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Physical review. E

Exact Markovian evolution of quantum systems with several degrees of freedom: Phase space representations

Aldo R. Fernandes Neto, Alfredo M. Ozorio de Almeida, Olivier Brodier

Published Oct 3, 2023• 1 citation
Publisher

Abstract

The exact solution of the Lindblad equation with a quadratic Hamiltonian and linear coupling operators was derived within the chord representation, that is, for the Fourier transform of the Wigner function, also known as the characteristic function. It is here generalized for several degrees of freedom, so as to provide an explicit expression for the reduced density operator of any subsystem, as well as moments expressed as derivatives of this evolving chord function. The Wigner function is then the convolution of its straightforward classical evolution with a widening multidimensional Gaussian window, eventually ensuring its positivity. Futher on, positivity also holds for the Glauber-Sundarshan P function, which guarantees separability of the components. In the context of several degrees of freedom, a full dissipation matrix is defined, whose trace is equal to twice the previously derived dissipation coefficient. This governs the rate at which the phase space volume of the argument of the Wigner function contracts, while that of the chord function expands. Examples of Markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed.

DOI
10.1103/physreve.108.044106

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The exact solution of the Lindblad equation with a quadratic Hamiltonian and linear coupling operators was derived within the chord representation, that is, for the Fourier transform of the Wigner function, also known as the characteristic function. It is here generalized for several degrees of freedom, so as to provide an explicit expression for the reduced density operator of any subsystem, as well as moments expressed as derivatives of this evolving chord function. The Wigner function is then the convolution of its straightforward classical evolution with a widening multidimensional Gaussian window, eventually ensuring its positivity. Futher on, positivity also holds for the Glauber-Sundarshan P function, which guarantees separability of the components. In the context of several degrees of freedom, a full dissipation matrix is defined, whose trace is equal to twice the previously derived dissipation coefficient. This governs the rate at which the phase space volume of the argument of the Wigner function contracts, while that of the chord function expands. Examples of Markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed.
The exact solution of the Lindblad equation with a quadratic Hamiltonian and linear coupling operators was derived within the chord representation, that is, for the Fourier transform of the Wigner function, also known as the characteristic function. It is here generalized for several degrees of freedom, so as to provide an explicit expression for the reduced density operator of any subsystem, as well as moments expressed as derivatives of this evolving chord function. The Wigner function is then the convolution of its straightforward classical evolution with a widening multidimensional Gaussian window, eventually ensuring its positivity. Futher on, positivity also holds for the Glauber-Sundarshan P function, which guarantees separability of the components. In the context of several degrees of freedom, a full dissipation matrix is defined, whose trace is equal to twice the previously derived dissipation coefficient. This governs the rate at which the phase space volume of the argument of the Wigner function contracts, while that of the chord function expands. Examples of Markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed.
We obtain analytic solutions to various models of dissipation of the quantum harmonic oscillator, employing a simple method in the Wigner function Fourier transform description of the system; and study as an exemplification, the driven open quantum harmonic oscillator. The environmental models we use are based on optical master equations for the zero and finite temperature bath and whose open dynamics are described by a Lindblad master equation, and also we use the Caldeira-Leggett model for the high temperature limit, in the the under damped an the over damped case. Under the Wigner Fourier transform or chord function as it has been called, it becomes particularly simple to solve the dynamics of the open oscillator in the sense that the dynamics of the system are reduced to the application of an evolution matrix related to the damped motion of the oscillator.
We use the characteristic function of the Wigner function (its double Fourier transform) to give solution to any generic open quantum linear systems (systems whose Hamiltonian is at most quadratic). The solution is carried out in terms of the application of the transition matrix of the dynamical evolution in the Fourier space. We address two cases: the time-independent coefficients for which we give the solutions for several dissipative models of the quantum harmonic oscillator and the one-dimensional free particle. In the latter, we also derive a heuristic model for a pure damped motion with suppression of diffusion. For the time-dependent coefficient problem, we give some particular cases that are integrable and derive a second order approximation to the generic case in which all parameters are time-dependent. We additionally explore the solutions of the system when the diffusion processes of the dissipative model lie beyond the weak coupling limit.
The time evolution of an open quantum system is governed by the Gorini-Kossakowski-Sudarshan-Lindblad equation for the reduced density operator of the system. This operator is obtained from the full density operator of the composite system involving the system itself, the bath, and the correlations between them, by performing a partial trace over the bath degrees of freedom. The entanglement between the system and the bath leads to a generalized Liouville evolution that involves, among other things, dissipation and decoherence of the system. In a similar fashion, the time evolution of a physical observable in a classically constrained dynamical system is governed by a generalization of the Liouville equation, in which the usual Poisson bracket is replaced by the so-called Dirac bracket. The generalization takes into account the reduction in the phase space of the system because of constraints that arise either because they are introduced by hand or because of singular Lagrangians. We derive an intriguing but precise classical-quantum correspondence between the aforementioned situations which connects the Lindblad operators to the constraints. The correspondence is illustrated in the ubiquitous example of coupled simple harmonic oscillators studied earlier in various contexts such as quantum optics, dissipative quantum systems, Brownian motion, and the area law for the entropy of black holes.
The connections between standard theoretical tools used to study open quantum systems can sometimes seem opaque. Whether it is a Lindblad master equation, the equation of motion for the Wigner function, or a dissipative Keldysh action, features evident in one formalism are often masked in another. Here, we reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. This symmetry can then be used to easily diagonalize these models, and it provides an intuitive way to demonstrate the separation of dissipation and fluctuations in linear systems. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ``wave functions'' of the density matrix in the eigenbasis of the third-quantized superoperators we introduce. The field-theory representation of the time-evolution operator in this basis is then the Keldysh path integral. To highlight the utility of our approach, we apply our version of third quantization to a dissipative nonlinear oscillator, and we use it to obtain new exact results.
The time evolution of an open quantum system is governed by the Gorini-Kossakowski-Sudarshan-Lindlad equation for the reduced density operator of the system. This operator is obtained from the full density operator of the composite system involving the system itself, the bath, and the interactions between them, by performing a partial trace over the bath degrees of freedom. The entanglement between the system and the bath leads to a generalized Liouville evolution that involves, amongst other things, dissipation and decoherence of the system. In a similar fashion, the time evolution of a physical observable in a classically constrained dynamical system is governed by a generalization of the Liouville equation, in which the usual Poisson bracket is replaced by the so-called Dirac bracket. The generalization takes into account the reduction in the phase space of the system because of constraints which arise either because they are introduced by hand, or because of singular Lagrangians. We derive an intriguing, but precise classical-quantum correspondence between the aforementioned situations which connects the Lindblad operators to the constraints. The correspondence is illustrated in the ubiquitous example of coupled simple harmonic oscillators studied earlier in various contexts like quantum optics, dissipative quantum systems, Brownian motion, and the area law for the entropy of black holes.