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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Phys Chem B. Author manuscript; available in PMC 2010 July 16.
Published in final edited form as:
PMCID: PMC2905168

Suppression of Photon-Echo As a Signature of Chaos


The paper discusses the effect of quantum chaos on photon-echo signals of two-electronic-state molecular systems. The temporal profile of photon-echo signals is shown to reveal key information about nuclear dynamics on the excited electronic state surface. Specifically, the suppression of echo signals at a particular value of the delay time τ1 between the first and second excitation pulses is demonstrated as a signature of quantum level statistics that corresponds to the classically chaotic nuclear motion in the excited electronic state surface.

I. Introduction

A great deal of theoretical work has been devoted to studying the signatures of chaos in quantum systems.18 It has been shown that systems with regular dynamics have a Poisson distribution of energy level spacings, while systems with chaotic dynamics have level statistics similar to that of the Gaussian orthogonal ensemble (GOE) of random matrices. Obtaining level statistics from an experimental spectrum has practical difficulties;911 thus, it is interesting to find effects of different level statistics on time domain signals, that is, quantum signatures of chaos in the time domain. Time domain experiments provide an opportunity to find the signatures of chaos without the necessity of resolving level statistics. In the present paper, we propose a photon-echo experiment which reveals an information on level statistics from a time domain echo signal.

Although the literature on the universal level statistics in strongly chaotic systems is controversial, the basic property of quantum chaos is the existence of energy level repulsion. One can think of level repulsion as being the result of the interaction between the “good quantum numbers” when the system changes its dynamics from regular to chaotic. For the Sinai billiard, which is a strongly ergodic classical system, it has been shown that its spectral fluctuations are similar to that of a random matrix of the Gaussian orthogonal ensemble.12 It is assumed that the same result remains valid for all chaotic systems. For the convenience of analytical derivations, we assume GOE statistics of eigenstates with classically chaotic dynamics in the present paper. Yet, in section IV, we show that any form of spectral correlation can be used to obtain the information about spectral fluctuations from the time domain photon-echo signal.

The dynamics (either regular or chaotic) that underlies particular energy level statistics is of interest to chemical physicists. In the present paper, we consider a model of a polyatomic molecule with two electronic states. Nuclear energy levels of the excited electronic state obey either Poisson or GOE nearest-neighbor statistics, corresponding to regular or chaotic dynamics, respectively. Nuclear dynamics of multidimensional motion on the ground electronic potential energy surface is assumed to be quasi-periodic with Poisson statistics of nuclear levels. Poisson statistics of vibrational energy levels in the ground electronic state was observed in a lower energy range, for instance, for the molecule of N2O.13 In general, two independent anharmonic spectra can be sufficient to form a Poisson statistics.

The basic idea in searching for a time-domain signature of level statistics lies in averaging over the ensemble of time-dependent superposition states. Consider a quantum state |ψright angle bracket, which is a superposition of two eigenstates |n1right angle bracket and |n2right angle bracket that correspond to eigenvalues En1 and En2, respectively; then, after coherent excitation of |ψright angle bracket, it will dephase due to the factor exp{i(En1En2)t/[variant Planck's over 2pi]}. The average over an ensemble of states |ψ(t)right angle bracket in some cases is equivalent to the average over level spacings En1En2, resulting in different time domain signals (because of the connection of time and level spacings in exp{i(En1En2)t/[variant Planck's over 2pi]}) for different level spacing statistics. Pechukas was the first to propose the idea that the average survival probability P(t) = |left angle bracketψ(0)|ψ(t)right angle bracket|2 behaves differently for systems with chaotic and regular dynamics.14 This idea was further developed by Wilkie and Brumer15,16 to show that the time-resolved fluorescence depends on the average survival probability and therefore carries signatures of quantum chaos. Yet, information from a fluorescence experiment is hidden behind a fluorescence decay due to radiative damping. In the present paper, we propose another type of optical experiment, a nonlinear photon-echo experiment, and show that it can avoid the effects of dephasing and reveal the necessary information about level statistics. A photon-echo technique is well-known for its capability to remove the effects of inhomogeneous line broadening. Homogeneous line broadening effects cannot be removed in a photon-echo experiment, resulting in a signal decay that hides the necessary information contained in the signal’s temporal profile, similarly to the fluorescence experiment. However, a nonlinear photon-echo experiment is an ultrafast experiment and allows one to resolve much smaller time scales than the fluorescence experiment discussed in ref 15. In this paper, we show that level statistics from a photon-echo experiment yields a universal time scale 4τ, where τ is a duration of the laser pulse. Given with the average signal decay rate [Gamma with macron] due to homogeneous line broadening mechanisms, one can always pick a laser pulse that will satisfy τ [double less-than sign] 1/[Gamma with macron] and thus obtain clean information about the signal at time 4τ, which is not possible in the fluorescence experiment.

