonding of the vibrational subsystem with the electronic transition, and characterizes the Condon perturbations of the electronic transition (unlike non-Condon perturbations which are determined by the dependence Di2(Q))- Thus, if the value u is a Gaussian one (intermolecular nonspecific interactions, linear electronic-vibrational coupling etc.), and also in the case of a weak electronic-vibrational coupling, irrespective of the nature of

238 u, the Condon contribution can be represented in the form [100, 39, 31]: A'FC(0,ti,t2,i3)=exp[5(i3-i2)+^i)+5(i2-ii)-^2)-^3-ii)+^3)] (18) where g(t)

= - / r 2 f dt\t - t')K{t')

(19)

Jo is the logarithm of the characteristic function of the spectrum of single-photon absorption after substraction of a term which is linear with respect to t and determines the first moment of the spectrum, K(t) = (u(0)u(t)) — (u) 2 is the correlation function of the value u.

4 4.1

Stochastic models in transient R F P S Non-Markovian relaxation effects in two-pulse RFPS with Gaussian random modulation of optical transition frequency

Let us consider the spatial parametric effect (SPE)[4, 68]. The SPE consists in the following. When the medium under study is acted upon by two short light pulses of frequency u> with wave vectors k x and k 2 separated by a time

239

interval i 2 ( s e e Eq.(4) for TO = 1,2 and wi = w2 = w), signals with wave vectors k 3 = 2k 2 — kj and k 4 = 2kx — k 2 are generated in this medium. The temporal characteristics of these signals provide information on the phase relaxation time of the optical transition studied. The problem of non-Markovian relaxation in the SPE was first discussed independently in Refs.

[14, 15]. Aihara[14] examined the transient SPE

in the case of a system with a linear and quadratic electron-phonon bond, and obtained numerical results illustrating a non-Markovian behavior. In Ref.[17] (see also Ref.[18]) a stochastic model was used to derive simple analytical relationships, which could serve as the basis for the spectroscopy of non-Markovian relaxations based on SPE. The model interpolates in a continuous way between the inhomogeneous broadening case and the homogeneous broadening case. The cubic polarization of the medium, corresponding to wave k 3 , is given by Eq.(13) for m = 1 and m' = TO" = 2. If pulses ki and k 2 are well separated in time, then only the first term in the curly brackets on the right hand side of Eq.(13) (~ i*\) makes a contribution. Let us assume that u(t) is a Gaussian-Markovian random process with correlation function K(t)

=

h2a2 exp(—|i|/r c ). Such a model of relaxation perturbation is highly realistic.

240

It corresponds to solvated systems in the case of the Debye spectrum of dielectric losses [102], concentration-dependent dephasing in mixed molecular crystals [103], Doppler-broadened lines in gases in weak collisions [104], and phase modulation by phonons in crystals [25]. The quantity Fi in the case of the Gaussian-Markovian random modulation of an electronic transition has been calculated in Refs.[17, 18]. In the Condon approximation, without taking into account tensor properties, the quantities Fit2 have the following form: ^,2(ri,r2,r3) =

£4exp{-p2[exp(-^) + ^ + e x p ( - ^ ) + ^ - 2 T

e x p ( - ^ ) ( e x p ( - ^ ) - l ) ( e x p ( - ^ ) - 1)]},

(20)

where p = arc. Considering the limit of short pulses and introducing the pulse areas e m = DH~l /f^ £m(t')dt',

we obtain the signal strength I3(t) ~ |p( 3 )+| 2 by

means of Eqs.(13) and (20): I3(t) = J B e x p { - T 1 - 1 t - 2 p 2 [ - + 2 e x p ( - ^ ) - e x p ( - - ) + 2 e x p ( - ^ ^ ) - 3 ] } , (21) where t\ = 0 is the moment of the appearance of the first pulse k i , t > t2 - the delay time of the pulse k 2 with respect k i , B = AA^O^O 2 , and A is

241

a proportionality factor. In the case of intense light fields, the quantity B should be replaced by B' = 16AN2 sin 4 (0 2 /2)sin 2 (0 1 ) [21]. In a more general case when u{t) is a Gaussian random process with an arbitrary correlation function K(t),

one can find K(t) on the basis of the

function Iz(t) [21]: -ft

2 — In I3(t) = 2h~-2r [K{t) - 2K(t - t2)}.

(22)

When £2 > Tc the second term on the right hand side of Eq.(22) makes the primary contribution. It follows directly from Eq.(21) that the maximum of signal h(t)

corre-

sponds to instant [17] tmar = rcln[p2(2exp(^)

- l)/(p2 +

(23)

T^TC/2)\.

The energy dependence of signal IC3 is equal to in

/•CO

Js(t2) = /

f„

h{t)dt = T c Bexp{-Tf 1 * 2 - 2 p 2 [ ^ + 2 e x p ( - ^ ) - 3]}

x{2p 2 [2 - e x p ( - ^ ) ] } - ( T r 1 - . + ^ ) 7 [ T - i T c

+

2p 2 ,2p 2 (2 - e x p ( - %

(24)

where 7(6,0) is an incomplete gamma function [105]. At the limit of fast modulation, which corresponds to a homogeneous

242 broadening (p < 1; t2,t — t2~> r c ), / 3 (t) = Bexp[-(T 1 - 1 + 27 0( ,)i], tmax _ , — In

To

Js(t2) =

, J-1

llad t2 ^2 ——i—;— ~r — ~ — i

Iad + T{ll2

B

TC

expKTf1 +

Tc'

2lad)t2],

+ ^7ati

where -jad = ap is the contribution of elastic (adiabatic) processes to the phase relaxation. At the limit of slow modulation, corresponding to inhomogeneous broadening (p > 1; t2,t - t2 < r c ), h(t) = 5 e x p ( - T f 4 ) e x p [ - a 2 ( i -

2t2)2],

Js(h) = ( S A / ( 2 a ) ) e x p ( ^ 7 -

2T~H2) T

x[l + $ ( a i 2

-i 1

" 2a "

As can readily be seen, a photon echo takes place in the latter case [71]. Fig.3 shows the signal I3(t) and Js(t2) in the case of intermediate modulation (p=l). Thus, as follows from the non-Markovian theory of SPE described in this subsection, the methods of transient RFPS (together with the spectra of single-photon absorption) make it possible to find the parameters of relax-

243

c

tf

+

CM

-

CJ

y

/ / /^

-

CJ -•—

^ /

1

1

1

tp/T,

t?/Tr

Figure 3: Time response ^ ( i ) (a) and pulse energy J,(<2) for T j - 1 ^ -C p = 1; £ — £2 = ro- The inset shows the instant of appearance of the maximum of the signal /3(f) as a function of the delay time £2 of the second pulse for TI1TC

echo.

-C p2 [Eq. (23)]; the tangent 2t2JTc corresponds to the case of photon

244

ation perturbation, i.e., the modulation amplitude a and correlation time TC, which, in the general case, determine the non-Markovian relaxation of the system studied. The model of the Gaussian-Markovian stochastic modulation for optical dephasing has been used for the description of a non-Markovian relaxation behavior in a number of two-pulse SPE experiments [72, 106, 59]. Fig.4 shows the result of the corresponding femtosecond experiment on resorufin in dimethylsulfoxide (DMSO) by Wiersma et al. [72]. The solid line is a fit based on the Gaussian-Markovian stochastic modulation model for optical dephasing. Wiersma et al. showed that optical dephasing of resorufin in DMSO can be described by using a stochastic modulation model. With GaussianMarkovian statistics, both the femtosecond photon-echo experiment and the steady-state absorption spectrum can be adequately simulated with the same values for the stochastic parameters [72]. Saikan et al. [106] observed non-Markovian relaxation in photon echos of iron-free myoglobin which was described by the Gaussian-Markovian stochastic modulation model for optical dephasing by using Eq.(20).

