On the Physical Content of Lindblad Form Master Equations

Abstract

We report on a solution of a quantum markovian equation of Lindblad form which yields, however, unphysical results. This is traced back to the assumptions made for the derivation of the equation which are valid only for a special class of interaction hamiltonians.

Completely Positive Quantum Dissipation

“…Indeed, CP appears somehow more natural if considered in the context of operations and quantum measurement in which it originally appeared in physics [6]. According to a famous paper by Lindblad [5], under suitable mathematical conditions the property of CP allows for the determination of the general structure of the generators of irreversible time evolutions, even though a thorough understanding of the physical limits of validity of this property is still on its way [7,8], so that satisfaction of CP by itself does not ensure physically meaningful results, and the connection to realistic microphysical models is strongly desirable. In fact recent work has stressed the connection between CP of the time evolution and weak coupling, so that, for example, an uncorrelated statistical operator can consistently be considered as an initial condition, as well as coarse graining in time [9].…”

Simple non-Markovian microscopic models for the depolarizing channel of a single qubit

“…However, it is convenient to interpret the first term of the right-hand side of (2) as the unitary evolution in the absence of interaction with the environment, and the second term as the environment-system coupling. Despite their mathematical properties, Lindblad master equations, under special conditions, can also show unphysical behavior, such as an unbounded increase of the system's energy [27].…”

<title>Dynamics of a Fermi system in a complex dissipative environment</title>

“…In the total Hamiltonian (5) we distinguish the potential of interaction with the electromagnetic field mode ii (8) and the potential of interaction with the field bath VB=AB,…”

“…while VG is the potential of interaction between these phonons and the system particles: Introducing (18), (19) in (8), (9), we derive also the particle-field potentials V(''), VB in the second quantization:…”

“…This equation represents a general form with phenomenological coefficients that guarantees the preservation of the quantum-mechanical conditions during the whole evolution of the system. However, Lindblad's form of the dissipative dynamics is not sufficient for a correct description of the phenomena, sometimes leading to unphysical results [8]. Other conditions for a correct description of the dissipative dynamics can be obtained only from explicit physical models.…”