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Physical Review A

Environmentally induced shift of the quantum arrival time

Michael Genkin, F. Ferro, Eva Lindroth

Published Nov 18, 2009• 3 citations
Publisher

Abstract

Using a simple model potential, we study the effects of weak Markovian dissipation on the quantum arrival time. The interaction with the environment is incorporated into the dynamics through a Markovian master equation of Lindblad type, which allows us to compare time-of-arrival distributions and approximate crossing probabilities for different dissipation strengths and temperatures. We also establish a connection to an earlier study where quantum tunneling with dissipation was investigated, which leads us to some conclusions concerning the formulation of the continuity equation in the Lindblad theory.

DOI
10.1103/physreva.80.052112

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Using a simple model potential, we study the effects of weak Markovian dissipation on the quantum arrival time. The interaction with the environment is incorporated into the dynamics through a Markovian master equation of Lindblad type, which allows us to compare time-of-arrival distributions and approximate crossing probabilities for different dissipation strengths and temperatures. We also establish a connection to an earlier study where quantum tunneling with dissipation was investigated, which leads us to some conclusions concerning the formulation of the continuity equation in the Lindblad theory.
The influence of a dissipative environment on scattering of a particle by a barrier is investigated by using the recently introduced bohmian mechanics with complex action [J. Chem. Phys. 125, 231103 (2006)]. An extension of this complex trajectory based formalism to include the interaction of the tunneling particle with an environment of harmonic oscillators with a continuous spectral density and at a certain finite temperature allows us to calculate transmission probabilities beyond the weak system bath coupling regime. The results display an increasing tunneling probability for energies below the barrier and a decreased transmission above the barrier due to the coupling. Furthermore, we demonstrate that solutions of a markovian master equation fail to do so in general.
In a unitary approach generally used in quantum optics, we consider Lindblad's Markovian master equation and the non- Markovian master equation of Ford, Lewis and O'Connell. We show that the second-order master equation in the hierarchy obtained from a Krylov-Bogoliubov expansion corresponds to the Born approximation. By time averaging, and neglecting the rapidly varying terms, Lindblad's master equation is obtained. With these two equations, we calculate the decay spectrum. We find that for rather low dissipated energies, only the non-Markovian master equation provides correct results. Based on the independent oscillator model of the dissipative coupling, explicit expressions of the dissipative coefficients are obtained.
An open quantum system is a quantum system that interacts with some environment whose degrees of freedom have been coarse grained away. This model describes non-equilibrium processes more general than scattering-matrix formulations. Furthermore, the microscopically-derived environment provides a model of noise, dissipation and decoherence far more general than Markovian (white noise) models. The latter are fully characterized by Lindblad equations and can be motivated phenomenologically. Non-Markovian processes consistently account for backreaction with the environment and can incorporate effects such as finite temperature and spatial correlations. We consider linear systems with bilinear coupling to the environment, or quantum Brownian motion, and nonlinear systems with weak coupling to the environment. For linear systems we provide exact solutions with analytical results for a variety of spectral densities. Furthermore, we point out an important mathematical subtlety which led to incorrect master-equation coefficients in earlier derivations, given nonlocal dissipation. For nonlinear systems we provide perturbative solutions by translating the formalism of canonical perturbation theory into the context of master equations. It is shown that unavoidable degeneracy causes an unfortunate reduction in accuracy between perturbative master equations and their solutions. We also extend the famous theorem of Lindblad, Gorini, Kossakowski and Sudarshan on completely positivity to non-Markovian master equations. Our application is primarily to model atoms interacting via a common electromagnetic field. The electromagnetic field contains correlations in both space and time, which are related to its relativistic (photon-mediated) nature. As such, atoms residing in the same field experience different environmental effects depending upon their relative position and orientation. Our more accurate solutions were necessary to assess sudden death of entanglement at zero temperature. In contrast to previous claims, we found that all initial states of two-level atoms undergo finite-time disentanglement. We were also able to access regimes which cannot be described by Lindblad equations and other simpler methods, such as near resonance. Finally we revisit the infamous Abraham-Lorentz force, wherein a single particle in motion experiences backreaction from the electromagnetic field. This leads to a number of well-known problems including pre-acceleration and runaway solutions. We found a more a more-suitable open-system treatment of the nonrelativistic particle to be perfectly causal and dissipative without any extraneous requirements for finite size of the particle, weak coupling to the field, etc..
We analyze the effect of a dissipative bosonic environment on the Landau-Zener-St\"uckelberg-Majorana (LZSM) level crossing model by using a microscopic approach to derive the relevant master equation. For an environment at zero temperature and weak dissipation, our microscopic approach confirms the independence of the survival probability on the decay rate that has been predicted earlier by the simple phenomenological LZSM model. For strong decay the microscopic approach predicts a notable increase of the survival probability, which signals dynamical decoupling of the initial state. Unlike the phenomenological model, our approach makes it possible to study the dependence of the system dynamics on the temperature of the environment. In the limit of very high temperature we find that the dynamics is characterized by a very strong dynamical decoupling of the initial state---the temperature-induced quantum Zeno effect.
In this paper we demonstrate how to generate the strong-coupling master equations for open quantum systems of continuous variables. These are the dissipative master equations of quantum Brownian particles for which the environmental noise is stronger than other system forces. Our strong-coupling master equations are very different from other so-called "strong-coupling" master equations (e.g. the quantum Smoluchowski equation) which are perturbing off a limit in which the system energy is taken to be perturbative and thus the dynamics is principally Markovian. Such approximations also require the system mass to be asymptotically large (even as compared to the ratio of noise and induced system frequencies) and thus they do not fully categorize the regime of what one might consider to be strong coupling. Our master equations are highly non-Markovian and radically different for different system potentials, admitting no apparent generic form. This result is quite exciting as it brings forth a new regime for theoretical exploration: the regime of strong noise and dissipation yet non-Markovian, such as strong coupling to a low-temperature environment with large 1/f fluctuations.
The dissipative dynamics of a quantum Brownian particle is studied for different types of environment. We derive analytic results for the time evolution of the mean energy of the system for Ohmic, sub-Ohmic, and super-Ohmic environments, without performing the Markovian approximation. Our results allow one to establish a direct link between the form of the environmental spectrum and the thermalization dynamics. This in turn leads to a natural explanation of the microscopic physical processes ruling the system time evolution both in the short-time non-Markovian region and in the long-time Markovian one. Our comparative study of thermalization for different environments sheds light on the physical contexts in which non-Markovian dissipation effects are dominant.