245

-40

0

40

80

120

160

pulse delay time (fs)

Figure 4: Photon-echo signal for resorufin dissolved in DMSO (dotted trace). The solid line is a fit based on the Gaussian-Markovian stochastic modulation model for dephasing, with parameters a = 41 THz and r - 1 = 27 THz [72].

246

4.2

Transient four-photon spectroscopy of near or over lapping resonances in t h e presence of spectral exchange

In Subsec.4.1 we considered non-Markovian relaxation effects in two-pulse RFPS with the Gaussian random modulation of the optical transition frequency. In this case the four-time correlation functions describing a fourphoton light-matter interaction can be expressed in terms of the correlation function K(t) (see Eqs.(18) and (19)). Another example of random modulation, which allows a detailed and general mathematical development, and has important physical applications, is the Markovian modulation [9, 11]. Let us return to the oscillator whose natural frequency is randomly modulated (see S e e l ) . For the sake of simplicity we will assume that the oscillator takes on any of two states a and b. The resonance frequency in states a and b will be wo ±^t, respectively [107]. In this model, the frequency stays at some value, say uia for a certain time T„, and changes suddenly to other value W&, remains constant for time period TJ, returns to ioa and so on. The quantities r~l and T^1 describe the phenomenon of spectral exchange in the system which is related to a coupled damping of oscillators [9, 11-13, 107].

247

The spectral exchange strongly influences the shape of an absorption spectrum of such an oscillator [9, 11,.108]. Let us introduce the time rc such as T~X = T~1 + T^1. Then in the limit TC —y 0 the spectrum will be a sharp line at the equilibrium average of the two frequencies coa and W&. As r c —> oo two sharp peaks will appear at wa and w& [9]. The model under consideration is interesting in relation to a spectroscopy of near or overlapping resonances [107]. The question of RFPS methods establishing the presence or absence of spectral exchange in a system, is of rather great current interest. Actually, one can distinguish two mechanisms of formation of spectra from superimposed lines. In the first mechanism the different lines belong to noninteracting transitions (for example, in different centers) and each transition decays independently of the others. In the second mechanism relaxation of transitions with different frequencies is connected with transfer of excitation between them, due to which relaxation of the transitions occurs in a coupled way. This is the case of spectral exchange. If, for example, one turns to doublets in the spectra of polyatomic molecules, the mechanism of line broadening due to spectral exchange is determined by isomeric transitions occurring in one center [109-111], while the mechanism of formation of doublets connected with the presence of different spatially

248

separated types of active centers, corresponds to the absence of spectral exchange [112]. Thus, revealing the nature of the overlap of near lines is of undoubted physical interest. To resolve the indicated question it has been proposed to use two-pulse RFPS [107]. In the case being considered, the four-time correlation functions describing a four-photon light-matter interaction can not be expressed in terms of the correlation function K{i) (see Eqs.(18) and (19)), as it was for the Gaussian modulation. Therefore, for solving the problem we used Burshtein's theory of sudden modulation [11]. We have calculated the intensity of the SPE signal in the limit of short pulses [107]: h{t)~\P^+\2 = B'\R^\\

(25)

where in the case of equal times ra — T& = r 0 the quantity |i?k 3 | 2

c a n De

represented in the form l^ksl 2 = ( 4 « T X e x p H T f 1 + 2r0'1)t)R2(t2)Sm2(Kt

R(t2)

+ v(t2)),

(26)

= fi{fi2 + T~2 + 2r0-1[«;sin (2«i 2 ) - T'1 cos (2/d 2 )]} 1 / 2 , sinp (t 2 ) = Br1 (i 2 ) [r0-2 - fx2 cos (2/ei 2 )],

cos tp (i 2 ) = -R-1 (t2) [KTO1 + /J,2 sin (2/rf2)], K = (A*2 - r 0 - 2 ) 1 / 2 .

(27)

249

Consider, for the sake of comparison, two noninteracting resonances, the frequency difference of which is 2A, decaying with the constant T 1 _1 /2 (system without spectral exchange - wse). In this situation IZ"{t) ~ e x p ( - T f 4 ) sin2[Af + (TT/2 - 2At2)}.

(28)

Eqs.(26),(27) for fj, > T^1 and Eq.(28) describe the beats of the intensity of the signal k 3 , which can be used to reveal the hidden structure of singlephoton spectra. This aspect of the problem has been well studied in photon echo spectroscopy [113]. Further, from a comparison of Eqs.(26),(27) and Eq.(28) it is seen directly that they are characterized by different dependences on t2 of the amplitude and phase of the corresponding signals. This can be used to reveal the mechanism responsible for the overlap of near lines [107]. As a matter of fact, the zeroes of intensity of the signal IC3 are realized when Kt + f(t2)

= n7r for the case of the presence of spectral exchange, and when

At + 7r/2 — 2At2 =

TITT

in its absence (n is an integer). Let, for example, the

delay t2 be chosen in such a way that Is(t2) = ^3/se(^2) = 0 for t — t2. Denote by t" the moment of time of the appearance of the next zero of intensity. Then it is not difficult to show that in a system with spectral exchange t" — = l + 7r/arccos( t2

1 ), \XTQ

(29)

250

while in the absence of spectral exchange is always t'^se/t2 = 3. When 1 (the case of slow modulation), arccos( — l/(fiT0)) while for (J,T0 ~ 1 the ratio t"jti t"/i2 = 2 | .

—>• n/2 and t"jti

JJLTQ

3>

—> 3,

^ 3. For example, if fir0 = \ / 2 , then

Thus, the nonstationary behavior of the SPE signal allows

distinguishing situations corresponding to the presence or absence of spectral exchange in the system investigated. It is worth noting that a manifestation of spectral exchange by a phase shift between components is the charactestic feature of four-photon beats spectroscopy. The proposed method of the establishment of the fact of the presence of spectral exchange has been used in Ref.[114] devoted to transient spectroscopy of coherent anti-Stokes Raman scattering (CARS) of thulium (Tm) atoms in a buffer gas. In this experiment, a slowing of Doppler dephasing and a spectral exchange effect have been detected for the first time in optical-range atomic spectroscopy. Ganikhalov et al. [114] observed the quantum beats stemmed from the hyperfine splitting of the 4F 5 / 2 and 4F 7 / 2 states of the thulium atoms. They found Tminlrmin

=

2-6 ± 0.1 ^ 3 for

thulium in xenon (see Fig. 5), where rm'in are the delay times which determine the positions of the first and second minima. Therefore, authors [114] concluded that they were dealing with a manifestation of spectral exchange.

251

0

.

1

•

T

i

i

!

min = 9 6 0 p s

•

2 -2 ^

••

3r

•

-3

I"

r * i n = 2500ps \

4

•

•

rnin=2880ps

• J

-

-6

i

0

1000

i

i

i

2000

3000

4000

T.ps

Figure 5: Temporal response at a xenon pressure of 825 torr. Shown here are the delay times r^'t2n, which determine the positions of the first and second minima of beats of the hyperfine-structure components, and the delay time 3T1 • , which determines the position of the second minimum of the beats in the absence of spectral exchange [114].

