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- Abstract
- I. Introduction
- II. Theory Section
- III. F(t) for Two Types of Statistics
- IV. Mixed Spectral Statistics
- V. Results and Discussion
- VI. Conclusions
- References and Notes

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J Phys Chem B. Author manuscript; available in PMC 2010 July 16.

Published in final edited form as:

PMCID: PMC2905168

NIHMSID: NIHMS212342

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

The publisher's final edited version of this article is available at J Phys Chem B

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 *τ*_{1} 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.

A great deal of theoretical work has been devoted to studying the signatures of chaos in quantum systems.^{1}^{–}^{8} 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;^{9}^{–}^{11} 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 *N*_{2}O.^{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 |*ψ*, which is a superposition of two eigenstates |*n*_{1} and |*n*_{2} that correspond to eigenvalues *E*_{n1} and *E*_{n2}, respectively; then, after coherent excitation of |*ψ*, it will dephase due to the factor exp{i(*E*_{n1} − *E*_{n2})*t*/}. The average over an ensemble of states |*ψ*(*t*) in some cases is equivalent to the average over level spacings *E*_{n1} − *E*_{n2}, resulting in different time domain signals (because of the connection of time and level spacings in exp{i(*E*_{n1}− *E*_{n2})*t*/}) for different level spacing statistics. Pechukas was the first to propose the idea that the average survival probability *P*(*t*) = |*ψ*(0)|*ψ*(*t*)|^{2} behaves differently for systems with chaotic and regular dynamics.^{14} This idea was further developed by Wilkie and Brumer^{15}^{,}^{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 4*τ*, where *τ* is a duration of the laser pulse. Given with the average signal decay rate due to homogeneous line broadening mechanisms, one can always pick a laser pulse that will satisfy *τ* 1/ and thus obtain *clean* information about the signal at time 4*τ*, 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-workers^{17} that classical nonlinear response functions are good indicators of chaotic dynamics since stability matrices diverge linearly in time^{18}^{–}^{22} for systems with quasi-periodic dynamics and exponentially for systems with chaotic dynamics. Chernyak and co-workers have recently shown^{23}^{,}^{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 *τ*_{1} = 4*τ* for chaotic systems.

We consider a system with two electronic states, ground |g and excited |e. The adiabatic Hamiltonian of the system is given by

$$H=\phantom{\rule{0.16667em}{0ex}}\mid \text{g}\rangle {H}_{\text{g}}\langle \text{g}\mid +\mid \text{e}\rangle ({H}_{\text{e}}+{\omega}_{\text{eg}})\langle \text{e}\mid $$

(1)

where *H*_{g} is the nuclear Hamiltonian on the ground electronic potential surface, *H*_{e} is the nuclear Hamiltonian on the excited electronic potential surface, and *ω*_{eg} is the electronic gap between the minima of both potentials (Figure 1). The nuclear dynamics of interest (either regular or chaotic) corresponds to Hamiltonian *H*_{e}, 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 *H*_{e} 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.

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 *τ*_{1} and *τ*_{2} between them. The measurement is done at time *t* after the third pulse (Figure 2). The electric field acting on a system is

$$E(\mathbf{r},t)={E}_{1}(t+{\tau}_{2}+{\tau}_{1})exp(\text{i}{\mathbf{k}}_{1}\mathbf{r}-\text{i}{\omega}_{1}t)+{E}_{2}(t+{\tau}_{2})exp(\text{i}{\mathbf{k}}_{2}\mathbf{r}-\text{i}{\omega}_{2}t)+{E}_{3}(t)exp(\text{i}{\mathbf{k}}_{3}\mathbf{r}-\text{i}{\omega}_{3}t)$$

(2)

where *ω _{j}* and

$${P}^{(3)}({\mathbf{k}}_{\text{s}}={\mathbf{k}}_{3}+{\mathbf{k}}_{2}-{\mathbf{k}}_{1},t)={\left(\frac{i}{\hslash}\right)}^{3}{\int}_{0}^{\infty}\text{d}{t}_{3}{\int}_{0}^{\infty}\text{d}{t}_{2}{\int}_{0}^{\infty}\text{d}{t}_{1}[{R}_{2}({t}_{3},{t}_{2},{t}_{1})+{R}_{3}({t}_{3},{t}_{2},{t}_{1})]\times {E}_{3}(t-{t}_{3}){E}_{2}(t+{\tau}_{2}-{t}_{3}-{t}_{2}){E}_{1}^{\ast}(t+{\tau}_{1}+{\tau}_{2}-{t}_{3}-{t}_{2}-{t}_{1})\times exp[i({\omega}_{0}-{\omega}_{\text{eg}})({t}_{3}-{t}_{1})]$$

