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arXiv:1302.6218 · Physical Review Letters

Feshbach Projection Formalism for Open Quantum Systems

Dariusz Chruściński, Andrzej Kossakowski

Published Jul 30, 2013• 36 citations
arXivHTMLPDF

Abstract

We provide a new approach to open quantum systems which is based on the Feshbach projection method. Instead of looking for a master equation for the dynamical map acting in the space of density operators we provide the corresponding equation for the evolution in the Hilbert space of the amplitude operators. Its solution enables one to construct a legitimate quantum evolution (completely positive and trace preserving). Our approach, contrary to the standard Nakajima-Zwanzig method, allows for a series of consistent approximations resulting in a legitimate quantum evolution. The new scheme is illustrated by the well-known spin-boson model beyond the rotating wave approximation. It is shown that the presence of counterrotating terms dramatically changes the asymptotic evolution of the system.

DOI
10.1103/physrevlett.111.050402

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We provide a new approach to open quantum systems which is based on the Feshbach projection method. Instead of looking for a master equation for the dynamical map acting in the space of density operators we provide the corresponding equation for the evolution in the Hilbert space of the amplitude operators. Its solution enables one to construct a legitimate quantum evolution (completely positive and trace preserving). Our approach, contrary to the standard Nakajima-Zwanzig method, allows for a series of consistent approximations resulting in a legitimate quantum evolution. The new scheme is illustrated by the well-known spin-boson model beyond the rotating wave approximation. It is shown that the presence of counterrotating terms dramatically changes the asymptotic evolution of the system.
The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath ϱB(t)\varrho_{B}(t)ϱB​(t) and the system ϱS(t)\varrho_{S}(t)ϱS​(t) are written down. A formal solution of the equation for ϱB(t)\varrho_{B}(t)ϱB​(t), having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in ϱS(t)\varrho_{S}(t)ϱS​(t), is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of ϱB(t)\varrho_{B}(t)ϱB​(t) and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.
The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.
One of the main methods for describing the dynamics of open quantum systems is the method of quantum master equations. These equations describe the dynamics of the reduced density operator of a system interacting with a reservoir. In this case, averaging is performed over the degrees of freedom of the reservoir, which does not allow one to describe the dynamics of reservoir observables. In this paper we show that applying the Zwanzig projection operator method, which is used in deriving quantum master equations, one can also derive dynamic equations for reservoir observables. As an example, we derive dynamic equations for the average number of quanta (photons, phonons) of a bosonic reservoir in the approximation of its weak coupling to the system in the case of the dipole interaction Hamiltonian.
We consider evolution of dynamical systems described by non-Hermitian Hamiltonians, using the density operator approach.The latter is formulated both at the level of the Hilbert space and the phase space, and adapted for applications to open quantum systems.We illustrate the formalism using a family of non-Hermitian system, which generators are quadratic with respect to both momentum and position.Despite the initial simplicity of a Hamiltonian, the structure of its solutions and spectral characteristics are nontrivial, and they can drastically change depending on parameters of the model and its symmetry in phase space.We present analytical solutions in L 2 (R) and in phase space, and an explicit form of the similarity transformation changing these generators into the corresponding normal operators.
We present a stochastic projection formalism for the description of quantum dynamics in bosonic or spin environments. The Schrödinger equation in the coherent state representation with respect to the environmental degrees of freedom can be reformulated by employing the Feshbach partitioning technique for open quantum systems based on the introduction of suitable non-Hermitian projection operators. In this picture the reduced state of the system can be obtained as a stochastic average over pure state trajectories, for any temperature of the bath. The corresponding non-Markovian stochastic Schrödinger equations include a memory integral over the past states. In the case of harmonic environments and linear coupling the approach gives a new form of the established non-Markovian quantum state diffusion stochastic Schrödinger equation without functional derivatives. Utilizing spin coherent states, the evolution equation for spin environments resembles the bosonic case with, however, a non-Gaussian average for the reduced density operator.
The dynamics of finite dimension open quantum systems is studied with the help of the simplest possible form of projection operators, namely the ones which project only onto one-dimensional subspaces. The simplicity of the action of the projection operators always leads to an analytical solution of the dynamical master equation, even in the non-Markovian case, in any perturbative order. The analytical solution correctly reproduces the short-time dynamics, and can be used to recursively recover the dynamics for an arbitrary time interval with arbitrary precision. The necessary number of relevant degrees of freedom to completely characterise an open quantum system is ( n − 1 ) ( n + 2 ) / 2 , where n is the dimension of the Hilbert space of the open system. The method is illustrated by two examples, the relaxation of a qubit in a thermal bath and the dynamics of two interacting qubits in a common environment.