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arXiv:1208.1891 · Physica Scripta

On the rotating wave approximation in the adiabatic limit

Jonas Larson

Published Mar 1, 2013• 7 citations
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Abstract

I revisit a longstanding question in quantum optics; when is the rotating wave approximation justified? In terms of the Jaynes–Cummings and Rabi models I demonstrate that the approximation in general breaks down in the adiabatic limit regardless of system parameters. This is explicitly shown by comparing Berry phases of the two models, where it is found that this geometrical phase is strictly zero in the Rabi model contrary to the non-trivial Berry phase of the Jaynes–Cummings model. The source of this surprising result is traced back to different topologies in the two models.

DOI
10.1088/0031-8949/2013/t153/014040

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I revisit a longstanding question in quantum optics; when is the rotating wave approximation justified? In terms of the Jaynes–Cummings and Rabi models I demonstrate that the approximation in general breaks down in the adiabatic limit regardless of system parameters. This is explicitly shown by comparing Berry phases of the two models, where it is found that this geometrical phase is strictly zero in the Rabi model contrary to the non-trivial Berry phase of the Jaynes–Cummings model. The source of this surprising result is traced back to different topologies in the two models.
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.
The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by extended Swain's ansatz with the conserved parity. The analytical approximations are then performed at different levels. The well-known rotating-wave approximation is naturally covered in the present zero and first approximations. The effect of the counter rotating-wave term emerges clearly in the second order approximation. The concise analytical expressions are given explicitly and can be applicable up to the ultra-strong coupling regime. The preliminary application to the vacuum Rabi splitting is shown to be very successful.
By using a wave packet approach, this paper reviews the Jaynes-Cummings model with and without the rotating wave approximation in a non-standard way. This gives new insight, not only of the two models themselves, but of the rotating wave approximation as well. Expressing the models by field quadrature operators, instead of the typically used boson ladder operators, wave packet simulations are presented. Several known phenomena of these systems, such as collapse-revivals, Rabi oscillation, squeezing and entanglement, are reviewed and explained in this new picture, either in an adiabatic or diabatic frame. The harmonic shape of the potential curves that the wave packets evolve on and the existence of a level crossing make these results interesting in a broader sense than only for models in quantum optics, especially in atomic and molecular physics.
The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by an extended Swain ansatz with conserved parity. Analytical approximations are then performed at different levels. The well-known rotating-wave approximation (RWA) is naturally covered in the present zeroth- and first-order approximations. A first-order correction to the RWA can be obtained in a second-order approximation, by which the effect of the counter-rotating-wave term emerges clearly. Concise analytical expressions are given explicitly and can be applicable up to the ultrastrong-coupling regime. A preliminary application to vacuum Rabi splitting is shown to be very successful.