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arXiv:2605.24478

Quantum Dynamics of Interacting oscillators in a thermal medium: A novel scheme

Ronak Moradi, Fardin Kheirandish

quant-phSubmitted May 23, 2026• Announced May 26, 2026• 0 citations• Revised Aug 7, 2026 (v2)
v1 · May 23, 2026v2 · Aug 7, 2026
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Abstract

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 $Q$-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 $n$ interacting oscillators, with analytical energy expressions provided and the case $n=3$ 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.

Comments
21 pages, 6 figures

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In this letter, we introduce a novel method for investigating dissipation (gain) and thermalization in an open quantum system. In this method, the quantum system is coupled linearly with a copy of itself or with another system described by a finite number of bosonic operators. The time-dependent coupling functions play a fundamental role in this scheme. To demonstrate the efficiency and significance of the method, we apply it to some ubiquitous open quantum systems. Firstly, we investigate a quantum oscillator in the presence of a thermal bath at the inverse temperature βββ, obtaining the reduced density matrix, the Husimi distribution function, and the quantum heat distribution function accurately. The results are consistent with existing literature by appropriate choices for the time-dependent coupling function. To illustrate the generalizability of this method to systems interacting with multiple thermal baths, we study the interaction of a quantum oscillator with two thermal baths at different temperatures and obtain compatible results. Subsequently, we analyze a two-level atom with energy or phase dissipation and derive the spontaneous emission and the pure dephasing processes consistently using the new method. Finally, we investigate the Markovianity in a dissipative two-level system.
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.
Abstract In this letter, we introduce a novel method for investigating dissipation (gain) and thermalization in an open quantum system. In this method, the quantum system is coupled linearly with a copy of itself or with another system described by a finite number of bosonic operators. The time-dependent coupling functions play a fundamental role in this scheme. To demonstrate the efficiency and significance of the method, we apply it to some ubiquitous open quantum systems. Firstly, we investigate a quantum oscillator in the presence of a thermal bath at the inverse temperature β, obtaining the reduced density matrix, the Husimi distribution function, and the quantum heat distribution function accurately. The results are consistent with existing literature by appropriate choices for the time-dependent coupling function. To illustrate the generalizability of this method to systems interacting with multiple thermal baths, we study the interaction of a quantum oscillator with two thermal baths at different temperatures and obtain compatible results. Subsequently, we analyze a two-level atom with energy or phase dissipation and derive the spontaneous emission and the pure dephasing processes consistently using the new method. Finally, we investigate the Markovianity in a dissipative two-level system.
Abstract In the framework of a novel dissipative scheme, we have investigated the quantum dynamics of an oscillating system interacting with two reservoirs with different temperatures trough different time-dependent coupling functions. The reduced density matrix, quantum optical characteristic functions, and (quasi) distribution functions like Husimi, Glauber–Sudarshan and Wigner functions on the phase space of the oscillator are obtained. The problem has been generalized to the case where the oscillator is interacting with n distinctive reservoirs, and a quantum current and an effective reservoir is introduced. Finally, the quantum dynamics of a two-level system interacting with two reservoirs has been investigated, and the exact reduced density matrix is obtained.
We consider the global thermal state of classical and quantum harmonic oscillators that interact with a reservoir. Ohmic damping of the oscillator can be exactly treated with a 1D scalar field reservoir, whereas general non-Ohmic damping is conveniently treated with a continuum reservoir of harmonic oscillators. Using the diagonalized Hamiltonian of the total system, we calculate a number of thermodynamic quantities for the damped oscillator: the mean force internal energy, mean force free energy, and another internal energy based on the free-oscillator Hamiltonian. The classical mean force energy is equal to that of a free oscillator, for both Ohmic and non-Ohmic damping no matter how strong the coupling to the reservoir. In contrast, the quantum mean force energy depends on the details of the damping and diverges for strictly Ohmic damping. These results give additional insight into the steady-state thermodynamics of open systems with arbitrarily strong coupling to a reservoir, complementing results for energies derived within dynamical approaches (e.g. master equations) in the weak-coupling regime.
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.