(3)

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

$${R}_{2}({t}_{3},{t}_{2},{t}_{1})=\langle \widehat{\mu}exp\left[\frac{\text{i}}{\hslash}{H}_{\text{e}}({t}_{1}+{t}_{2})\right]\widehat{\mu}exp\left[\frac{\text{i}}{\hslash}{H}_{\text{g}}{t}_{3}\right]\widehat{\mu}\times exp\left[-\frac{\text{i}}{\hslash}{H}_{\text{e}}({t}_{2}+{t}_{3})\right]\widehat{\mu}exp\left[-\frac{\text{i}}{\hslash}{H}_{\text{g}}{t}_{1}\right]{\rho}_{\text{g}}\rangle $$

(4)

$${R}_{3}({t}_{3},{t}_{2},{t}_{1})=\langle \widehat{\mu}exp\left[\frac{\text{i}}{\hslash}{H}_{\text{e}}{t}_{1}\right]\widehat{\mu}exp\left[\frac{\text{i}}{\hslash}{H}_{\text{g}}({t}_{2}+{t}_{3})\right]\widehat{\mu}\times exp\left[-\frac{\text{i}}{\hslash}{H}_{\text{e}}{t}_{3}\right]\widehat{\mu}exp\left[-\frac{\text{i}}{\hslash}{H}_{\text{g}}({t}_{1}+{t}_{2})\right]{\rho}_{\text{g}}\rangle $$

(5)

Here, is an electronic dipole moment operator,
${\rho}_{\text{g}}={\mathrm{\sum}}_{t=1}^{N}{a}_{n}\mid {\text{g}}_{n}\rangle \langle {\text{g}}_{n}\mid $ is a ground-state nuclear density operator, with *a _{n}* as the population of the

Assuming that pulses do not overlap, that is, *t*, *τ*_{2}, *τ*_{1} > *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|*n**n*| = 1 in eqs 4 and 5 repeatedly, we obtain

$${R}_{2}({t}_{3},{t}_{2},{t}_{1})=\sum _{n,k,u,v}\langle {\text{g}}_{n}\mid \widehat{\mu}\mid {\text{e}}_{u}\rangle \times exp\left[\frac{\text{i}}{\hslash}{E}_{u}^{\text{e}}({t}_{1}+{t}_{2})\right]\langle {\text{e}}_{u}\mid \widehat{\mu}\mid {\text{g}}_{k}\rangle exp\left[\frac{\text{i}}{\hslash}{E}_{k}^{\text{g}}{t}_{3}\right]\langle {\text{g}}_{k}\mid \widehat{\mu}\mid {\text{e}}_{v}\rangle \times exp\left[-\frac{\text{i}}{\hslash}{E}_{v}^{\text{e}}({t}_{2}+{t}_{3})\right]\langle {\text{e}}_{v}\mid \widehat{\mu}\mid {\text{g}}_{n}\rangle exp\left[-\frac{\text{i}}{\hslash}{E}_{n}^{\text{g}}{t}_{1}\right]exp[-\beta {E}_{n}^{\text{g}}]$$

(6)

$${R}_{3}({t}_{3},{t}_{2},{t}_{1})=\sum _{n,k,u,v}\langle {\text{g}}_{n}\mid \widehat{\mu}\mid {\text{e}}_{u}\rangle exp\left[\frac{\text{i}}{\hslash}{E}_{u}^{\text{e}}{t}_{1}\right]\langle {\text{e}}_{u}\mid \widehat{\mu}\mid {\text{g}}_{k}\rangle \times exp\left[\frac{\text{i}}{\hslash}{E}_{k}^{\text{g}}({t}_{2}+{t}_{3})\right]\langle {\text{g}}_{k}\mid \widehat{\mu}\mid {\text{e}}_{v}\rangle \times exp\left[-\frac{\text{i}}{\hslash}{E}_{v}^{\text{e}}{t}_{3}\right]\langle {\text{e}}_{v}\mid \widehat{\mu}\mid {\text{g}}_{n}\rangle \times exp\left[-\frac{\text{i}}{\hslash}{E}_{n}^{\text{g}}({t}_{1}+{t}_{2})\right]exp[-\beta {E}_{n}^{\text{g}}]$$