Consideration of a nonlinear experiment to extract information about chaos is also interesting in the context of recent studies of the effect of chaos in classical response theory. It was suggested by Mukamel and co-workers17 that classical nonlinear response functions are good indicators of chaotic dynamics since stability matrices diverge linearly in time1822 for systems with quasi-periodic dynamics and exponentially for systems with chaotic dynamics. Chernyak and co-workers have recently shown23,24 that classical nonlinear response functions exhibit frequency domain signatures of chaotic motions.

The present paper is organized as follows. In section II, we describe the nonlinear experiment and analytically derive the expression for the third-order polarization. In section III, we consider the differences in a photon-echo signal for systems with regular and irregular dynamics. In section IV, we discuss the effects of impurities of the spectral level statistics on the photon-echo signal. In section V, we discuss the suppression of the photon-echo signal at time τ1 = 4τ for chaotic systems.

II. Theory Section

We consider a system with two electronic states, ground |gright angle bracket and excited |eright angle bracket. The adiabatic Hamiltonian of the system is given by


where Hg is the nuclear Hamiltonian on the ground electronic potential surface, He is the nuclear Hamiltonian on the excited electronic potential surface, and ωeg is the electronic gap between the minima of both potentials (Figure 1). The nuclear dynamics of interest (either regular or chaotic) corresponds to Hamiltonian He, and thus, the statistics of nuclear energy levels in the excited electronic state is assumed to be either random (Poisson ensemble) or correlated (Gaussian orthogonal ensemble). Physically, only particular areas of the energy level spectrum of He obey particular level statistics; at low energies, nuclear dynamics is mostly quasiperiodic, and thus, the corresponding level statistics should be that of the Poisson ensemble, while at high energies, it can be chaotic with the corresponding statistics of the GOE. By changing the carrier frequency of the excitation pulse, we can select the energy region of interest.

Figure 1
The molecular level scheme for a two-level system.

The most common technique in nonlinear spectroscopy is a three-pulse photon-echo experiment. In this experiment, a system is irradiated with three subsequent pulses with delay periods of τ1 and τ2 between them. The measurement is done at time t after the third pulse (Figure 2). The electric field acting on a system is

Figure 2
Three-pulse photon-echo experiment depicted in (a) space and (b) time.25


where ωj and kj are frequencies and wave vectors of the incident waves and Ej(t) denotes the temporal envelope. We assume that all three pulses have the same frequencies ω1 = ω2 = ω3 = ω0 and temporal envelopes E(t) = E0 exp(−t2/2τ2), although they have different wavevectors kj. The photon-echo signal is measured in the direction ks = k3 + k2k1.25 The corresponding nonlinear polarization is given by25


where the two response terms in the photon-echo signal are


Here, [mu] is an electronic dipole moment operator, ρg=t=1Nangngn is a ground-state nuclear density operator, with an as the population of the nth vibrational level |gnright angle bracket of the ground electronic state and N as the total number of initially populated ground vibrational states. The distribution of populations is Boltzmann, that is, an = exp(−βEn), where β = 1/kT.

Assuming that pulses do not overlap, that is, t, τ2, τ1 > τ, (which is actually the necessary condition for deriving eq 3), we can set the lower limit for the integrals in eq 3 to −∞. Using a completeness relation Σ|nright angle bracketleft angle bracketn| = 1 in eqs 4 and 5 repeatedly, we obtain


By plugging eqs 6 and 7 into eq 3 and performing integrations, we get


where we denote


Here, Eng and |gnright angle bracket are the nth eigenvalue and the nth eigenstate of the Hamiltonian Ĥg, respectively; Eve and |evright angle bracket are the vth eigenvalue and the vth eigenstate of Ĥe.