252 An examination of the non-Markovian effects on the relaxation processes in the excited levels between which beating occurs has been also made in Ref.[115].

4.3

Non-Markovian relaxation effects in three-pulse RFPS

Let us consider three-pulse time-dependent four-wave-mixing experiments (Fig.2a) [51-64]. A stochastic theory of these experiments for Gaussian random modulation of a frequency of an optical transition has been developed in Ref.[116]. Let us consider the signal corresponding to wave k 3 = k3 + k2 — kt in the limit of short pulses. Then using Eqs.(13),(14),(17),(18) and (19), one can show that the correlation function K(t) is obtained from the time dependence of the signal power Is(t) [116]: — In /.(<) = -2H-2{K(t

-T-T)

+ [K(t - r) - K(t)}}.

(30)

When T 3> TC, the first term on the right-hand side of Eq.(30) makes the main contribution. To carry out further calculations, we need to specify the form of the correlation function K(t).

We assumed that u(t) is a Gaussian-Markovian

253

random process [116]. Then using Eq.(20), one can show that the maximum of signal Is(t) occurs at the instant time [116] r T tmax = r c ln{exp(—)[1 +exp(—)] - 1}.

(31)

One can see that for T,T
(32)

q = 2p 2 {l + e x p ( - T / r c ) [ l - e x p ( - r / r c ) ] } .

(33)

where

Here / ( r ) = exp[gr(r)] is the characteristic function of the resonance absorption spectrum F(ui — w 2 i) and is expressed by / ( r ) = /f^

F(LJ')

exp(iw'r)rfw',

where o;2i is the frequency of the corresponding transition 1 —>• 2. It is wellknown that the characteristic function is the relaxation function that de-

254

scribes the relaxation of the response of a system after removal of the outer disturbance [118, 9]. Interesting aspects of the influence of non-Markovian effects in a threepulse time-dependent four-wave-mixing experiment have been noted by Lavoine and Villaeys [119]. They showed that the energy of the diffracted light can be expressed as a square of the relaxation function / ( r ) :

Mr)

~ |/(r)|2,

(34)

when the medium is excited by very short pulses. Their calculation does not make assumptions about the analytical form of / ( r ) . For this reason it is possible to consider result (34) as general, and this result is interesting from the point of view of learning about the dynamics of the bath. Eqs.(32) and (33) enable to define more precisely the conditions for the correctness of Eq.(34) [117]. One can see that Eqs.(32) and (33) reduce to Eq.(34) only for T » TC. That is to say, the delay time T of the probe pulse must be much larger than the correlation time. Further, in the slow modulation limit (inhomogeneously broadened transition) the dependence JS(T) for T 3> r c is the following (Ref.[116], Eq.(17)): JS(T)

~ exp(—a 2 r 2 ). In other words, in this case, formula (34) does not

255

provide information about the correlation time of the interaction with the surrounding bath. Such information can be obtained by the modification of a three-pulse four-wave-mixing experiment to the population grating configuration [Fig.2b] (see [120] and Subsec.7.1 below). Thus, formula (34) is of value in the case of fast or intermediate modulation of the frequency of a transition under study when T 3> r c .

5 5.1

N o n - M a r k o v i a n t h e o r y of s t e a d y - s t a t e R F P S Introduction and the cubic susceptibility in the case of Gaussian-Markovian random modulation of an electronic transition

A non-Markovian theory of steady-state RFPS has been developed in Refs.[19, 16, 20, 21, 39, 121, 40-43]. In Ref.[19] only non-Markovian corrections to the Markovian approximation were considered: a situation not appropriate to the broad inhomogeneously broadened bands of dyes. In Refs.[16, 121] the factorization approximation was proposed to calculate the cubic susceptibility. This approximation enables us to express the cross section for an

256

arbitrary multiphoton process in terms of ordinary single-photon line-shape functions. It is exact in the Markovian limit, however it is not a good approximation in the extreme non-Markovian case corresponding to the broad inhomogeneously broadened dye bands. A stochastic theory of steady-state RFPS describing in a continuous fashion, a transition from the Markovian limit (a homogeneously broadened optical spectrum) to the extreme non-Markovian case (an inhomogeneously broadened optical transition), has been developed in Refs.[20, 21]. Eq.(13) for the steady-state case can be written as

Vi3)+ = l E mm'm"

where Xabidi^s) X$L(u.)

1S

^

e cu

£xiLK)£m«>£m
(35)

bed

bic susceptibility,

= 2NL4h-3{D:DbDeD-d)or

£

Q(un,uj'm,^),

(36)

mm'tri"

the quantities Q (u>m, uj'm, w^) determine the frequency dependences of the susceptibility. In the case of the Gaussian-Markovian random modulation of an electronic transition, the quantities ( ^ ( w m , ^ , ^ ) have the following form [20, 21]: r\ I

Q (u>m,um.,um.,)

\

• 2 V~* P

= -irc

^ n=o

ft™

\xs)

— s^frp-i. , \ n - i (wm - um>) + (7\ + n/Tc) x[i? n (a; m ) + ( - l ) n J R n ( a ; m 0 ] ,

(37)

257

where Xj -

[T^1TC/2

+ p2J + ir c Awj; j = s, m,

TO';

Aus = w2i - w s , Aum,

=

u.'2i — w m /, Aa; m = w m — u>2i;

#„

(XJ)

= n!$ (ra + l , i j + n + l;p2)

[XJ

(x3 + 1)... (XJ + n)]~\ $(n + l,Xj + n + l;p2)

is a confluent hypergeometric function [122]. Eq.(37) is convenient for calculations for the cases of fast (p 1) modulation [21]. In the last case its convenience is confined to detunings

|fi| « ( « V ) 1 / 3

(38)

(fi = wi — w2) as applied to spectroscopy based on a resonance mixing of the Rayleigh type [35, 1, 36, 5]. For detunings T~1 <

|fi| <

(aVj" 1 ) 1 / 3 ,

\Q\ ~ Ifil -1 / 2 . For larger detunings fi, the quantity Q can be calculated for a more general and more realistic model of a complex molecule in a solution than the model of Gaussian-Markovian modulation (see Subsec. 5.3).

5.2

Model for frequency modulation of electronic transition of complex molecule in solution

The effect of the vibrational subsystem of a molecule and a solvent on the electronic transition can be represented as a modulation (a quantum

258

modulation, in the general case) of the frequency of the electronic transition. According to Eqs.(18) and (19)), when the quantity u is Gaussian, a four-photon light-matter interaction can be completely described by the correlation function K(t)

or the corresponding power spectrum $(w) =

(2w)~1 Jf^ A'(£)exp( —iu>t)dt, expressing the Fourier transform of K(t).

It

is obvious that $(w) has maxima in the regions corresponding to the optically active (OA) vibrations i.e., vibrations which change their equilibrium positions when the electronic transition occurs (see Fig. 6). The molecular electronic transition model under consideration includes two groups of OA vibrations [39, 40, 101, 43, 120]: low-frequency (LF) high-frequency (HF) (huh > kT).

