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Journal of Physics Conference Series

Thermodynamics of a qubit undergoing dephasing

Stefano Marcantoni

Published May 1, 2017• 4 citations
Publisher

Abstract

The thermodynamics of a qubit undergoing dephasing due to the coupling with the external environment is discussed. First of all, we assume the dynamics of the system to be described by a master equation in Lindblad form. In this framework, we review a standard formulation of the first and second law of thermodynamics that has been known in literature for a long time. After that, we explicitly model the environment with a set of quantum harmonic oscillators choosing the interaction such that the global dynamics of system and bath is analytically solvable and the Lindblad master equation is recovered in the weak-coupling limit. In this generalized setting, we can show that the correlations between system and bath play a fundamental role in the heat exchange. Moreover, the internal entropy production of the qubit is proven to be positive for arbitrary coupling strength.

DOI
10.1088/1742-6596/841/1/012019

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The thermodynamics of a qubit undergoing dephasing due to the coupling with the external environment is discussed. First of all, we assume the dynamics of the system to be described by a master equation in Lindblad form. In this framework, we review a standard formulation of the first and second law of thermodynamics that has been known in literature for a long time. After that, we explicitly model the environment with a set of quantum harmonic oscillators choosing the interaction such that the global dynamics of system and bath is analytically solvable and the Lindblad master equation is recovered in the weak-coupling limit. In this generalized setting, we can show that the correlations between system and bath play a fundamental role in the heat exchange. Moreover, the internal entropy production of the qubit is proven to be positive for arbitrary coupling strength.
A consistent theory of non-equilibrium thermodynamics for Markovian open quantum systems has been developed in the late seventies in analogy with Classical Irreversible Thermodynamics. The time-evolution of these open systems is usually described by means of effective master equations in Lindblad form, that turn out to be reliable when there is a separation of time-scales between system and environment, such that memory effects are negligible and the so-called Markovian approximation is justified. In this framework, the variations of energy and entropy in the system are consistently described, distinguishing between heat and work contributions and providing a statement of the second law of thermodynamics as positivity of the entropy production.\nHowever, there is empirical evidence that many physical systems like photosynthetic complexes,\nopto-mechanical resonators and superconducting qubits, just to mention a few, experience more general non-Markovian dynamics. The formulation of the laws of thermodynamics in a non-Markovian setting is matter of current research and represents the main topic of the present work. As a first step, we show that the entropy production defined as in the Markovian case can be negative for a class of non-Markovian dynamics. We argue that this outcome should not be interpreted as a violation of the second law of thermodynamics because the environment must be explicitly taken into account in the balance of entropy in the non-Markovian setting. In order to justify this claim we adopt a more general point of view, studying a closed bipartite quantum system, such that each of the two subsystems plays the role of a finite environment out of equilibrium for the other one. We concentrate on the balance of energy first and construct an effective Hamiltonian for each subsystem using physically reasonable requirements; then we define heat and work as in the standard Markovian treatment, with the effective Hamiltonian replacing the free Hamiltonian. It turns out that, in our framework, the work power is perfectly balanced between subsystems, while the correlations can store a part of energy locally inaccessible and exchange it with both subsystems in the form of heat.\nConcerning the balance of entropy, a quite general formulation of the second law of thermodynamics can be given as follows: under the assumption of a factorized initial state for the compound system, the sum of the total variations of the entropies in the two subsystems is always nonnegative. We show with an explicit example that this general formulation does not correspond to the statement presented in the framework of Markovian master equations, which should not be considered a priori the second law of thermodynamics.\nIn the last part of the thesis we concentrate of the so-called fluctuation relations, that are results extending the thermodynamic formalism beyond the behavior of average quantities. After reviewing the main theoretical outcomes, such as the Jarzynski equality and the Crooks fluctuation theorem, we describe a proposal to access experimentally the work performed on an ensemble of diatomic molecules by a time-dependent electric field coupled with their vibrational degree of freedom. This procedure could then be used to test the quantum Jarzynski equality. With respect to the results so far appeared in the literature, in which the left-hand side of the equality is inferred from an experiment and the right-hand side is computed according to a model, in our proposed setting we should be able to estimate from the experiment both the left-hand side and the right-hand side of the equality, independently.
We study a model of dephasing (decoherence) in a two-state quantum system (qubit) coupled to a bath of harmonic oscillators. An exact analytic solution for the reduced dynamics of a two-state system in this model has been obtained previously for factorizing initial states of the combined system. We show that the model admits exact solutions for a large class of correlated initial states which are typical in the theory of quantum measurements. We derive exact expressions for the off-diagonal elements of the qubit density matrix, which hold for an arbitrary strength of coupling between the qubit and the bath. The influence of initial correlations on decoherence is considered for different bath spectral densities. Time behavior of the qubit entropy in the decoherence process is discussed.
Abstract Decoherence has been observed to be one of the main limitations to build a practical quantum computer. This is because the qubits that conform it rapidly lose the entanglement and correlations existent among them due to interactions with the environment. In this work, the relaxation time of a Transmon qubit interacting with an environment is predicted by using the steepest-entropy-ascent quantum thermodynamics framework and the Lindblad master equation. For the study it is considered that the environment behaves as a thermal bath composed of a collection of harmonic oscillators. Results show that the qubit processes from a non-equilibrium state to a state of stable equilibrium, exchanging energy with the thermal bath during the process. In addition, the prediction of the relaxation time of the qubit is in agreement with experimental results obtained from the literature. The decoherence time is also accurately predicted by the models.
We present analytical results for the time-dependent information entropy in exactly solvable two-state (qubit) models.The first model describes dephasing (decoherence) in a qubit coupled to a bath of harmonic oscillators.The entropy production for this model in the regimes of "complete" and "incomplete" decoherence is discussed.As another example, we consider the damped Jaynes-Cummings model describing a spontaneous decay of a two-level system into the field vacuum.It is shown that, for all strengths of coupling, the open system passes through the mixed state with the maximum information entropy.
We study the bath dynamics in the dephasing model of a two-state quantum system (qubit) coupled to an environment of harmonic oscillators.This model was shown [Morozov et al., Phys.Rev. A, 2012 85, 022101] to admit the analytic solution for the qubit and environment dynamics.Using this solution, we derive the exact expression for the bath reduced density matrix in the presence of initial qubit-environment correlations.We obtain the non-equilibrium phonon distribution function and discuss in detail the time behavior of the bath energy.It is shown that only the inclusion of dynamic correlations between the qubit and the bath ensures the proper time behavior of the quantity which may be interpreted as the "environment energy".
Abstract There is a current interest in quantum thermodynamics in the context of open quantum systems. An important issue is the consistency of quantum thermodynamics, in particular the second law of thermodynamics, i.e. the flow of heat from a hot reservoir to a cold reservoir. Here recent emphasis has been on composite system and in particular the issue regarding the application of local or global master equations. In order to contribute to this discussion we discuss two cases, namely as an example a single qubit and as a simple composite system two coupled qubits driven by two heat reservoirs at different temperatures, respectively. Applying a global Lindblad master equation approach we present explicit expressions for the heat currents in agreement with the second law of thermodynamics. The analysis is carried out in the Born–Markov approximation. We also discuss issues regarding the possible presence of coherences in the steady state.