### Abstract

- Top of page
- Abstract
- Introduction
- Formalism
- Application to Vibrational Dephasing
- Decoherence in an Nonequilibrium Environment: An Analytic Model
- Summary

In this article, we describe recent work on investigating the role of the environment in influencing coherent quantum dynamics. We review numerical methodology based on a semiclassical limit of the quantum Liouville equation for simulating quantum coherent processes using classical-like molecular dynamics simulation and ensemble averaging and apply the approach to simulate vibrational dephasing of I_{2} in cryogenic rare gas matrices and the quantum vibrations of OH stretches of HOD in D_{2}O. We then describe a simple analytic and numerical model that highlights novel behavior that can be exhibited by quantum coherent processes in the presence of an environment that is not at thermal equilibrium. © 2012 Wiley Periodicals, Inc.

### Introduction

- Top of page
- Abstract
- Introduction
- Formalism
- Application to Vibrational Dephasing
- Decoherence in an Nonequilibrium Environment: An Analytic Model
- Summary

Quantum interference and coherence are phenomena that set the dynamics of molecular scale systems in distinct contrast with the behavior of the macroscopic classical world.1 In chemical physics applications, the creation, evolution, and destruction of quantum coherence plays a central role in a range of physical processes, such as the harvesting and transport of electronic energy in photobiological systems,2–8 the design and interpretation of nonlinear spectroscopies,9 the coherent control of molecular processes,10–12 the manipulation and storage of quantum information,13, 14 and many others.15–17

Quantum coherence exists and is most pronounced in simple few body systems. Decoherence, the irreversible destruction of quantum coherence, is a phenomenon that is associated with complex systems and the resulting interactions between a coherent subsystem and a multidimensional environment or bath. System–bath interactions can never be eliminated completely, and so decoherence is in principle always at work eroding quantum superpositions to their incoherent classical statistical limits. In most formal approaches to dissipative quantum dynamics, the assumption is made that the environment is in thermal equilibrium characterized by a Boltzmann distribution at temperature *T*.18, 19 This is a reasonable approximation in most contexts and greatly simplifies the theoretical analysis.

There are physical situations, however, where nonequilibrium bath effects may be important. For example, light-induced ultrafast coherent electronic processes in chemical or biological systems may occur on time scales that are sufficiently short that initial nonequilibrium states induced in the bath by the excitation may not have a chance to regress to equilibrium. The transient nonequilibrium bath dynamics may undergo nontrivial interplay with the coherent quantum evolution occurring on comparable time scales. On these time scales, the environment has the opportunity to influence the quantum evolution in a manner that is more rich and complex than simply acting to dissipate energy and randomize and destroy quantum phases. Indeed, recent experiments have suggested that the environmental protein dynamics in light harvesting complexes may play an essential role in enhancing quantum energy transport.2–8 In the proposed picture, bath fluctuations aid quantum energy flow by overcoming localization due to energy site inhomogeneities, while at the same time acting to destroy quantum phase coherence.

In this article, we describe recent work on investigating the role of the environment in influencing coherent quantum dynamics. We review numerical methodology developed in our group, based on a semiclassical limit of the quantum Liouville equation, for simulating quantum coherent processes using classical-like molecular dynamics simulation and ensemble averaging, and apply the approach to simulating vibrational dephasing of I_{2} in cryogenic rare gas matrices and the quantum vibrations of OH stretches of HOD in D_{2}O. We then describe a simple analytic and numerical model that highlights novel behavior that can be exhibited by quantum coherent processes in the presence of an environment that is not at thermal equilibrium.

