Completely positive post-Markovian master equation via a measurement approach

Abstract

A new post-Markovian quantum master equation is derived, that includes bath memory effects via a phenomenologically introduced memory kernel k(t). The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of k(t) is presented, in addition to a prescription for the experimental determination of k(t). The formalism is illustrated with examples. An open quantum system is one that is coupled to an external environment [1,2]. Such systems are of fundamental interest, as the notion of a closed system is always an idealization and approximation. Open quantum systems tend to decohere, and for this reason have recently received intense consideration in quantum information science, where decoherence is viewed as fundamental obstacle to the construction of quantum information processors [3]. It is possible to write down an exact dynamical equation for an open system, but the resultan integro-differential equation [4] -is mostly of formal interest, as such an exact equation can almost never be solved analytically or even numerically. In contrast, when one makes the Markovian approximation, i.e., when one neglects all bath memory effects, the resulting Lindblad master equation [2,5] is formally solvable and amenable to numerical treatment. Moreover, the desirable property of complete positivity [6] is maintained (see, however, [7] for a debate on the importance of this property). A coveted goal of the theory of open quantum systems [1, 2] is a "post-Markovian" master equation that (i) generalizes the Markovian Lindblad equation so as to include bath memory effects, at the same time (ii) remains both analytically and numerically tractable, and (iii) retains complete positivity. A variety of post-Markovian master equations have been proposed and analyzed, e.g., [1,8,9,10,11,12,13,14,15,16]. However, one of the desirable properties (i)-(iii) above is typically lost: e.g., in the case of time-convolutionless master equations (e.g., [10]) one may lose complete positivity, while in the case of nonlocal stochastic Schrodinger equations (e.g., [13]) one loses analytical solvability. In this work we propose a new post-Markovian master equation that satisfies all of the desirable properties (i)-(iii) above. The key idea we introduce is an interpolation between the generalized measurement interpretation of the exact Kraus operator sum map [6], and the continuous measurement interpretation of Markovian-limit dynamics [16,18].Review of quantum measurements approach to open system dynamics.-Consider a quantum system S coupled to a bath B (with respective Hilbert spaces H S , H B ), evolving unitarily under the total system-bath Hamiltonian H SB . The exact system dynamics is given by tracing over the bath degrees of freedom [1, 2, ...

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