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Physica A Statistical Mechanics and its Applications

Quantum Markovian master equation for a system of identical particles interacting with a heat reservoir

Boris Volodumurovich Bondarev

Published Aug 1, 1991• 17 citations
Publisher
DOI
10.1016/0378-4371(91)90294-m

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We present a non-Markovian master equation for a qubit interacting with a general reservoir, which is derived according to the Nakajima—Zwanzig and the time convolutionless projection operator technique. The non-Markovian solutions and Markovian solution of dynamical decay of a qubit are compared. The results indicate the validity of non-Markovian approach in different coupling regimes and also show that the Markovian master equation may not precisely describe the dynamics of an open quantum system in some situation. The non-Markovian solutions may be effective for many qubits independently interacting with the heated reservoirs.
Markovian transport is often described by a master equation for the system state. The thermodynamic information measured in transport experiments, however, is carried by reservoir-resolved transfer records, such as particle currents, heat currents, entropy production, and current noise. We identify a thermodynamic incompleteness of state dynamics: a Markovian state generator can fix the occupation probabilities, stationary response, and relaxation without specifying how the underlying transitions are assigned to reservoirs and energy filters. We study a multi-terminal Coulomb-blockaded quantum dot coupled to energy-filtered reservoirs, for which different assignments of reservoir channels can generate the same state master equation. These assignments give identical occupation dynamics, stationary state, and linear response of the dot, but different heat currents, entropy production, and current noise. We formulate a thermodynamic completeness criterion: a transport observable can be reconstructed from state dynamics only when it is invariant under all changes of reservoir-channel assignments that leave the state generator unchanged. The criterion gives a practical diagnostic for Markovian transport models and a measurable prediction: state tomography can be insufficient to predict heat-noise and cross-correlation measurements, even when the full Markovian state dynamics is known. The analysis identifies a concrete limitation of state-only Markovian thermodynamics and shows which additional transport records must be specified to make thermodynamic predictions experimentally complete.
We derive a Markovian master equation for driven open quantum systems based on the Lewis-Riesenfeld-invariants theory, which is available for arbitrary driving protocols. The role of the Lewis-Riesenfeld invariants is to help us bypass the time-ordering obstacle in expanding the propagator of the free dynamics, such that the Lindblad operators in our driven Markovian master equation can be determined easily. We also illustrate that, for the driven open quantum systems, the spontaneous emission and the thermal excitation induce the transitions between eigenstates of the Lewis-Riesenfeld invariant, but not the system Hamiltonian's. As an example, we present the driven Markovian master equation for a driven two-level system coupled to a heat reservoir. By comparing to the exactly solvable models, the availability of the driven Markovian master equation is verified. Meanwhile, the adiabatic limit and inertial limit of the driven Markovian master equation are also discussed, which result in the same Markovian master equations as those presented before in the corresponding limits.
We build an exact framework to evaluate heat, energy, and particle transport between Gaussian reservoirs mediated by a quadratic quantum system. By combining full counting statistics with newly developed non-Markovian master equation approaches, we introduce an effective master equation whose solution can be used to generate arbitrary moments of the heat statistics for any number of reservoirs. This theory applies equally to fermionic and bosonic systems, holds at arbitrarily strong coupling, and resolves out-of-equilibrium transient dynamics determined by the system's initial state. In the steady-state, weak-coupling limit, we recover results analogous to those of the well-known Landauer-Büttiker formalism. We conclude our discussion by demonstrating an application of the method to a prototypical fermionic system. Our results uncover a regime of transient negative heat conductance contingent upon the initial system preparation, providing a clear signature of non-trivial out-of-equilibrium dynamics.