(7)

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

$${P}^{(3)}(t)={\left(\frac{\text{i}}{\hslash}\right)}^{3}{(\sqrt{2\pi}\tau {E}_{0})}^{3}\sum _{n,k,u,v}{\text{e}}^{-\hslash \beta {\epsilon}_{n}}\langle {\text{g}}_{n}\mid \widehat{\mu}\mid {\text{e}}_{u}\rangle \langle {\text{e}}_{u}\mid \widehat{\mu}\mid {g}_{k}\rangle \times \langle {\text{g}}_{k}\mid \widehat{\mu}\mid {\text{e}}_{v}\rangle \langle {\text{e}}_{v}\mid \widehat{\mu}\mid {\text{g}}_{n}\rangle \times {\text{e}}^{-{({\epsilon}_{k}-{\epsilon}_{v})}^{2}{\tau}^{2}/2}{\text{e}}^{-{({\epsilon}_{n}-{\epsilon}_{u})}^{2}{\tau}^{2}}\times {\text{e}}^{({\epsilon}_{k}-{\epsilon}_{n})({\epsilon}_{u}-{\epsilon}_{v}){\tau}^{2}}\times \{{\text{e}}^{\text{i}({\epsilon}_{u}-{\epsilon}_{v}){\tau}_{2}}+{\text{e}}^{\text{i}({\epsilon}_{k}-{\epsilon}_{n}){\tau}_{2}}\}{\text{e}}^{\text{i}({\epsilon}_{k}-{\epsilon}_{v})t}{\text{e}}^{\text{i}({\epsilon}_{u}-{\epsilon}_{n}){\tau}_{1}}$$

(8)

where we denote

$$\begin{array}{l}{\epsilon}_{n}\equiv {E}_{n}^{\text{g}}/\hslash \\ {\epsilon}_{k}\equiv {E}_{k}^{\text{g}}/\hslash \\ {\epsilon}_{u}\equiv ({E}_{u}^{\text{e}}/\hslash )-({\omega}_{0}-{\omega}_{\text{eg}})\\ {\epsilon}_{v}\equiv ({E}_{v}^{\text{e}}/\hslash )-({\omega}_{0}-{\omega}_{\text{eg}})\end{array}$$

(9)

Here,
${E}_{n}^{\text{g}}$ and |g* _{n}* are the

Matrix elements e* _{i}*||g

$${P}_{\text{a}}^{(3)}(t)={\left(\frac{\text{i}}{\hslash}\right)}^{3}{(\sqrt{2\pi}\tau {E}_{0})}^{3}\sum _{n=1}^{N}{\text{e}}^{-\hslash \beta {\epsilon}_{n}}\sum _{u,v}\prime \mid \langle {\text{g}}_{n}\mid \widehat{\mu}{\mid {\text{e}}_{v}\rangle \mid}^{2}\mid \langle {\text{g}}_{n}\mid \widehat{\mu}{\mid {\text{e}}_{v}\rangle \mid}^{2}\times {\text{e}}^{-{({\epsilon}_{n}-{\epsilon}_{v})}^{2}{\tau}^{2}/2}{\text{e}}^{-{({\epsilon}_{n}-{\epsilon}_{u})}^{2}{\tau}^{2}}\times \{{\text{e}}^{\text{i}({\epsilon}_{u}-{\epsilon}_{v}){\tau}_{2}}+1\}{\text{e}}^{\text{i}({\epsilon}_{n}-{\epsilon}_{v})t}{\text{e}}^{\text{i}({\epsilon}_{u}-{\epsilon}_{n}){\tau}_{1}}$$

(10)

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

In the Condon approximation, matrix elements g* _{n}*||e

$${P}_{\text{a}}^{(3)}(t)={\left(\frac{\text{i}}{\hslash}\right)}^{3}{(\sqrt{2\pi}\tau {E}_{0})}^{3}{\mu}_{0}^{2}{\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}\times \sum _{u,v}\prime \sum _{n=1}^{N}{\text{e}}^{-\hslash \epsilon {\beta}_{n}}{\text{e}}^{-{({\epsilon}_{n}-{\epsilon}_{v})}^{2}{\tau}^{2}/2}{\text{e}}^{-{({\epsilon}_{n}-{\epsilon}_{u})}^{2}{\tau}^{2}}\times \{{\text{e}}^{\text{i}({\epsilon}_{u}-{\epsilon}_{v}){\tau}_{2}}+1\}{\text{e}}^{\text{i}({\epsilon}_{n}-{\epsilon}_{v})t}{\text{e}}^{\text{i}({\epsilon}_{u}-{\epsilon}_{n}){\tau}_{1}}$$

(11)

where
${\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}$ is the average of products of squared Franck–Condon factors for the vertical transitions from the *N* ground vibrational states, *u* ≠ *v*.