(HUJS

< 2kT) and

Accordingly, K[t) = Ks(t) + Kh(t)

and

g(t) = gh{t) + 9s{t)- The corresponding contributions to the spectrum $(u>) are $s(u>) and

$A(U;): $ ( W )

= $ s (w) + $/,(w). It follows from the relationship

$ ( - w ) = $(w) exp[-/»w/(fcT)]

(39)

that the HF part of the spectrum $ji(w) is localized mainly in the region, corresponding to the frequency of the HFOA vibrations: 1000 — 1500cm - 1 . As to the OALF vibrations, the value of Ks(0) — /f^ $a(u})dw = h2a2s is determined by the area included between the curve $ 4 (w) and the LO axis

259

3>(OJ)

$saM

<S>h(cu) *sC|(w)

0|q"

aj h

QJ

Figure 6: Approximate power spectrum of the vibrational perturbation for the model of the electronic transition of a complex molecule in solution. \q"\ =
The methods of RFPS al-

low us to find such parameters of the distribution $(w) as q1. T and q", determining the vibronic relaxation of the optical transition. In the special case of shifted adiabatic potentials, cr^s — Hi Si coth[/iu;„7{2kT)}u]2si,

\q"\=Y.iSi^Ja23.

260

(Fig.6). The quantity a2s is the contribution from the OALF vibrations to the second central moment of the absorption spectrum. In the situation considered, one can represent Ks(t) in the form of two contributions: a classical Kaci(t) and a quantum Ksq(t):

Ks(t)

= Ksc\{i) + Ksq(t)

and, correspond-

ingly, 4>s(w) = 3>sc/(w) + $ s ? (w). The classical part (huis -C 2kT) corresponds to intermolecular motions and also to the quadratic electron-phonon interaction [123, 39, 40]. The quantum contribution arises from the LF intramolecular vibrations [124, 43] and may stem from molecular librations, for which huis ~ 2kT.

It follows from Eq.(39) that the spectrum

$SCI(LU)

is

approximately symmetric relative to the frequency to — 0. An approximate dependence $(w) is shown in Fig. 6 for the model being considered in this part. Using the properties of the Fourier transform, one can show that the correlation time for Ks{t) (which determines the characteristic decay time of Ks(t)) is rs where r s _1 ~ max(q', \q"\) and \q"\ ~ LOS [43]. We consider that the condition of "strong heat generation" [123, 21, 39, 40, 43] (a2s S> ^)

is realized for the LF system {w s }. If the system {u)s}

is purely classical, then the fulfillment of the latter condition is guaranteed by the inequality hus
If the system {w s } is basically quantum,

then large shifts of the minima of the adiabatic potentials upon electronic

261

excitation (J2iSi 2> 1, where 5,- are the dimensionless parameters of the shift) are necessary for fulfillment of this condition. Because of the inequality <72aT* S> 1 in the case under consideration, there is a large parameter in the exponents in Eq.(18). This makes it possible to limit the expansion of these exponents to power series at the extremum points T\ = r 3 = 0 with an accuracy up to the second order terms with respect to T\ and r 3 [21, 116, 39, 40, 101, 43, 120, 99]:

Kfc{0, r3, n + T2 + T3, T2 + r3) = exp[G?(T 1 ,T 2) T 3 )], c

Kf * (0, T3,T2

Kf c ( 0 , r 2 + r 3 ,

+

T, + T2 +

r3, n + r2 + r3)

r3,

T3)

exp{~i2Tjmgs(T2) c

Kf *(0, Tl +

T2

+ r3,

(40)

+

G?{TUT2,

r3)](41)

+ r3, r3)

T2

where Gr(r 1 ? T 2 ,r 3 ) = - ^ [ r gs(r2) = dgs/dT2 and

2

T/>S(T2)

+ r2

T

2r 1 r 3 (EeV' s (r 2 ) ± z / m ^ ( r 2 ) ) ] ,

(42)

= A' s (r 2 )/A' s (0) is the normalized correlation

function of the system {w s }. If the system {HJS} is classical, then the term —2Imgs(T2) = ujst[l —

T/>.J(T2)]

describes the dynamical Stokes shift [120, 99]

where uost is the contribution of the LFOA vibrations to the Stokes shift between steady-state absorption and luminescence spectra.

262

5.3

Cubic susceptibility for detunings larger than reciprocal correlation time

5.3.1

Classical s y s t e m of LF motions

It follows from Eq.(13) that when detuning is \u>m — w m /| ^> r s _1 , we need to consider only the behavior of ij)a[T2) at small values of r 2 . Then for the classical system {w3} (hua -C 2kT) we obtain an approximate analytic expression, using Eqs.(40), (41),(42) [39, 40]:

Q(w r a ,w;,u;^)~-t/3 0 - 1 /(A)Mo), where /J,0 = [i(u>m - w3) + T^1]/'y/ff^, 01 = [i(um - w m ') + T^/q',

(43)

q' =

—i?e0 s (+O), Ref3o > 0; f(z) = ci{z) sinz — si(z) cos z, ci(z) and si(z) are the integral cosine and sine, respectively [122]. An approximate analytic expression (43) practically coincides with the rigorous dependence in the region important for comparison with experiment [43]. We will now consider the dependence |Q(w2,Wi,u>i)| of Eq.(43) (Fig. 7). If 2|fi|3/2/(<72s
\Q(W2,LO\,U}I)\

OC

(44)

Ifil - 1 / 2 and reversal of t h e above inequality

263

[Q(aj2,cu,,ai, >[rel. unit*

5

10

I<JL»|—LU2I/C

Figure 7: Dependence of \Q(u2,u)i,Ui)\

(curve 1) on the detuning for the

LF system (s) compared with the experimental results of Ref.[5] (points) obtained for rhodamine B in ethanol when c = 182cm - 1 . The open circles do not satisfy the condition \Cl\/q' > 1 and can not be compared with the theoretical curve 1. Curve 2 is the theoretical approximation to the experimental data of Ref.[5j obtained using two fitting parameters T[ and T2.

264 yields \Q{u>2,uii,uJi)\ oc |fi| - 2 . The inequality of Eq.(44) is also the condition of validity of the solution of Subsec.5.1 [see Eq.(38)], but the latter is not limited by the condition \tom — iomi\Tc 3> 1. Therefore, in the range where the solutions of Subsec.5.1 and this subsection are valid (this range is characterized by the dependence |Q(u;2,w1,a;1)[ oc | f i | - 1 ' 2 ) , we can match the solutions. The fact that the characteristic scale of variation of |<5(oj'2, u;i, OJJ) | with detuning wj — w2 is considerably greater than r c - 1 can be explained by the fact that, in addition to the Rayleigh scattering, a RFPS signal also includes contributions from the processes of multiphonon Raman scattering by the LF system (s) [21, 39, 40]. This follows directly from Eq.(37), which represents expansions typical of the theory of multiphonon processes. Experimental data [5] for a solution of rhodamine B in ethanol are in satisfactory agreement with the present theory by using only one adjustable parameter c = (iq'o^s) 1 '' 3 = 182cm - 1 (Fig.7), while the theoretical treatment in Ref.[5] requires at least two fitting parameters T2 and T[ for the same range 1 . Within the framework of the theory of this subsection, the ratio T2/T{ ~ 1 reported in Ref.[5] (and also in the experiments on rhodamine 6G x

In the case of rhodamine B one can take into account only the system (s) [39, 40].

