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arXiv:quant-ph/0110155 · Journal of Physics A Mathematical and General

Matter waves from quantum sources in a force field

Tobias Kramer, Christian Bracher, M Kleber

Published Sep 24, 2002• 40 citations
arXivHTMLPDF

Abstract

Localized scattering phenomena may result in the formation of stationary matter waves originating from a compact region in physical space. Mathematically, such waves are advantageously expressed in terms of quantum sources that are introduced into the Schrödinger equation. The source formalism yields direct access to the scattering wavefunction, particle distribution and total current. As an example, we study emission from three-dimensional Gaussian sources into a homogeneous force field. This model describes the behaviour of an atom laser supplied by an ideal Bose-Einstein condensate under the influence of gravity. We predict a strong dependence of the beam profile on the condensate size and the presence of interference phenomena recently observed in photodetachment experiments.

DOI
10.1088/0305-4470/35/40/301

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Localized scattering phenomena may result in the formation of stationary matter waves originating from a compact region in physical space. Mathematically, such waves are advantageously expressed in terms of quantum sources that are introduced into the Schrödinger equation. The source formalism yields direct access to the scattering wavefunction, particle distribution and total current. As an example, we study emission from three-dimensional Gaussian sources into a homogeneous force field. This model describes the behaviour of an atom laser supplied by an ideal Bose-Einstein condensate under the influence of gravity. We predict a strong dependence of the beam profile on the condensate size and the presence of interference phenomena recently observed in photodetachment experiments.
Abstract The Schrödinger equation is homogeneous reflecting the linearity of quantum mechanics. However, we show that adding a source term provides a very convenient and flexible tool for the description of processes where an electron is generated by detachment or by ionization and subsequently propagates in the presence of an external field, which may include the detaching or the ionizing field. The source term provides the mathematical formulation of a separation of the process into two steps, detachment and subsequent propagation. We present the source formalism and illustrate it by various examples: photodetachment in the presence of a uniform electric field, the conductivity of an ideal wire, scanning tunneling microscopy, the velocity map of an electron liberated from a bound state in an atom or ion by an intense laser field and its features such as the carpet pattern, and a novel realization of the time-honored double-slit experiment by means of temporal slits afforded by an atom in a few-cycle laser pulse. All examples compare the theoretical source formulation with experimental data.
A formalism is introduced to describe a number of physical processes that may break down the coherence of a matter wave over a characteristic length scale l. In a second-quantized description, an appropriate master equation for a set of bosonic "modes" (such as atoms in a lattice, in a tight-binding approximation) is derived. Two kinds of "localizing processes" are discussed in some detail and shown to lead to master equations of this general form: spontaneous emission (more precisely, light scattering), and modulation by external random potentials. Some of the dynamical consequences of these processes are considered: in particular, it is shown that they generically lead to a damping of the motion of the matter-wave currents, and may also cause a "flattening" of the density distribution of a trapped condensate at rest. The possible role played by effects of this sort in the recently-observed damping of the motion of one-dimensional Bose-Einstein condensates is briefly discussed.
We present a three-dimensional quantum-mechanical theory of radio-frequency-outcoupled atom lasers from trapped atomic gases in the presence of the gravitational force. Predictions for the total outcoupling rate as a function of the radio frequency and for the beam wave function are given. We establish a sum rule for the energy-integrated outcoupling, which leads to a separate determination of the coupling strength between the atoms and radiation field. For a noninteracting Bose-Einstein condensate analytic solutions are derived which are subsequently extended to include the effects of atomic interactions. The interactions enhance interference effects in the beam profile and modify the outcoupling rate of the atom laser. We provide a complete quantum-mechanical solution which is in line with experimental findings and allows us to determine the validity of commonly used approximative methods. We also extend the formalism to a fermionic atom laser and analyze the effect of superfluidity on the outcoupling of atoms.
Summary form only given. Bose-Einstein condensed atomic gases are a new class of quantum fluids with many novel properties. We have studied collective modes and dissipation by perturbing the condensate with focused laser beams. The scattering of photons imparts momentum to the condensate and creates excitations. Consequently, the coherence and collective behavior of the condensate can strongly affect optical properties. We used light scattering to demonstrate that the coherence length of a condensate is equal to its physical size, and obtained dramatic evidence for the presence of correlated momentum excitations in the many-body condensate wavefunction. At higher laser intensity, a new form of superradiance was found which provided a gain mechanism for the generation of directed beams of atoms and light. This process amplified matter waves in a phase coherent way.
An alternative description of quantum scattering processes rests on inhomogeneous terms amended to the Schr\"odinger equation. We detail the structure of sources that give rise to multipole scattering waves of definite angular momentum, and introduce pointlike multipole sources as their limiting case. We obtain results for ballistic scattering in an external uniform force field, where we provide analytical solutions for both the scattering waves and the integrated particle flux. Our theory directly applies to p-wave photodetachment in an electric field. Furthermore, illustrating the effects of extended sources, we predict some properties of vortex-bearing atom laser beams outcoupled from a rotating Bose-Einstein condensate under the influence of gravity.