### Formalism

- Top of page
- Abstract
- Introduction
- Formalism
- Application to Vibrational Dephasing
- Decoherence in an Nonequilibrium Environment: An Analytic Model
- Summary

In this section, we briefly review our semiclassical approach to solve Liouville equation describing dynamics on coupled quantum states.20–24 The formal development starts with the quantum mechanical Liouville equation for the density operator,

- (1)

where is the Hamiltonian of the system. The classical limit of Eq. (1) becomes the classical Liouville equation of nonequilibrium statistical mechanics25:

- (2)

where ρ = ρ(**q**, **p**, *t*) and *H* = *H*(**q**, **p**) are functions of the classical phase space variables (**q**, **p**), and we assume that the Hamiltonian does not depend explicitly on time. We consider here general multidimensional systems, and so **q** and **p** are vectors in *F*-dimensional spaces, where *F* is the number of degrees of freedom. The Poisson bracket is expressed as:

- (3)

This result can be derived through the Wigner–Moyal expansion of the quantum mechanical Liouville equation.9, 26–28 To the lowest order in , the quantum commutator is replaced by the corresponding classical Poisson bracket:

- (4)

This classical limit can be extended to the case where is an *N* × *N* matrix of operators, representing an *N* state “quantum” subsystem for which the classical limit is not taken.20–24 For simplicity, our development will consider the case *N* = 2. The Hamiltonian and density operator for the system can then be represented in a given quantum basis as:

- (5)

and

- (6)

respectively. The quantum mechanical Liouville equation then becomes three coupled equations for the operator matrix elements,

- (7)

- (8)

- (9)

where . We consider below the special case of individually Hermitian off-diagonal Hamiltonians, so Ĥ_{12} = Ĥ_{21} = V̂. When the quantum mechanical operators are replaced by the corresponding classical phase space functions in the semiclassical limit, the multistate semiclassical analog can be written as20–24:

- (10)

- (11)

- (12)

The Hamiltonian matrix elements *H*_{ij}(**q**, **p**, *t*) and density matrix elements ρ_{ij} = ρ(**q**, **p**, *t*) are now functions of the many-body classical variables (**q**, **p**) describing the classical bath degrees of freedom. We have defined a set of classical Liouville operators in terms of Poisson brackets with the corresponding Hamiltonians,

- (13)

The function *H*_{0} = (*H*_{11}+*H*_{22})/2 is the average of the two uncoupled surface Hamiltonians, and the frequency term ω(*q*) is defined as the difference potential divided by Planck's constant,

- (14)

where **q** = (*q*_{1},…,*q*_{F}) and ▵*V*(**q**) = *V*_{11}(**q**) − *V*_{22}(**q**). The first term on the right-hand side of Eqs. (10) and (11) corresponds to classical dynamics of the densities, governed by the classical Liouville operators and . The remaining terms that involve the coupling matrix element *V* and the coherence ρ_{12} correspond to the nonclassical quantum coupling between the states and act as sink and source terms on the evolving phase space probability densities ρ_{11} and ρ_{22}. In Eq. (13), the first term on the right-hand side corresponds to a “generalized” classical dynamics, involving both the propagation under the average Liouvillian and accumulation of a nonclassical phase factor resulting from the imaginary frequency term. Unlike the diagonal probability densities, the coherence ρ_{12} is thus a complex valued function. The densities ρ_{11} and ρ_{22} come in through the inhomogeneous terms that create and modify the evolving coherence.

In our previous work,15–17, 20–23, 29–31 we considered the coupled two-state semiclassical Liouville equations for the general case of population transfer and dephasing. Here, we consider only the special case of “pure dephasing,” where population transfer is excluded, and thus *H*_{12} ≡ *V* = 0, and the diagonal density matrix elements ρ_{11} and ρ_{22} have conserved phase space traces. The equation of motion for the off-diagonal element ρ_{12} becomes decoupled from the populations and is given in the semiclassical limit by15–17, 20–23, 29–31

- (15)

The formal solution of this equation is

- (16)

where

- (17)

The generalized phase space density ρ_{12}(**q**, **p**, *t*) is a complex valued function. It can nonetheless be represented by a classical ensemble of *N* trajectories (**Q**_{j}(*t*), **P**_{j}(*t*)), *j* = 1,2,…,*N*, propagated under Hamilton's equations for the average Hamiltonian *H*_{ave}.15–17, 20–23, 29–31 Expressing Eq. (16) in terms of a sampling with trajectories yields

- (18)

where the nonclassical phase factor χ_{j}(*t*) is the time integral of the difference Hamiltonian divided by along the *j*th trajectory:

- (19)

A measure of the evolution and decay of coherence that can be directly related to experimental measurements of dephasing is the trace of ρ_{12}. It is given in terms of the trajectory ensemble by

- (20)

Diverging histories of the phase factors χ_{j}(*t*) across the ensemble of trajectories cause cancellation in the sum defining Tr ρ_{0}_{n}(*t*); this is the ensemble-level semiclassical manifestation of the decay of quantum coherence.

