Initial correlations in open quantum systems: constructing linear dynamical maps and master equations

Abstract

We investigate the dynamics of open quantum systems which are initially correlated with their environment. The strategy of our approach is to analyze how given, fixed initial correlations modify the evolution of the open system with respect to the corresponding uncorrelated dynamical behavior with the same fixed initial environmental state, described by a completely positive dynamical map. We show that, for any predetermined initial correlations, one can introduce a linear dynamical map on the space of operators of the open system which acts like the proper dynamical map on the set of physical states and represents its unique linear extension. Furthermore, we demonstrate that this construction leads to a linear, time-local quantum master equation with generalized Lindblad structure involving time-dependent, possibly negative transition rates. Thus, the general non-Markovian dynamics of an open quantum system can be described by means of a time-local master equation even in the case of arbitrary, fixed initial system-environment correlations. We present some illustrative examples and explain the relation of our approach to several other approaches proposed in the literature.

Commutation simulator for open quantum dynamics

“…As shown in appendix section 'A brief derivation of the Lindblad master equation with time step δt' , the derivation of the Lindblad master equation can be described by a series of short time periods δt. At the start of the simulation, the initial density matrix is given in ρ(0) = |ψ 0 ⟩⟨ψ 0 | at t = 0, which is not correlated with other systems [19,20]. During the later period between t and t + δt, the time evolution description of the quantum system is split into two terms: a unitary term with the commutation relation (the same as the von Neumann case) and a sum of the Lindblad terms including the anti-commutation relation [21].…”

“…Thus, all parts of the expectation values ⟨ L ρ(t) L † ⟩ can be computed as above. Finally, we choose the combination of three controlled operators and the parameter χ to compute the second Lindblad term in equation ( 21), based on equations ( 17)- (19). The diagonal elements are given with the set { N, M, Â} = { L † L, 1, 1}.…”

Estudo de superoperadores exatos na forma de Lindblad para dinâmicas de sistemas quânticos abertos correlacionados com o ambiente

“…Uma outra abordagem para encontrar equações mestras exatas na forma de Lindblad para estados correlacionados inicialmente é dada em (12). Essa proposta é mais restritiva do que a anterior, de (11), porque pressupõe a existência de um mapa inversível para a dinâmica sem correlações.…”

“…Ainda que V (t, t 0 ) tenha um domínio de validade restrito, será interessante que ele possua características que preservam as propriedades de operadores densidade que estão além desse domínio. Superoperadores na forma de Lindblad, mesmo quando têm um domínio físico restrito como em (11,12), serão HPTA. O que revela que eles aniquilam traço e preservam hermiticidade de todos os operadores operadores densidade do espaço em que atuam.…”

“…No capítulo 5 descrevemos a derivação e o resultado recente a respeito da generalidade da equação de Lindblad para qualquer dinâmica física. Também expomos os resultados obtidos em (12) de uma equação para o sistema reduzido fazendo uso apenas de informações iniciais a respeito do sistema e do ambiente para dinâmicas inversíveis.…”