265

reported in Ref.[6]) becomes understandable. The value of c just given and an estimate of the half-width of the subband of rhodamine B in ethanol 5u> = 1070cm- 1 (y/a^ = 454cm" 1 ) of Ref.[125] yield q' « 60cm" 1 (r s « 0.09psec). The inclusion of the HFOA vibrations enables us to explain the increase, observed in Ref.[5], of the frequency difference, for which

\Q(U)2,LOI,UJI)\

OC

(UJI — w
5.3.2

Quantum s y s t e m of LFOA vibrations

When detuning is large (|wm — wm#| S> T S _ 1 ), one can also obtain an analytic expression for the quantum system {u>s} (hu>s ~ 2kT) [42, 43]:

Q^LJUUH) where (32 = {9. + i2Y0)a^/3'',

~ att'^Kfro),

Reft > 0; a3s = id3gsq(0)/dt3

(45) is the contribution

of the LFOA vibrations to the third central moment of the absorption spectrum; I/Q = (T\ - 1 — iCl)
Jo

= p-'iciipvo)

21 o — i{il + vzsx*)

s i n ( ^ o ) - si(Pvo) cos(/3z/0)].

(46)

266

Eq.(45) is written for the special case 2r 0 — T^1 = 0 (see Refs. [42, 43] for general case). Let us discuss the dependence For detunings (|H| 3 /|a- 3s ) 1/ ' 2 <

1,

\Q{UI2,UI,UJ1)\, \Q(LJ2,U>I,U}I)\

determined by Eq.(45). CC

l ^ l - 1 / 2 and reversal of

the above inequality yields \Q(u;2,u>i,iVi)\ oc |fi| - 2 for detunings fi > 0. On the basis of the form of the integrand in Eq.(46), one should expect a more rapid decrease of the quantity

\Q(LO2,LOI,U>I)\

for positive detunings Q > 0

in comparison with the case of negative detunings fi < 0 since resonances arise when Q, < 0, in particular when D, = —a3sx2.

Such a behavior is

explained by the predominant contribution to the intensity of the us signal from the multiphonon Raman processes when the excitation frequency is LO2 and scattering frequency is u>\ [43]. It is clear that for such processes the corresponding probabilities will be larger for u>2 > U\.

One can note the

related mechanisms leading to the discussed behavior of \Q(ui2,UJI,LOI)\ when ui2 — u>i S> T 3 _1 and to the appearance of a red wing of the multiphonon Stokes Raman scattering [24]. The aforesaid conclusions are illustrated in Fig. 8. Experimentally, the effect has been observed for ethanol solutions of malachite green [44] (Fig. 9). Fig. 9 shows that the dependence \Q(u>2,u>i,u)i)\

267

I g j Q ( c a 2 , <JU ,, cu,) j

/

/

/

\ \ \

0

0

- I

- I

0

igdni/c,)

Figure 8: Dependences of

\Q(U2,(JJI,OJI)\

on the detuning for quantum LF

system (s) when Q > 0 (curve 1) and Q, < 0 (curve 2) for Toc^ Tr1^3 units.

= 0.25,

= 0.1, d = {
268

2

3

log ((lill/27rc)-cm) Figure 9: The experimental behavior of |x^ 3 '(n)| for a solution of malachite green in ethanol (Ref.[44]).

U1/{2TTC)

= 16077cm -1 for 1,2; w x /(2-c) =

17361cm - 1 for 3,4; ft < 0 for 1,3 and ft > 0 for 2,4. | x ( 3 ) ( n ) l units.

is in

arbitrary

269

is asymmetrical with respect to the sign of Q (the corresponding curves are not parallel). The values of responding values of

\Q(CJ2,^I,^I)\

\Q(U>2,LOI,U)I)\

for Q < 0 are larger than the cor-

for fi > 0. This circumstance indicates

the presence of a nonclassical system of HFOA vibrations. The structure observed near |fi|/(27rc) ss 230cm - 1 corresponds to the vibration of 230cm - 1 defining beats in the photoinduced changes in the transmission of the malachite green solution [79, 80, 101]. For a quantitative estimate of the vibronic relaxation parameters, the experimental curve 2 (Fig. 9) was compared with theoretical dependence for the classical LFOA system [43] (Fig. 10a) and Eq.(45) (Fig. 10b). With the assumption about the greatest contribution to the RFPS signal of the LF system (for excitation in the region of the 0 — 0 transition with respect to the OAHF vibration) the following estimates were obtained:^' ~ 100cm - 1 (rs ss 0.05psec), if one considers the LF system classical, and \q"\ = cr3s/a2s ~ 50crn - 1 (TS « O.lpsec), if one considers the LF system quantum. In these estimates we used the value a2s » 500cm - 1 , determined from the spectrum of the single- photon absorption of malachite green in ethanol. We note that the estimates obtained agree with the estimate TS ~ O.lpsec, which was obtained by a treatment within the framework of the theory

270

X

>s

S

-2

"V

\i

C3i O

\

\

-3

3

2

log[(|ft|/27Tc)-cnn]

Figure 10: Comparison of the theoretical dependences, for the classical LFOA system [43]-a and Eq.(45)-b with the part of the experimental curve 2 of Fig.9 for detunings ft > 90cm - 1 . |x ( 3 ) ( f i )l is in arbitrary units.

271

Ref.[101] of the results of direct femtosecond experiments [79, 80].

6

Four-photon spectroscopy of superconductors

Recently a number of papers have been published devoted to laser-induced grating spectroscopy [45-49, 127, 50] and femtosecond ultrashort pulse spectroscopy [128-135] of metallic and high- temperature superconductors.

A

new method has been proposed in Refs.[45, 47, 48] for the investigation of electron-phonon interaction in metals and superconductors on the basis of laser-induced moving gratings. Shuvalov et al. observed a well-defined dip in the nonlinear spectroscopy signal of superconducting Y-Ba-Cu-0 thin films based on a biharmonic pumping technique [46, 49, 50] (see Sec.2). The upper limit of the region of this dip corresponded to a value of 2A of the superconducting energy gap. It has been proposed in Refs.[128, 129] to use an impulsive stimulated light scattering [136-139] in order to study the energy gap in superconductors, since in the usual Raman spectrum the superconducting energy gap is very

272

weakly displayed. The corresponding method presents a real time optical spectroscopy of superconducting-gap excitation, and therefore it is essentially non-Markovian. In Refs.[128, 129] the signal was been calculated in scheme corresponding to an impulsive stimulated light scattering when ultrashort light pulses interact with a superconducting film (see Fig.2b; below we use r instead of T). In this method the energy Js of the signal k s generated due to the four-photon interaction of the type k s = k 3 + kj — k2, is measured. The beats have been predicted in the dependence of the energy Js on the delay time r of the probe pulse k 3 with respect to pump pulses ki and k 2 . The beats are due to oscillations of the charge density, and their doubled period determines the value of the superconducting band 2A. The examination was performed on the basis of the phenomenological BCS model [140] for an isotropic superconductor with a large correlation length. The signal electromagnetic wave with vector potential As is generated by a nonlinear current j^ 3 '(r, t) in the medium, created by waves Aj(j = 1,2,3):

A3{r,t)

= (l/2)a$(t)exp{-i[wt - k,(n + ib)r]} + c.c,

(47)

where |kj| = k = co/c. One can obtain for the positive frequency component

273

of the nonlinear current [128, 129]: / 3 > + (r, t) = j ( 3 ) + (t) V exp [iksr (n + *&)],

(48)

where

J' (3)+ W = -JT^T^ / ^ [J" (* " a ) + 7 " (* + ^ ^ ( s A ) a 3 (*) ^ P 4 (27rfic) Jo

(-iut) (49)

determines the time dependence of the current, the factor U describes the nonlocal character of the interaction [128, 129], Ip(t)=l-Re[a'1(t)a'2*(t)],

f (sA) = J0 (sA) Y0 (sA) - J x (sA) Yx

(sA),

Jn (sA) and Yn (sA) are Bessel functions of the first and second kind, respectively. Solving the Maxwell equations for the signal wave with current (48) we find the amplitude of the signal wave after the passage of the superconducting sample of thickness / A

TI

j (3)+ (*)[/exp(«;/) - K'1 sinhUl)},

as(l, t) = CK

where « = — kb + ikn.