Matrix elements left angle bracketei|[mu]|gjright angle bracket can be positive or negative depending on i and j. For systems with a chaotic classical limit, the distribution of matrix elements is shown to be Gaussian and centered at around zero.2628 We assume that near-symmetrical distribution of matrix elements around zero also holds for systems with regular dynamics; we illustrate this, in particular, in the Appendix on the example of a two-dimensional square-well potential. The result of summation (eq 8) is therefore determined by terms that contain squares of coefficients left angle bracketei|[mu]|gjright angle bracket, that is, by terms with {k = n, vu}, {v= u, kn}, and {k = n, v= u}. Let us consider these three cases separately, denoting contributions from each of them as Pa(3)(t),Pb(3)(t), and Pc(3)(t), respectively. The contribution from summation {k = n, vu} is


where Σ′ indicates the exclusion of terms with u = v.

In the Condon approximation, matrix elements left angle bracketgn|[mu]|euright angle bracket can be represented as a product of an electronic dipole matrix element μ0 [equivalent] left angle bracketg|[mu]|eright angle bracket, which is a constant, and a multidimensional Franck–Condon factor Snu [equivalent] left angle bracketg, νn|e, νuright angle bracket, which is an overlap between multidimensional nuclear wave functions. We now consider the question whether a multidimensional Franck–Condon factor Snu can be considered as an independent random variable in summation over n and v, that is, independent of the difference of the corresponding eigenvalues εnεu. In the case of chaotic motion, that was shown to be true;28 for any given eigenstate |g, νnright angle bracket, the Franck–Condon factor left angle bracketg, νn|e, νuright angle bracket is a Gaussian random variable, independent of eigenvalue. The reason is that an eigenstate of a classically chaotic system can be represented as a superposition of plane waves with random phase,28 which is intrinsically independent of the eigenvalue, thus resulting in the random overlap integrals, that is, Snu, independent of eigenvalues. In the case of regular motion, the independence of Snv and εnεv is not obvious. In the Appendix, we consider an example of the simplest multidimensional regular system, a two-dimensional infinite square well, which allows analytical treatment. There, we show that the model of a square-well potential allows one to consider Franck–Condon factors as independent random variables. We therefore can assume similar independence of the Franck–Condon factors for the generic regular system. The independence of the Franck–Condon factors means that the summation (eq 10), which is an average ΣM… = Mleft angle bracketright angle bracket, will result in the product of averages left angle bracketf(Sij)g(εiεj)right angle bracket = left angle bracketf(Sij)right angle bracketleft angle bracketg(εiεj)right angle bracket


where Snu2Snv2 is the average of products of squared Franck–Condon factors for the vertical transitions from the N ground vibrational states, uv.

The last summation over n in the above equation is the averaging of the expression under the summation sign over the different values of εn, n=1N=N. However, since εn is a random spectrum, its distribution density is known; it is a uniform distribution with the density ρ(ε) = [variant Planck's over 2pi]/Nleft angle bracketΔEright angle bracket0, 0 < ε < Nleft angle bracketΔEright angle bracket0, where left angle bracketΔEright angle bracket0 is the mean level spacing in the ground potential. Therefore n=1N=N(/NΔE0)dε=(/ΔE0)dε, and we can integrate εn out (setting the upper limit of integration to infinity because of the decay coefficient exp[−[variant Planck's over 2pi]βε]) to have


where we have introduced a new variable Δr [equivalent] Δuv= εuεv, which stands for the distance between nearest r levels (rth nearest-neighbor distance). We can neglect a nonconstant behavior of the error function (boundary effects) in the very small region εv [set membership] {−2π/τ, 2π/τ} and consider the error function in eq 12 as a step function, which equals 1 in the interval εv [set membership] {0,∞} and 0 outside. Equation 12 thus takes the form


