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Europhysics Letters (EPL)

Non-Markovian quantum dissipation in the Kraus representation

A. J. van Wonderen, K. Lendi

Published Jul 26, 2005• 15 citations
Publisher

Abstract

In order to describe the evolution of a quantum system that is coupled to a reservoir, a non-phenomenological Kraus map is constructed. At time zero, system and reservoir are not entangled. In the perturbative series for the density operator of the system all reservoir correlation functions are factorised into products of pair-correlation functions. This allows for a resummation of the perturbative series up to infinite order. The density operator can be expressed in terms of an auxiliary system operator that satisfies an analytically tractable integral equation. Hence, the difficulties caused by integral kernels of Nakajima-Zwanzig type are circumvented. Assuming an interaction between system and reservoir of the Jaynes-Cummings form, one shows that the Kraus map is capable of generating Rabi oscillations of a two-level atom. If the reservoir is a continuum, the Kraus map reproduces the Wigner-Weisskopf theory of spontaneous emission.

DOI
10.1209/epl/i2005-10147-6

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In order to describe the evolution of a quantum system that is coupled to a reservoir, a non-phenomenological Kraus map is constructed. At time zero, system and reservoir are not entangled. In the perturbative series for the density operator of the system all reservoir correlation functions are factorised into products of pair-correlation functions. This allows for a resummation of the perturbative series up to infinite order. The density operator can be expressed in terms of an auxiliary system operator that satisfies an analytically tractable integral equation. Hence, the difficulties caused by integral kernels of Nakajima-Zwanzig type are circumvented. Assuming an interaction between system and reservoir of the Jaynes-Cummings form, one shows that the Kraus map is capable of generating Rabi oscillations of a two-level atom. If the reservoir is a continuum, the Kraus map reproduces the Wigner-Weisskopf theory of spontaneous emission.
Starting from unitary dynamics we study the evolution in time of a non-relativistic quantum system that exchanges energy with a thermal reservoir of harmonic oscillators. System and reservoir are assumed to be initially decorrelated. Reservoir correlation functions are factorized by means of a Kraus compliant version of Wick's theorem. As a result, the non-Markovian Kraus map for the system density operator can be completely expressed in terms of system potentials and reservoir pair correlation functions. An infinite hierarchy for the evolution operators of the Kraus map is derived. The system density operator is obtained as a time-ordered exponential containing a non-Markovian counterpart of the standard Markovian generator for dissipative dynamics. We specify a condition on this non-Markovian generator for return to thermal equilibrium. We set up a non-Markovian perturbation theory that preserves both trace and positivity of the system density operator.
Abstract Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir. By performing factorizations in the Kraus map a perturbation theory is set up that conserves in arbitrary perturbative order both positivity and probability of the density matrix. The latter is determined by an integral equation for a bitemporal matrix and a finite hierarchy for Kraus matrices. In the lowest perturbative order this hierarchy reduces to one equation for one Kraus matrix. Its solution is given by a continued fraction of a much simpler structure as compared to the non-perturbative case. In the lowest perturbative order our non-Markovian evolution equations are applied to the damped Jaynes–Cummings model. From the solution for the atomic density matrix it is found that the atom may remain in the state of maximum entropy for a significant time span that depends on the initial energy of the radiation field.
Starting from a prescribed Hamiltonian, we construct a non-Markovian evolution equation for a non-relativistic quantum system that exchanges energy with a large reservoir. In order to create sufficient mathematical freedom, the density operator is replaced by a more flexible entity that depends on two times. If these times are chosen equal, the density operator is recovered. In deriving a non-Markovian integral equation for our bitemporal operator, it is assumed that initially system and reservoir are completely uncorrelated. Furthermore, in employing Wick's theorem for factorization of reservoir correlation functions, only those Wick contractions between reservoir potentials are retained that belong to a generalized nearest-neighbour class. The latter is established by subjecting the set of plain nearest-neighbour contractions to any cyclic permutation of reservoir potentials. Through generalizing the notion of nearest-neighbour contraction, it is ensured that the trace of the density operator is conserved. By construction, our bitemporal evolution equation agrees with the Kraus map for quantum dissipation. Moreover, a sound Markovian limit exists that reproduces the complete van Hove–Davies theory. By making use of a rotating-wave approximation and Laplace transformation, the density operator of a damped N -level atom can be computed. For large times and moderate coupling to the reservoir, the atom ends up near the state of thermal equilibrium. At zero temperature, our non-Markovian integral equation gives an exact solution for the atomic density operator.
Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir.
We propose a way to understand the evolution of an open quantum system using a description that dispenses a continuous evolution in time, by discrete operators entangled states, in its most direct and fundamental way. We show that the successive application of these operators in very small time intervals reproduce continuous evolution. It describes and compares the temporal evolution of an open quantum system of three levels, for which the Lindblad equation is solved to obtain the density matrix function of time, a method is developed to find Kraus operators time dependent and also finds constant Kraus operators differentials operate computationally $ n $ times evolving in discrete-time $ Δt = τ= t / n $. It is seen in the example of atom cavity inside a reservoir that by calculating the distance and the relative error we define is a relationship with the evolution of discrete steps and physical variables as $ ω$ which presuppose a natural discretization found time.
We study the decoherence properties of an entangled bipartite qubit system, represented by two two-level atoms that are individually coupled to non-Markovian reservoirs. This coupling ensures that the dynamical equations of the atoms can be treated independently. The non-Markovian reservoirs are described by a model which leads to an exact non-Markovian master equation of the Nakajima-Zwanzig form [J. Salo, S. M. Barnett, and S. Stenholm, Optics Commun. 259, 772 (2006)]. We consider the evolution of the entanglement of a two-atom state that is initially completely entangled, quantified by its concurrence. Collapses and revivals in the concurrence, induced by the memory effects of the reservoir, govern the dynamics of the entangled quantum system. These collapses and revivals in the concurrence are a strong manifestation of the non-Markovian reservoir.