The last summation over *n* in the above equation is the averaging of the expression under the summation sign over the different values of *ε _{n}*,
${\sum}_{n=1\cdots}^{N}=N\langle \dots \rangle $. However, since

$${P}_{\text{a}}^{(3)}(t)={\left(\frac{\text{i}}{\hslash}\right)}^{3}{(\sqrt{2\pi}\tau {E}_{0})}^{3}{\mu}_{0}^{2}{\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}\times \sqrt{\frac{\pi}{6}}\frac{\hslash}{\tau {\langle \mathrm{\Delta}E\rangle}_{0}}{\text{e}}^{{(\hslash \beta )}^{2}/6{\tau}^{2}}{\text{e}}^{-[[{(t-{\tau}_{1})}^{2}/6{\tau}^{2}]-\text{i}[\hslash \beta (t-{\tau}_{1})/3{\tau}^{2}]]}\times \sum _{v}{\text{e}}^{-\hslash \beta {\epsilon}_{v}}\sum _{r=\pm 1,\pm 2,\dots}{\text{e}}^{-[({\mathrm{\Delta}}_{r}^{2}{\tau}^{2}/3)-(2\hslash \beta {\mathrm{\Delta}}_{r}/3)]}{\text{e}}^{(\text{i}{\mathrm{\Delta}}_{r}/3)(2t+{\tau}_{1})}\times \{{\text{e}}^{\text{i}\mathrm{\Delta},{\tau}_{2}}+1\}\times \left(1+\text{erf}\left[\frac{\text{i}(t-{t}_{1})-\hslash \beta}{\sqrt{6\tau}}+\frac{2}{\sqrt{6}}{\mathrm{\Delta}}_{r}\tau +\frac{3}{\sqrt{6}}{\epsilon}_{v}\tau \right]\right)$$

(12)

where we have introduced a new variable Δ* _{r}* Δ

$${P}_{\text{a}}^{(3)}(t)={\left(\frac{\text{i}}{\hslash}\right)}^{3}{(\sqrt{2\pi}\tau {E}_{0})}^{3}{\mu}_{0}^{2}{\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}\times \sqrt{\frac{\pi}{6}}\frac{\hslash}{\tau {\langle \mathrm{\Delta}E\rangle}_{0}}{\text{e}}^{{(\hslash \beta )}^{2}/6{\tau}^{2}}{\text{e}}^{-[[{(t-{\tau}_{1})}^{2}/6{\tau}^{2}]-\text{i}[\hslash \beta (t-{\tau}_{1})/3{\tau}^{2}]]}\times 2\sum _{v=0}^{\infty}{\text{e}}^{-\hslash \beta {\epsilon}_{v}}\sum _{r=\pm 1,\pm 2,\dots}{\text{e}}^{-[({\mathrm{\Delta}}_{r}^{2}{\tau}^{2}/3)-(2\hslash \beta {\mathrm{\Delta}}_{r}/3)]}{\text{e}}^{(\text{i}{\mathrm{\Delta}}_{r}/3)(2t+{\tau}_{1})}\times \{{\text{e}}^{\text{i}{\mathrm{\Delta}}_{r}{\tau}_{2}}+1\}$$

(13)

Again, the summation over *v* is an averaging over the variable *ε _{v}*. The position in spectrum

$${P}_{\text{a}}^{(3)}(t)={\left(\frac{\text{i}}{\hslash}\right)}^{3}{(\sqrt{2\pi}\tau {E}_{0})}^{3}{\mu}_{0}^{2}{\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}\times \sqrt{\frac{\pi}{6}}\frac{\hslash}{\tau {\langle \mathrm{\Delta}E\rangle}_{0}}{\text{e}}^{{(\hslash \beta )}^{2}/6{\tau}^{2}}{\text{e}}^{[-[{(t-{\tau}_{1})}^{2}/6{\tau}^{2}]-\text{i}[\hslash \beta (t-{\tau}_{1})/3{\tau}^{2}]]}\times 2(\sum _{v=0}^{\infty}{\text{e}}^{-\hslash \beta {\epsilon}_{v}})\times \sum _{r=\pm 1,\pm 2,\dots}\langle {\text{e}}^{[-({\mathrm{\Delta}}_{r}^{2}{\tau}^{2}/3)-(2\hslash \beta {\mathrm{\Delta}}_{r}/3)]}{\text{e}}^{(\text{i}{\mathrm{\Delta}}_{r}/3)(2t+{\tau}_{1})}\times \langle \{{\text{e}}^{\text{i}{\mathrm{\Delta}}_{r}{\tau}_{2}}+1\}\rangle $$

