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Annual Meeting Optical Society of America

Single-photon observation of Berry’s phase

Paul G. Kwiat, R. Y. Chiao

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Berry's phases in optics have been studied extensively, and examples are typically discussed in quantum-mechanical terms. To our knowledge, however, experimental demonstrations have been performed only at the classical level. We report the first observation of a Berry's phase at the single-photon level. In particular, we investigated Pan-charatnam's phase,1 which is acquired whenever light is taken through a closed cycle of polarization states, e.g., with various retardation plates. Correlated pairs of linearly-polarized photons were produced by parametric fluorescence in a UV-pumped KDP crystal. One of each pair was directed into a photomultiplier tube (PMT1). The other was directed into a Michelson interferometer, one arm of which contained two quarter-wave plates, that was used to generate a Berry's phase.

DOI
10.1364/oam.1990.mr2

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Berry's phases in optics have been studied extensively, and examples are typically discussed in quantum-mechanical terms. To our knowledge, however, experimental demonstrations have been performed only at the classical level. We report the first observation of a Berry's phase at the single-photon level. In particular, we investigated Pan-charatnam's phase,1 which is acquired whenever light is taken through a closed cycle of polarization states, e.g., with various retardation plates. Correlated pairs of linearly-polarized photons were produced by parametric fluorescence in a UV-pumped KDP crystal. One of each pair was directed into a photomultiplier tube (PMT1). The other was directed into a Michelson interferometer, one arm of which contained two quarter-wave plates, that was used to generate a Berry's phase.
Coincidence detection of photon pairs produced in parametric fluorescence, in conjunction with a Michelson interferometer in which one member of each pair acquired a geometrical phase due to a cycle in polarization states, has allowed the first observation of Berry's phase at the single-photon level. We have verified that each interfering photon was essentially in an n=1 Fock state, by means of a beam splitter following the interferometer combined with a triple-coincidence technique. The results can be interpreted in terms of a nonlocal collapse of the wave function.
Photon pairs produced by type-I and type-II parametric down-conversion are passed through a Michelson interferometer. Two quarter-wave plates in one arm of the interferometer allow variation of the polarization state of the photons. We investigate experimentally the geometric phase or Pancharatnam phase acquired by single photons and photon pairs dependent on the solid angle that is subtended by the circuit that represents the varying state of polarization on the Poincar\'e sphere. It is found that the geometric phase acquired by the pair depends on the initial polarization state of the two photons. If both photons are in equal states of linear polarization, we observe a doubling of the geometric phase compared to single photons; in the case of orthogonal states of linear polarization, the geometric phase is completely canceled. Our results show the role of Pancharatnam's phase in nonclassical two-photon interference phenomena and the interplay between the geometric phase and the dynamical phase in these phenomena.
The contract was for performing experiments on a recently discovered topological phase factor known as 'Berry's phase.' We proposed to examine this phase at both the quantum and classical levels, in its various manifestations. The goals of this contract were successfully accomplished; in particular, we succeeded in demonstrating this phase at the single photon level, thereby demonstrating that this phase did not originate at the classical level. Possible applications of this phase were explored. A two-photon light source, in the form a parametric down-converter of uv laser photons into highly correlated red photons was successfully constructed, and coincidence counting techniques were successfully applied to the detection of the conjugate signal and idler red photons. This formed the experimental basis for the examination of not only Berry's phase, but of the energy-time uncertainty relations, the Franson experiment, the measurement of the tunneling time of the photon, the quantum eraser, and other optical phenomena closely related to the Einstein-Podolsky- Rosen 'paradox.'
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
The wave function of a quantum system may acquire an additional phase factor (the geometric phase), when its parameters undergo a cyclic change. An optical manifestation of the geometric phase is the change in the phase of a light wave produced when its state of polarization is taken around a circuit on the Poincare sphere. This phase change can be verified by interferometric measurements. We describe an interferometer in which the two beams traverse the same path in opposite directions, so that their optical paths are always equal, and the phase difference between them is varied by operating on the geometric phase. Measurements made with this interferometer, at a light level at which the probability of more than one photon being present simultaneously in the interferometer was negligibly small, confirmed that the interference effects observed due to the geometric phase persist down to the single photon level.
A simple definition of Berry phases based on the Pancharatnam phase convention is stated in the language of ordinary quantum mechanics. It is shown to be equivalent to previous definitions for a cycle of transformations built continuously from infinitesimal transformations. It also applies to discrete transformations and partial cycles. The phase for a partial cycle can be the same as for the complete cycle. This suggests possibilities for simpler experiments. In an example considered here, each part of a split beam of light is put through two half-wave plates. The light is not brought back to the original polarization state, but interference produced by the Berry phase can be observed because the different parts of the beam are brought to the same polarization state.