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arXiv:quant-ph/0406018 · Physical Review A

Geometric phase for an adiabatically evolving open quantum system

I. Kamleitner, James D. Cresser, Barry C. Sanders

Published Oct 22, 2004• 29 citations
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Abstract

We derive a solution for a two-level system evolving adiabatically under the influence of a driving field, which includes open system effects. This solution, which is obtained by working in the representation corresponding to the eigenstates of the time-dependent Hermitian Hamiltonian, enables the dynamic and geometric phases of the evolving density matrix to be separated. The dynamic phase can be canceled in the limit of weak coupling to the environment, thereby allowing the geometric phase to be readily extracted both mathematically and operationally.

DOI
10.1103/physreva.70.044103

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We derive a solution for a two-level system evolving adiabatically under the influence of a driving field, which includes open system effects. This solution, which is obtained by working in the representation corresponding to the eigenstates of the time-dependent Hermitian Hamiltonian, enables the dynamic and geometric phases of the evolving density matrix to be separated. The dynamic phase can be canceled in the limit of weak coupling to the environment, thereby allowing the geometric phase to be readily extracted both mathematically and operationally.
Geometric phase for novel analytical solutions (Barnes and Das Sarma) of time-dependent two-level quantum systems is discussed, specically for a general single-axis driving term, which is represented by a function J(t) in the Hamiltonian, and its corresponding evolution operator.It is demonstrated how general results for corresponding phases (total, dynamic and geometric) can be obtained.Using a specic case, it was found that over time in which the driving eld is appreciably dierent from zero, the corresponding geometric phase changes (in the specic example by ∆β ≈ 0.8 radians) thus enabling detection.The results are relevant to qubit control and to quantum computing applications.
Abstract We present an analytical framework to investigate phase dynamics in discrete-time quantum walks, with a focus on geometric phases in both single- and two-particle systems. Leveraging the Cayley-Hamilton theorem, we construct the evolution operator in momentum space and derive exact closed-form solutions for the full time evolution. This allows us to explicitly extract total, dynamic, and most notably, non-adiabatic geometric phases throughout the walk. In the two-particle case, we further examine how initial entanglement influences the accumulation and interplay of these phases. Our results offer a unified and exact approach to understanding the geometric structure underlying quantum walk dynamics, with particular emphasis on non-adiabatic effects.
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.
We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-1∕21∕21∕2 particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.
We introduce an operational framework to analyze nonadiabatic Abelian and non-Abelian, cyclic and noncyclic, geometric phases in open quantum systems. In order to remove the adiabaticity condition, we generalize the theory of dynamical invariants to the context of open systems evolving under arbitrary convolutionless master equations. Geometric phases are then defined through the Jordan canonical form of the dynamical invariant associated with the superoperator that governs the master equation. As a by-product, we provide a sufficient condition for the robustness of the phase against a given decohering process. We illustrate our results by considering a two-level system in a Markovian interaction with the environment, where we show that the nonadiabatic geometric phase acquired by the system can be constructed in such a way that it is robust against both dephasing and spontaneous emission.
We study the adiabatic approximation of the dynamics of a bipartite quantum system with respect to one of its components, when the coupling between the two components is perturbative. We show that the density matrix of the considered component is described by adiabatic transport formulae exhibiting operator-valued geometric and dynamical phases. The present results can be used to study the quantum control of the dynamics of qubits and of open quantum systems where the two components are the system and its environment. We treat two examples, the control of an atomic qubit interacting with another one and the control of a spin in the middle of a Heisenberg spin chain.