(14)

where the last averaging is due to the summation over *v*. Since Δ* _{r}* has different values at different parts of spectrum, then the averaging over

$$\sum _{r=\pm 1,\pm 2,\dots}\langle {\text{e}}^{-[({\mathrm{\Delta}}_{r}^{2}{\tau}^{2}/3)-(2\hslash \beta {\mathrm{\Delta}}_{r}/3)]}{\text{e}}^{(\text{i}{\mathrm{\Delta}}_{r}/3)(2t+{\tau}_{1})}\{{\text{e}}^{\text{i}{\mathrm{\Delta}}_{r}{\tau}_{2}}+1\}\rangle =2{\int}_{0}^{\infty}{\text{e}}^{-[({\mathrm{\Delta}}^{2}{\tau}^{2}/3)-(2\hslash \beta {\mathrm{\Delta}}_{r}/3)]}\left\{cos\left[\mathrm{\Delta}\left(\frac{2t+{\tau}_{1}}{3}+{\tau}_{2}\right)\right]+cos\left[\mathrm{\Delta}\left(\frac{2t+{\tau}_{1}}{3}\right)\right]\right\}\times \sum _{r=1,2,\dots}{\rho}_{r}(\mathrm{\Delta})\text{d}\mathrm{\Delta}$$

(15)

One can see that the statistics energy spectrum enters the above expression as a sum *ρ*(Δ) = Σ* _{r}ρ_{r}*(Δ), which is just a two-level density of states.

$${P}_{\text{a}}^{(3)}(t)={C\text{e}}^{-[[{(t-{\tau}_{1})}^{2}/6{\tau}^{2}]-\text{i}[\hslash \beta (t-{\tau}_{1})/3{\tau}^{2}]]}F(t)$$

(16)

where
$C={(\text{i}/\hslash )}^{3}{[{(2\pi )}^{1/2}\tau {E}_{0})}^{3}{\mu}_{0}^{2}{\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}{(\pi /6)}^{1/2}[2\hslash /\tau {\langle \mathrm{\Delta}E\rangle}_{0}][1/\beta \langle \mathrm{\Delta}E\rangle ]{\text{e}}^{{(\hslash \beta )}^{2}/6{\tau}^{2}}$ is a constant and where we have used
${\sum}_{v=0}^{\infty}{\text{e}}^{-\hslash \beta {\epsilon}_{v}}\approx 1/\beta \langle \mathrm{\Delta}E\rangle $ with Δ*E* being the mean level spacing in the excited electronic potential surface.

Let us now consider ${P}_{\text{b}}^{(3)}(t)$; it reads

$${P}_{\text{b}}^{(3)}(t)={\left(\frac{\text{i}}{\hslash}\right)}^{3}{(\sqrt{2\pi}\tau {E}_{0})}^{3}\sum _{n=1}^{N}{\text{e}}^{-\hslash \beta {\epsilon}_{n}}\sum _{k,v}{\mid \langle {\text{g}}_{n}\mid \widehat{\mu}\mid {\text{e}}_{v}\rangle \mid}^{\text{2}}{\mid \langle {\text{g}}_{k}\mid \widehat{\mu}\mid {\text{e}}_{v}\rangle \mid}^{\text{2}}\times {\text{e}}^{-{({\epsilon}_{k}-{\epsilon}_{v})}^{2}{\tau}^{2}/2}{\text{e}}^{-{({\epsilon}_{n}-{\epsilon}_{v})}^{2}{\tau}^{2}}\times \{1+{\text{e}}^{\text{i}({\epsilon}_{k}-{\epsilon}_{n}){\tau}_{2}}\}{\text{e}}^{\text{i}({\epsilon}_{n}-{\epsilon}_{v})t}{\text{e}}^{\text{i}({\epsilon}_{v}-{\epsilon}_{n}){\tau}_{1}}$$