Again, the summation over v is an averaging over the variable εv. The position in spectrum εv and the distance to its nearest neighbor Δr(v) are independent variables and therefore can be averaged out separately. This results in


where the last averaging is due to the summation over v. Since Δr has different values at different parts of spectrum, then the averaging over v results in the average over Δr. The latter can be done using nearest-neighbor distribution functions, which are known functions for both Poisson and GOE statistics.15 Thus, we have


One can see that the statistics energy spectrum enters the above expression as a sum ρ(Δ) = Σrρr(Δ), which is just a two-level density of states.29 For now, we postpone a further consideration of the expression in eq 15 until the next section and continue with the remaining contributions Pb(3)(t) and Pc(3)(t). Denoting the expression in eq 15 with F(t), we get the final expression for Pa(3)(t) in the form of


where C=(i/)3[(2π)1/2τE0)3μ02Snu2Snv2(π/6)1/2[2/τΔE0][1/βΔE]e(β)2/6τ2 is a constant and where we have used v=0eβεv1/βΔE with left angle bracketΔEright angle bracket being the mean level spacing in the excited electronic potential surface.

Let us now consider Pb(3)(t); it reads


Using the same assumptions as those in the derivation of Pa(3)(t) and replacing summations n=1N and k=1 with the integral (/ΔE0)0dε, we get


where C=(i/)3[(2π)1/2τE0)3μ02Snv2Skv2π(2)1/2[(/τΔE0)2][1/βΔE]e(β)2/4τ2. Obviously, the contribution of this term to the overall nonlinear polarization is negligible when the conditions of pulse nonoverlapping t, τ1, τ2 > τ are satisfied.

The last term to consider is Pc(3)t


which simplifies to


The overall third-order nonlinear polarization reads


Here, we did not substitute for a small contribution of Pb(3)(t) in order not to overload the formula. One can see that at t = τ1, we have an echo.

III. F(t) for Two Types of Statistics

Obviously, F(t) carries the information about level statistics in the excited electronic state. We now consider the two cases of statistics separately.

A. Poisson Statistics

Systems with regular dynamics possess Poisson nearest-neighbor energy level statistics. For this statistics, energy levels are uncorrelated, and the two-level density of states is uniform


where left angle bracketΔEright angle bracket is a mean level spacing. Thus, F(t) for systems with regular dynamics reads




Here, we have introduced dimensionless parameters [alpha] = τleft angle bracketΔEright angle bracket/[variant Planck's over 2pi] and beta = β[variant Planck's over 2pi]/τ. The necessary conditions for the photon-echo experiment described in this paper look very simple in terms of these parameters; they are


The first condition means that the spectral width of the laser pulse should be greater than the mean level spacing in order to excite at least two states to form an excited superposition state, as discussed in the Introduction. The second condition in eq 25, which is βleft angle bracketΔEright angle bracket [double less-than sign] 1, defines the obvious requirement for the allowed temperature; it should be greater than the mean level spacing to populate several levels to form a statistics of levels.

A photon-echo signal measured in experiments is given by25


Substituting eqs 21 and 23 into the above integral results in monotonically decaying signals shown in Figure 3a,b.

Figure 3
Photon-echo signals for regular systems (a,b) and irregular systems (c,d) at a temperature of βleft angle bracketΔEright angle bracket = 0.05. Inset (d) contains plots for increasing values of τ2 from the bottom curve to the top curve; τ ...

B. GOE Statistics

For GOE statistics, the first nearest-neighbor distribution function is given by the Wigner distribution15


where left angle bracketΔEright angle bracket is a mean level spacing. In this case, the two-level density of states reads29


with c(ω) = sin[π[variant Planck's over 2pi]ω/left angle bracketΔEright angle bracket]/(π[variant Planck's over 2pi]ω/left angle bracketΔEright angle bracket). Numerical integration of eq 15 with eq 28 gives


where functions f[alpha]beta in the range of parameters given by eq 25 can be well-approximated by analytic functions

The plots of f[alpha]beta(x) and g[alpha]beta(x) are shown in Figure 4 for different temperatures. One can see that f[alpha]beta(x) has a clear minimum. Its

Figure 4
The functions f[alpha]beta(x) (solid line) and g[alpha]beta(x) (dashed line) for the different values of the inverse temperature β = 1/kT. The values of the parameters are [alpha] = ...


position is a nonlinear function of parameters [alpha] and beta, which can be found numerically; in the range of beta < 0.3, 0.4 < [alpha] < 1, it can be given by the approximate formula xmin = 1.25(beta/[alpha]1.39) + (2.61/[alpha]0.2) ≈ 2.61/[alpha]0.2. F(t) for the echo condition t = τ1 thus has the minimum at t ~ 2.6τ0.8.

