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Quantum Optics Journal of the European Optical Society Part B

Berry phases and Rabi oscillations

D J Moore

Published Apr 1, 1992• 2 citations
Publisher

Abstract

The Berry phases for a spherically symmetric atom in a circularly polarized semiclassical radiation field are calculated using an operator decomposition scheme. The author then takes the two-level atomic limit and recovers the Berry phases already calculated by other means. His attention is then focused on the two-level atom. He shows that in the semiclassical problem the Rabi oscillations do not arise from cyclic wavefunctions. Hence any comparison of the phases of the initial and final states must use the Pancharatnam connection. In the quantum model he no longer gets perfect Rabi oscillations as there are partial collapses and revivals. These collapses and revivals are exploited in alternative quantum optical models, the Raman coupling model and intensity-dependent coupling model, for which they are exactly cyclic. Again he finds that he must invoke the Pancharatnam connection.

DOI
10.1088/0954-8998/4/2/006

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The Berry phases for a spherically symmetric atom in a circularly polarized semiclassical radiation field are calculated using an operator decomposition scheme. The author then takes the two-level atomic limit and recovers the Berry phases already calculated by other means. His attention is then focused on the two-level atom. He shows that in the semiclassical problem the Rabi oscillations do not arise from cyclic wavefunctions. Hence any comparison of the phases of the initial and final states must use the Pancharatnam connection. In the quantum model he no longer gets perfect Rabi oscillations as there are partial collapses and revivals. These collapses and revivals are exploited in alternative quantum optical models, the Raman coupling model and intensity-dependent coupling model, for which they are exactly cyclic. Again he finds that he must invoke the Pancharatnam connection.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects
In 1950, Ramsey and Hahn separately described a method using separated oscillatory magnetic fields and electronic levels to measure interference in atomic states. Today, similar physics is exploited to make atom interferometer sensors utilizing what is called the spin-echo sequence. In 1956, Pancharatnam described the geometric phase difference between two polarized states of light. In 1984 Berry described the phase accrued in cyclic and adiabatic quantum systems. A Pancharatnam-Berry phase experiment involving a quarter-wave – half-wave – quarter-wave plate stack of optics and an experiment using the spin-echo sequence are based on seemingly different phenomena in optics versus atomic optics, respectively. However, the mathematical representations used to describe the Pancharatnam-Berry phase in optics are identical to those used to describe the spin-echo sequence. Therefore, the relationship between the spin-echo sequence and the Pancharatnam-Berry phase was explored. Experiments to test the comparisons using a pre-existing cold atom experiment, ~150 μK, were performed. The mathematical connection between the two types of experiments was explained more deeply.
Recent interest in the collapse and revival of Rabi oscillations in the Jaynes-Cummings model was stimulated by a paper in 1980 by Eberly et al.1 These authors presented an analytic expression for the inversion of a two-level atom interacting with a single mode of the radiation field taken to be initially in a coherent state. Their analytic expression agreed well with a numerical evaluation of the standard summation formula for the inversion. Since the publication of this analytic result various discussions of its proof have appeared. Existing versions of the proof all convert the summation formula into an integral and evaluate this integral using the method of stationary phase.2 This approach is mathematically intricate and leaves one with little physical intuition for the approximations that are made. I offer an alternative proof, which, in compact form, takes only a few lines. The proof is built up in a series of approximations based on an expansion of the troublesome square root (n) in the Jaynes-Cummings summation formula in a Taylor series about the mean photon number: n=n¯[1+12n−n¯n¯−18(n−n¯n¯)2+…]. The analytical expression of Eberly ef al. is recovered using standard sums and integrals and this Taylor expansion to second order.
Interaction of a lossless two-level atom with a monochromatic (classical) field of radiation is considered, as the atom initially possesses a translational state with a number of equidistant and discrete momenta. It is shown that the Rabi oscillations in such an atom evolve as a sequence of collapses and revivals, if the coupling wave deeply saturates the optical transition. Between revivals, the populations undergo subrevivals. Approximate analytical formulae are obtained taking the initial momentum distribution in the form of two shifted Gaussians or a Besselian. A possible experimental realization of such revivals is discussed.
The semiclassical interaction of a two-level atom with a singlemode field, taking into account the atomic reaction onto the field, is considered. Analytical expressions for the field and atomic dynamics are found. An expansion of this solution in terms of the inverse of the field intensity is performed. To zeroth order, the classical Rabi solution is recovered, while first-order corrections give rise to a slow modulation of the atomic dipole, in agreement with a fully quantum model. The evolution of the atomic dipole is studied in the framework of the reduced Q function, both in the semiclassical and quantum regimes. PACS 42.50.Dv — 03.65.-w.