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arXiv:1405.1227 · Physical Review A

Open-system geometric phase based on system-reservoir joint state evolution

Shi‐Biao Zheng

Published Jun 27, 2014• 6 citations
arXivHTMLPDF

Abstract

The geometric phase is of fundamental interest and plays an important role in quantum information processing. However, the definition and calculation of this phase for open systems remain problematic due to the lack of agreement on generalizations of the parallel transport condition to mixed-state nonunity evolutions. Here we tackle this problem by associating the open-system geometric phase with the parallel transport of the joint system-reservoir state and find that the associated geometric phase can be calculated when the total Hamiltonian is time independent. Our approach not only provides a way around the nonunitary evolution obstacle, but also sheds light on the relation between the geometric phase and the system-reservoir entanglement. Based on this approach, we calculate the geometric phase of different quantum systems subject to energy decay, showing that it is robust against decoherence, which is in distinct contrast with previous results.

DOI
10.1103/physreva.89.062118

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The geometric phase is of fundamental interest and plays an important role in quantum information processing. However, the definition and calculation of this phase for open systems remain problematic due to the lack of agreement on generalizations of the parallel transport condition to mixed-state nonunity evolutions. Here we tackle this problem by associating the open-system geometric phase with the parallel transport of the joint system-reservoir state and find that the associated geometric phase can be calculated when the total Hamiltonian is time independent. Our approach not only provides a way around the nonunitary evolution obstacle, but also sheds light on the relation between the geometric phase and the system-reservoir entanglement. Based on this approach, we calculate the geometric phase of different quantum systems subject to energy decay, showing that it is robust against decoherence, which is in distinct contrast with previous results.
The problem of a geometric phase for an open quantum system is reinvestigated in a unifying approach. Two of the existing methods to define geometric phase, one by Uhlmann's approach and the other by a kinematic approach, which have been considered to be distinct, are shown to be related in this framework. The method is based upon purification of a density matrix by its uniform decomposition and a generalization of the parallel transport condition obtained from this decomposition. It is shown that the generalized parallel transport condition can be satisfied when Uhlmann's condition holds. However, it does not mean that all solutions of the generalized parallel transport condition are compatible with those of Uhlmann's. It is also shown how to recover the earlier known definitions of geometric phase as well as how to generalize them when degeneracy exists and varies in time.
We calculate the geometric phase for an open system (spin-boson model) which interacts with an environment (ohmic or nonohmic) at arbitrary temperature. However there have been many assumptions about the time scale at which the geometric phase can be measured, there has been no reported observation of geometric phases for mixed states under nonunitary evolution yet. We study not only how they are corrected by the presence of the different type of environments but estimate the corresponding times at which decoherence becomes effective as well. These estimations should be taken into account when planning experimental setups to study the geometric phase in the nonunitary regime, particularly important for the application of fault-tolerant quantum computation.
The quantum jump method for the calculation of geometric phase is reviewed. This is an operational method to associate a geometric phase to the evolution of a quantum system subjected to decoherence in an open system. The method is general and can be applied to many different physical systems, within the Markovian approximation. As examples, two main source of decoherence are considered: dephasing and spontaneous decay. It is shown that the geometric phase is to very large extent insensitive to the former, i.e. it is independent of the number of jumps determined by the dephasing operator.
This thesis consists of several studies performed over different few-dof quantum systems exposed to the effect of an uncontrolled environment. The primary focus of the work is to explore the relation between decoherence and environmentally-induced dissipative effects, and the concept known as geometric phases. The first mention of such an object in the context of quantum mechanics goes back to the seminal work by Berry. He demonstrated that the phase acquired by an eigenstate of a time-dependent Hamiltonian in an adiabatic cycle consists of two distinct contributions: one termed 'geometric' and the other known as the dynamical phase. Since Berry's work, the notion of geometric phase has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. These geometric phases naturally arise in the geometric description of Hilbert space, where they manifest as holonomies and possess significance in the fundamental understanding of quantum mechanics and its mathematical framework, and in explaining various physical phenomena, including the Fractional Hall Effect. Moreover, from a modern perspective, geometric phases hold promise for practical applications, such as constructing geometric gates for quantum information processing and storage. However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Consequently, characterizing these geometric phases encompase motivations from fundamental aspects of quantum mechanics to technological applications.
Based on a generic quantum open system model, we study the geometric nature of decoherence by defining a complex-valued geometric phase through stochastic pure states describing nonunitary, noncyclic, and nonadiabatic evolutions. The ensemble average of the complex geometric phases for the pure stochastic states yields a conventional geometric phase together with an amplitude factor. We show that the decoherence process described by the decaying amplitude can be a geometric quantity independent of the system's dynamics. It is a remarkable fact that the geometric phase of a quantum system can serve as an ideal realization of quantum gates due to its robustness against dynamical errors; however, in this paper we show that, for some open quantum systems, a desirable geometric phase may be accompanied by an unwanted robust geometric decoherence factor. Two exactly solvable models are studied to demonstrate that, while the decoherence is a purely dynamical effect for a dephasing two-level model, the decoherence in a dissipative two-level model can be a geometric process. Finally, we show that such a geometric decoherence effect may be eliminated by a nonperturbative control scheme.