Michael V Berry

Quantal phase factors accompanying adiabatic changes

A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.

Semiclassical approximations in wave mechanics

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynmans path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.

The Adiabatic Phase and Pancharatnam's Phase for Polarized Light

Abstract In 1955 Pancharatnam showed that a cyclic change in the state of polarization of light is accompanied by a phase shift determined by the geometry of the cycle as represented on the Poincare sphere. The phase owes its existence to the non-transitivity of Pancharatnams connection between different states of polarization. Using the algebra of spinors and 2 × 2 Hermitian matrices, the precise relation is established between Pancharatnams phase and the recently discovered phase change for slowly cycled quantum systems. The polarization phase is an optical analogue of the Aharonov-Bohm effect. For slow changes of polarization, the connection leading to the phase is derived from Maxwells equations for a twisted dielectric. Pancharatnams phase is contrasted with the phase change of circularly polarized light whose direction is cycled (e.g. when guided in a coiled optical fibre).

Optical vortices evolving from helicoidal integer and fractional phase steps

The evolution of a wave starting at z = 0a s exp(iαφ) (0 φ 0as trength n optical vortex, whose neighbourhood is described in detail. Far from the axis, the wave is the sum of exp{i(αφ + kz)} and a diffracted wave from r = 0. The paraxial wave and the wave far from the vortex are incorporated into a uniform approximation that describes the wave with high accuracy, even well into the evanescent zone. For fractional α ,n o fractional-strength vortices can propagate; instead, the interferenc eb etween an additional diffracted wave, from the phase step discontinuity, with exp{i(αφ + kz)} and the wave scattered from r = 0, generates a pattern of strength-1 vortex lines, whose total (signed) strength Sα is the nearest integer to α .F or small|α − n| ,t heselines are close t ot hez axis. As α passes n +1 /2, Sα jumps by unity, so a vortex is born. The mechanism involves an infinite chain of alternating-strength vortices close to the positive x axis for α = n +1 /2, which annihilate in pairs differently when α> n +1 / 2a nd when α< n +1 /2. There is a partial analogy between α and the quantum flux in the Aharonov–Bohm effect.

Transitionless quantum driving

For a general quantum system driven by a slowly time-dependent Hamiltonian, transitions between instantaneous eigenstates are exponentially weak. But a nearby Hamiltonian exists for which the transition amplitudes between any eigenstates of the original Hamiltonian are exactly zero for all values of slowness. The general theory is illustrated by spins driven by changing magnetic fields, and implies that any spin expectation history, including those where the spin never precesses, can be generated by infinitely many driving fields, here displayed explicitly. Asymptotically, the absence of transitions is explained by continuation to complex time, where the complex degeneracies in the transitionless driving fields have a nongeneric structure for which there is no Stokes phenomenon; this is analogous to the explanation of reflectionless potentials.

Physics of Nonhermitian Degeneracies

A summary, with references and additional comments, of a talk delivered at the Second International Workshop on Pseudohermitian Hamiltonians in Quantum Physics (Prague, 14-16 June 2004). After explaining some general features of nonhermitian degeneracies (‘exceptional points’), several applications are outlined: to multiple reflections in a pile of plates, linewidths of unstable lasers, atom diffraction by light, and crystal optics.

Anticipations of the Geometric Phase

In science we like to emphasize the novelty and originality of our ideas. This is harmless enough, provided it does not blind us to the fact that concepts rarely arise out of nowhere. There is always a historical context, in which isolated precursors of the idea have already appeared. What we call “discovery” sometimes looks, in retrospect, more like emergence into the air from subterranean intellectual currents.

Histories of Adiabatic Quantum Transitions

The way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation, for a 2-state quantum system. By transforming to a series of superadiabatic bases, clinging ever closer to the exact evolving state, it is shown that transition histories renormalize onto a universal one, in which the amplitude grows to its final value as an error function (rather than via large oscillations as in the ordinary adiabatic basis). The time for the universal transition is of order √ (ћ/δ) where δ is the small adiabatic (slowness) parameter. In perturbation theory the pre-exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value ⅓π (for the ordinary adiabatic basis) to the correct value unity. The various histories could be observed in spin experiments.

Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue

The authors study the wavefronts (i.e. the surfaces of constant phase) of the wave discussed by Aharonov and Bohm, representing a beam of particles with charge q scattered by an impenetrable cylinder of radius R containing magnetic flux Phi . Defining the quantum flux parameter by alpha =q Phi /h, they show that for the case R=0 the wave psi AB possesses a wavefront dislocation on the flux line, whose strength (i.e. the number of wave crests ending on the dislocation) equals the nearest integer to alpha . When alpha passes through half-integer values, the strength changes, by wavefronts unlinking and reconnecting along a nodal surface. In quantum mechanics this phase structure is unobservable, but they devise an analogue where surface waves on water encounter an irrotational bathtub vortex; in this case alpha depends on the frequency of the waves and the circulation of the vortex. Experiments show dislocation structures agreeing with those predicted. psi AB is an unusual function in which incident and scattered waves cannot be clearly separated in all asymptotic directions; they discuss its properties using a new asymptotic method.

Quantum Phase Corrections from Adiabatic Iteration

The phase change γ acquired by a quantum state |ψ(t)> driven by a hamiltonian H0(t), which is taken slowly and smoothly round a cycle, is given by a sequence of approximants γ(k) obtained by a sequence of unitary transformations. The phase sequence is not a perturbation series in the adiabatic parameter ∊ because each γ(k) (except γ(0)) contains ∊ to infinite order. For spin-½ systems the iteration can be described in terms of the geometry of parallel transport round loops Ck on the hamiltonian sphere. Non-adiabatic effects (transitions) must cause the sequence of γ(k) to diverge. For spin systems with analytic H0(t) this happens in a universal way: the loops Ck are sinusoidal spirals which shrink as ∊k until k ~ ∊-1 and then grow as k!; the smallest loop has a size exp{-1/∊}, comparable with the non-adiabaticity.