Correlation-function approach to the momentum diffusion of atoms moving in standing waves

Abstract

We consider the momentum diffusion of atoms moving in a standing-wave laser field. We show how the correlation-function approach as originally applied to atoms at rest can be generalized to derive the velocity dependence of the momentum diffusion coefficient in standing waves, and that it gives results in agreement with the transport-equation approach to laser cooling. As an example we apply our calculations to determine the achievements of laser cooling in intense fields where cooling may occur around a nonvanishing velocity. Here we obtain temperatures which are 30'% lower than the corresponding minimum obtained around zero velocity. Our explicit calculations involve usage of the optical Bloch equations, the quantum regression theorem, and the matrix continued-fraction method.

Laser cooling and magneto-optical trapping of molecules analyzed using optical Bloch equations and the Fokker-Planck-Kramers equation

“…However, this method is not straightforward to apply to our case since, in the presence of several laser beams of different frequencies, there is no longer a stationary solution to the OBEs. A later paper [25] provides a method for calculating the first order velocity dependence of the diffusion tensor for a simpler system, which could potentially be adapted to our case. Instead, we choose to approximate the diffusion tensor by a simple expression that ignores stimulated emission and only considers the random nature of spontaneous emission and the corresponding absorption events.…”

Cooling of a single atom in an optical trap inside a resonator

“…In order to find friction and diffusion coefficients we apply a simple procedure, which yields these coefficients at zero velocity: this is sufficient for our purposes as the atom we are interested in, Cs, is relatively heavy. More precisely, the relevant dimensionless parameters determining the velocity dependence of friction and diffusion coefficients are kv/Γ and kv/κ (see for instance [20]), and both are very small in all our simulations. In particular, Γ/k ∼ 4.3 m/s and κ/k ∼ 3.4 m/s, while velocities in the trapping regime we are interested in (where atoms are localized in wells at low temperatures for times ≫ κ −1 , Γ −1 ) are around the Doppler limit velocity…”

Trapping of single atoms with single photons in cavity QED

“…Thus, as assumed in earlier work, calculating the quasiclassical motion of the atom in a cavity field only requires that the force and its correlation function be evaluated for the full atom-cavity master equation. Such prior treatments assumed that the atom is motionless; however, they can be extended to atoms moving at some velocity under the same conditions [30,31]. The diffusion coefficients may be found by first calculating the correlation functions via the quantum regression theorem and numerical integration, or directly via matrix-continued fraction techniques [30,31].…”

“…Such prior treatments assumed that the atom is motionless; however, they can be extended to atoms moving at some velocity under the same conditions [30,31]. The diffusion coefficients may be found by first calculating the correlation functions via the quantum regression theorem and numerical integration, or directly via matrix-continued fraction techniques [30,31]. A matrix-continued fraction calculation requires that the field mode be periodic, and as such it only works along the standing-wave axis of the cavity mode.…”