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arXiv:0807.4066 · Europhysics Letters (EPL)

Geometric potentials in quantum optics: A semi-classical interpretation

Marc Cheneau, Steffen Patrick Rath, Tarik Yefsah, Kenneth Günter, Gediminas Juzeliūnas, Jean Dalibard

Published Aug 22, 2008• 37 citations
arXivHTMLPDF

Abstract

We propose a semi-classical interpretation of the geometric scalar and vector potentials that arise due to Berry's phase when an atom moves slowly in a light field.Starting from the full quantum Hamiltonian, we turn to a classical description of the atomic centre-of-mass motion while still treating the internal degrees of freedom as quantum variables.We show that the scalar potential can be identified as the kinetic energy of an atomic micro-motion caused by quantum fluctuations of the radiative force, and that the Lorentz-type force appears as a result of the motion-induced perturbation of the internal atomic state.For a specific configuration involving two counter-propagating Gaussian laser beams, we relate the geometric forces to the radiation pressure and dipole forces known from quantum optics.The simple physical pictures provided by the present analysis may help for the design and the implementation of novel geometric forces.

DOI
10.1209/0295-5075/83/60001

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We propose a semi-classical interpretation of the geometric scalar and vector potentials that arise due to Berry's phase when an atom moves slowly in a light field.Starting from the full quantum Hamiltonian, we turn to a classical description of the atomic centre-of-mass motion while still treating the internal degrees of freedom as quantum variables.We show that the scalar potential can be identified as the kinetic energy of an atomic micro-motion caused by quantum fluctuations of the radiative force, and that the Lorentz-type force appears as a result of the motion-induced perturbation of the internal atomic state.For a specific configuration involving two counter-propagating Gaussian laser beams, we relate the geometric forces to the radiation pressure and dipole forces known from quantum optics.The simple physical pictures provided by the present analysis may help for the design and the implementation of novel geometric forces.
It has been known for some time that the interaction of laser light with a free atom can give rise to electromagnetic pressure forces that act on its centre of mass. These forces have been extensively examined both theoretically and experimentally: see, for example Metcalf and van der Straten (5.1) or Adams and Riis (5.2). A simple model of a two-level atom subject to a plane electromagnetic wave may be shown to give rise to two kinds of force acting on the atom centre of mass. These are, a dissipative force arising from the absorption of the light by the atom and its subsequent spontaneous emission in a random direction and a dipole force arising from the non-uniformity of the field distribution which can attract the atom to the regions of intense field. These forces underpin the manipulation of atoms by lasers in a variety of beam configurations. The dissipative force has been exploited in the Doppler cooling of the atomic motion (5.3, 5.4) and the dipole force used for trapping (5.5), while a combination of these effects has led to the realisation of Bose-Einstein condensation (5.6). These forces are, of course, closely related to the ones discussed in Section 4.
We consider the dynamics of a particle carrying a magnetic moment in a strong magnetic field whose direction varies slowly in space. In particular, we discuss the geometric Lorentz-type and electric-type forces that were discovered in studies of Berry's phase. We show that the Lorentz-type force felt by the particle is caused by a small misalignment of the magnetic moment with respect to the magnetic field while the electric-type force is the time average of a strong oscillatory force induced by the precession of the magnetic moment.
The usual multipolar Hamiltonian for atom-light interaction features a nonrelativistic moving atom interacting with electromagnetic fields which inherently follow Lorentzian symmetry. This combination can lead to situations where atoms appear to experience a friction force, when in fact they only change their internal mass-energy due to the emission or absorption of a photon. Unfortunately, the simple Galilean description of the atom's motion is not sufficient to distinguish between a change in momentum due to acceleration and a change in momentum due to a change in internal mass-energy. In this work we show how a low-order relativistic correction can be included in the multipolar atom-light Hamiltonian. We also give examples how this affects the most basic mechanical interactions between atoms and photons.
Quantum motion of atoms in light fields is described on the basis of adiabatic internal states. Forces arise due to the spatial variation of these states, which is determined by the electric field polarization. In a dark state, these are the only forces present. They are described by a geometric vector and a scalar potential. We give analytical expressions for the geometric potentials in the dark states of a driven $\stackrel{\ensuremath{\rightarrow}}{j}j\ensuremath{-}1$ transition and the dark state in the $\stackrel{\ensuremath{\rightarrow}}{1}1$ system, for arbitrary electromagnetic fields. For systems with velocity selective trapping states, the scalar geometric potential is inversely proportional to the field intensity squared. When the field has nodes the potential diverges. In one dimension, this constitutes an exact realization of the Kronig-Penney model.
To examine dynamical effects on the geometric phase at the semi-classical level, we apply the technique of the time-dependent variational principle to simple models which typically exhibit the geometric phase effects discovered by Berry, namely, the Jaynes-Cummings models and the generalized harmonic oscillator model. Parameters are themselves dynamical variables in our models. Instead of integrating out the “fast” variables at the first stage, we derive the canonical equations of motion both for “fast” and “slow” varibales without the adiabatic assumption. Using special solutions of the canonical equations, we requantize the systems by Einstein's condition, in which Berry's phase appearing as a correction for the Bohr-Sommerfeld condition is modified by the dynamical effects related to the slow degree of freedom.
The mechanical action of light on atoms is nowadays a tool used ubiquitously in cold atom physics. In the semiclassical regime where the atomic motion is treated classically, the computation of the mean force acting on a two-level atom requires in the most general case numerical approaches. Here we show that this problem can be tackled in a pure analytical way. We provide an analytical yet simple expression of the mean force that holds in the most general case where the atom is simultaneously exposed to an arbitrary number of lasers with arbitrary intensities, wave vectors, and phases. This yields a novel tool for engineering the mechanical action of light on single atoms.