### Application to Vibrational Dephasing

- Top of page
- Abstract
- Introduction
- Formalism
- Application to Vibrational Dephasing
- Decoherence in an Nonequilibrium Environment: An Analytic Model
- Summary

In this section, we apply the formalism developed above to the simulation of vibrational dephasing in the condensed phase. We consider two applications: (1) calculation of the state- and temperature-dependent vibrational dephasing rates of I_{2} in a solid Kr matrix,17 for which detailed experimental data has recently been published by Apkarian and coworkers32 and (2) the dephasing of the quantum OH vibration of HOD in D_{2}O.33

In both cases, the method is implemented using a simple approach to define many-body classical potentials describing the interaction of a molecule in a particular quantum vibrational state with its surroundings.31 The method is tailored for efficient use in simulations of mixed classical–quantum dynamics in condensed phases and is based on an adiabatic separation between the quantum molecular vibration and the classical many-body bath. First-order perturbation theory is used to describe the dependence of the quantum vibrational energies on the bath configuration.31 We briefly review the method here.

Consider a multidimensional system described by the Hamiltonian

- (21)

where *V*(**q**, **Q**) is the potential, and a separation between the quantum mechanical “system” with coordinates **q** and the classical “bath” with coordinates **Q** is made. We write the full potential as *V*(**q**, **Q**) = *V*_{sys.}(**q**) + *V*_{bath}(**Q**) + *V*_{s-b}(**q**, **Q**). In Eq. (21), p̂_{i} is the momentum operator of the *i*th quantum degree of freedom, and *P*_{j} is the canonical momentum conjugate to the *j*th classical coordinate. This defines the zeroth-order Hamiltonian operator for the quantum subsystem degrees of freedom:

- (22)

The eigenstates and eigenvalues of this zeroth-order Hamltonian are the solutions of

- (23)

for *n* = 0,1,2,…. The goal is to calculate the many-body bath potential *U*_{n}(**Q**) as a function of **Q** for the system in quantum state *n*. In general, this could be accomplished by solving the Schrödinger equation in the Born–Oppenheimer approximation34:

- (24)

*E*_{n}(**Q**) is then the state-dependent many-body potential. In our method, we evaluate its dependence on **Q** using first-order perturbation theory,35 which assumes that the presence of a weakly coupled bath affects the system “energies” while leaving the “states” |*n*〉 unaffected. In this approximation, the potential becomes

- (25)

where

- (26)

and the |*n*〉 are solutions of the zeroth-order Schrödinger equation, Eq. (23).

In evaluating the expectation value in Eq. (26), we approximate the probability distribution |ψ_{n}(**q**)|^{2} of the system coordinates with a finite sampling. For a one-dimensional system, the sampled probability distribution can thus be written as

- (27)

where *K* denotes the number of sampling points chosen, and normalization dictates that ∑_{k} *w*_{k} (*n*) = 1. Then, the interaction term *U*_{n}(**Q**) becomes

- (28)

By increasing *K*, the quantum density |ψ_{n}(**q**)|^{2} can be represented with arbitrary numerical accuracy. Our goals are simplicity, numerical efficiency, and ease of implementation, rather than maximum accuracy, and so we adopt the smallest nontrivial value, *K* = 2, in the work presented here. Thus,

- (29)

where *w*_{+}(*n*) + *w*_{-}(*n*) = 1. The three independent parameters in Eq. (29) are determined, so that the first three moments of the distribution agree with the moments of the *n*th vibrational state of the potential describing the quantum vibration:

- (30)

for *k* = 1–3. Details of the procedure are given in Ref. 31.

