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arXiv:1510.03212

A Gaussian density matrix under decoherence and friction

János Polonyi

Published Oct 12, 2015• 0 citations
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Abstract

The time evolution of a Gaussian density matrix of a one dimensional particle, generated by a quadratic, ${\cal O}(\partial_t^2)$ effective Lagrangian, describing a harmonic potential, a friction force and decoherence, is studied within the Closed Time Path formalism. The density matrix converges to an asymptotic form, given by a completely decohered thermal state with an ${\cal O}(\hbar)$ temperature in the translation invariant case. The time evolution of the state of a harmonic oscillator is followed numerically. The asymptotic density matrix, the fixed point of the master equation, is found analytically and its dependence on the oscillator frequency, the friction constant and the decoherence strength is explored.

DOI
10.48550/arxiv.1510.03212

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The time evolution of a Gaussian density matrix of a one dimensional particle, generated by a quadratic, O(∂t2){\cal O}(\partial_t^2)O(∂t2​) effective Lagrangian, describing a harmonic potential, a friction force and decoherence, is studied within the Closed Time Path formalism. The density matrix converges to an asymptotic form, given by a completely decohered thermal state with an O(ℏ){\cal O}(\hbar)O(ℏ) temperature in the translation invariant case. The time evolution of the state of a harmonic oscillator is followed numerically. The asymptotic density matrix, the fixed point of the master equation, is found analytically and its dependence on the oscillator frequency, the friction constant and the decoherence strength is explored.
In this paper we study the dissipative effects and decoherence induced on a particle moving at constant speed in front of a dielectric plate in quantum vacuum, developing a closed-time-path (CTP) integral formulation in order to account for the corrections to these phenomena generated by finite temperatures. We compute the frictional force of the moving particle and find that it contains two different contributions: a pure quantum term due to quantum fluctuations (even present at vanishing temperatures) and a temperature-dependent component generated by thermal fluctuations (the bigger the contribution, the higher the temperature). We further estimate the decoherence timescale for the internal degree of freedom of the quantum particle. As expected, decoherence time is reduced by temperature; however, this feature is stronger for large velocities and for resonant situations. When the particle approaches relativistic speed, decoherence time becomes independent of temperature. The finite temperature corrections to the force or even in the decoherence timescale could be used to track traces of quantum friction through the study of the velocity dependence since the sole evidence of this dependence provides an indirect testimony of the existence of a quantum frictional force.
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Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of Egs+kBTE_{gs}+k_BTEgs​+kB​T. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.
The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. We find that the dynamics must include a nonvanishing noise term to yield physical results for the purity and that the oscillator decoheres in time such that the late-time density operator is thermal. We show that the frequency spectrum or unequal-time correlations can, however, distinguish between the damped oscillator and an isolated oscillator in thermal equilibrium and obtain a generalized fluctuation-dissipation relation for the damped oscillator. We briefly consider time-nonlocal dissipation as well, to show that the fluctuation-dissipation is satisfied for a specific choice of dissipation kernels. Lastly, we develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results for time-local dissipation are in fact nonperturbative in the initial state.