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Physical Review A

Geometric-phase effects in laser dynamics

Vladislav Yu. Toronov, Vladimir Leonardovich Derbov

Published Jul 1, 1994• 10 citations
Publisher

Abstract

We show that such phenomena of laser dynamics as mean-phase-slope jumps and temporal phase jumps at resonance between the cavity and spectral line frequencies are intrinsically connected with the topology of attractors in the space of rays and can be interpreted as the manifestations of the geometric-phase properties of the evolution operator.

DOI
10.1103/physreva.50.878

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We show that such phenomena of laser dynamics as mean-phase-slope jumps and temporal phase jumps at resonance between the cavity and spectral line frequencies are intrinsically connected with the topology of attractors in the space of rays and can be interpreted as the manifestations of the geometric-phase properties of the evolution operator.
Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.
We investigate geometric-phase control in resonantly driven two-level atoms exposed to near-degenerate dichromatic laser pulses. In contrast to conventional two-color schemes based on widely separated frequencies, the closely spaced frequency components generate a slowly varying beating envelope that repeatedly reverses the pseudo-spin rotation on the Bloch sphere. This enables coherent control of the geometric phase accumulated during Rabi dynamics and strongly modifies the resulting photoelectron spectra. Using an exactly solvable model for flat-top pulse envelopes, we derive the essential-state dynamics analytically and analyze photoionization induced both by an auxiliary field and by the dichromatic driving field itself. We show that the formation of Autler--Townes doublets can be interpreted in terms of destructive interference associated with geometric phases acquired during completed pseudo-spin rotations. Beyond the canonical Autler--Townes regime, beat-induced reversal dynamics lead to qualitatively different spectral structures, including re-emergent central peaks and higher-order sidebands. These effects originate from the nonuniform temporal evolution induced by the beating envelope, which modifies the balance between positive and negative excitation amplitudes. Our results establish beating-field control as a route toward engineering geometric-phase interference in ultrafast light--matter interactions and suggest broader applications in coherent control of atoms, molecules, and x-ray-driven systems.
We propose a cohomological modelling schema of quantum state spaces and their connectivity structures in relation to the formulation of global geometric phase phenomena. In the course of this schema, we introduce the notion of Hermitian differential line sheaves or unitary rays and classify their gauge equivalence classes in terms of a global differential invariant given by the de Rham cohomology class of the curvature. Furthermore, we formulate and interpret physically the curvature recognition integrality theorem for unitary rays. Using this recognition theorem, we define the notion of a quantum spectral beam and show that it has an affine space structure with structure group given by the characters of the fundamental group.