In Figure 1, we show a comparison of the density *P*_{n}(*q*) = *w*_{+}(*n*)δ(*q*-*q*_{+}(*n*)) + *w*_{-}(*n*)δ(*q*-*q*_{-}(*n*)) with the exact quantum probability density |ψ_{n}(*q*)|^{2} for the Morse oscillator state *n* = 5 of the ground electronic state of the I_{2} diatomic molecule. The asymmetry of the locations and magnitudes of the δ functions resulting from the corresponding anharmonic and asymmetrical shape of the Morse potential is apparent in the figure. The *K* = 2 ansatz can be seen to be a crude but qualitatively correct representation of the full quantum density.

Also shown in the Figure 1 is an illustration of the correspondence between the quantum vibrational distribution and the spatial locations of the quantum particles in three dimensions. For the *K* = 2 case, each atom is represented by “two” quantum particles frozen at the appropriate relative distances corresponding to *q*_{+} and *q*_{-}. These quantum particles interact with solvent atoms via potentials whose classical well depths ε are scaled by the weights *w*_{+} and *w*_{-}. The interaction of the diatomic molecule in the *n* = 5 quantum vibrational state, represented by “four” atoms, with a single solvent atom is depicted in the Figure.

In Ref. 32, the results of time-resolved coherent anti-Stokes Raman scattering (TRCARS) measurements on I_{2} in a cryogenic Kr matrix were reported. The diatomic iodine occupies a double substitution site in these high-quality samples. Temperatures in the range *T* = 7 − 45 K were considered, and dephasing rates of coherent vibrational superposition states on the ground *X* electronic state between the ground *n* = 0 vibrational state and states with higher *n* were determined for *n* ≤ 19. The dephasing rates Γ_{n} ≡ γ_{0}_{n} were computed by fitting the TRCARS signal *S*(*t*) to a sum of damped trigonometric functions, of the form

- (31)

The authors of Ref. 32 found that the results of all experiments conducted could be summarized by the expression

- (32)

where

- (33)

The numerical values determined by the fit are τ_{1} = 353 ps, τ_{2} = 1550 ps, and Ω = 27 cm^{−1}. See Ref. 32 for details of the experimental method and data analysis. Here, we use this function to represent the experimental results presented in that article.

Simulations covering the range of conditions considered by Apkarian and coworkers were performed using the methodology outlined above. For determining numerical values of *q*_{±}(*n*) and *w*_{±}(*n*) appearing in the two δ function ansatz of Eq. (29), we use the ground *X* state I_{2} Morse oscillator parameters ω_{e} = 211.3 cm^{−1} and ω_{e}*x*_{e} = 0.6523 cm^{−1}.36 The Kr–Kr and I–Kr interactions are represented by Lennard–Jones potentials with parameters ε_{Kr-Kr} = 138.9 cm^{−1}, σ_{Kr-Kr} = 3.85 Å, ε_{I-Kr} = 162.3 cm^{−1}, and σ_{I-Kr} = 3.74 Å.37 The average and difference potentials are determined using the Kr–Kr interactions combined with the appropriate *w*_{±} scaled Kr–I interactions between the four quantum particles representing the diatomic molecule, as described above.

We simulate the evolution of the quantity |Trρ_{0}_{n}(*t*)| for all coherences between the vibrational ground state |0〉 and the excited states |*n*〉 for *n* ≤ 20. This quantity decays in a manner that is usually well represented by an exponential: |Trρ_{0}_{n}(*t*)| ≃ *e*. The dephasing rates γ_{0}_{n} extracted by fitting the simulation data to an exponential are directly comparable to the data measured by Apkarian and coworkers.32 Several initial temperatures of the system were considered, ranging from *T* = 5 to 70 K. Molecular dynamics simulations were performed using 108 atoms in a simulation box with periodic boundary conditions.38 In all simulations, we propagated ensembles of *N* = 500 trajectories. The ensembles are equilibrated using the ground vibrational state potential at the appropriate temperature. At the initial time *t* = 0, the potential governing the ensemble dynamics is switched to the appropriate *H*_{ave}, as described before. From the ensemble, the time evolution of |Trρ_{0}_{n}(*t*)| is determined, and the best fit to an exponential is found. This defines the corresponding dephasing rate γ_{0}_{n}.

