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arXiv:quant-ph/0606036 · Physical Review A

Geometric phases in open systems: A model to study how they are corrected by decoherence

Fernando C. Lombardo, Paula I. Villar

Published Oct 12, 2006• 86 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.74.042311

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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 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.
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.
Geometric phases, arising from cyclic evolutions in a curved parameter space, appear in a wealth of physical settings. Recently, and largely motivated by the need of an experimentally realistic definition for quantum computing applications, the quantum geometric phase was generalized to open systems. The definition takes a kinematical approach, with an initial state that is evolved cyclically but coupled to an environment--leading to a correction of the geometric phase with respect to the uncoupled case. We obtain this correction by measuring the nonunitary evolution of the reduced density matrix of a spin one-half coupled to an environment. In particular we are interested in baths near a quantum phase transition, which are known to induce strong decoherence. The experiments are done with a NMR quantum simulator, where we emulate qualitatively the influence of a critical environment using a simple one-qubit model.
We report on recent results showing that the geometric phase can be used as a tool in the analysis of many different physical systems, as mixed boson systems, CPT and CP violations, Unruh effects and thermal states. We show that the geometric phases appearing in the time evolution of mixed meson systems like Bs0−Bˉs0B_{s}^{0}-\bar{B}_{s}^{0}Bs0​−Bˉs0​ and the K0−Kˉ0K^{0}-\bar{K}^{0}K0−Kˉ0 are linked to the parameter zzz describing the CPTCPTCPT violation. A non zero phase difference between particle and antiparticle arises only in the presence of CPTCPTCPT symmetry breaking. Then the geometric phase can represent a completely new test for the CPTCPTCPT invariance. Moreover, we study the geometric phase of systems represented by mixed state and undergoing a nonunitary evolution and propose the realization of interferometers which can prove the existence of the Unruh effect and can allow very precise measurements of temperature. We are glad to dedicate this paper to Professor Gaetano Vilasi in the occasion of his 70th birthday.