(50)

274

In the experiment it is convenient to record the energy Js of signal k s , which is proportional to f dt\as(l,t)\2,

as a function of delay r . In the case

of sufficiently short pump and probe pulses tp
is determined by the function / ' :

JS(T)

~

/,2(TA).

This function for

large values of the argument has the asymptotic representation

/'(rA) ~

- [ 2 / ( T T T A ) ] COS(2TA).

The dependence / , 2 ( r A ) is shown in Fig.ll, in which the beats with frequency 4A are clearly seen. Using Eqs.(49) and (50), one can obtain an estimate of the ratio of the intensities of the signal and probe fields [128, 129]. For characteristic values of the parameters n ~ b ~ 3, I ~ 10~ 5 cm, A/UJ ~ 1 0 - 3 , this ratio is \as(l,t)\2/\a'3(t)\2

« l O - 1 5 ^ ^ , where Eit2 are the energy densities of the

pump pulses in terms of

J/m2.

275

Figure 11: Plot of fn versus r A .

276

7

Ultrafast spectroscopy with pulses longer t h a n reciprocal bandwidth of t h e absorption spectrum

The consideration of the transient RFPS up till now (see Sec.4) has been based upon the assumption that exciting pulses are very short with respect to all relaxation times in a system. In the case of inhomogeneously broadened transitions, one must distinguish two dephasing times [98]: reversible (~ &2

) a n d irreversible T" (T" >
of the absorption spectrum. Therefore, for the electronic spectra of complex organic molecules in solutions the shortest relaxation time corresponds to the reversible dephasing of the electronic transition which is about equal to the reciprocal bandwidth of the absorption spectrum (~ 10/sec). At present, pulses of such durations are used in ultrafast spectroscopy [55, 60, 61, 76, 75, 62, 63, 59], and they provide unique information concerning ultrafast intraand intermolecular processes. However, there are situations when the pulses long compared with a reciprocal bandwidth of the absorption spectrum have decisive advantages [81,

277

141, 142]. In the four-photon spectroscopy methods with pulses long compared with reciprocal bandwidth of the solvent contribution to the absorption spectrum ( tv 3> a2a

where <723 is the solvent contribution to the central

second moment of the absorption spectrum) [65-67, 81, 120], pump pulses of the frequency u create light-induced changes in the sample under investigation, which are measured with a time delayed probe pulse. Due to condition — 1/2

tp S> (T2s , pump pulses have a relatively narrow bandwidth and therefore create a narrow hole in the initial thermal distribution with respect to a generalized solvation coordinate in the ground electronic state (Fig. 12) and, simultaneously, a narrow spike in the excited electronic state. These distributions tend to the equilibrium point of the corresponding potentials over time. By varying the excitation frequency w, one can change the spike and the hole position on the corresponding potential. The rates of the spike and the hole movements depend on their position. The changes related to the spike and the hole are measured at the same or another frequency u>\ by the delayed probe pulse. Therefore, one can control relative contribution of the ground state (a hole) and the excited state (a spike) to an observed signal. i

This property of the spectroscopy with pulses tv ^> a2s

In

can be used for

278

U

U2(QS)

U|(QS)

Solvation

Cordinate •Q.

Figure 12: Potential surfaces of the ground and the excited electronic states of a solute molecule in a liquid: one dimensional potential surfaces as a function of a generalized solvent polarization coordinate.

279

the nonlinear solvation study, when the breakdown of linear response for solvation dynamics occurs [81, 141, 142] (see Sec.9). A theoretical description of the interaction of pulses comparable with relaxation times is an essentially more complex problem than that of pulses short with respect to all relaxations in a system. However, it is possible to develop a method of solving of such problems even in strong electromagnetic field (without using four-photon approximation) for pulses long compared with the electronic dephasing [143, 91]. This issue will be considered in Subsec.7.2. We will also present recent experimental results obtained by pulses long compared with the electronic dephasing [83] (Sec.8).

7.1

Theory of transient RFPS with pulses long compared with reversible electronic dephasing

We will use the general theory of Sec.3 for the model described in Subsec.5.2. The transient nonlinear optical response strongly depends on the relations between the intramolecular chromophore relaxation and solvation dynamics.

Numerous experiments [144-147, 65, 148, 62] show that the Franck-

280 Condon molecular state achieved by an optical excitation, relaxes very fast and the relaxed intramolecular spectrum forms within 0.1 ps.

Therefore,

we shall consider that the intramolecular relaxation takes place within the pump pulse duration. Such a picture corresponds to a rather universal dynamical behavior of large polar chromophores in polar solvents, which may be represented by four well-separated time scales [62]: an intramolecular vibrational component, and intermolecular relaxation which consists of an ultrafast (~ 100/s),l ~ 4ps, and 10 ~ lOOps decay components. As to the interactions with the solvent, they satisfy the slow modulation limit [99, 66, 120, 117, 86] in the spirit of Kubo's theory of the stochastic modulation [9] (see S e e l ) : T>25 » 1

(51)

As a consequence of condition (51), times Ti and T3 (see Eq.(13)) become fast [21, 39, 116, 40, 43]. Therefore, we can use Eqs.(40),(41),(42) and integrate the right-hand side of Eq.(13) with respect to them if the exciting pulses are Gaussian of frequency w [120]:

Sm{t) = S0exp[-{52/2){t

- tmf

+ iutm]

281

with pulse duration of tp = 1.665/6 » <72~1/2

(52)

As a result, Eq.(13) is strongly simplified [120, 99]:

mm'm"

oca

(53) where xlfcL^) *> r 2) *s the cubic susceptibility. It can be represented as a sum of products of "Condon" X^Cc,*^^,^)

an<

i

a

"non-Condon"

Babcd'v(T2)

parts x i L ( " , ' , r2) = ^ x j & a > , *, T2)B»bTcr{r2) where indices a, v? of XFC

an

(54)

d B%>Id show that the corresponding values are

related to nonequilibrium processes in the absorption (a) or emission {ip) (for more details see below). The "Condon" factors XFCa, v ( w i *> r 2) depend on the excitation frequency u,t and r 2 , but they do not depend on the polarization states of exciting beams. The "non-Condon" terms Babc^'v{r2)

do not depend

on to, but depend on r 2 and the polarizations of the exciting beams. The origin of the "non-Condon" terms Babc£,ip stems from the dependence of the dipole moment of the electronic transition on the nuclear coordinates D i 2 ( Q ) that is explained by the Herzberg-Teller (HT) effect i.e., mixing different

282 electronic molecular states by nuclear motions (see Sec.3). We are interested mainly in non-Condon effects in solvation. Therefore, we shall consider for simplicity high-frequency "intramolecular" vibrations as Condon ones. We consider the translational and the rotational motions of liquid molecules as nearly classical at room temperatures, since their characteristic frequencies are smaller than the thermal energy kT. Here we do not consider the rotational motion of a solute molecule as a whole. The corresponding times are in the range of several hundreds picoseconds for complex molecules and are not important for ultrafast investigations (< 10 — lOOps). In the ultrafast range such effects are only important for small molecules. One can take into account the influence of the rotational motion of an impurity molecule on P ' 3 ' by using approach [149].