In Figure 2 we show the time dependence of |Trρ_{0}_{n}(*t*)| for *n* = 5 at *T* = 32 K. The simulation data are shown, along with the fit to an exponential used to determine the vibrational dephasing rate γ_{0}_{n}. As can be seen, the evolution of the vibrational coherence of the system indeed decays exponentially. Other values of *n* exhibit similar behavior.

In Figure 3, we show a comparison of the simulated vibrational dephasing rates as a function of the upper state *n* with the fit of the experimental data from Ref. 32 using Eqs. (32) and (33). Results are given for two temperatures: *T* = 7 and 32 K. The simulations do an excellent job of capturing both the absolute magnitude of the dephasing rates and the functional dependence on the vibrational quantum number *n*. The agreement is nearly quantitative for the higher temperature *T* = 32 K results, while the theoretical predictions are approximately a factor of 2 too low for much of the *T* = 7 K case. Our purely classical treatment of the lattice dynamics does not capture quantum zero point contributions to the coherent quantum dynamics and vibrational dephasing, and so more error is expected at lower temperatures.

In Figure 4, the temperature dependence of the dephasing rates for coherences between *n* = 0 and three excited vibrational states, *n* = 8, *n* = 14, and *n* = 18, are shown. The simulations represent the qualitative dependence of the experimentally determined dephasing rates on temperature for all the states considered and are in excellent quantitative agreement for temperatures that are above about *T* = 20 K.

We now consider a second application to the vibrational dephasing of the OH stretch in liquid water.33 The structure and dynamics of the hydrogen bond network in liquid water is a problem of current interest. The properties of this system can be probed by measuring the ultrafast dephasing of OH vibrational coherences. Recent experimental studies have used ultrafast techniques such as pump-probe,39 infrared photon echo,40 vibration echo peak shift,41 and 2D infrared42 spectroscopy to investigate the hydrogen bond network characterizing this strongly associated liquid. Spectroscopic measurables can be related to characteristic dynamical processes, such as molecular reorientation and hydrogen bond stretching, breaking, and re-formation by detailed analysis of the data. An important tool for forging these connections is molecular dynamics simulation. The quantum nature of the OH chromophore must be combined with the classical trajectories representing the many-body system dynamics in the simulation methodology (see, e.g., Ref. 43–48). Here, we show that the semiclassical Liouville methodology provides an alternate and efficient computational approach to vibrational dephasing in water.33 We demonstrate that reasonable and qualitatively accurate results can be obtained using a simple and efficient model that requires modest computational effort, not to obtain quantitatively accurate results for this particular physical problem.

We approximate the quantum state-dependent many-body potential to modeling vibrational dephasing of OH stretches of HOD in D_{2}O using the approach outlined before. Simulations were performed by integrating an ensemble of 500 trajectories each for 1 ps and evaluating the difference potential Δ*V*(*t*) = *V*_{1}(**Q**(*t*))-*V*_{2}(**Q**(*t*)) between the *n* = 0 and *n* = 1 vibrational states along the trajectories. We used a perturbative approach to calculate the ground and excited state energies of the quantum OH stretch along a classical trajectory **Q**(*t*) of HOD held rigid at its classical equilibrium geometry in a solution of rigid D_{2}O. We performed our simulations with a modified version of the TINKER package49 using the TIP3P water potential.50 Water molecules were held rigid in the simulations using the Rattle algorithm.51 In the integration, the H atom was treated as a single point particle in the TIP3P HOD equilibrium geometry to determine the forces governing the classical dynamics. The energies of the quantum states were then estimated by representing the H atom as a pair of quantum particles, as described above, and evaluating the resulting intermolecular energy using the TIP3P potential. The quantum displacements of H atom from the classical geometry are small, and so the perturbative approach is expected to be valid. The difference between the energies for the two vibrational states with the equilibrium average value subtracted then gives instantaneous fluctuation of the difference potential δ*V*(*t*).

