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Quantum Brownian Motion in the Heisenberg picture: derivation of the adjoint master equation and applications

Matteo Carlesso, Angelo Bassi

Published Feb 16, 2016• 3 citations
OpenAlex

Abstract

Open quantum system theory is fundamental for a proper description of quantum systems in different contexts such as chemistry, condensed matter physics, bio-physics and opto-mechamics. Quantum brownian motion which is analytically described by the Hu-Paz-Zhang (HPZ) model represents one of the most important examples of an open quantum system. In this paper we propose a novel derivation of its master equation, based on the Heisenberg picture. We provide an exact and analytic equation both for the operators as well as for the states. The result, equivalent to the one derived originally by HPZ, is expressed in terms of the spectral density, regardless the strength of the coupling between the system and the environment and allows to compute the time evolution of physically relevant quantities in a much easier way, since it is directly expressed in terms of the spectral density. An example is explicitly studied.

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We revisit the model of a system made up of a Brownian quantum oscillator under the influence of an external classical force and linearly coupled to an environment made up of many quantum oscillators at zero or finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] with coefficients obtained from solutions of integro-differential equations can assume closed functional forms for a fairly general class of spectral densities of the environment at arbitrary temperature and coupling strength. As an illustration of these new results we solve the corresponding master equation and calculate, among other physical quantities, the uncertainty function whose late time behavior can be obtained fully. This produces a formula for investigating the standard quantum limit which is central to addressing many theoretical issues in macroscopic quantum phenomena and experimental concerns related to low temperature precision measurements. We find that any initial state always settles down to a Gaussian density matrix whose covariance is determined by the thermal reservoir and whose mean is determined by the external force. For more general spectra we show that the solution of the master equation can be reduced to solving for the motion of a classical parametric oscillator with parametric frequency determined by the unsolved for master equation coefficients. States in these systems experience evolution that is parametrically similar to the simpler evolution explicitly determined for in the case of Laurent-series spectra.

arXiv:0705.2766•May 18, 2007•3 citations
Open quantum system theory is fundamental for a proper description of quantum systems in different contexts such as chemistry, condensed matter physics, bio-physics and opto-mechamics. Quantum brownian motion which is analytically described by the Hu-Paz-Zhang (HPZ) model represents one of the most important examples of an open quantum system. In this paper we propose a novel derivation of its master equation, based on the Heisenberg picture. We provide an exact and analytic equation both for the operators as well as for the states. The result, equivalent to the one derived originally by HPZ, is expressed in terms of the spectral density, regardless the strength of the coupling between the system and the environment and allows to compute the time evolution of physically relevant quantities in a much easier way, since it is directly expressed in terms of the spectral density. An example is explicitly studied.
Quantum Brownian motion is a fundamental model for a proper understanding of open quantum systems in different contexts such as chemistry, condensed-matter physics, biophysics, and optomechanics. In this paper we propose a different approach to describe this model. We provide an exact and analytic equation for the time evolution of the operators and we show that the corresponding equation for the states is equivalent to well-known results in the literature. The dynamics is expressed in terms of the spectral density, regardless of the strength of the coupling between the system and the bath. Our derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations. An example is explicitly studied.
The master equation for a linear open quantum system in a general environment is derived using a stochastic approach. This is an alternative derivation to that of Hu, Paz and Zhang, which was based on the direct computation of path integrals, or to that of Halliwell and Yu, based on the evolution of the Wigner function for a linear closed quantum system. We first show by using the influence functinal formalism that the reduced Wigner function for the open system coincides with a distribution function resulting from averaging both over the initial conditions and the stochastic source of a formal Langevin equation. The master equation for the reduced Wigner function can then be deduced as a Fokker-Planck equation obtained from the formal Langevin equation.
In this paper we generalize the quantum Brownian motion (QBM) to include the momentum-dependent system-environment couplings. The resulting Hamiltonian in the particle number representation is given by ${H}_{t}ot=\ensuremath{\hbar}{\ensuremath{\omega}}_{\mathrm{S}}{a}^{\ifmmode\dagger\else\textdagger\fi{}}a+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}{\ensuremath{\omega}}_{k}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}({V}_{k}{a}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}+{V}_{k}^{*}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}a)+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}({W}_{k}{a}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}+{W}_{k}^{*}{b}_{k}a)$. The conventional QBM model corresponds to the special case Wk=Vk{W}_{k}={V}_{k}Wk​=Vk​. The generalized QBM can capture more physical applications because the single-particle transition and the two-particle pair production between the system and the environment represent two very different physical processes that usually do not have the same coupling strengths. We discuss the physical realizations of the generalized QBM in different physical systems, and derive its exact master equation for both the initial decoupled states and initial correlated states. The Hu-Paz-Zhang master equation of the conventional QBM model is reproduced as a special case. We find that the renormalized system Hamiltonian after tracing out all the environmental states induced naturally a momentum-dependent potential, which shows the consistency of including the momentum-dependent coupling in the QBM Hamiltonian. In the Hu-Paz-Zhang master equation, such a renormalized potential is misplaced so that the correct renormalization Hamiltonian has not been found. With the exact master equation for both the initial decoupled and initial correlated states, the issues about the initial jolt are also reexamined. We find that the so-called ``initial jolt,'' which has been thought to be an artificial effect due to the use of the initial decoupled system-environment states, has nothing to do with the initial decoupled state. The exact master equation for the generalized QBM also has potential applications to photonics quantum computing.
An original method to exactly solve the non-Markovian master equation describing the interaction of a single harmonic oscillator with a quantum environment in the weak-coupling limit is reported. By using a superoperatorial approach, we succeed in deriving the operatorial solution for the density matrix of the system. Our method is independent of the physical properties of the environment. We show the usefulness of our solution deriving explicit expressions for the dissipative time evolution of some observables of physical interest for the system, such as, for example, its mean energy.
Stochastic Schr\"odinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schr\"odinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most employed exactly soluble models of open system dynamics. We show explicitly how to establish the direct connection between the exact convolutionless master equation of QBM and the corresponding convolutionless exact stochastic Schr\"odinger equation.
We revisit the model of a system made up of a Brownian quantum oscillator under the influence of an external classical force and linearly coupled to an environment made up of many quantum oscillators at zero or finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] with coefficients obtained from solutions of integro-differential equations can assume closed functional forms for a fairly general class of spectral densities of the environment at arbitrary temperature and coupling strength. As an illustration of these new results we solve the corresponding master equation and calculate, among other physical quantities, the uncertainty function whose late time behavior can be obtained fully. This produces a formula for investigating the standard quantum limit which is central to addressing many theoretical issues in macroscopic quantum phenomena and experimental concerns related to low temperature precision measurements. We find that any initial state always settles down to a Gaussian density matrix whose covariance is determined by the thermal reservoir and whose mean is determined by the external force. For more general spectra we show that the solution of the master equation can be reduced to solving for the motion of a classical parametric oscillator with parametric frequency determined by the unsolved for master equation coefficients. States in these systems experience evolution that is parametrically similar to the simpler evolution explicitly determined for in the case of Laurent-series spectra.