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arXiv:1203.5357 · Physical Review A

Dynamics of parametric matter-wave amplification

Robert Bücker, Ulrich Hohenester, Tarik Berrada, Sandrine van Frank, A. Perrin, Stephanie Manz, Thomas Betz, Julian Grond, Thorsten Schumm, Jörg Schmiedmayer

Published Jul 23, 2012• 17 citations
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Abstract

We develop a model for parametric amplification, based on a density matrix approach, which naturally accounts for the peculiarities arising for matter waves: significant depletion and explicit time dependence of the source state population, long interaction times, and spatial dynamics of the amplified modes. We apply our model to explain the details in an experimental study on twin-atom beam emission from a one-dimensional degenerate Bose gas.

DOI
10.1103/physreva.86.013638

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We develop a model for parametric amplification, based on a density matrix approach, which naturally accounts for the peculiarities arising for matter waves: significant depletion and explicit time dependence of the source state population, long interaction times, and spatial dynamics of the amplified modes. We apply our model to explain the details in an experimental study on twin-atom beam emission from a one-dimensional degenerate Bose gas.
The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the PPP representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and nnn-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative PPP representation for the density operator for a single mode of oscillation.
Spin-changing collisions may lead under proper conditions to the parametric amplification of matter waves in spinor Bose-Einstein condensates. Magnetic dipole-dipole interactions, although typically very weak in alkali-metal atoms, are shown to play a very relevant role in the amplification process. We show that these interactions may lead to a strong dependence of the amplification dynamics on the angle between the trap axis and the magnetic-field orientation. We analyze as well the important role played by magnetic-field gradients, which also modify strongly the amplification process. Magnetic-field gradients, hence, must be carefully controlled in future experiments, in order to observe clearly the effects of the dipolar interactions in the amplification dynamics.
Bose-Einstein condensates in designed potentials represent a model system for the investigation of nonlinear matter wave dynamics, since both the nonlinearity and the potentials can be very well controlled experimentally. The nonlinearity arising from the atom-atom interaction can be adjusted through the atom number. The potentials can be readily realized by employing the atom-light interaction leading to potentials which are proportional to light intensity. Since the techniques for manipulation of light are very well developed almost arbitrary potentials can be realized. In our experiments we employ the most simple light intensity configurations such as focused laser beam and standing light waves
The aim of this thesis is to find a model that describes the generation of twinatom beams emitted from a quasi Bose-Einstein condensate.Our work is motivated by an experiment performed by a group from the Institute of Atomic and Subatomic Physics at the Vienna University of Technology and was published in [1].Starting point for the generation of twin-atom beams is a one-dimensional, degenerate Bose gas trapped in the ground state of an elongated magnetic potential.This quasi-condensate is transferred to the first radial excited state by shaking the trap.Subsequently, pairs of atoms with opposite momenta are emitted from the source and propagate in longitudinal direction.To model this behavior, we use a density matrix approach which is first applied to a simplified two-mode model of the emission process.We probe different factorization schemes, a pseudospin-operator ansatz, and a coupled-cluster approach, and compare the results to the one obtained by the exact solution of the full Schrödinger equation.Then we use the first-order approximation within the density matrix approach and generalize it to a multi-mode model.We allow several twin-modes to be populated and also imply the pumping-process in our simulation, finding good agreement with the experiment.
The analysis of the quantum-mechanical parametric amplifier begun in the preceding paper is extended to describe the joint quantum state of the two interacting modes of oscillation of the system. Equations are derived for the time evolution of the Schr\"odinger density operator. Comparison with classical theory is made by describing the state of the quantum-mechanical system by means of the PPP representation. The time evolution of the weight function $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ for this representation, and of the corresponding Wigner distribution function $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$, is found. A decoupled form for the equations of motion for the field variables is exhibited which simplifies the solutions for these functions. It is shown that the amplification process gives rise to a correlation between the two mode amplitudes which becomes exceedingly strong with increasing time. The relation of this correlation to the time development of the functions $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ and $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ is discussed.
We develop a Green's function based mean-field theory for coherent mixing of matter- and light-waves. To demonstrate the utility of this approach, we analyse a co-propagating Raman matter-wave amplifier. We find that for a given laser intensity, a significantly faster amplification process can be achieved employing resonant rather than off-resonance driving. The ratio of the matter-wave gain to atom loss-rate due to spontaneous emission is given by the optical depth of the sample, and is the same both on- and off-resonance. Furthermore, we show that for short-times, the single-mode approximation for the matter-waves gives exact agreement with the full spatial dynamics. For long times, the off-resonant case shows suppressed amplification due to a spatially inhomogenous AC Stark shift associated with laser depletion. This suppression is absent on-resonance, where the AC Stark shift is absent.