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Journal of Physics Condensed Matter

The Berry phase in GaAs semiconductor with a quantized field

Gang Chen, Zidong Chen, Lixian Yu

Published Feb 13, 2007• 1 citation
Publisher

Abstract

In this paper we investigate the Berry phase in GaAs semiconductor with a quantized magnetic field in the rotating wave approximation. The eigenfunctions of the nuclear spin in the quantized external field are obtained and thus the Berry phase is evaluated explicitly in terms of the introduction of the phase shift. It is shown that the Berry phase can be easily controlled by the coupling strength, the anisotropy constant and the frequency of the electromagnetic wave, which can be important in applications in geometric quantum computing.

DOI
10.1088/0953-8984/19/9/096005

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In this paper we investigate the Berry phase in GaAs semiconductor with a quantized magnetic field in the rotating wave approximation. The eigenfunctions of the nuclear spin in the quantized external field are obtained and thus the Berry phase is evaluated explicitly in terms of the introduction of the phase shift. It is shown that the Berry phase can be easily controlled by the coupling strength, the anisotropy constant and the frequency of the electromagnetic wave, which can be important in applications in geometric quantum computing.
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.
The geometric phase, in particular the Berry phase, in an entangled state of five spin-1/2 particles is studied. A time-dependent magnetic field is applied to control the time evolution of the cluster. Using the method of algebraic dynamics, we calculate the non-adiabatic geometric phase or Berry phase and the degeneracy energy levels when the magnetic rotates around Z axis. Based on the exact analytical solutions, we show how the Berry phase of the entangled state of this cluster depends on the external magnetic field parameters ω (the angular velocity of the rotating magnetic field) and θ (the angle between the magnetic field and Z axis).
We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].
We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].
The time-dependent Schr\"odinger equation for a spin particle in a rotating magnetic field is solved analytically by the cranking method, and the exact solutions are employed to study the nonadiabatic Berry's phase. An alternative expression for Berry's phase is given, which shows that Berry's phase is related to the expectation value of spin along the rotating axis and gives Berry's phase a physical explanation besides its gauge geometric interpretation. This expression also presents a simple algorithm for calculating the nonadiabatic Berry's phase for Hamiltonians that are nonlinear functions of the SU(2) generators. It is shown that nonadiabaticity alters the time evolution ray and in turn changes its Berry's phase. For the SU(2) dynamical group, the nonadiabatic effect on Berry's phase manifests itself as spin alignment (a phenomenon in nuclear physics), and spin-alignment quantization (observed recently in high-spin nuclear physics) is related to Berry's-phase quantization.
Due to the potential application in quantum information process, geometric phase of interacting system arouse many interests. Some physicists concentrate on the system in pure classical envi- ronment, while others study the system in pure quantized environment. So a natural question is asked: how about an interacting system in composite environments made up of both classical and quantized field. In this letter, we analyze a quantum system composed of two interacting spins, of which one is in classical magnetic field and the other is in quantized field. First, classical magnetic field driven Berry phases for the whole system and subsystem are studied. The effect of couplings between particles and photon on these phases are analyzed. In comparison with the dynamical quantized field, We find that even a static quantized field in its vacuum state can also have an effect on Berry phase. Second, quantized field driven Berry phases for the whole system and sub- system are formulated, including both one and two mode of this field. The vacuum induced effects are elaborated, moreover compared with the constant vacuum induced phases in former papers, the counterpart in this letter varies according to classical magnetic field, couplings and other pa- rameters. For the two mode quantized field, the rigorous relationship between the concurrence and Berry phase for subsystem are built up.