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Canadian Journal of Physics

Explicit Kraus operator-sum representations for time-evolution of Fermi systems in amplitude- and phase-decay processes

Xianfeng Chen, Li-Li Hou

Published May 6, 2015• 2 citations
Publisher

Abstract

We set up the fermionic thermal entangled state approach to address Fermi operator master equations for the amplitude- and phase-decay processes and find the explicit Kraus operator-sum representations (KOSR) describing time-evolution of Fermi systems. As an application of KOSR, the evolution law of the two-level atomic system interacting with a photon field in these two decay processes is analytically derived, which shows that the interaction system has a quantum jump as a result of dissipation, but it always remains unchanged in the phase-decay process.

DOI
10.1139/cjp-2014-0718

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We set up the fermionic thermal entangled state approach to address Fermi operator master equations for the amplitude- and phase-decay processes and find the explicit Kraus operator-sum representations (KOSR) describing time-evolution of Fermi systems. As an application of KOSR, the evolution law of the two-level atomic system interacting with a photon field in these two decay processes is analytically derived, which shows that the interaction system has a quantum jump as a result of dissipation, but it always remains unchanged in the phase-decay process.
We solve the fermionic master equation for a thermal bath to obtain its explicit Kraus operator solutions via the fermionic state approach. The normalization condition of the Kraus operators is proved. The matrix representation for these solutions is obtained, which is incongruous with the result in the book completed by Nielsen and Chuang [ Quantum Computation and Quantum Information , Cambridge University Press, 2000]. As especial cases, we also present the Kraus operator solutions to master equations for describing the amplitude-decay model and the diffusion process at finite temperature.
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The necessity and utility of considering the interaction of a quantum system with its environment when describing its time evolution have been recognized in several branches of physics and of other sciences. The Kraus' representation is a general and succinct approach to describe such open system dynamics in a wide range of relevant physical scenarios. In this article, by abdicating from the generality of the formalism of quantum operations and with this avoiding its associated complications, we show in a simple manner how we can obtain the Kraus' representation using basically the closed system (system plus environment) unitary dynamics and the partial trace function. The example of a two-level atom interacting with the vacuum of the electromagnetic field is regarded for the sake of instantiating this formalism, which is then applied to study the time evolution of the atom's quantum coherence.
We consider the exact time-evolution of a broad class of fermionic open quantum systems with both strong interactions and strong coupling to wide-band reservoirs. We present a nontrivial fermionic duality relation between the evolution of states (Schr\"odinger) and of observables (Heisenberg). We show how this highly nonintuitive relation can be understood and exploited in analytical calculations within all canonical approaches to quantum dynamics, covering Kraus measurement operators, the Choi-Jamio{\l}kowski state, time-convolution and convolutionless quantum master equations and generalized Lindblad jump operators. We discuss the insights this offers into the divisibility and causal structure of the dynamics and the application to nonperturbative Markov approximations and their initial-slip corrections. Our results underscore that predictions for fermionic models are already fixed by fundamental principles to a much greater extent than previously thought.
We show that any arbitrary time-dependent density operator of an open system can always be described in terms of an operator-sum representation (Kraus representation) regardless of its initial condition and the path of its evolution in the state space, and we provide a general expression of Kraus operators for arbitrary time-dependent density operator of an NNN-dimensional system. Moreover, applications of our result are illustrated through several examples.