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arXiv:1111.0154 · Physical Review A

Vacuum-induced Berry phase beyond the rotating-wave approximation

Tao Liu, Mang Feng, Kelin Wang

Published Dec 13, 2011• 13 citations
arXivHTMLPDF

Abstract

With reference to the vacuum-induced Berry phase (VIBP) obtained in the interaction of a spin-1/2 particle with quantized irradiation field under rotating-wave approximation (RWA), we present a completely different treatment for the VIBP by a fully quantum-mechanical treatment beyond the RWA, which gives different definition of the VIBP and indicates the validity of RWA to be relevant not only to the energy conservation and comparison of characteristic parameters, but also to more subtle physics, such as the geometric property of the state evolution. Our result is of conceptual importance and also of significant relevance to quantum-information processing.

DOI
10.1103/physreva.84.062109

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With reference to the vacuum-induced Berry phase (VIBP) obtained in the interaction of a spin-1/2 particle with quantized irradiation field under rotating-wave approximation (RWA), we present a completely different treatment for the VIBP by a fully quantum-mechanical treatment beyond the RWA, which gives different definition of the VIBP and indicates the validity of RWA to be relevant not only to the energy conservation and comparison of characteristic parameters, but also to more subtle physics, such as the geometric property of the state evolution. Our result is of conceptual importance and also of significant relevance to quantum-information processing.
We address Berry phases that have been predicted to appear when a two-level system interacts with a quantized field, including the vacuum state. The issue became controversial after it was argued that the appearance of such Berry phases is an artifact of the rotating-wave approximation (RWA). As is widely believed, whenever the RWA applies, one may replace the Rabi model of a two-level system interacting with a quantized field by the analytically solvable Jaynes-Cummings model. Conflicting predictions of these two models under conditions that validate the RWA would signal a serious inconsistency of this approximation. We show that this is not the case and that claims to the contrary are inconsistent with analytical results concerning the Rabi model. We provide also numerical evidence supporting our analytical approach. Furthermore, we argue that the appearance of Berry phases in the addressed cases does not depend on adiabatic conditions nor on any particular Hamiltonian, but on the underlying vector space. Thus, the appropriate framework is given by the kinematic approach to geometric phases, which contains Berry's phase as a special case.
We revisit earlier studies on Berry phases suggested to appear in certain cavity QED settings. It has been especially argued that a nontrivial geometric phase is achievable even in the situation of no cavity photons. We, however, show that such results hinge on imposing the rotating wave approximation (RWA), while without the RWA no Berry phases occur in these schemes. A geometrical interpretation of our results is obtained by introducing semiclassical energy surfaces which in a simple way brings out the phase-space dynamics. With the RWA, a conical intersection between the surfaces emerges and encircling it gives rise to the Berry phase. Without the RWA, the conical intersection is absent and therefore the Berry phase vanishes. It is believed that this is a first example showing how the application of the RWA in the Jaynes-Cummings model may lead to false conclusions, regardless of the mutual strengths between the system parameters.
Since a pioneer work on vacuum-induced Berry phase (VIBP) was done by Fuentes-Guridi et al. [Phys. Rev. Lett. 89 (2002) 220404], much attention has been paid to the geometric phase effects of vacuum field. However, all the so-called VIBPs investigated previously are not purely vacuum-induced (i.e. the nonvacuum components of the field are also involved). In this paper, we discuss how to deliver geometric phases from the evolution of a genuine vacuum field in a standard cavity quantum electrodynamics (QED) system. First, we design a cyclic evolution of an atom–field system with the atom being initially prepared at the excited state and the field at the genuine vacuum. Then, we calculate the geometric phases acquired during such a cyclic evolution. It is found that such geometric phases are really induced by an evolution of the genuine vacuum field. Specifically, our generic proposal is demonstrated with both the one- and two-mode Jaynes–Cummings model interactions (JCM).
The Berry phase of the Rabi model beyond the rotating-wave approximation is investigated. By use of two unitary transformations, the analytical expressions of the Berry phase are obtained approximately for the Rabi model. The obtained analytical solutions reproduce accurately the high-accuracy numerical results of the Berry phase in the strong coupling regime. It is found that the Berry phase of the ground state is relevant to the second-order terms of the coupling strength, and the Berry phases of the first and second excited states are mainly relevant to the first-, second- and third-order terms of the coupling strength when the effect of the counter-rotating-wave terms is taken into account.
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].