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Physical Review A

Phase avalanches in near-adiabatic evolutions

Tamás Vértesi, R. Englman

Published Feb 2, 2006• 5 citations
Publisher

Abstract

In the course of slow, nearly adiabatic motion of a system, relative changes in the slowness can cause abrupt and high magnitude phase changes, ``phase avalanches,'' superimposed on the ordinary geometric phases. The generality of this effect is examined for arbitrary Hamiltonians and multicomponent $(>2)$ wave packets and is found to be connected (through the Blaschke term in the theory of analytic signals) to amplitude zeros in the lower half of the complex time plane. Motion on a nonmaximal circle on the Poincar\'e-sphere suppresses the effect. A spectroscopic transition experiment can independently verify the phase-avalanche magnitudes.

DOI
10.1103/physreva.73.022103

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In the course of slow, nearly adiabatic motion of a system, relative changes in the slowness can cause abrupt and high magnitude phase changes, ``phase avalanches,'' superimposed on the ordinary geometric phases. The generality of this effect is examined for arbitrary Hamiltonians and multicomponent (>2)(>2)(>2) wave packets and is found to be connected (through the Blaschke term in the theory of analytic signals) to amplitude zeros in the lower half of the complex time plane. Motion on a nonmaximal circle on the Poincar'e-sphere suppresses the effect. A spectroscopic transition experiment can independently verify the phase-avalanche magnitudes.
Abstract This book is an introduction to explicitly time-dependent physics. We describe the theory of the dynamic regime that we refer to as “quasi-adiabatic”. It pertains to time-dependent evolution that, while slow, is not genuinely adiabatic. It is rich in effects that can be understood even in quantum systems with complex many-body interactions. Examples from research in quantum computing, phase transitions, Bose-Enstein condensates, and quantum control will be used to illustrate these effects. Prerequisites include undergraduate introductory courses in quantum and classical mechanics, as well as complex analysis. Our book aims to fill the gap in the literature by providing an accessible, for early graduate and advanced undergraduate students, introduction to both geometric phases and nonadiabatic transitions at a level sufficient to consider numerous applications.
Abstract The exponentially small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian Ĥ = H(δt). Ŝ ( where δ is a small adiabatic parameter and Ŝ is the Vector spin -½ operator), contains an additional factor exp{ᴦg} of purely geometric origin (that is, independent of δ and ħ). For ᴦg to be non-zero, Ĥ must be complex hermitian rather than real symmetric, and the hamiltonian curve H(ז) must not lie in a plane through the origin nor be a helix identical (up to rigid motion) with its time reverse. An expression is given for ᴦg, as an integral from the real t axis to the complex time of degeneracy of the two states. Explicit examples are given. The geometric effect could be observed in experiments with spinning particles.
If the time evolution of a quantum state leads back to the initial state, a geometric phase is accumulated that is known as the Berry phase for adiabatic evolution or as the Aharonov-Anandan (AA) phase for nonadiabatic evolution. We evaluate these geometric phases using Floquet theory for systems in time-dependent external fields with a focus on paths leading through a degeneracy of the eigenenergies. Contrary to expectations, the low-frequency limits of the two phases do not always coincide. This happens as the degeneracy leads to a slow convergence of the quantum states to adiabaticity, resulting in a nonzero finite or divergent contribution to the AA phase. Steering the system adiabatically through a degeneracy provides control over the geometric phase as it can cause a π shift of the Berry phase. On the other hand, we revisit an example of degeneracy crossing proposed by AA. We find that, at suitable driving frequencies, both geometric-phase definitions give the same result and the dynamical phase is zero due to the symmetry of time evolution about the point of degeneracy, providing an advantageous setup for manipulation of quantum states.