Communication: Decoherence in a nonequilibrium environment: An analytically solvable model

We describe an analytically solvable model of quantum decoherence in a nonequilibrium environment. The model considers the effect of a bath driven from equilibrium by, for example, an ultrafast excitation of a quantum chromophore. The nonequilibrium response of the environment is represented by a nonstationary random function corresponding to the fluctuating transition frequency between two quantum states coupled to the surroundings. The nonstationary random function is characterized by a Fourier series with the phase of each term starting initially with a definite value across the ensemble but undergoing random diffusion with time. The decay of the off-diagonal density matrix element is shown to depend significantly on the particular pattern of initial phases of the terms in the Fourier series, or equivalently, the initial phases of bath modes coupled to the quantum subsystem. This suggests the possibility of control of quantum decoherence by the detailed properties of an environment that is driven from thermal equilibrium.

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,^2–9 the design and interpretation of nonlinear spectroscopies,¹⁰ the coherent control of molecular processes,^11–13 the manipulation and storage of quantum information,^14,15 and many others.^16–18 

Quantum coherence exists and is most pronounced in few simple 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 many-dimensional thermal environment or bath.^19,20 Most previous theoretical investigations of environmental effects on quantum coherent dynamics have taken a description of the environment as a thermal reservoir, usually with Markovian statistical properties,^10,19–21 although bath correlations can also be included.²² 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. 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. If this is indeed the case, living systems could have exploited these effects in the design of quantum processes by evolution.

In this paper, we investigate the dynamics of quantum decoherence in nonequilibrium environments. We consider a novel model of pure dephasing consisting of a two level quantum system in a nonequilibrium bath, represented by random perturbations with nonstationary statistics. Our model allows an approximate analytic solution for the time evolution of the off-diagonal density matrix element

$ρ12$ 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. This model treats the process of pure dephasing in the absence of any population relaxation. This more general process could be incorporated at the expense of the simplicity of the model.

The model we consider consists of a two level quantum system described by density operator

$ρ̂(t)$^10,19,20 with energy gap

$E_2(t)-E_1(t)=\hbar \omega (t)$

$E2(t)−E1(t)=ℏω(t)$ that fluctuates due to the effect of the environment, where

$E_j(t)$

$Ej(t)$ (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,^10,19,20 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. Such a bath is no longer at thermal equilibrium. The distribution of phases of an ensemble of realizations then spreads with time over the interval (0, 2π) under a diffusion equation as the environment returns to equilibrium. 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

$1|ρ̂(t)|2=ρ12(t)$⁠, which will decay due to the environment according to the expression^10,19,20

where

$ω(t)=ωo+δω(t)$ and 〈⋅⋅⋅〉 represents a nonequilibrium average over the nonstationary random bath. The term

$\omega _o$

$ωo$ 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

$ω(t)=ωo+δω(t)$⁠, where

The Fourier components

$ck$ 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

$\theta _k(t)$

$θk(t)$⁠. These phases are given by

$\theta _k(t)\break = \theta _k(0) + x_k(t)$

$θk(t)=θk(0)+xk(t)$⁠. The random function

$x_k(t)$

$xk(t)$ is described by a time-dependent probability distribution

$P_k(x_k,t)$

$Pk(xk,t)$ that obeys a diffusion equation

where

$Dk$ is the diffusion constant. For simplicity, the initial state consists of a distribution localized at x = 0:

$P_k(x,0)\break = \delta (x)$

$Pk(x,0)=δ(x)$⁠; 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

The time-dependent probability distribution for component k that solves Eq. (3) with this initial condition is

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

$Dk$⁠. The bath is thus not initially at equilibrium.

In Fig. 1 we show for illustration an example of a nonstationary random function δω(t) described by the phase diffusion model in Eq. (2). 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

We consider a typical factor

$fk(t)=exp(−∫otδωk(s)ds)$⁠. Performing the time integral of the fluctuating frequency gives

where

$ck/ωk≡zk$⁠. This is an approximate expression, due to the dependence of the integrand on the random function

$x_k(t)$

$xk(t)$⁠; we adopt it here for simplicity. Alternatively, we could take Eq. (7) as the definition of our nonstationary stochastic time series representing the evolution of the phase.

We now evaluate the average of

$exp(−izksin(ωkt+θk(0)+xk))$ over the probability distribution

$P_k(x_k,t)$

$Pk(xk,t)$

which gives

where

$Jn(z)$ is a Bessel function of order n and argument z. As t → 0 we see that

$f_k(0)=1$

$fk(0)=1$⁠, as it should. For t → ∞ we find that

$f_k(t) \rightarrow J_0(z_k) e^{i z_k \sin (\theta _k(0))}$

$fk(t)→J0(zk)eizksin(θk(0))$⁠. This is a number whose absolute value is less than unity, so the product of factors

$\prod _{k=1}^{\infty } f_k(\infty) \rightarrow 0$

$∏k=1∞fk(∞)→0$ as the number of factors goes to infinity, as expected for a correlation function.

By performing a Taylor series expansion of

$fk(t)$ in powers of

$z_k$

$zk$ and keeping only the most slowly decaying terms, a simple but accurate approximation can be derived:

Here,

and

where

We note that the more rapidly decaying term with time dependence of

$e−4Dt$ must be retained to give

$|f_k(0)|=1$

$|fk(0)|=1$⁠.

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

where

Here, g(ω) is the spectral density of the environment, and a continuum limit

$∑k...→∫0∞g(ω)...dω$ 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 Gaussian approximation to the product of density of states and squared coupling, and take

where

$A=N eff z eff 2$⁠; here

$N_{\mathrm{eff}}$

$N eff$ is the effective number of bath modes,

$z_{\mathrm{eff}}$

$z eff$ is the effective coupling, and

$\omega _c$

$ωc$ is a measure of the frequency range of the bath modes. We also take D(ω) = D, a constant independent of ω. We choose the relative strength of the interaction

$z_{\mathrm{eff}}$

$z eff$ and diffusion constant D so that the time scales for electronic dephasing and the loss of phase fidelity of the initial nonequilibrium bath excitation are roughly the same. This is consistent with experiments and previous work: the electronic dephasing time scale observed in Refs. 2–9 is of the order of picoseconds, as is the time scale for the decay of mechanical coherence in a bath of vibrational modes.^22,23

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 θ(ω) = −λω. 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

This result demonstrates an element of controllability of the coherence

$ρ12(t)$⁠, whose modulus |F(t)| = exp ( − β(t; λ)). The modulus drops from its initial value of unity toward its asymptotic value

$|F(t\rightarrow \infty)|=\exp \left(-\frac{1}{8} N_{\mathrm{eff}} z_{\mathrm{eff}}^2\right)$

$|F(t→∞)|=exp−18N eff z eff 2$ at the intermediate time t = λ, but then rephases back to the slowly decaying envelope

$\exp \left(-\frac{1}{8} N_{\mathrm{eff}} z_{\mathrm{eff}}^2 \,[1-e^{-2 D t}]\right)$

$exp−18N eff z eff 2[1−e−2Dt]$⁠. 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 Fig. 2. Nonexponential behavior in the decay of quantum coherence has been observed in full many-particle simulations of quantum coherent dynamics under nonequilibrium conditions.^16–18 

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.

In this work we treated a simple model of a special case to illustrate the general phenomenon of dephasing in a nonequilibrium bath. 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. The nonequilibrium bath coherence described by these initial phase relations then decays with a time scale related to the decay time for bath correlations, modeled here by vibrational mode phase diffusion. The possible 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. The results here suggest that, 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.^11–13 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.

This work was supported by the National Science Foundation.