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Journal of Physics A Mathematical and General

Path integral, semiclassical and stochastic propagators for Markovian open quantum systems

Walter T. Strunz

Published Jun 7, 1997• 24 citations
Publisher

Abstract

We develop the path integral theory for master equations of general Lindblad form (positive semigroups), describing Markovian open quantum systems. First the Hamiltonian path integral expression for the propagator is derived, which exhibits nicely the decoherence of pairs of phase space histories. A very appealing picture arises in the semiclassical limit where the degree of decoherence is expressible in terms of a phase space decoherence distance functional. For the important class of (effective) Hamiltonians quadratic in the momenta, we derive the Lagrangian version of the path integral propagator. We then evaluate the path integral approximately in a stationary phase approximation, leading to a Van Vleck-type propagator valid under semiclassical conditions. We also derive the propagator for the soluble damped harmonic oscillator in closed form from path integrals. Finally, connections to the active field of stochastic pure-state descriptions of open quantum systems are established, here in particular to linear quantum state diffusion.

DOI
10.1088/0305-4470/30/11/030

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We develop the path integral theory for master equations of general Lindblad form (positive semigroups), describing Markovian open quantum systems. First the Hamiltonian path integral expression for the propagator is derived, which exhibits nicely the decoherence of pairs of phase space histories. A very appealing picture arises in the semiclassical limit where the degree of decoherence is expressible in terms of a phase space decoherence distance functional. For the important class of (effective) Hamiltonians quadratic in the momenta, we derive the Lagrangian version of the path integral propagator. We then evaluate the path integral approximately in a stationary phase approximation, leading to a Van Vleck-type propagator valid under semiclassical conditions. We also derive the propagator for the soluble damped harmonic oscillator in closed form from path integrals. Finally, connections to the active field of stochastic pure-state descriptions of open quantum systems are established, here in particular to linear quantum state diffusion.
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A semiclassical approximation for an evolving density operator, driven by a 'closed' Hamiltonian and 'open' Markovian Lindblad operators, is reviewed. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian is a quadratic function and the Lindblad operators are linear functions of positions and momenta. The semiclassical formulae are interpreted within a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra 'open' term in the double Hamiltonian is generated by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by the definition of a propagator, here developed in both representations. Generalized asymptotic equilibrium solutions are thus presented for the first time.
A semiclassical approximation for an evolving density operator, driven by a 'closed' Hamiltonian operator and 'open' Markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of Hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra 'open' term is added to the double Hamiltonian by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. The particular case of generic Hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase space, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further 'small-chord' approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.
A semiclassical approximation for an evolving density operator, driven by a “closed” hamiltonian operator and “open ” markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra “open ” term is added to the double Hamiltonian by the non-hermitian part of the Lindblad operators in the general case of dissipative markovian evolution. The particular case of generic hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighborhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further “small-chord” approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.
Consistent dynamics which couples classical and quantum degrees of freedom exists. This dynamics is linear in the hybrid state, completely positive and trace preserving. Starting from completely positive classical-quantum master equations, we derive a general path integral representation for such dynamics in terms of a classical-quantum action, which includes the necessary and sufficient conditions for complete positivity and trace preservation. The path integral we study is a generalization of the Feynman path integral for quantum systems, and the stochastic path integral used to study classical stochastic processes, allowing for interaction between the classical and quantum systems. When the classical-quantum Hamiltonian is at most quadratic in the momenta we are able to derive a configuration space path integral, providing a map between master equations and covariant classical-quantum path integrals.