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Journal of Physics A Mathematical and Theoretical

Thermodynamics of dissipative coherent states

A.R. Moura, P D Mesquita

Published Nov 4, 2021• 0 citations
Publisher

Abstract

Abstract Almost all traditional physical formalisms are developed by using conservative forces, and the microscopic implementation of dissipation involves a sort of unusual process, mainly in quantum systems. In this work, we study the quantum harmonic model endowed with a non-Hermitian term responsible for dissipation. In addition, we also include an oscillating field that drives the model to a coherent state, which is dominated by fluctuation in a specific frequency, while regular thermal states are lowly occupied. The usual coherent state formalism at zero temperature is extended to treat dissipative models at finite temperature. We define a generating function that is used in the evaluation of the most relevant statistical averages, such as the particle distribution. Then, we successfully employ the developed formalism to discuss two well-known applications; the damped quantum harmonic oscillator, and the precession magnetization in a ferromagnetic sample.

DOI
10.1088/1751-8121/ac367b

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Abstract Almost all traditional physical formalisms are developed by using conservative forces, and the microscopic implementation of dissipation involves a sort of unusual process, mainly in quantum systems. In this work, we study the quantum harmonic model endowed with a non-Hermitian term responsible for dissipation. In addition, we also include an oscillating field that drives the model to a coherent state, which is dominated by fluctuation in a specific frequency, while regular thermal states are lowly occupied. The usual coherent state formalism at zero temperature is extended to treat dissipative models at finite temperature. We define a generating function that is used in the evaluation of the most relevant statistical averages, such as the particle distribution. Then, we successfully employ the developed formalism to discuss two well-known applications; the damped quantum harmonic oscillator, and the precession magnetization in a ferromagnetic sample.
The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.
This paper is concerned with the dynamics of coupled quantum oscillators in the presence of a classical harmonic driving force. By means of a theorem which describes the evolution of Gaussian distributions of coherent states, the solutions of the equations of motion are presented in a very compact form. Specific applications deal with forced oscillation and parametric amplification. The theoretical models include systems of damping oscillators with a continuous frequency distribution. The approach to thermal equilibrium in the absence of a driving force is discussed as a special case.
We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi QQQ-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to nnn interacting oscillators, with analytical energy expressions provided and the case n=3n=3n=3 examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. We find that the dynamics must include a nonvanishing noise term to yield physical results for the purity and that the oscillator decoheres in time such that the late-time density operator is thermal. We show that the frequency spectrum or unequal-time correlations can, however, distinguish between the damped oscillator and an isolated oscillator in thermal equilibrium and obtain a generalized fluctuation-dissipation relation for the damped oscillator. We briefly consider time-nonlocal dissipation as well, to show that the fluctuation-dissipation is satisfied for a specific choice of dissipation kernels. Lastly, we develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results for time-local dissipation are in fact nonperturbative in the initial state.