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Treatment of open systems via bosonic and fermionic thermal reservoirs

Dannilo Iágor Evangelista de Araújo

Published Apr 15, 2019• 0 citations
OpenAlex

Abstract

We developed this work at the moment with the objective to resolver the master equation for both thermal reservoir: bosonic and fermionic, as well as the master equation in reservoirs of vacuum and of Fock compressed . the bosonic thermal reservoir is modeled by a collection of coupled harmonic oscillators and the fermionic reservoir by a collection of two-level atoms. We made use of the Jaynes-Cummings model and the Bohr, Bohr-Markov and Markov approaches. In the end we discussed the application to the master equation for two distinct phenomena, Cooling by Heating (CBH) analyzing average energy graphs to the atomic system versus number of photons, and the average energy of the bosonic system as a function of the mean number of phonons, we observed the negative temperature in the absolute scale due to the inversion of population of states in function of cooperativity, we also analyzed a signature of negative response in a three level system interacting in a optical cavity, that comes as a result of decrease in the output field intensity when the input field intensity is increased.

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We developed this work at the moment with the objective to resolver the master equation for both thermal reservoir: bosonic and fermionic, as well as the master equation in reservoirs of vacuum and of Fock compressed . the bosonic thermal reservoir is modeled by a collection of coupled harmonic oscillators and the fermionic reservoir by a collection of two-level atoms. We made use of the Jaynes-Cummings model and the Bohr, Bohr-Markov and Markov approaches. In the end we discussed the application to the master equation for two distinct phenomena, Cooling by Heating (CBH) analyzing average energy graphs to the atomic system versus number of photons, and the average energy of the bosonic system as a function of the mean number of phonons, we observed the negative temperature in the absolute scale due to the inversion of population of states in function of cooperativity, we also analyzed a signature of negative response in a three level system interacting in a optical cavity, that comes as a result of decrease in the output field intensity when the input field intensity is increased.
The quantum reservoirs commonly considered in open-quantum systems theory are those modeled by quantum harmonic oscillators, which are called bosonic reservoirs. Recently, quantum reservoirs modeled by two-level systems, the so-called fermionic reservoirs, have received attention due to their features. Given that the components of these reservoirs have a finite number of energy levels, unlike bosonic reservoirs, some studies are being carried out to explore the advantages of using this type of reservoir, especially in the operation of heat machines. In this paper, we carry out a case study of a quantum refrigerator operating in the presence of bosonic or fermionic thermal reservoirs, and we show that fermionic baths have advantages over bosonic ones.
The master equation currently used to describe a two level atom interacting with a single mode field damped by contact with a thermal reservoir (the damped Jaynes-Cummings model) is shown not to have, as its steady state solution, the expected canonical density operator prescribed by the general principles of statistical mechanics for a system in thermal equilibrium. A modified master equation is derived here which satisfies this requirement. Except for a reservoir at zero temperature, this master equation differs from that which is currently used in that the damping terms contain contributions due to the atom-field interaction.
We study the behavior of two weakly coupled quantum systems, described by a separable density operator. One of them is a single oscillator, representing a microscopic system, while the other is a set of oscillators, which perform the role of a reservoir in thermal equilibrium. From the Liouville-Von Neumann equation for the reduced density operator, we devise the master equation that governs the evolution of the microscopic system, incorporating the effects of temperature via Thermofield Dynamics formalism and suitably redefining the vacuum of the macroscopic system. As applications, we study an oscillator interacting with a set of boson oscillators at finite temperature, usually employed in quantum mechanics to describe a Brownian particle; and investigate the behavior of an oscillator in the presence of a heat bath consisting of a set of fermion oscillators.
A master equation for the Jaynes-Cummings (JC) system interacting with a bosonic reservoir through a conserved quantity is derived analytically without any approximation.The derivation is valid for any initial JC state that has no correlation with the initial reservoir at any temperature and yields a Gorini-Kossakowski-Lindblad-Sudarshan-type master equation with time-dependent coefficients.Several physical quantities of the open JC system are examined and the validity or limit of some of the standard approximations used to derive the master equation is discussed on the basis of the exact results obtained here.
We propose physical interpretations, also valid for temperatures different from zero, for stochastic methods which have been developed recently to describe the evolution of a quantum system interacting with a reservoir. As opposed to the usual reduced density operator approach, which refers to ensemble averages, these methods deal with the dynamics of single realizations, and involve the solution of stochastic Schrödinger equations. These procedures have been shown to be completely equivalent to the master equation approach when ensemble averages are taken over many realizations. We show that these techniques are not only convenient mathematical tools for dissipative systems, but may actually correspond to concrete physical processes, for any temperature of the reservoir. We consider a mode of the electromagnetic field in a cavity interacting with a beam of two- or three-level atoms, the field mode playing the role of a small system and the atomic beam standing for a reservoir at finite temperature, the interaction between them being given by the Jaynes-Cummings model. We show that the evolution of the field states, under continuous monitoring of the state of the atoms which leave the cavity, can be described in terms of either the Monte Carlo wavefunction (quantum jump) method or a stochastic Schrödinger equation, depending on the system configuration. We also show that the Monte Carlo wavefunction approach leads, for finite temperatures, to localization into jumping Fock states, while the diffusion equation method leads to localization into states with a diffusing average photon number, which for sufficiently small temperatures are close approximations to mildly squeezed states. We prove analytically that, in the quantum jump situation, the system evolves in the mean towards a Fock state, even if an infinite number of photon-number amplitudes is present in the initial state.