(17)

Using the same assumptions as those in the derivation of ${P}_{\text{a}}^{(3)}(t)$ and replacing summations ${\mathrm{\sum}}_{n=1}^{N}$ and ${\mathrm{\sum}}_{k=1}^{\infty}$ with the integral $(\hslash /{\langle \mathrm{\Delta}E\rangle}_{0}){\int}_{0}^{\infty}\text{d}\epsilon $, we get

$${P}_{\text{b}}^{(3)}(t)={C}^{\prime}{\text{e}}^{-[[2{t}^{2}+{\tau}_{1}^{2}/4{\tau}^{2}]+\text{i}(\hslash \beta {\tau}_{1}/2{\tau}^{2})]}\times (1+{\text{e}}^{-[{\tau}_{2}(4t+2{\tau}_{1}+3{\tau}_{2})/4{\tau}^{2}]+\text{i}(\hslash \beta {\tau}_{2}/2{\tau}^{2})]})$$

(18)

where
${C}^{\prime}={(\text{i}/\hslash )}^{3}{[{(2\pi )}^{1/2}\tau {E}_{0})}^{3}{\mu}_{0}^{2}{\langle {S}_{nv}^{2}{S}_{kv}^{2}\rangle}^{\prime}\pi {(2)}^{1/2}[{(\hslash /\tau {\langle \mathrm{\Delta}E\rangle}_{0})}^{2}][1/\beta \langle \mathrm{\Delta}E\rangle ]{\text{e}}^{{(\hslash \beta )}^{2}/4{\tau}^{2}}$. Obviously, the contribution of this term to the overall nonlinear polarization is negligible when the conditions of pulse nonoverlapping *t*, *τ*_{1}, *τ*_{2} > *τ* are satisfied.

The last term to consider is ${P}_{\text{c}}^{(3)}t$

$${P}_{\text{c}}^{(3)}(t)=2{\left(\frac{\text{i}}{\hslash}\right)}^{3}{(\sqrt{2\pi}\tau {E}_{0})}^{3}\sum _{n=1}^{N}{\text{e}}^{-\hslash \beta {\epsilon}_{n}}\sum _{v}\mid \langle {\text{g}}_{n}\mid \widehat{\mu}{\mid {\text{e}}_{v}\rangle \mid}^{4}\times {\text{e}}^{-(3{\tau}^{2}/2){({\epsilon}_{n}-{\epsilon}_{v})}^{2}}{\text{e}}^{\text{i}({\epsilon}_{n}-{\epsilon}_{v})(t-{\tau}_{1})}$$

(19)

which simplifies to

$${P}_{\text{c}}^{(3)}(t)=2C\frac{\langle {S}_{nv}^{4}\rangle}{{\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}}{\text{e}}^{-[[{(t-{\tau}_{1})}^{2}/6{\tau}^{2}]-i[\hslash \beta (t-{\tau}_{1})/3{\tau}^{2}]]}$$

(20)

The overall third-order nonlinear polarization reads

$$\begin{array}{l}{P}^{(3)}(t)={P}_{\text{a}}^{(3)}(t)+{P}_{\text{b}}^{(3)}t+{P}_{\text{c}}^{(3)}(t)\\ ={C\text{e}}^{-[[{(t-{\tau}_{1})}^{2}/6{\tau}^{2}]-\text{i}[\hslash \beta (t-{\tau}_{1})/3{\tau}^{2}]]}\left(F(t)+2\frac{\langle {S}_{nv}^{4}\rangle}{{\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}}\right)+{P}_{\text{b}}^{(3)}(t)\end{array}$$

(21)

Here, we did not substitute for a small contribution of
${P}_{\text{b}}^{(3)}(t)$ in order not to overload the formula. One can see that at *t* = *τ*_{1}, we have an echo.

