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Physical Review A

Bragg scattering of an atomic beam by a standing laser wave with time-periodic amplitude modulation

Andrey R. Kolovsky, H. J. Korsch

Published May 1, 1998• 4 citations
Publisher

Abstract

We study the scattering of an atomic beam slowly passing through a standing laser wave with an amplitude periodically modulated in time. The formal analysis of the problem involves the theory of interacting nonlinear resonances, developed earlier in the field of quantum chaos. Two different diffraction regimes are distinguished: a quasiregular regime, where the atoms are Bragg-like scattered by a primary or secondary nonlinear resonance, and a chaotic regime, where the scattering can be regarded as (quasi)random.

DOI
10.1103/physreva.57.3763

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We study the scattering of an atomic beam slowly passing through a standing laser wave with an amplitude periodically modulated in time. The formal analysis of the problem involves the theory of interacting nonlinear resonances, developed earlier in the field of quantum chaos. Two different diffraction regimes are distinguished: a quasiregular regime, where the atoms are Bragg-like scattered by a primary or secondary nonlinear resonance, and a chaotic regime, where the scattering can be regarded as (quasi)random.
Bragg diffraction of atoms at thick standing light waves requires that the wave-matching condition is fulfilled. This usually means that the atomic beam crosses the light wave exactly at the Bragg angle. Nevertheless, our experiments also demonstrate Bragg diffraction at detuned angles if the amplitude of the standing light wave is temporally modulated with an appropriate frequency. If, on the other hand, the phase of the light wave is modulated no diffraction is observed. Both modulation processes produce frequency sidebands which set up `slowly travelling standing waves' in front of a retro-reflection mirror. Atoms are diffracted at these `almost standing' light waves in a similar way to photons at the travelling sound waves in an acousto-optic modulator. The frequency of the diffracted de Broglie waves is assumed to be shifted by the intensity modulation frequency. The different results using amplitude- and phase-modulated light waves are due to interference between the diffraction contributions of the individual frequency sidebands contained in the standing light wave.
A vector quantum field theory is developed to describe the interaction of an ultracold atomic ensemble with a laser field. The ultracold atomic ensemble is composed of either bosonic atoms or fermionic atoms. The photon exchanges between different ultracold atoms result in a long-range interatom interaction. Under appropriate conditions, such a many-body interaction leads to an effective ``Kerr-type'' nonlinearity of atomic waves. In terms of the nonlinearity, we construct a general formalism of nonlinear atom optics for an ultracold atomic beam. Applying the formalism to the diffraction of an ultracold atomic beam composed of bosonic atoms by a standing-wave laser field, we find that the many-body nonlinearity induces self-phase and cross-phase modulation of diffracted atomic waves. As a result, a standing-wave laser acts as a nonlinear atomic grating which diffracts atoms in the same way as a nonlinear periodic medium diffracts photons.
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 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.
Slow atoms from a magneto-optical trap are used to study diffraction from a near-resonant standing-wave light field. Long interaction times make it possible to observe unidirectional Bragg scattering. In particular, the Pendellösung interference effect between two resonant momentum states is demonstrated for first- and second-order diffraction. The oscillation frequency of second-order scattering shows a transition from a four-photon–type process at low light intensity to a two-photon–type process at higher intensity. The data are in quantitative agreement with a theoretical description which takes into account lowest-energy momentum states.