In Figure 5, we show the correlation function of the transition frequency 〈δω(0)δω(*t*)〉 = 〈δ*V*(0)δ*V*(*t*)〉/ averaged over the evolving ensemble. Two temperatures are considered, *T* = 150 and 298 K. The result for 298 K exhibits two characteristic decay times of 33 and 362 fs. This is in qualitative agreement with results of previous experiments and simulations,43–48 and in particular compares with results in Refs. 41, 48. The short and long time scales are thought to correspond physically to the stretching dynamics of the hydrogen bond involving the OH and the more collective reorganization of the hydrogen bond network, respectively.43–48 This agreement with previous results suggests that the approximations used in propagating the molecular dynamics trajectories and estimating the difference potential are accurate. The lower temperature *T* = 150 K result shows a smaller initial variance 〈δω(0)^{2}〉 and more pronounced time-dependent coherent behavior, consistent with expectations.

In Figure 6, we show the time evolution of the trace of ρ_{01}, calculated by direct simulation of the evolving density matrix element and using Eq. (20). Results for *T* = 150 and 298 K are shown and compared with the result obtained using a cumulant expansion9 of the quantity and the transition frequency correlation function shown in Figure 5. Both of these quantities are determined by the same dynamics as simulated by the classical trajectory integration and ensemble averaging. Agreement is thus a test of the Gaussian approximation for this theoretical system. It can be seen that the initial decay is captured quantitatively by the cumulant result, which, in addition, does a reasonably good job overall of representing the decay of quantum coherence in this system. However, the direct simulation reveals structure that is not captured by the cumulant approximation, showing that the assumption of Gaussian statistics is a good, but not exact, approximation. A similar conclusion regarding accuracy of the Gaussian approximation was arrived at by Lawrence and Skinner.46

### Decoherence in an Nonequilibrium Environment: An Analytic Model

- Top of page
- Abstract
- Introduction
- Formalism
- Application to Vibrational Dephasing
- Decoherence in an Nonequilibrium Environment: An Analytic Model
- Summary

In this section, we explore the dynamics of quantum decoherence in nonequilibrium environments.52 Most previous works on environmental effects in quantum coherent dynamics have taken a description of the environment as a thermal reservoir, usually with Markovian statistical properties.9, 18, 19, 53 This is an important limiting case that undoubtedly describes many dissipative quantum processes quite adequately. However, nonequilibrium bath effects leading to nonstationary statistics are a richly expanded yet unexplored range of possible dynamical effects in some cases of ultrafast quantum transport.

Quantum dynamics in nonequilibrium environments has been considered previously in a number of contexts. Schriefl et al. have studied dephasing of model two-level systems coupled to nonstationary 1/*f*^{μ} noise modeling interacting defects54 and nonstationary classical intermittent noise.55 Gordon et al.56 discuss the control of quantum coherence and the inhibition of dephasing by using stochastic control fields. Beer and Lutz57 discuss decoherence in a general nonequilibrium environment consisting of several equilibrium baths at different temperatures, described as a single effective bath with a time-dependent temperature. Myatt et al.58 study the decoherence of single trapped ions coupled to engineered reservoirs, where the internal state and coupling can be controlled. Agarwal59 describes slowing of decoherence by modulation of system–bath coupling. Emary60 describes a formalism for treating the dynamics of a quantum system coupled to a nonequilibrium environment and applies it to a charge qubit coupled to nonequilibrium electron transport through quantum dots. Clausen et al.61 describe a bath-optimized minimal energy control scheme to use arbitrary time-dependent perturbations to slow decoherence of quantum systems interacting with non-Markovian but stationary environments. Verstraete et al.62 describe the engineering of quantum states that exploits environmental dissipation in accomplishing quantum computation.