Straightforward numerical calculation of the summation in eq 11 was also performed to check the obtained analytical results. For the energy spectrum εn, 400 levels with the mean level spacing of left angle bracketΔEright angle bracket = 1 were randomly generated on the interval (0, 400left angle bracketΔEright angle bracket) using a uniform distribution function. For the energy spectrum εv,u, in the case of regular motion, the same generation of the spectrum as that above was used. In the case of chaotic motion, 400 level spacings {Δi} were generated using the Metropolis algorithm with the Wigner distribution function (eq 27) and left angle bracketΔEright angle bracket = 1; the spectrum εv was then obtained as εv=l=1vΔi. The result of the summation in eq 11 is given in Figure 5 as a function of t for τ = 0.5, τ1 = t (echo condition), and different values of the inverse temperature β and τ2. The variation of τ2 does not significantly affect the position of the minimum of F(t); yet, it helps to average out the fluctuations of the numerical results due to a limited number of spectral lines, which may effectively become even smaller at lower temperatures. One can see that for the Wigner nearest-neighbor distribution, F(t) in Figure 5 has a minimum at t = 2.6τ0.8 = 1.5, in accordance with the analytical predictions.

Figure 5
The results of the numerical calculation of F(t) by evaluating the summations in eq 11. The top four curves correspond to the Poisson nearest-neighbor statistics; the bottom four curves correspond to the GOE nearest-neighbor statistics. Solid symbols ...

Calculation of a signal (eq 26) with eqs 29 and 30 is shown in Figure 3c,d. The χ(τ1, τ2) has a minimum at τ1 ~ 4τ for any given value of τ2, and its location along τ1 axis is independent of τ2. We call this minimum a suppression of photon-echo signal.

IV. Mixed Spectral Statistics

It is interesting that the minimum of the photon-echo signal at τ1 ≈ 4τ is not sensitive to the purity of the GOE spectral statistics or the type of correlated level statistics. The level statistics enters the expression for the photon-echo signal (eq 21) as a two-level correlation function ρ(ω). In some sense, the minimum of the photon-echo signal is directly related to the dip in ρ(ω) at ω = 0. As long as ρ(ω) is different from the uniform distribution of completely uncorrelated levels (eq 22), there will be a minimum in the photon-echo signal. For instance, if the spectrum is a mixture of the correlated and the uncorrelated levels, its nearest-neighbor statistics can be described with a Brody distribution, which is an intermediate between the Poisson and the Wigner distributions. The two-level correlation functions will look then as an intermediate between the expressions in eqs 22 and 28. As an example, the two-level density of states shown in the inset of Figure 6 results in the function F(t) shown in Figure 6. One can see that F(t) still has a clear minimum in the region of ~4τ. The depth of the minimum is proportional to the strength of the spectral correlation.

Figure 6
F(t) for the two-level density of weakly correlated states ρ(ω) shown in the inset. The magnitudes of the parameters are τ = 0.5, left angle bracketΔEright angle bracket = 1, and [variant Planck's over 2pi] = 1.

Interestingly, the type of the spectral level statistics in the ground electronic state is not important. In the present analysis, we have made an assumption about the random level statistics in the ground electronic state for easier analytical derivations, yet as one can see, the spectroscopic signal, and in particular its minimum, depends only on the term Pa(3)(t), which includes only the differences between energy levels εu and εv in the excited electronic state and therefore effectively probes only the level statistics of the excited electronic state. We will publish the results on the general level statistics elsewhere.

