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The Journal of Chemical Physics

Coherent state representation of thermal correlation functions with applications to rate theory

Eli Pollak, Sameernandan Upadhyayula, Jian Liu

Published Jun 2, 2022• 8 citations

Abstract

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

DOI: 10.1063/5.0088163

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We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

On the basis of a coherent state representation of quantum noise operator and an ensemble averaging procedure a scheme for quantum Brownian motion has been proposed recently [Banerjee {\it et al}, Phys. Rev. E {\bf65}, 021109 (2002); {\bf66}, 051105 (2002)]. We extend this approach to formulate reactive flux theory in terms of quantum phase space distribution functions and to derive a time dependent quantum transmission coefficient - a quantum analogue of classical Kramers'-Grote-Hynes coefficient in the spirit of Kohen and Tannor's classical formulation. The theory is valid for arbitrary noise correlation and temperature. The specific forms of this coefficient in the Markovian as well as in the non-Markovian limits have been worked out in detail for intermediate to strong damping regime with an analysis of quantum effects. While the classical transmission coefficient is independent of temperature, its quantum counterpart has significant temperature dependence particularly in the low temperature regime.

Exact quantum thermal flux–flux correlations are obtained for the inverted parabolic barrier and free particle by smearing and scaling corresponding two-surface classical correlations. The smearing width is identified as a dynamical uncertainty in x associated with imaginary time evolution by −iħβ/2. The scaling is just the ratio of the high temperature asymptotic form of this dynamical uncertainty to the dynamical uncertainty itself. It represents a sort of quantum amplification. This quantum adaptation of classical flux–flux correlation is generalized to generic one-dimensional barrier systems, and applied to the symmetric and asymmetric Eckart barriers. Good agreement between model and converged numerical results is obtained down to temperatures where tunnelling begins to predominate above-barrier transmission. The model does, however, fail further into the tunnelling regime.

We investigate the dynamics of a driven quantum field coupled to a finite-temperature reservoir. The corresponding master equation is solved using superoperator techniques, yielding an analytical expression for the density operator. To obtain a compact and physically transparent description of the dynamics, we adopt a phase-space representation based on the Husimi Q-function. For an initially coherent state, we derive a closed-form Gaussian expression for the Husimi Q-function whose stationary limit corresponds to a displaced thermal state. This approach also enables an analytical study of quantum-interference dynamics for an initial superposition of coherent states. Furthermore, we derive the corresponding Fokker–Planck equation for the Husimi Q-function and obtain closed-form expressions for relevant statistical quantities, including the mean photon number, the photon-number standard deviation, and the Mandel parameter. We also investigate the Wehrl and linear entropies, which quantify the loss of phase-space information and purity induced by the thermal environment. The framework provides a complete analytical characterization of the phase-space dynamics, photon statistics, and entropic properties of driven–dissipative quantum fields while avoiding the explicit manipulation of the density operator.

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.

A rigorous and practical methodology for evaluating thermal-equilibrium density matrices, finite-temperature time-dependent expectation values, and time-correlation functions is described. The method involves an extension of the matching-pursuit/split-operator-Fourier-transform method to the solution of the Bloch equation via imaginary-time propagation of the density matrix and the evaluation of Heisenberg time-evolution operators through real-time propagation in dynamically adaptive coherent-state representations.