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Journal of Experimental and Theoretical Physics

The Effective Hamiltonian Method in the Thermodynamics of Two Resonantly Interacting Quantum Oscillators

Andrew I. Trubilko, Ashat M. Basharov

Published Sep 1, 2019• 12 citations
Publisher

Abstract

Abstract We investigate the classical problem of two resonantly interacting oscillators each of which is coupled to “its own” heat bath based on the effective Hamiltonian method and the quantum stochastic differential equation (in contrast to the well-known “global” and “local” approaches). We show that in the second order of the algebraic perturbation theory, each of the oscillators turns out to be also coupled to the “foreign” heat bath. We calculate the steady-state heat flows and prove that there is no heat flow from the cold heat bath to the hot one, as evidenced by some results of the local approach.

DOI
10.1134/s1063776119080090

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We consider a situation where an NNN-level system (NLS) is coupled successively to two heat baths with different temperatures without being necessarily thermalized and approaches a steady state. For this situation we apply a general Jarzinski-type equation and conclude that heat and entropy is flowing from the hot bath to the cold one. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities. Our approach is illustrated by an analytical example. For the linear regime, i.~e., for small temperature differences between the two heat baths we derive an expression for the heat conduction coefficient.
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Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.