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Modern Physics Letters B

BERRY PHASE IN COUPLED TWO-LEVEL SYSTEMS

X. Y. Zhang, J. H. TENG, X. X. Yi

Published Apr 18, 2013• 5 citations
Publisher

Abstract

The application of geometric phases into robust control of quantal systems has triggered exploration of the geometric phase for coupled subsystems. Earlier studies have mainly focused on the situation where the external control parameters are in the free Hamiltonian of the subsystems, i.e. the controls exert only on the individual subsystems. Here we consider another circumstance that we can control the coupling geiϕ between the subsystems. By changing only the phase ϕ in the coupling constant, we derive the Berry phase acquired by the system and compare it to the geometric phase acquired by changing the coupling strength g. We find that the asymptotic behavior of the Berry phase depends on the relative Rabi frequency of the two subsystems, and it approaches π when the amplitude of the coupling tends to infinity.

DOI
10.1142/s0217984913500887

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The Berry phase has universal applications in various fields. Here, we explore the coupling effect on the Berry phase of a two-level system adiabatically driven by a rotating classical field and interacting with a single quantized mode. Our simulations clearly reveal that the Berry phase change is quadratic proportional to the coupling constant if it is less than the level spacing between neighboring instantaneous eigenstates. Remarkably, if the nearest neighbouring level spacing is comparable with the coupling constant, this simple quadratic dependence is lost. Around this resonance, the Berry phase can be significantly tuned by slightly adjusting the parameters, such as the coupling constant, the frequency of the quantized mode, and the transition frequency. These numerical results, agreeing well with the perturbation theory calculations, provide an alternative approach to tune the Berry phase near the resonance, which is useful in quantum information science, i.e. designing quantum logic gates.
We apply the perturbation theory to investigate the behavior of the Berry phase of a composite system. The composite system consists of two weakly-coupled spin-½ particles, in which each particle is driven by a varying magnetic field. The result shows that the Berry phase for one subsystem is controlled by the state of the other subsystem. The method can also be used to deal with the effect of a spin environment on a single-particle system if the particle is weakly coupled to the spin environment.
In this work, we study the geometric phases and quantum entanglement of a composite system consisting of two interacting spin-1/2 particles in the presence of an external magnetic field. The conditions for real and complex eigenvalues are derived and discussed. Based on the non-Hermitian Berry phase, we propose the geometric phase of projective quantum states, discuss their relationship, and analyze their geometric properties. We expound on how the geometric phase of projective quantum states is contingent upon the coupling strength and the nonreciprocity between two subsystems, illustrating the connection between the geometric phase of the composite system and the geometric phases of its subsystems. By studying the quantum entanglement of the subsystems, the role of the correlation between the subsystem and the nonreciprocity is discussed. The results show that the quantum entanglement is influenced and determined by the nonreciprocity of the coupling. These schemes can provide a way to manipulate the nonreciprocal correlation between the subsystems to control the non-Hermiticity, geometric phase, and entanglement of the subsystems.
The effect of intersubsystem coupling on the Berry phase of a composite system as well as that of its subsystem is investigated. We analyze two coupled spin-12\frac{1}{2}21​ systems with one driven by a quantized field as an example. The pure-state geometric phase of the composite system as well as the mixed-state geometric phase for the subsystem is calculated and discussed.
We consider a two-level system coupled to a highly non-Markovian environment when the coupling axis rotates with time. The environment may be quantum (for example a bosonic bath or a spin bath) or classical (such as classical noise). We show that an Anderson orthogonality catastrophe suppresses transitions, so that the system's instantaneous eigenstates (parallel and antiparallel to the coupling axis) can adiabatically follow the rotation. These states thereby acquire Berry phases; geometric phases given by the area enclosed by the coupling axis. Unlike in earlier proposals for environment-induced Berry phases, here there is little decoherence, so one does not need a decoherence-free subspace. Indeed we show that this Berry phase should be much easier to observe than a conventional one because it is not masked by either the dynamic phase or the leading nonadiabatic phase. The effects that we discuss should be observable in any qubit device where one can drive three parameters in the Hamiltonian with strong man-made noise.