The Lorentz-Dirac equation for a nonrelativistic particle in a blackbody radiation field is solved by a stationary trial solution formed by a slowly varying part depending on the coordinate, which is supposed to be separable, plus a rapidly fluctuating part with zero average and constant variance, depending essentially on time only. Then an equation is obtained for the slowly varying part, including the stochastic field parameters, and this dependence is evaluated explicitly for special cases. The diffusion equation in coordinate space is established for this system in a general stationary state. It is shown that the condition for the probability density P(chi,t) to be representable in the same form as a quantum-mechanical pure state, is a certain relation between the diffusion coefficient in any stationary state and the value of the mean squared fluctuation of the momentum.
We investigate the semiclassical Kepler/Coulomb problem using the classical constants of the motion in the framework of Nelson's stochastic mechanics. This is done by considering the eigenvalue relations for a family of coherent states (known as the atomic elliptic states) whose wave functions are concentrated on the elliptical orbit corresponding to the associated classical problem. We show that these eigenvalue relations lead to identities for the semiclassical energy, angular momentum and Hamilton–Lenz–Runge vectors in the elliptical case. These identities are then extended to include the cases of circular, parabolic and hyperbolic motions. We show that in all cases the semiclassical wave function is determined by our identities and so our identities can be seen as defining a semiclassical Kepler/Coulomb problem. The results are interpreted in terms of two dynamical systems: one a complex valued solution to the classical mechanics for a Coulomb potential and the other the drift field for a semiclassical Nelson diffusion.
The momentum diffusion coefficient, as a function of position and velocity, is discussed for two-state atoms travelling through a standing-wave laser field. Equations leading to the force and diffusion coefficient have been obtained by Minogin. The authors present an alternative derivation, based on moments of the Wigner function and give numerical solutions in terms of matrix continued fractions. In strong fields, the discussion is supplemented by a derivation of the diffusion coefficient within the dressed-atom basis, and a picture of the diffusion mechanism emerges which is quite different from the one at low intensities. Calculated forces and diffusion coefficients are used in a Fokker-Planck equation to obtain stationary atomic velocity distributions.
We assume that an N -level moving atom interacts with an ( N − 1) mode electromagnetic cavity field. The model includes the detuning and the intensity-dependent coupling. The constants of motion and the wave function for the atomic system is obtained. Some statistical aspects of the considered atomic system such as momentum increment and momentum diffusion have been calculated. The influence of the initial mean photon number, the detuning parameter and the intensity dependent coupling on the momentum increment are investigated when the atom is initially in excited state and the field is in a squeezed state. It is found that the momentum of the atom is unchanged for large initial mean photon number.
The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c
)2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/
c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/
c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2
e2/3
c3)(d〈v〉/dt
)2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.
A semiclassical model of the force and momentum diffusion on a point particle is used to describe the motion of a two-level atom strongly coupled to a single Gaussian cavity mode. The effects of the momentum diffusion on the motion of an atom in a cavity are investigated in a regime similar to that of the experiment performed by Mabuchi et al. [Opt. Lett. 21, 1393 (1996)]. It is found that a slow atom quickly develops significant velocities along the cavity axis. The limited bandwidth in the experiment of Mabuchi et al. means that the full intensity signal due to atomic motion in the standing wave is filtered leading to the apparently smaller velocities observed. It is shown that a negative detuning of the laser and cavity from the atomic resonance would lead to nonzero dipole forces and significantly reduced velocities along the standing wave. An analysis of the intensity signal with a larger bandwidth is proposed, which would track the velocity of the atom along the cavity axis. These results are compared with a Monte Carlo wave-function simulation similar to that used to treat Doppler cooling.