It is shown that Berry's phase appears in a more general context than realized so far. The evolution of the quantum system need be neither unitary nor cyclic and may be interrupted by quantum measurements. A key ingredient in this generalization is the use of some ideas introduced by Pancharatnam in his study of the interference of polarized light, which, when carried over to quantum mechanics, allow a meaningful comparison of the phase between any two nonorthogonal vectors in Hilbert space.
It is shown that Berry's phase appears in a more general context than realized so far. The evolution of the quantum system need be neither unitary nor cyclic and may be interrupted by quantum measurements. A key ingredient in this generalization is the use of some ideas introduced by Pancharatnam in his study of the interference of polarized light, which, when carried over to quantum mechanics, allow a meaningful comparison of the phase between any two nonorthogonal vectors in Hilbert space.
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
According to Pancharatnam [S. Pancharatnam, in Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989)], two light beams of differing elliptical polarization can be said to be ``in phase'' if their interference is maximally constructive. In the same spirit, the relative phase between two beams a and b is the amount by which the phase of beam b needs to be changed to put it in phase with beam a. Berry has established a simple relation between Pancharatnam's concept and a phase relation between two quantum-mechanical states: If two nonorthogonal states \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉 and \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 are phased in such a way that \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉+\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 has maximum norm, i.e., such that 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 is real and positive, they can be said to be in phase or parallel, and in general the relative phase between them is the phase angle of 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉. This concept plays a role in the theory of the Abelian geometric phase. In the non-Abelian case, one has two n-dimensional subspaces instead of two single states, and instead of choosing the phase of each state one has to choose an orthonormal basis (including phase) in each subspace. We show that the concept of ``block diagonalization'' introduced by Cederbaum, Shirmer, and Meyer [J. Phys. A 22, 2427 (1989)] in the context of a molecular problem provides a natural generalization of the Pancharatnam-Berry phase relation.
The geometric phase was first anticipated 1956 by Pancharatnam, in the context of polarised light. Much later, in 1984, Berry made an analogue discovery, but this time in the context of quantum mechanics. It turns our that the geometric phase, or Berry's phase, or the Pancharatnam phase, is a very general phenomenon. It is a phase shift that can be acquired by a system, when it has undergone a adiabatic process, bringing it back to its original initial conditions. If the parameter space of the involved Hamiltonian is curved, the final condition may involve a phase change. This effect has been demonstrated in a range of different interferometry experiments. Although the Pancharatnam phase has been seen, this has always been in studies involving the first order correlation, g(1), this is in the intereferogramm of two or more wave amplitudes. Here, we demonstrate experimental proofs of the effect of the Pancharatnam phase in intensity interferometry, or in other words, where the second order correlation factor g(2) is the relevant description of the interference.
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.
While Pancharatnam discovered the geometric phase in 1956, his work was not widely recognized until its endorsement by Berry in 1987, after which it received wide appreciation. However, because Pancharatnam's paper is unusually difficult to follow, his work has often been misinterpreted as referring to an evolution of states of polarization, just as Berry's work focused on a cycle of states, even though this consideration does not appear in Pancharatnam's work. We walk the reader through Pancharatnam's original derivation and show how Pancharatnam's approach connects to recent work in geometric phase. It is our hope to make this widely cited classic paper more accessible and better understood.
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.