Here, we consider a simple model consisting of a two-level quantum system in a nonequilibrium bath, represented by random perturbations with nonstationary statistics.52 Our model allows an approximate analytic solution for the time evolution of the off-diagonal density matrix element ρ_{12} of the density operator describing the two-level quantum system interacting with the environment. By introducing a simple and specific ansatz for the dependence of the initial oscillator phases on frequency in terms of a single adjustable parameter, we demonstrate that significant modification of the decoherence process can result from variations of this parameter.

Our model consists of a two-level quantum system described by density operator 9, 18, 19 with energy gap *E*_{2}(*t*) − *E*_{1}(*t*) = ω(*t*) that fluctuates due to the effect of the environment, where *E*_{j}(*t*) (*j* = 1,2) is the instantaneous energy of state *j* as perturbed by the surroundings. The bath is represented by a random function of time corresponding to the transition frequency of the two-state quantum system ω(*t*), and the Fourier components of the time series represent the modes of motion of the bath. In contrast with the usual treatment,9, 18, 19 the statistical properties of this random function are nonstationary, corresponding physically to impulsively excited phonons of the environment with initial phases that are not random, but which have sharply defined values at *t* = 0. The distribution of phases of an ensemble of realizations then spreads with time over the interval (0,2π) under a diffusion equation. Well-defined initial phases of a phonon bath could result physically, for instance, by an ultrafast excitation of a quantum system that abruptly changed its size or charge distribution at *t* = 0, leading to systematic and reproducible mode-specific short time bath response. For example, if electronic excitation causes the size of a molecule in a condensed phase to increase, then a component of the initial short time motion of the surrounding atoms will be to move away from the chromophore due to the sudden increase in repulsive interactions. Each member of an initial thermal ensemble will share this component of motion, leading to an initial nonequilibrium statistical bias in the initial bath mode phases.

An initially prepared coherence between the two states |1〉 and |2〉 is described by the off-diagonal density matrix element 〈1| (*t*)|2〉 = ρ_{12}(*t*), which will decay due to the environment according to the expression9, 18, 19

- (34)

where ω(*t*) = ω_{o} + δω(*t*) and 〈…〉 represents a nonequilibrium average over the nonstationary random bath. The term ω_{o} represents the average frequency difference, while the average of the fluctuating term δω(*t*) is zero. This defines the function *F*(*t*), which we will use in our analysis.

In our nonequilibrium bath model, the time-dependent frequency is written in the form ω(*t*) = ω_{o} + δω(*t*), where

- (35)

The Fourier components *c*_{k} are positive constants related to the spectral density of the environment and the coupling of the bath modes to the quantum system. In this model, the randomness enters only through the “nonstationary” distribution of random phases θ_{k}(*t*). These phases are given by

- (36)

The random function *x*_{k}(*t*) is described by a time-dependent probability distribution *P*_{k}(*x*_{k},*t*) that obeys a diffusion equation

- (37)

where *D*_{k} is the diffusion constant. For simplicity, the initial state consists of a distribution localized at *x* = 0: *P*_{k}(*x*,0) = δ(*x*); the effect of thermal equilibrium of the bath before optical excitation could be incorporated by a broader initial distribution of *x*. The quantity *x* is an angle, so *P*(*x* + 2π,*t*) = *P*(*x*,*t*) is a periodic function of *x* with period 2π. A 2π-periodic δ function can be written in Fourier series form as

- (38)

The time-dependent probability distribution for component *k* that solves Eq. (37) with this initial condition is

- (39)

Physically, the phase of each component of the random force is not random at *t* = 0, when an impulsive excitation creates a quantum coherence in the system but decays to a uniform 1/2π distribution under diffusive evolution with diffusion constant *D*_{k}. The bath is thus not initially at equilibrium.

