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Physical Review A

Analysis of atomic mirrors based on light-induced forces

Sze Meng Tan, Daniel F. Walls

Published Aug 1, 1994• 18 citations
Publisher

Abstract

We describe a numerical method for solving the problem of the interaction of a monoenergetic atomic beam represented by a plane wave with a classical light field at glancing incidence. In the approximation where spontaneous emission is simply treated as a loss, this leads to a vector Schr\"odinger equation which is solved using a matrix technique. This allows the analysis of arbitrary optical intensity profiles rather than the rectangular profile previously considered by Zhang and Walls [Phys. Rev. Lett. 68, 3287 (1992)]. Selected numerical results show that the method describes a variety of phenomena including state-selective reflection and velocity-tuned (doppleron) resonances.

DOI
10.1103/physreva.50.1561

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We describe a numerical method for solving the problem of the interaction of a monoenergetic atomic beam represented by a plane wave with a classical light field at glancing incidence. In the approximation where spontaneous emission is simply treated as a loss, this leads to a vector Schr"odinger equation which is solved using a matrix technique. This allows the analysis of arbitrary optical intensity profiles rather than the rectangular profile previously considered by Zhang and Walls [Phys. Rev. Lett. 68, 3287 (1992)]. Selected numerical results show that the method describes a variety of phenomena including state-selective reflection and velocity-tuned (doppleron) resonances.
We numerically solve the time-dependent Schr\"odinger equation for a two-level atom interacting with an evanescent light field. The atom may be reflected or diffracted. Using the experimental parameter values we quantitatively model the evanescent field dopplerons (velocity-tuned resonances) observed by Stenlake et al. [Phys. Rev. A 49, 16 (1994)]. Besides successfully modeling the experiment, our approach provides complementary insights to the usual solution of the time-independent Schr\"odinger equation. We neglect spontaneous emission.
We present a model to describe the behavior of an atomic beam in a focusing experiment and discuss its implementation. Such modeling is relevant to the development of atom lithography. For simplicity, the light-atom interaction may be treated as a conserving potential (neglecting spontaneous emission) so that the Schrödinger equation is sufficient for a description of the problem. We start from a two-dimensional time-dependent Schrödinger equation that is reduced to a one-dimensional equation by a paraxial approximation, and then solved iteratively via Padé approximation. The solution is implemented in an interactive routine with Matlab(5.x) to compute and visualize the atomic wave packet development as it propagates through the laser light optical potential. We present examples of our implementation, starting with cases that have known analytic solutions such as propagation of the atomic wave packet through free space and through a parabolic potential. We also present a case that does not have a simple analytic solution, namely the propagation of the atomic wave packet through a focusing optical potential, which has a Gaussian profile in the propagation direction and a parabolic profile in the perpendicular direction. © 1998 American Institute of Physics.
The diffraction of a coherent, stationary, beam of atoms from a standing wave of light is considered. Particular attention is paid to long interaction times for which the motion of the atoms inside the light field must be taken into account. This is the regime of dynamical diffraction, and is treated within the framework of a set of differential difference equations due to Raman and Nath. The popular Raman-Nath approximation, in which the kinetic energy imparted to the atoms by the light wave is ignored, is no longer valid and the full equations must be solved. When the interaction between the atoms and the light is very weak then few beams are produced. To describe spontaneous emission from the atom one can invoke a complex potential as a model for dissipation. If the atom beam is obliquely incident one finds anomalously high transmission close to the Bragg angles which is the physical manifestation of degeneracies in non-Hermitian matrices. The classical limit is approached when the atom-light interaction is strong, producing many beams. The classical dynamics (geometrical rays) reveal that the farfield intensity pattern is dominated by caustics. These proliferate with increasing interaction distance. Two methods are presented for analytically treating the semiclassical limit of the wave mechanics. The first begins with W.K.B. solutions to the continuised Raman-Nath equations. Mapping onto parabolic cylinder functions removes the spurious divergences of the W.K.B. solutions, giving eigenvectors which are uniformly valid for all scattering angles and may be summed to give the wavefunction. The second method transforms the original W.K.B. eigensum, using the Poisson summation formula, giving a new sum, each term of which has classical significance. This method shows that wave theory decorates the classical caustics with an Airy function intensity profile, as predicted by catastrophe theory.
We study theoretically the diffraction of atoms by a standing-wave light field. Conceptually the text is divided into two parts. In the first part we deal with the coherent interaction only. We use a band theoretical approach to describe the laser-atom interaction, and to characterize two-beam resonances (Dopplerons and Bragg resonances). These two-beam resonances are major candidates to develop effective atom beam splitters. We study the interaction with classical light fields as well as with quantum fields. In the second part we take incoherent processes, i.e. spontaneous emission, into account and present a numerical scheme for the solution of the generalized optical Bloch equations. This scheme is based on the split-operator technique, and we use it to study numerically the influence of spontaneous emission on the diffraction process. We compare our results with recent experimental data, and investigate the impact of spontaneous emission on the performance of two-beam resonances.
In this paper we consider the effects of spontaneous emission on the problem of atomic reflection from an optical evanescent travelling wave. The atoms are assumed to be travelling so slowly that the external degrees of freedom of the atom need to be treated quantum mechanically. Spontaneous emission is modelled using a fully quantum mechanical model based on the quantum Monte Carlo technique which includes the effects of atomic recoil. The results of the simulation are compared with a semi-classical Monte Carlo simulation in which the atomic motion is treated classically and the internal state of the atom is assumed to follow the light-induced quasipotentials adiabatically between spontaneous emission events. The spontaneous emission leads to a broadening of the momentum distribution of the outgoing wave packet and mechanisms for this broadening are discussed.
This poster presents a first analysis of the laws of reflection of cold atoms by an evanescent wave atomic mirror. We demonstrate that when the laser beam is tuned to the blue of the atomic transition, i.e, for a positive frequency detuning compared to the frequency resonance Δ = ωL−ωa, there is a reflection without a contact with the surface of the prism (no adsorption). The condition for an internal total reflection is θ>θc = arcsin (1/n) where θ is the incidence angle of the laser wave on the internal face of the prism and n is its refraction index. The evanescent wave is parallel to the surface and propagates over few wavelengths only for θ = θc. The atomic trajectory is such as the incidence angle φi is different from the reflection angle φr (non respect of the first law of reflection in photonic optics). We’ll have to demonstrate the reflection law in atom optics and give numerical examples as an illustration [1]. At its interaction with the evanescent wave, the atom is submitted to a pressure radiation force Fx parallel to the surface of separation which communicates a quantity of motion in the direction of the x axis. It is also submitted to a dipolar force Fy perpendicular to the separation surface responsible of the repulsion of cold atoms in the case where we neglect the Van der Waals forces. We define R as the ratio of two forces Fx and Fy. Their expressions, function of the detuning, are given by the Ehrenfest theorem and the Bloch equations [2]. When the ratio R tends towards zero (spontaneous emission negligible for a detuning important compared to natural linewidth of the excited state), the first law of Descartes reflection is thus satisfied idem φi = φr Thus, it is simple to demonstrate that the atomic trajectories obey to the general reflection law: tg φr = tg φi−2R where R is independent of the spatial component. Its expression is given by the ratio of the velocity transfers Δvx to Δvy. The case of retro‐reflection seen in quasi normal incidence and for values of the ratio R different from zero (φi = −φr) is also examined. In the case where the value of the ratio R is non null, we may demonstrate that the incidence plan is different from the reflection plan (second law of Descartes reflection).