7.1.1

Condon contributions t o cubic susceptibility

At first, let us consider a case of one optically active (OA) intramolecular vibration of frequency wo- Then the Condon contributions XFCa^i^i ^ ri) t ° the cubic susceptibility (54) can be written in the form [120]:

283 OO

x

J2

In(So/s'mh6o)Ik{So/s\nhdo)exp[-2Socoth.Oo

+ (n + k)60}

n,k=—oo

Here So is the dimensionless parameter of the shift of the equilibrium point for the intramolecular vibration u0 under electronic excitation, 0o = In(x)

is the modified Bessel function of first kind [122], F^

we/) = (2ira2s)~1/2exp[— (w — nw0 — ioei — {us)/h)2/(2a2s)}

hui0/(2kT), (u — nui0 —

is the equilibrium

absorption spectrum of a chromophore corresponding to a n-th member of a progression with respect to the vibration co0, us — W2s — Wis, w(z) = exp{-z2)[l

+ (2i/y/n)

exp(t2)dt]

T Jo

is the error function of the complex argument [122], za,v = {iS2[r2(2 +

S(T2))

- *(3 + 5(r 2 )) + tm„ + tm, + tm(l + S(r2))] + +u - w a , v (r 2 ) +

Uo(Tk

+ n5(r 2 ))}/(2

(57)

is the time-dependent central second moment of the changes related to nonequilibrium processes in the absorption and the emission spectra , at the active pulse frequency u>,

uwfo) = ^ ±

'Y

+ S(T2)[U - (We/ ± ^ ) ]

(58)

284

are the first moments related to the solvent contribution to transient absorption (a) and emission (
= (us(0)us(t))

— (u s ) 2 , S(t) is the normalized

solute-solvent correlation function,
— (us)2) is the solvent

contribution to the second central moment of both the absorption and the luminescence spectra. The terms w(za^)

on the right-hand side of Eq.(55)

describe contributions to the cubic polarizations of the nonequilibrium absorption and emission processes, respectively. The third term on the right-hand side of Eq.(57) which is proportional to S2/cr2s, plays the role of the pulse width correction to the hole or spike width. This term is important immediately after the optical excitation when r 2 w 0 and, therefore, Sfa)

ss 1. The first term on the right-hand side of Eq.(56)

which is proportional to 52 ~ l/t2, takes into account the contribution of the electronic transition coherence. It is worth noting that Eqs.(53), (55),(56),(57),(58) describe in a continuous fashion a transition from the time frame in which coherent effects like photon echo exist to the time range where reversible dephasing disappears

285 [120, 117]. For acting pulse durations tv satisfying the condition

<72~1/2 « tP « K/<x 2 *) 1/3 = T'

(59)

Eqs.(53), (55),(56),(57),(58) describe the effects of two-pulse and three-pulse (stimulated) photon echo [120, 117]. For example, ignoring the vibration wo, one can obtain from these equations at the specified conditions for two-pulse excitation [120]: V^+

~ exp[— (S2/6)(t —

2T)2],

i.e., a photon echo appears

in the system. Here r is the delay time between the first and the second pulses. When tv » V

(60)

and the pump and the probe pulses do not overlap in time, one can ignore terms ~ S1 in Eqs.(56) and (57) [120, 117]. In the last case, Eq.(53) can be used for any pulse shape, and the cubic susceptibilities Xabcd(UJ^^T'^) and XFCa,
286 Thus, the quantity T" = (r 5 /cr 23 ) 1/ ' 3 plays the role of the irreversible dephasing time in the system under consideration [120, 117, 91]. Such an interpretation of T" is consistent with the behavior of the four-photon scattering signal excited by biharmonic pumping (see Eq.(44)).

7.1.2

Nonlinear polarization in a Condon case for nonoverlapping p u m p and probe pulses

The consideration of Subsec. 7.1.1 is confined by the Gaussian character of the value us = W2s — WXs. For nonoverlapping pump and probe pulses when condition (60) is satisfied, a nonlinear polarization in a Condon case can be expressed by the formula [150, 67]:

PNL+(r,t)

=^ND12(-D21E{T,t)){i[Fa(LJ,u,t)

-

Fv(u>,u,t)]

+[$a(w',w,t)-$„(<","',*)]}

(61)

for any us. Here 3

E(r,t)

= J2

Sm{t)exp{ikmr),

771 = 1 OO

/

^ ' F Q , V M ( W ' ) ^ C , I ¥ . « ( W I - ujei ~ w', w, t) -OO

(62)

287 are the spectra of the non-equilibrium absorption (a) or luminescence (ip) of a molecule in solution, Fa>lfi3(ij',u,t)

1 = —

r00

Z7T J-oo

dT1fc,tV,s(Ti,t)exp(-iuj'T1)

(63)

and 1 f°° Fa,vM(u') = — / ^ri/ a , ¥ , M (T-i)exp(-2u;'T 1 ) ZlT

(64)

J-oo

the corresponding "intermolecular" (s) and "intramolecular" (M) spectra; $ a , v (wi,u;,i) = 7r X P /

du'-^j

J — oo

'(J

(6o)

— CJj

are the non-equilibrium spectra of the refraction index which are connected to the corresponding spectra Fa^(u>i,ui,t)

by the Kramers-Kronig formula,

P is the symbol of the principal value. fa,VM(Ti) = TrM[exp(±(i

/h)W2iiMTi)

exp(^(i /h)W1>2MTi)pi,2M]

(66)

are the characteristic functions (the Fourier transforms) of the "intramolecular" absorption (a) or emission (
exp(-/W1)2M)

is the equilibrium density matrix of the solute molecule,

fa,Vs(ri,t)

= Tra[exp((i/h)u3Ti)pias(t)]

(67)

288 are the characteristic functions of the "intermolecular" absorption (a) or the emission (
A T ( r ) ~ -u>[Fa(u>,u,T) -

F^(UJ,U,T)]

(68)

and Aa(u')~-[Fa(u

+ Lj',u},T)-Fv{u>

+ u]',u,T)]

(69)

Eqs.(68) and (69) have been obtained for pump pulse duration shorter than the solute-solvent relaxation time. The formulae of this subsection are not limited by the four-photon approximation because they are based on the approach of Refs. [143, 91] (see Subsec. 7.2), which has been developed for solving problems related to the interaction of vibronic transitions with strong fields.

289

7.1.3

Non-Condon terms

Let us consider the non-Condon terms in Eq.(54) for Xabidi^^i^)-

They

have the following forms [99]:

BlT\T2)

= J

Jdpdv(aab(v)adc(t))orexp{-2j2[(Ql3(0)) 3

x ( ^ 2 + v) + 2fijisjysj{T2))

+ i5mtfidsju3{l

- *„-(T2))]}

(70)

where m — a,
M

J dQsaab(Qs)exp(-i{/Qs)

(71)

is the Fourier-transformation of the tensor
(72) (Qsj{0)Qsj{T2))/(Q2sj{0))is

the correlation function, corresponding to coordinate Qsj- If this vibration is an OA one, then the solvation correlation function S(T2) is related to the correlation functions ^sj(r2).

