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Exact Density Matrix Elements for a Driven Dissipative Bosonic Mode in a Kerr-Like Medium

Sh. Saedi, Fardin Kheirandish

Published May 19, 2021• 0 citations
Publisher

Abstract

Abstract For a prototype Hamiltonian describing a driven, dissipative bosonic mode in a Kerr-like medium, the exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations and operator calculus. The special and limiting cases are discussed.

DOI
10.21203/rs.3.rs-513805/v1

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Abstract For a prototype Hamiltonian describing a driven, dissipative bosonic mode in a Kerr-like medium, the exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations and operator calculus. The special and limiting cases are discussed.
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We develop a systematic projection-operator technique for constructing Gaussian approximations and their perturbative corrections in bosonic nonlinear models. As a case study, we apply it to the driven dissipative Kerr oscillator. In the absence of external driving, the model can be solved exactly within a low-dimensional Fock subspace, leading to strongly non-Gaussian states. Nevertheless, we demonstrate that the evolution of first- and second-order moments is captured by our Gaussian scheme with high accuracy even in this regime, providing a natural benchmark. For the general case with external driving, our approach reduces the equations of motion to a closed system for means and covariances and allows one to compute systematic corrections beyond the Gaussian level in closed form. We also calculate the dynamics of linear and quadratic combinations of creation and annihilation operators in both weak- and strong-drive regimes.
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.
We present a method to perform a variational analysis of the quantum master equation for driven-disspative bosonic fields with arbitrary large occupation numbers. Our approach combines the PPP representation of the density matrix and the variational principle for open quantum system. We benchmark the method by comparing it to wave-function Monte Carlo simulations and the solution of the Maxwell-Bloch equation for the Jaynes-Cummings model. Furthermore, we study a model describing Rydberg polaritons in a cavity field and introduce an additional set of variational parameters to describe correlations between different modes.
The exact form of a quantum propagator of a quantum oscillator, interacting with a bosonic bath consisting of N -distinguished quantum oscillators with different frequencies, is obtained in the Heisenberg picture. A reduced density matrix for the oscillator is obtained. The kernel or Greenʼs function connecting the initial density matrix of the oscillator to the density matrix in an arbitrary time is obtained, and its connection to the Feynman–Vernon influence functional is discussed. A weak coupling regime and squared mean values for position, momentum, and energy of the oscillator are obtained in equilibrium.