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Physical Review A

Diffraction of an atomic beam by a phase-fluctuating standing light wave

Benno H. W. Hendriks, Gerard Nienhuis

Published Dec 1, 1987• 4 citations
Publisher

Abstract

We consider an atomic beam at normal incidence with a standing wave of light with phase fluctuations and we evaluate the momentum distribution of the diffracted atoms after passing through the light beam. We consider the independent-increment model or the random-jump model for the phase of the light, and the random-jump model for the light frequency. In each of these three cases we distinguish the situation of independent or identical phases of the two counter-running light waves that compose the standing wave. For identical phases, the momentum distribution seems to be sensitive mainly to the spectrum of the light, whereas for independent phases higher correlation functions are also important.

DOI
10.1103/physreva.36.5615

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Physical Review A•Nov 14, 2000•6 citations
We consider an atomic beam at normal incidence with a standing wave of light with phase fluctuations and we evaluate the momentum distribution of the diffracted atoms after passing through the light beam. We consider the independent-increment model or the random-jump model for the phase of the light, and the random-jump model for the light frequency. In each of these three cases we distinguish the situation of independent or identical phases of the two counter-running light waves that compose the standing wave. For identical phases, the momentum distribution seems to be sensitive mainly to the spectrum of the light, whereas for independent phases higher correlation functions are also important.
The role of the initial-state preparation in the coherent diffraction of atoms in the field of a standing wave is considered. It is shown that the time evolution of the atomic wave packet with a Gaussian initial momentum distribution of amplitudes, due to the constructive quantum-mechanical interference, occurs practically without transformation of the initial Gaussian form. Depending on the phases of the initial momentum distribution components, the increment of the atomic momentum can vary from a maximum possible value (determined by the limiting speed of stimulated photon re-emission) to zero. Consequently, the standing wave presents an effective mirror for Gaussian atoms, the reflection angle being controllable.
The role of the initial state preparation of the coherent diffraction of atoms in the field of a standing wave is considered. It is shown that the time evolution of the atomic wave packet with a Gaussian initial momentum distribution of amplitudes, due to the constructive quantum- mechanical interference, occurs practically without transformation of the initial Gaussian form. Depending on the phases of the initial momentum distribution components, the increment of the atomic momentum can vary from the maximum possible amount (determined by the limiting speed of stimulated photon re-emission acts) to zero. Consequently, for Gaussian atoms the standing wave presents an effective mirror with controllable reflection angle.
We consider the interference of atoms caused by the wave nature of their motion. Coherently interfering beams of particles may be obtained by using the diffraction of atoms from separated standing waves resonant with the transition between the ground and excited states of the atoms. Interference arises with and without spatial separation of the diffracted atomic beams. A problem concerning two-quantum perturbation of the wave function of an atom in the ground state by a standing-wave field has been solved. Both amplitude and phase perturbation are taken into account. We show that in spite of the absence of coherence in the incident beam and a rapid decay of the coherence induced by the standing wave for scattered spatially overlapping beams, the interference can be observed for the conditions of an echo. We first consider the phenomenon of an echo under the quantum-mechanical action of light on the translational degrees of freedom. We show that owing to this phenomenon phase memory can be transferred through the ground state over unlimitedly large distances. Owing to the interference of the density harmonics, a periodic structure in the spatial distribution of the atomic density with a period of less than one wavelength of the light is localized at distances of the order of the distance between the fields. The harmonic amplitude is calculated for instantaneous excitation in the case in which the duration of the excitation is comparable with the backward Doppler linewidth in the beam and for scattering under Bragg conditions. In the latter case we show that the effective temperature of the atoms participating in the interference is less than one recoil energy.
We have observed both diffractive and diffusive regimes in the transfer of momentum to an atomic beam by a standing-wave radiation field. The momentum transfer was measured by passing a well-collimated beam of sodium atoms through a focused standing-wave laser waist. A narrow velocity distribution beam was obtained by utilizing a supersonic expansion and a two-state system was realized by optically pumping the beam prior to the interaction. The transverse momentum transferred to the atomic beam by the radiation field was measured by scanning a detector downstream from the interaction. The overall momentum resolution of the apparatus is less than the momentum of a single photon. The transition from diffractive to diffusive behavior is associated with an increase in the average number of spontaneous decays occurring during the interaction. In the diffractive regime, spontaneous decay is insignificant and momentum transfer proceeds by pairs of absorption/stimulated emission processes. This produces momentum exchange quantized in even multiples of photon momentum, the so-called Kapitza-Dirac effect; the deflection patterns show oscillatory structure. In the diffusive regime, spontaneous decay is important and causes fluctuations in the radiative forces. In addition, the recoil distribution associated with spontaneous decay results in a washing out of the diffraction peaks associated with stimulated processes, resulting in a smooth deflection pattern.
Atoms undergo two types of momentum changes as they pass across a near-resonant standing wave. First, their transverse momentum spreads. Second, if the atoms cross the field at an angle, they undergo Bragg scattering from the periodic structure provided by the wave. Classically, this behavior can be understood in terms of traveling waves: the atom catalyzes the transfer of momentum from one traveling wave to the other, thereby changing its own transverse momentum. In the quantum regime, however, there is an essential difference between scattering by two running waves and by a standing wave. With running waves, we can in principle know which of the two waves has exchanged a unit of momentum with the atom. In contrast, a standing wave is an inseparable quantum unit whose average momentum remains zero at all times. This unity is imposed by the fixed mirrors that establish the standing wave. These act as infinite sinks of momentum. Quantum mechanics forbids one even in principle to determine, via a field measurement from which traveling wave the atom picked up its momentum. To demonstrate this point, we discuss the diffraction and deflection of an atom in both a standing wave and a superposition of two counter-propagating running waves.
We consider the dynamical Bragg reflection of an atomic matter wave from a finite optical lattice; the resulting energy dependence of the transmittivity allows us to filter out the velocities of the incident atoms. In particular, we show how the coherence length of the incident beam can be inferred from the sharpness of the transmittivity as a function of the lattice intensity. For incident frequencies well inside the reflecting window, the interference of incident and reflected matter waves gives rise to an oscillatory density profile in front of the lattice which can be observed by means of light diffraction. The angular width of the diffracted peaks also provides information on the coherence length of the incident atomic beam.