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arXiv:1004.0652 · Physical Review A

Nonlinear thermodynamic quantum master equation: Properties and examples

Hans Christian Öttinger

Published Nov 29, 2010• 45 citations
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Abstract

The quantum master equation obtained from two different thermodynamic arguments is seriously nonlinear. We argue that, for quantum systems, nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature and consequences of the nonlinear contribution. The thermodynamic nonlinearity naturally leads to canonical equilibrium solutions and extends the range of validity to lower temperatures. We discuss the Markovian character of the thermodynamic quantum master equation and introduce a solution strategy based on coupled evolution equations for the eigenstates and eigenvalues of the density matrix. The general ideas are illustrated for the two-level system and for the damped harmonic oscillator. Several conceptual implications of the nonlinearity of the thermodynamic quantum master equation are pointed out, including the absence of a Heisenberg picture and the resulting difficulties with defining multitime correlations.

DOI
10.1103/physreva.82.052119

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The quantum master equation obtained from two different thermodynamic arguments is seriously nonlinear. We argue that, for quantum systems, nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature and consequences of the nonlinear contribution. The thermodynamic nonlinearity naturally leads to canonical equilibrium solutions and extends the range of validity to lower temperatures. We discuss the Markovian character of the thermodynamic quantum master equation and introduce a solution strategy based on coupled evolution equations for the eigenstates and eigenvalues of the density matrix. The general ideas are illustrated for the two-level system and for the damped harmonic oscillator. Several conceptual implications of the nonlinearity of the thermodynamic quantum master equation are pointed out, including the absence of a Heisenberg picture and the resulting difficulties with defining multitime correlations.
The quantum master equation obtained by generalizing the geometric formulation of nonequilibrium thermodynamics to dissipative quantum systems is seriously nonlinear. We argue that nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature of the nonlinear contribution. The thermodynamic nonlinearity leads to proper equilibrium solutions and improves the dynamic behavior of dissipative quantum systems. We also discuss the Markovian character of the thermodynamic quantum master equation. The general ideas are illustrated for two-level systems.
We investigate the zero-temperature limit of thermodynamic quantum master equations that govern the time evolution of density matrices for dissipative quantum systems. The quantum master equations for T=0T=0T=0 and for T>0T>0T>0 possess completely different structures: (i) the equation for T=0T=0T=0 is linear in the deviation from the ground-state density matrix, whereas the equation for T>0T>0T>0, in general, is seriously nonlinear, and (ii) the Gibbs state is obtained as the steady-state solution of the nonlinear equation for T>0T>0T>0, whereas the ground state cannot be found from the equation for T=0T=0T=0. Nevertheless, the equation for T=0T=0T=0 can reproduce the behavior for $T\ensuremath{\gtrsim}0$ remarkably well. We discuss some implications of that observation for dissipative quantum field theory.
The different conditions that could be imposed on a Markovian master equation for a nonlinear oscillator weakly coupled to a thermal reservoir are formulated. They concern preservation of trace and positivity of a density matrix, return to a proper equilibrium state, and the detailed balance condition. It is shown that only one of the known master equations satisfies all of these conditions. Then the validity of the Markovian approximation is reanalyzed using certain non-Markovian weak-coupling approximations, and the existence of different stages of evolution associated with different time scales of the Hamiltonian dynamics is predicted. The consequences of these facts for the description of a damped nonlinear oscillator are discussed.
Open quantum systems are studied from the thermodynamical point of view unifying the principle of maximum informational entropy and the hypothesis of relaxation times hierarchy. The result of the unification is a non-Markovian and local-in-time master equation that provides a direct connection for dynamical and thermodynamical properties of open quantum systems. The power of the approach is illustrated by the application to the damped harmonic oscillator and the damped driven two-level system, resulting in analytical expressions for the non-Markovian and nonequilibrium entropy and inverse temperature.
Based on the standard system-heat-bath model, we have formulated a theory of the non-Markovian master equation for linear and nonlinear systems for a correlation time that is short but finite. We have applied the theory to linear systems, such as harmonic oscillators and two-level atoms and calculated the band-shape functions for transient optical absorption in the pump-probe experiments. The significant interplay of nonlinearity and non-Markovian decay has been illustrated by another application of the theory to a Morse oscillator. The theory is valid for arbitrary nonlinearity and finite temperatures but within the weak-coupling limit and has been derived without ad hoc implementation of a phenomenological memory function.
We simulate a piecewise deterministic Markovian jump process that represents an unraveling of a nonlinear quantum master equation describing the evolution of a quantum subsystem in contact with a heat bath. This process relies on running ensemble averages and state vectors normalized only on average. The latter causes the normalization of a small subset of trajectories to grow exponentially, resulting in numerical instability. We propose an efficient solution to this problem and illustrate the general ideas for a harmonic oscillator and a two-level system, each of them being weakly coupled to a heat reservoir. Our findings lay the foundation for a powerful simulation technique for dissipative quantum systems surrounded by general classical nonequilibrium environments.