V. Results and Discussion

The main result of the present analysis is that the photon-echo experiment carried out with the conditions in eq 25 should result in the suppression of echo-signals at τ1 ~ 4τ for chaotic systems, where τ is a pulse duration. The time interval between second and third laser pulses, τ2, does not influence the location of the signal’s minimum along the τ1 axis. The suppression can be considerable; the general formula for the ratio χ(τ1, τ2)/χ(∞,∞) near the global minimum τ1 = 4τ and τ2 = 0 at high temperatures, βleft angle bracketΔEright angle bracket [double less-than sign] 1, is


where τ has dimensionless units of [variant Planck's over 2pi]/left angle bracketΔEright angle bracket. We can estimate the ratio assuming |Snu| and |Snv| are uncorrelated, uniformly distributed variables; then, Snu2Snv2/Snv4=S2S2/S4=5/9, which results in [χ(4τ,0)/χ(∞,∞)] → 0.36 and τ → 0. Thus, the suppression of the photon-echo signal can be up to 50%.

On the other hand, the photon-echo signal of regular systems χ(t1, t2) does not have any minima (Figures 3a,b). Thus, the following conditions always hold: χ(4τ, τ2)/χ(∞,∞) ≥ 1 for regular systems and χ(4τ, τ2)/χ(∞,∞) < 1 for irregular systems. In real experiments, χ(τ1, τ2) decays to zero due to different broadening mechanisms, but on the time scale of an ultrafast experiment, we can neglect broadening effects and thus consider the long time limit of χ(τ1, τ2) as a constant, which we plot in Figure 3 as χ(∞,∞). Since the location of the correlation minimum at τ1 = 4τ does not depend on τ2 (Figure 3d), we can make the above inequalities stronger by averaging over some interval of τ2. The latter averaging can remove experimental nonideality and thus provide more conclusive measurements.

The physics of the observed suppression of the echo signal is similar to the physics for the suppression of the averaged survival probability |left angle bracketψ(0)|ψ(t)right angle bracket|2 discussed in refs 14 and 15. The main idea is that since the energy levels obeying GOE statistics are correlated on the energy scale left angle bracketΔEright angle bracket, the superposition state |ψ(t)right angle bracket = Σ exp(−iEnt/[variant Planck's over 2pi])|nright angle bracket would remember its initial conditions on the time scale of Δt = [variant Planck's over 2pi]/left angle bracketΔEright angle bracket. This time scale defines the interval of quantum coherence, which will “survive” after the averaging over initial conditions and energy level statistics. During this time, |left angle bracketψ(0)|ψ(t)right angle bracket|2 would behave as a typical quantum dephasing process with oscillatory behavior around its average value due to quantum coherence effects. As a result, |left angle bracketψ(0)|ψ(t)right angle bracket|2 can go below its long time limit (it could have made several oscillations around its long time limit; however, the time of coherence Δt ends up earlier than the second oscillation). For the regular motion, however, the energy levels do not have any correlation, and thus, no time interval Δt of quantum coherence exists after the averaging over the ensemble of levels. Therefore, |ψ(t)right angle bracket is not correlated with its initial conditions and decays to its statistical average.

In the proposed experiment, the time domain signature of chaotic motion at 4τ arises from the minimum of the integrated signal and not from the response function itself. Although the response function is more fundamental since it is invariant under experimental conditions, only its convolution with the electric field yields the results of the present theory. The reason lies in a particular physical mechanism in which we are interested. Each eigenstate in the ground potential surface being irradiated by a laser pulse forms a superposition state in the excited electronic potential surface. It is the time evolution of a superposition state (when averaged over many mutually incoherent superposition states) that reveals energy level statistics in the excited electronic potential surface. To form a superposition state, we need an explicit presence of a shaped laser field in our theory. Since the information about level statistics is determined by a superposition state and the latter is determined by the parameters of a laser pulse, the information on level statistics should be determined by the parameters of the laser pulse. This is exactly what we have in our theory; the minimum of the photon-echo signal is located at approximately four pulse durations. The total number of the excited eigenstates Ne in the excited electronic potential is equal to the number of initially populated eigenstates Ng in the ground electronic potential multiplied by the ratio of the mean level spacings in the ground electronic potential and that in the excited electronic potential, Ne = Ngleft angle bracketΔEright angle bracket0/left angle bracketΔEright angle bracket. To have good statistics, we need to populate a considerable number of ground states Ng [dbl greater-than sign] 1, which suggests performing the experiment at high temperatures, that is, βleft angle bracketΔEright angle bracket0 [double less-than sign] 1.

