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arXiv:quant-ph/0210182 · Journal of Physics A Mathematical and General

The quantum mechanical geometric phase of a particle in a resonant vibrating cavity

K W Yuen, H T Fung, K M Cheng, M-C Chu, K Colanero

Published Oct 22, 2003• 4 citations
arXivHTMLPDF

Abstract

We study the general-setting quantum geometric phase acquired by a particle in a vibrating cavity. Solving the two-level theory with the rotating-wave approximation and the SU (2) method, we obtain analytic formulae that give excellent descriptions of the geometric phase, energy and wavefunction of the resonating system. In particular, we observe a sudden π-jump in the geometric phase when the system is in resonance. Similar behaviour is found in the geometric phase of a spin- particle in a rotating magnetic field. Our result suggests that the π-jump is a general feature for two-level systems, for which we developed a geometrical model to help visualize its evolution.

DOI
10.1088/0305-4470/36/44/010

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We study the general-setting quantum geometric phase acquired by a particle in a vibrating cavity. Solving the two-level theory with the rotating-wave approximation and the SU (2) method, we obtain analytic formulae that give excellent descriptions of the geometric phase, energy and wavefunction of the resonating system. In particular, we observe a sudden π-jump in the geometric phase when the system is in resonance. Similar behaviour is found in the geometric phase of a spin- particle in a rotating magnetic field. Our result suggests that the π-jump is a general feature for two-level systems, for which we developed a geometrical model to help visualize its evolution.
We study the general-setting quantum geometric phase acquired by a particle in a vibrating cavity. Solving the two-level theory with the rotating-wave approximation and the SU(2) method, we obtain analytic formulae that give excellent descriptions of the geometric phase, energy, and wavefunction of the resonating system. In particular, we observe a sudden pi\\pipi-jump in the geometric phase when the system is in resonance. We found similar behaviors in the geometric phase of a spin-1/2 particle in a rotating magnetic field, for which we developed a geometrical model to help visualize its evolution.
Abstract We investigate the effects of counterrotating terms on geometric phase and its relation to the resonance of the Rabi model. We apply the unitary transformation with a single parameter to the Rabi model and obtain the transformed Hamiltonian involving multiple harmonic terms. By combining the counter-rotating-hybridized rotating-wave method with time-dependent perturbation theory, we solve systematically time evolution operator and then obtain the geometric phase of the two-level system. Our results are beyond adiabatic approximation and rotating-wave approximation (RWA). Higher-order harmonic resonance happens when driving frequency is equal to higher-order subharmonic of the Rabi frequency. In comparison with numerically exact results, our calculated results are accurate over a wide range of parameters space, especially in higher-order harmonic resonance regimes. In these regimes we demonstrate geometric phases change dramatically while those of the RWA are smooth. The RWA is thoroughly invalid even if the driving strength is extremely weak. We find it is the higher-order harmonic terms that play an important role on the cyclic state and demonstrate the characters of geometric phase in higher-order harmonic resonance regime. We also present analytical formalism of the change rate of geometric phase and quasienergies, which agree well with numerically exact ones even in the strong driving case. The developed method can be applied to explore the dynamics of strongly driven qubits and physical properties of higher-order harmonic processes.
We follow a generalized kinematic approach to compute the geometric phases acquired in both unitary and dissipative Jaynes-Cummings models, which provide a fully quantum description for a two-level system interacting with a single mode of the (cavity) electromagnetic field, in a perfect or dissipative cavity, respectively. In the dissipative model, the nonunitary effects arise from the outflow of photons through the cavity walls and the incoherent pumping of the two-level system. Our approach allows to compare the geometric phases acquired in these models, leading to an exhaustive characterization of the corrections introduced by the presence of the environment. We also provide geometric interpretations for the observed behaviors. When the resonance condition is satisfied, we show the geometric phase is robust, exhibiting a vanishing correction under a nonunitary evolution. This fact is supported with a geometrical explanation as well.
In the present paper, we study the geometric phase by considering a model that closely describes a realistic experimental scenario. We present an active way to control and quench the variation of the geometric phase using ultra-short strong pulses for a two-level quantum system with an exact analytical solution, i.e. beyond the rotating wave approximation. In particular, we study the variation of the geometric phase with a few cycles pulse and a smooth phase jump over a finite time intervals. The control of the variation rate of the geometric phase is obtained by acting on the shape of the phase transient and other parameters of the considered system. These features make two-level systems incorporated in ultra-short, of-resonant and gradually changing phase good candidates for implementation of schemes for the quantum computation and the coherent information processing.
We investigate the geometric phase of a two-level atom (qubit) coupled to a bosonic reservoir with Lorentzian spectral density and find that for the non-Markovian dynamics in which the rotating-wave approximation (RWA) is performed, the geometric phase has a $\ensuremath{\pi}$-phase jump at the nodal point. However, the exact result without the RWA given by the hierarchical equations of motion method shows that there is no such phase jump or nodal structure in the geometric phase. Thus our results demonstrate that the counterrotating terms significantly contribute to the geometric phase in the multimode Hamiltonian under certain circumstances.
In this paper, we investigate the behaviour of the geometric phase of a more generalized nonlinear system composed of an effective two-level system interacting with a single-mode quantized cavity field. Both the field nonlinearity and the atom–field coupling nonlinearity are considered. We find that the geometric phase depends on whether the index k is an odd number or an even number in the resonant case. In addition, we also find that the geometric phase may be easily observed when the field nonlinearity is not considered. The fractional statistical phenomenon appears in this system if the strong nonlinear atom–field coupling is considered. We have also investigated the geometric phase of an effective two-level system interacting with a two-mode quantized cavity field.