In Figure 7, we show for illustration an example of a nonstationary random function described by the phase diffusion model in Eq. (35). An ensemble of 500 realizations of the random time series is generated, and the minimum and maximum resulting functions span the shaded region. The width of this region is initially zero but grows with time, illustrating the diffusive loss of initial phase memory.

We now evaluate the time evolution of the off-diagonal density matrix element. The nonequilibrium averaged coherence is given by a product of single mode factors:

- (40)

We consider a typical factor *f*_{k}(*t*) = exp(−∫ δω_{k}(*s*)*ds*). Performing the time integral gives

- (41)

where *c*_{k}/ω_{k} ≡ *z*_{k}. This is an approximate expression, due to the dependence of the integrand on the “random” function *x*_{k}(*t*); we adopt it here for simplicity. Alternatively, we could take Eq. (41) as the “definition” of our nonstationary stochastic time series representing the evolution of the phase.

By performing a Taylor series expansion of *f*_{k}(*t*) in powers of *z*_{k} and keeping only the most slowly decaying terms, a simple but accurate approximation can be derived:

- (44)

Here,

- (45)

and

- (46)

where

- (47)

We note that the more rapidly decaying term with time dependence of *e*^{−4Dt} must be retained to give |*f*_{k}(0)| = 1.

The modulus function |*F*(*t*)| is given by

- (48)

where

- (49)

Here, *g*(ω) is the spectral density of the environment, and a continuum limit ∑_{k} … ∫ *g*(ω) … *d*ω has been taken.

We now investigate the dependence of the coherence dynamics and decoherence on the detailed nature of the initial bath excitation. To explore the general question of sensitivity of dephasing to these initial phases in a concrete example, we consider one simple model. We make a simple Gaussian approximation to the product of density of states and squared coupling and take

- (50)

where *A* = *N*_{eff} *z*; here *N*_{eff} is the effective number of bath modes, *z*_{eff} is the effective coupling, and ω_{c} is a measure of the frequency range of the bath modes. We also take *D*(ω) = *D*, a constant independent of ω.

The key quantity to consider in terms of the effect of the initial phases of the bath modes is the function θ(ω). There are of course a wide range of possible forms this can take. We adopt a very simple one parameter linear dependence of θ(ω) on ω and take

- (51)

Within this narrow set of possible phase relations, we explore the ability to control dephasing by varying the “single parameter” λ.

Evaluating the integral for β(*t*) for this model yields

- (52)

This result demonstrates an element of controllability of the coherence ρ_{12}(*t*), whose modulus |*F*(*t*)| = exp(−β(*t*; λ)). The modulus drops from its initial value of unity toward its asymptotic value at the intermediate time *t* = λ, but then “rephases” back to the slowly decaying envelope . As λ becomes large and positive, or assumes negative values, the decay approaches the envelope function without the nonmonotomic “dip.” This behavior is illustrated in Figure 8.

The simple relation θ(ω) = −λω is an idealized and minimalistic model allowing the nature of the relative oscillator phases to be varied systematically. Much more rich and variable relations can be contemplated, which in turn will undoubtedly allow more elaborate control of the decoherence dynamics. This will be explored in future work.

We have shown that the decoherence behavior of a two-state quantum system interacting with an initially “nonequilibrium” bath can be controlled by manipulating the nature of the relative initial phases of the bath modes. To demonstrate this possibility, we treated a simple model of a special case for illustration. The general phenomenon of the role played by environmental modes prepared initially with a well-defined initial phase by optical excitation of a system chromophore in many-body ultrafast quantum dynamics has received little attention. By engineering these initial phases, the character, and in particular, the dephasing, of the subsequent quantum evolution can potentially be controlled, in a manner reminiscent of a coherent control experiment using shaped pulses.10–12 Here, however, the control field is derived not from a shaped laser pulse but rather from the well-defined phase relations between the modes of the many-body bath. In living systems, ultrafast quantum dephasing in a nonequilibrium environment provides another possible handle on biophysical processes that could be exploited by natural selection, and it is of interest to explore whether the signature of this optimization can be found in the quantum dynamics of, for example, biological light harvesting systems. This question will be explored in future work.