VI. Conclusions

In this paper, we have shown that information about level statistics can be extracted from a time domain signal of the photon-echo experiment. Correlated (GOE) statistics of level spacings results in a suppressed photon-echo signal at τ1 = 4τ, whereas Poisson level statistics does not show a dip in the intensity of a signal. The main advantage of the proposed experiment is the implication of a nonlinear photon-echo technique which may overcome both homogeneous and inhomogeneous energy level broadening. The possibility to conduct the experiment for thermal ensembles makes it easy for practical applications.


We thank the anonymous reviewers for their valuable comments and suggestions and professor Bob Field for stimulating discussions. The research reported here is supported by Camille & Henry Dreyfus Foundation and by the U.S. Army through the Institute of Soldier Nanotechnologies at MIT.


In this appendix, we consider the simplest case of a multidimensional anharmonic system with regular dynamics, a two-dimensional infinite square-well potential, −Lx/2 < x < Lx/2 and −Ly/2 < y < Ly/2. Its eigenstates and eigenvalues are known to be


For the irrational ratio of Lx and Ly, the simple quadratic spectrum (eq 32) forms a random spectrum and results in a Poisson nearest-neighbor distribution. In the present analysis, we took Lx = 21/8Ly since it yields a Poisson distribution for the first 100 levels.

In application to the two-electronic-state problem considered in the present paper, we represent the ground electronic potential surface as a two-dimensional square well and the excited electronic surface with a two-dimensional square well twice the size, Lx = 2Lx and Ly = 2Ly; see Figure 7. We numerate energy levels Ei [equivalent] Enx,ny, i = 1, 2, 3,… in the ground well and Ej [equivalent] Enx,ný, j = 1, 2, 3,… in the upper well in ascending order. The Franck–Condon factors Sij of the overlap of states Ψnx,ny and Ψnx,ný that correspond to the two eigenvalues Ei and Ej, respectively, are

Figure 7
The model of a two-electronic-state system with two-dimensional square-well potential surfaces. Only one dimension, x, is shown in the figure. Numbers along the vertical axis numerate energy levels. A laser pulse with a spectral width of ΔΩ ...




If initially an ith eigenstate with an eigenvalue Ei is populated in the ground potential well, a laser pulse with a spectral width ΔΩ and frequency ω0 will excite eigenstates in the upper potential well with eigenvalues Ej that fit into the energy range of Ei + [variant Planck's over 2pi]ω0 ± [variant Planck's over 2pi]ΔΩ/2; see Figure 7. We can thus predict all of the Franck–Condon factors Sij which are involved in the excitation from the state |iright angle bracket; there will be, on average, [variant Planck's over 2pi]ΔΩ/left angle bracketΔEright angle bracket of them, where left angle bracketΔEright angle bracket is the mean level spacing in the upper potential well. In numerical analysis, we used Lx = 1, π2[variant Planck's over 2pi]2/2m = 1, left angle bracketΔEright angle bracket = 1.8, left angle bracketΔEright angle bracket = 0.45, and ΔΩ = 4, and the difference between [variant Planck's over 2pi]ω0 and the energy gap between the bottoms of potential wells is equal to 10. The Franck–Condon factors involved in the process of excitation from the first 200 states of the ground potential well are shown in Figure 8. Only the overlap integrals between the states which have eigenvalues within the spectral window ΔΩ of the laser pulse are considered. From Figure 8a,c, one can see that there is no correlation between the eigenvalues, or indices i and j, and the values of the Franck–Condon factors Sij. Figure 8c also shows that, for the considered square-well model, the values of the overlap integrals are symmetrically distributed around zero. We thus can consider eigenvalues and Franck–Condon factors to be uncorrelated.

Figure 8
Franck–Condon factors Sij for the pair of ground and excited two-dimensional infinite square-well potentials. (a) The overlap integral between the ith eigenfunction of the ground potential well and the jth eigenfunction of the upper potential ...


Part of the “Karl Freed Festschrift”.

Contributor Information

Maksym Kryvohuz, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.

Jianshu Cao, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.

Shaul Mukamel, Department of Chemistry, University of California, Irvine, California 92697-2025.

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