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

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

$$\rho (\omega )=\sum _{r=1,2,\dots}^{\infty}{\rho}_{r}(\omega )=\frac{1}{\langle \mathrm{\Delta}E\rangle /\hslash}$$

(22)

where Δ*E* is a mean level spacing. Thus, *F*(*t*) for systems with regular dynamics reads

$$F(t)=2\left\{{g}_{\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}}\left(\frac{2t+{\tau}_{1}+3{\tau}_{2}}{3\tau}\right)+{g}_{\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}}\left(\frac{2t+{\tau}_{1}}{3\tau}\right)\right\}$$

(23)

where

$${g}_{\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}}(x)=Re\left\{\frac{\sqrt{3\pi}}{\stackrel{\sim}{\alpha}}{\text{e}}^{-(3/4){(x+\text{i}(2\stackrel{\sim}{\beta}/3))}^{2}}\times \left(1+\text{i}\phantom{\rule{0.16667em}{0ex}}\text{erfi}\left[\frac{\sqrt{3}}{2}(x+\text{i}(2\stackrel{\sim}{\beta}/3))\right]\right)\right\}$$

(24)

Here, we have introduced dimensionless parameters = *τ*Δ*E*/ and = *β*/*τ*. The necessary conditions for the photon-echo experiment described in this paper look very simple in terms of these parameters; they are

$$\begin{array}{l}\stackrel{\sim}{\alpha}<1\\ \stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}\ll 1\end{array}$$

(25)

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 *β*Δ*E* 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 by^{25}

$$\chi ({\tau}_{1},{\tau}_{2})={\int}_{0}^{\infty}{\mid P(t)\mid}^{2}\text{d}t$$

(26)

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

For GOE statistics, the first nearest-neighbor distribution function is given by the Wigner distribution^{15}

$${\rho}_{1}(\omega )=\frac{\pi}{2}\frac{\omega \hslash}{\langle \mathrm{\Delta}E\rangle}\frac{\hslash exp\{-(\pi /4){[\langle \omega \hslash /\langle \mathrm{\Delta}E\rangle \rangle ]}^{2}\}}{\langle \mathrm{\Delta}E\rangle}$$

(27)

where Δ*E* is a mean level spacing. In this case, the two-level density of states reads^{29}

$$\rho (\omega )=\frac{\hslash}{\langle \mathrm{\Delta}E\rangle}\left(1-{c}^{2}(\omega )-\frac{\text{d}c(\omega )}{\text{d}\omega}{\int}_{\omega}^{\infty}c({\omega}^{\prime})\text{d}{\omega}^{\prime}\right)$$

(28)

with *c*(*ω*) = sin[*π**ω*/Δ*E*]/(*π**ω*/Δ*E*). Numerical integration of eq 15 with eq 28 gives

$$F(t)=2\left\{{f}_{\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}}\left(\frac{2t+{\tau}_{1}+3{\tau}_{2}}{3\tau}\right)+{f}_{\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}}\left(\frac{2t+{\tau}_{1}}{3\tau}\right)\right\}$$

(29)

where functions *f*_{}* _{}* in the range of parameters given by eq 25 can be well-approximated by analytic functions

The plots of *f*_{}* _{}*(

The functions *f*_{}_{}(*x*) (solid line) and *g*_{}_{}(*x*) (dashed line) for the different values of the inverse temperature *β* = 1/*kT*. The values of the parameters are = **...**

$${f}_{\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}}(x)={g}_{\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}}(x)-\frac{8}{\pi}Re\left[\frac{{\text{erf}}^{2}[(\stackrel{\sim}{\alpha}x+\text{i}(2\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}/3))/3]}{{(\stackrel{\sim}{\alpha}x+\text{i}(2\stackrel{\sim}{\alpha}\stackrel{\sim}{\beta}/3))}^{2}}\right]$$

(30)

position is a nonlinear function of parameters and , which can be found numerically; in the range of < 0.3, 0.4 < < 1, it can be given by the approximate formula *x*_{min} = 1.25(/^{1.39}) + (2.61/^{0.2}) ≈ 2.61/^{0.2}. *F*(*t*) for the echo condition *t* = *τ*_{1} thus has the minimum at *t* ~ 2.6*τ*^{0.8}.

Straightforward numerical calculation of the summation in eq 11 was also performed to check the obtained analytical results. For the energy spectrum *ε _{n}*, 400 levels with the mean level spacing of Δ

Calculation of a signal (eq 26) with eqs 29 and 30 is shown in Figure 3c,d. The *χ*(*τ*_{1}, *τ*_{2}) has a minimum at *τ*_{1} ~ 4*τ* for any given value of *τ*_{2}, and its location along *τ*_{1} axis is independent of *τ*_{2}. We call this minimum a suppression of photon-echo signal.

It is interesting that the minimum of the photon-echo signal at *τ*_{1} ≈ 4*τ* 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 ~4*τ*. The depth of the minimum is proportional to the strength of the spectral correlation.

