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Physical Review A

Gauge-independent theory of light pressure in classical fields

Juha Javanainen

Published Oct 1, 1988• 1 citation
Publisher

Abstract

We present a quantum theory of light pressure in external classical fields, consistently incorporating both the electric and the magnetic components of the radiation. We start with the equation of motion of the gauge-independent Wigner function governing the individual charged constituents of the atom, separate the center of mass and internal degrees of freedom in the presence of the external field with the aid of an expansion that goes one order beyond the dipole approximation, and finally specialize to a near-resonantly driven two-state atom. The ensuing light-pressure theory turns out to be equivalent to the standard approach, even though in our treatment it is imperative to retain the magnetic field which does not appear explicitly in the conventional theory.

DOI
10.1103/physreva.38.3522

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We present a quantum theory of light pressure in external classical fields, consistently incorporating both the electric and the magnetic components of the radiation. We start with the equation of motion of the gauge-independent Wigner function governing the individual charged constituents of the atom, separate the center of mass and internal degrees of freedom in the presence of the external field with the aid of an expansion that goes one order beyond the dipole approximation, and finally specialize to a near-resonantly driven two-state atom. The ensuing light-pressure theory turns out to be equivalent to the standard approach, even though in our treatment it is imperative to retain the magnetic field which does not appear explicitly in the conventional theory.
A theory for a bound system of charges interacting with electromagnetic fields is developed, with special emphasis on the radiation-pressure effects imposed by the fields on the gross motion of the charges. The theory is particularly simple and transparent for the case of two opposite charges of finite mass M in the electric-dipole approximation. This enables a rigorous description to be given for the quantum electrodynamics of the problem with the various interaction terms affecting the gross motion being easily identified. Here, too, it becomes clear that, even in the dipole approximation, the conventional interaction term should be supplemented by the so-called R\"ontgen term. Besides ensuring the consistency of the theory for the overall (charges plus fields) system regarding conservation laws, the R\"ontgen term has dynamical consequences. This is established by explicit calculations of the expectation values of the mechanical momentum 〈MR\ifmmode \dot{}\else \.{}\fi{}〉 and the radiation pressure force 〈MR\ifmmode\ddot\else\textasciidieresis\fi{}〉 on a two-level atom in the dipole approximation. Results of calculations are displayed for the case of a single light beam and for counterpropagating beams. The feasibility of experimentally observing the transverse force is explored. Generalizations of the well-known friction force arising in one-dimensional optical molasses are given and discussed.
We introduce a gauge-invariant Wigner operator (GIWO) and a gauge-independent Wigner function (GIWF) that allow for both quantized and classical electromagnetic fields. If only classical fields are present, a Weyl transform analogous to the one associated with ordinary Wigner functions can be defined for any operator; in the case of quantized fields this is at least possible for Weyl-ordered functions of the position and kinetic momentum operators. We show how the evolution of the operators having a Weyl transform can be followed with the aid of the GIWF defined as the Heisenberg-picture expectation value of the GIWO, and derive for the GIWO the Heisenberg equation of motion which only involves the physical electric and magnetic fields. Some aspects of the case of entirely classical fields are discussed in more detail. (i) The GIWF in conjunction with the postulate that physical observables can be measured without referring to the gauge permits a quantum-mechanical treatment of a full experimental run without the problem of relating the measured values to the gauge-dependent density operator. (ii) A closed equation of motion for the GIWF is obtained. (iii) In the dipole approximation quantum features of the dynamics are lost. (iv) Quantum corrections to dynamics are associated with recoil effects.
We investigate the effects of quantized radiation reaction fields on the motion of a charged particle using the gauge-independent Wigner operator (GIWO) and gauge-independent Wigner function (GIWF) introduced earlier [Phys. Rev. A 33, 2913 (1986)]. To complement the equation of motion of the GIWO, the Heisenberg equations of motion of the quantized electromagnetic fields are solved within the Markov approximation. After considering the operator orderings and orders of magnitude of the radiation reaction terms, we eliminate the quantum fields from the evolution equation of the GIWO, and obtain for the GIWF a closed equation containing relaxation terms. As an example of the formalism we derive a Fokker-Planck equation (FPE) for the GIWF of a particle in a constant magnetic field. To the order \ensuremath{\Elzxh} 0^{0}0 the classical radiation damping ensues, and the first quantum correction proportional to \ensuremath{\Elzxh} emerges as diffusion. The diffusion operator turns out to be indefinite and the FPE consequently defies our attempts at a complete analysis, but we demonstrate that at least the coherent states constructed from the Landau levels exhibit a manifestly physical time evolution under the FPE. We point out that the GIWF calculated with quantized electromagnetic fields is divergent even if the fields are in the vacuum state, and suggest that the GIWF should be associated with the particle state by ignoring the quantized fields altogether.
A theory of radiation pressure and its effects on the gross motion of an aggregrate of charges is established within a canonical framework. The theory indicates the existence of radiation-pressure effects arising from an additional interaction--the R\"ontgen interaction--whose origin is the classical R\"ontgen current. A careful development of the formalism reveals the importance of distinguishing between the canonical and mechanical momenta of the gross motion of the aggregate and of incorporating the R\"ontgen-type interaction terms when calculating the gross-motion dynamics. A form of the R\"ontgen interaction is present even in the dipole approximation: an effect which has previously been ignored but which is necessary to ensure gauge invariance of the radiation-induced mechanical force. Explicit calculations of the rates of change of canonical and mechanical momenta are presented for a general atomic dipole, with specializations to a two-level atom, revealing the presence of velocity-dependent terms in a natural way. The formalism is consistent with the Minkowski form of the classical momentum density of an electromagnetic field.
We revisit the interaction of a first-quantized atomic system (consisting of two charged quantum particles) with the quantum electromagnetic field, pointing out the subtleties related to the gauge nature of electromagnetism and the effect of multipole approximations. We connect the full minimal-coupling model with the typical effective models used in quantum optics and relativistic quantum information such as the Unruh-DeWitt (UDW) model and the dipole coupling approximation. We point out in what regimes different degrees of approximation are reasonable and in what cases effective models need to be refined to capture the features of the light-matter interaction. This is particularly important when considering the center of mass (COM) of the atom as a quantum system that can be delocalized over multiple trajectories. For example, we show that the simplest UDW approximation with a quantum COM fails to capture crucial R\"ontgen terms coupling COM and internal atomic degrees of freedom with each other and the field. Finally we show how effective dipole interaction models can be covariantly prescribed for relativistically moving atoms.
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.