In the classical case this relation is [83]

S(T2) = J2"st,3ysj(T2)/ust

(73)

3

where uistj is the contribution of the j-th. intermolecular motion to the whole "intermolecular" Stokes shift wsi (iost = Ylj^st^)-

S(T2) can be considered

290

as an average of the values

^/(T2)

distributed with the density w s i j/u; s t . If

the non-Condon contribution is due to a non-OA vibration which does not contribute to the Stokes shift w st , then \P(r 2 ) is not related to

S(T2).

The second addend in the square brackets in Eq.(70) describes the interference of the Franck-Condon and Herzberg-Teller contributions. The value of the parameter dsj can be expressed by the following equation [83]:

\dtj\ = (aw^) 1 ' 2 /^

(74)

For freely orientating molecules, the orientational averages {d'ab{v)<Jdc{iJ'))or can be expressed by the tensor invariants a0, hs and ha [99] (see Appendix A). In the last case the values Babcj

;

can be expressed by the values [99, 83]

-\-i8mifidsjVj(l - * s j(r 2 ))]}

h.fav) ha(&, v)

related to the tensor invariants.

( 75 )

291

7.2

Nonlinear polarization and spectroscopy of vibronic transitions in the field of intense ultrashort pulses

The four-photon approximation used up till now is inadequate in a number of cases. These are the application of intense ultrashort pulses to femtosecond spectroscopy [152], the transmission of strong pulses through a saturable absorber and an amplifier of a femtosecond laser and so on. In Refs.[143, 91] the problem of calculating the non-linear polarization of electronic transitions in a strongly broadened vibronic system in a field of intense ultrashort pulses of finite duration, has been solved. This problem is of interest as it involves two types of nonperturbative interactions: light-matter and relaxation (nonMarkovian) ones. This problem is similar to that of calculating chemical reactions under strong interaction [102, 153]. Let us consider a molecule with two electronic states (Eq.(3)) which is affected by electromagnetic radiation of frequency u:

E(r, t) = -E(r, t) exp(-zu;i) + c.c.

One can describe an electronic optical transition as an electron-transfer re-

292

action between photonic 'replication' 1' of state 1 and state 2 2 (or between state 1 and photonic 'replication' 2' of state 2) induced by the disturbance V(t) = — r>2i-E(t)/2.

The problem of electron transfer for strong interaction

has been solved by the contact approximation [102, 153], according to which the transition probability is taken as proportional to S(Q — Qo) where Q0 is the intersection of terms. The contact approximation enables one to reduce the problem to balance equations. A similar approximation can be used in the problem under consideration. One can describe the influence of the vibrational subsystems of a molecule and a solvent on the electronic transition within the range of definite vibronic transition 0 —>• k related to HFOA vibration (f« 1000 — 1500cm - 1 ) as a modulation of this transition by LFOA vibrations {w s } (see Subsec.5.2). In accordance with the Franck-Condon principle, an optical electronic transition takes place at a fixed nuclear configuration. Therefore, the highest probability of optical transition is near the intersection Qo of 'photonic replication' 2

The wave function of the system can be expanded in Fourier series due to the periodic

dependence of the disturbance on time: ^{x,t) = Y^aa Vn{x,t) exp[—i(s + nu)t], where
293

Figure 13: The adiabatic potentials corresponding to electronic states 1,2 and their photonic 'replications' 1', 2'.

294

and the corresponding term (Fig. 13) and rapidly decreases as \Q — Q0\ increases. The quantity u„(Q) = W-23(Q) — Wi 3 (Q) is the disturbance of nuclear motion under electronic transition. Electronic transition relaxation stimulated by LFOA vibrations is described by the correlation Ks(t) =

(us(0)us(t))

of the corresponding vibrational disturbance with characteristic attenuation time TS (see Sec.3 and Subsec.5.2). <J2srs2 3> 1 for broad vibronic spectra satisfying the 'slow modulation' limit, where <72s = Ks(0)h~2

is the LFOA

vibration contribution to a second central moment of an absorption spectrum. According to Ref.

[120], the following times are characteristic for

the time evolution of the system under consideration: a2s where cr^

and X" =

(T^/O^S) 1 / 3

< X" -C rs,

are the times of reversible and irreversible

dephasing of the electronic transition, respectively (Subsec. 7.1.1). Their characteristic values are a^s1/2 s» 10 _ 1 4 5, V « 2.2 x 10 - 1 4 5, rs sa 10 - 1 3 s for complex molecules in solutions. The inequality r s 3> T" implies that optical transition is instantaneous and the contact approximation is correct. Thus, it is possible to describe vibrationally non-equilibrium populations in electronic states 1 and 2 by balance equations for the intense pulse excitation (pulse duration tp > T"). This procedure enables us to solve the problem for strong fields.

295

7.2.1

Classical nature of the LF vibration s y s t e m and the e x p o nential correlation function

We suppose that Huis <§C kT.

Thus {uis} is an almost classical system

and operators Wns are assumed to be stochastic functions of time in the Heisenberg representation. ua can be considered as a stochastic Gaussian variable.

We consider the case of the Gaussian-Markovian process when

Ks(t)/Ks(0)

= S(t) = exp(—\t\/r s ).

Using Burshtein's theory of sudden

modulation [11, 102, 153], one can obtain the approximate balance equations for the density matrix of this system when it is excited with pulses of duration tp » (r s /<7 23 ) 1/3 [143, 91]: j f P j j (a',t) = (-l)jh-2

(ir/2)8(u21

-u> -a')

\V21E(t)\2A(a>,t)+L33p33

(a',t) (76)

where j = 1,2; a' = —u/H, u>2\ is the frequency of Franck-Condon transition 1 —y 2, A (a', t) = p\\ (a', t) — p22 (<*', t). The operator L33 is determined by the equation:

Ljj

— Ts

\ , / r d 1 + (a - Sj2u.t) d{a'-5 a/ysJ 32utsti) i

N

+

d2 2 , r j2Lost,)2 ' d{a'-5

^2*^7-;s

(77)

5ij is the Kronecker delta, wst is the Stokes shift of the equilibrium absorption and luminescence spectra. The partial density matrix of the system p33 (a', t)

296

describes the system distribution in states 1 and 2 with a given value at time t. Eq.(76) corresponds to the contact approximation [153]. The complete density matrix averaged over the stochastic process which modulates the system energy levels, is obtained by integration of pjj (a', t) over a':

Pn(t) = Jft,K<)

da'.

(78)

The positive frequency component of the nonlinear polarization is expressed in terms of A = pu — p22 [143, 91]:

P ^ + (t) = -±-ND12aa in xf

drlD2l

(u;21) (2rr
£

exp{-i<7 (r) r\ - i [Uj (r) - w] n},(79)

where a (r) = a2s [1 — S2 (r)] is the time-dependent central second moment of spectra, aa (w2i) is the cross section at the maximum of the absorption band, / (t) is the power density of the exciting radiation, uij (T) = w21 — $jv>ust + (ui — w2i + $j, t — r ) is the solution of the integral equation:

A' (t) = 1 - aa (OJ21 ) f drI(r)A'(T)R(t-r), Jo

(80)

297

where A' (t) = A (w21 - to, t) / A (w2i - w, 0), R (t) = [a (t) /<7 2 s ] - 1 / 2 Ej=a,