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
${P}_{\text{a}}^{(3)}(t)$, which includes only the differences between energy levels *ε _{u}* and

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 *τ*_{1} ~ 4*τ* for chaotic systems, where *τ* is a pulse duration. The time interval between second and third laser pulses, *τ*_{2}, does not influence the location of the signal’s minimum along the *τ*_{1} axis. The suppression can be considerable; the general formula for the ratio *χ*(*τ*_{1}, *τ*_{2})/*χ*(∞,∞) near the global minimum *τ*_{1} = 4*τ* and *τ*_{2} = 0 at high temperatures, *β*Δ*E* 1, is

$$\frac{\chi (4\tau ,0)}{\chi (\infty ,\infty )}\approx {\left|1-\frac{8}{\pi}\frac{{\langle {S}_{nu}^{2}{S}_{nv}^{2}\rangle}^{\prime}}{\langle {S}_{nv}^{4}\rangle}\left[2\frac{{\text{erf}}^{2}\left(\frac{4\tau}{3}\right)}{{(4\tau )}^{2}}\right]\right|}^{2}$$

(31)

where *τ* has dimensionless units of /Δ*E*. We can estimate the ratio assuming |*S _{nu}*| and |

On the other hand, the photon-echo signal of regular systems *χ*(*t*_{1}, *t*_{2}) does not have any minima (Figures 3a,b). Thus, the following conditions always hold: *χ*(4*τ*, *τ*_{2})/*χ*(∞,∞) ≥ 1 for regular systems and *χ*(4*τ*, *τ*_{2})/*χ*(∞,∞) < 1 for irregular systems. In real experiments, *χ*(*τ*_{1}, *τ*_{2}) 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*(*τ*_{1}, *τ*_{2}) as a constant, which we plot in Figure 3 as *χ*(∞,∞). Since the location of the correlation minimum at *τ*_{1} = 4*τ* does not depend on *τ*_{2} (Figure 3d), we can make the above inequalities stronger by averaging over some interval of *τ*_{2}. 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 |*ψ*(0)|*ψ*(*t*)|^{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 Δ*E*, the superposition state |*ψ*(*t*) = Σ exp(−i*E _{n}t*/)|

In the proposed experiment, the time domain signature of chaotic motion at 4*τ* 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 *N*_{e} in the excited electronic potential is equal to the number of initially populated eigenstates *N*_{g} 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, *N*_{e} = *N*_{g}Δ*E*_{0}/Δ*E*. To have good statistics, we need to populate a considerable number of ground states *N*_{g} 1, which suggests performing the experiment at high temperatures, that is, *β*Δ*E*_{0} 1.

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 *τ*_{1} = 4*τ*, 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, −*L _{x}*/2 <

$$\begin{array}{l}{\mathrm{\Psi}}_{{n}_{x},{n}_{y}}=\sqrt{\frac{4}{{L}_{x}{L}_{y}}}sin\left[\frac{\pi {n}_{x}(x-{L}_{x}/2)}{{L}_{x}}\right]sin\left[\frac{\pi {n}_{y}(y-{L}_{y}/2)}{{L}_{y}}\right]\\ {E}_{{n}_{x},{n}_{y}}=\frac{{\pi}^{2}{\hslash}^{2}}{2m}\left[{\left(\frac{{n}_{x}}{{L}_{x}}\right)}^{2}+{\left(\frac{{n}_{y}}{{L}_{y}}\right)}^{2}\right]\end{array}$$

(32)

For the irrational ratio of *L _{x}* and

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, *L _{x}*

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 **...**

$${S}_{ij}=2s({n}_{x},{n}_{x}^{\prime})s({n}_{y},{n}_{y}^{\prime})$$

(33)

where

$$s(n,{n}^{\prime})=\{\begin{array}{ll}\frac{1}{2}cos\left(\frac{\pi n}{2}\right)\hfill & \text{if}\phantom{\rule{0.16667em}{0ex}}{n}^{\prime}=2n\hfill \\ \frac{4n(sin(\pi {n}^{\prime}/4)-cos(\pi n)sin(3\pi {n}^{\prime}/4))}{\pi (4{n}^{2}-{{n}^{\prime}}^{2})}\hfill & \text{if}\phantom{\rule{0.16667em}{0ex}}{n}^{\prime}\ne 2n\hfill \end{array}$$

(34)

If initially an *i*th eigenstate with an eigenvalue *E _{i}* is populated in the ground potential well, a laser pulse with a spectral width ΔΩ and frequency

^{†}Part of the “Karl Freed Festschrift”.

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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