The Effective Hamiltonian Method in the Thermodynamics of Two Resonantly Interacting Quantum Oscillators
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The Effective Hamiltonian Method in the Thermodynamics of Two Resonantly Interacting Quantum Oscillators A. I. Trubilkoa,* and A. M. Basharovb,c,** a
St. Petersburg University of State Fire Service of Emercom of Russia, St. Petersburg, 196105 Russia Research Center “Kurchatov Institute,” pl. Akademika Kurchatova 1, Moscow, 123182 Russia c Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia *e-mail: [email protected] **e-mail: [email protected] bNational
Received March 15, 2019; revised March 15, 2019; accepted April 2, 2019
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
1. INTRODUCTION In recent years, there has been interest in a field called quantum thermodynamics. On the one hand, it stems from a fundamental aspect related to the possibility of directly obtaining the thermodynamic laws from the quantum dynamical equations of motion. On the other hand, it is also determined by purely practical interests—the description of energy transfer in quantum devices is a fundamental question for the purposes of new technologies and the creation of thermodynamic devices determined by the manifestation of the quantum nature of interactions. The study of quantum transport in physical systems, for instance, in ultracold atoms [1–4], quantum dots [5], molecular, ionic, and solid-state systems [6–8] can be cited as examples. Such systems of a different physical nature presumably can serve as the main basic elements of quantum networks in which the connection between the elements is made through the organization of various kinds of interactions between them. They can also serve as a basis for quantum refrigerators [9] and quantum heat engines [10–13]. Each element of such a system also irreversibly interacts with its environment. Therefore, the problem of deriving the master equation for an open quantum system that corresponds to the realized physical conditions is the central one in the description. The approaches of mathematicians and physicists should be distinguished in the derivation/application of the master equation. Mathematicians begin with the abstract general concepts of the master equation and
invoke particular model interaction operators [14–17]. Physicists construct an appropriate effective Hamiltonian and interaction operators in each specific physical problem and then derive the master equations for the problem under consideration [18, 19].
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