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arXiv:1703.06864 · Atoms

A Wigner Function Approach to Coherence in a Talbot-Lau Interferometer

Eric Imhof, James Stickney, Matthew B. Squires

Published Jun 22, 2016• 1 citation
arXivHTMLPDF

Abstract

Using a thermal gas, we model the signal of a trapped interferometer. This interferometer uses two short laser pulses, separated by time T, which act as a phase grating for the matter waves. Near time 2 T , there is an echo in the cloud’s density due to the Talbot-Lau effect. Our model uses the Wigner function approach and includes a weak residual harmonic trap. The analysis shows that the residual potential limits the interferometer’s visibility, shifts the echo time of the interferometer, and alters its time dependence. Loss of visibility can be mitigated by optimizing the initial trap frequency just before the interferometer cycle begins.

DOI
10.3390/atoms4020018

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Using a thermal gas, we model the signal of a trapped interferometer. This interferometer uses two short laser pulses, separated by time T, which act as a phase grating for the matter waves. Near time 2 T , there is an echo in the cloud’s density due to the Talbot-Lau effect. Our model uses the Wigner function approach and includes a weak residual harmonic trap. The analysis shows that the residual potential limits the interferometer’s visibility, shifts the echo time of the interferometer, and alters its time dependence. Loss of visibility can be mitigated by optimizing the initial trap frequency just before the interferometer cycle begins.
Weyl functions conveniently describe the evolution of wave coherences in periodic or quadratic potentials. In this work we use Weyl functions to study the ``Talbot-Lau effect'' in a time-domain matter-wave interferometer. A ``displacement diagram'' is introduced to analyze and calculate the matter-wave interference for an atomic cloud in a quadratic potential that interacts with a sequence of short optical standing wave pulses producing an atomic grating echo. Unlike previous treatments, this new approach allows the atomic ensemble to have an arbitrary initial phase-space distribution, and the standing wave grating vectors to span three dimensions. Several examples are discussed to illustrate the convenience of the diagrammatic technique including the following: a two-dimensional Talbot-Lau effect, the shift in the echo time and the recoil phase for the interferometer perturbed by a quadratic potential; and the realization of a time-domain ``Lau effect'' using a pulsed harmonic potential. The diagrammatic technique is applicable to diffraction gratings with arbitrary grating transmission functions. We conclude the paper with a general discussion on the Weyl function representations of matter-wave coherence, and relate the conservation of matter-wave coherence with the conservation of purity that distinguishes decoherence effects from dephasing effects.
We present a theoretical framework to describe the effects of decoherence on matter waves in Talbot-Lau interferometry. Using a Wigner description of the stationary beam the loss of interference contrast can be calculated in closed form. The formulation includes both the decohering coupling to the environment and the coherent interaction with the grating walls. It facilitates the quantitative distinction of genuine quantum interference from the expectations of classical mechanics. We provide realistic microscopic descriptions of the experimentally relevant interactions in terms of the bulk properties of the particles and show that the treatment is equivalent to solving the corresponding master equation in paraxial approximation.
This thesis describes theoretical work carried out in connection with the high-mass matter-wave interference experiments at the University of Vienna. Its scope ranges from the simulation and analysis of experimental data to the proposal of a trapped interference scheme applicable to a variety of matter-wave platforms. The first main result of the thesis is the phase-space description of a near-field interferometer with any number of gratings. This theory is used to analyze a four-grating Talbot-Lau interferometer, in which passive compensation of low-frequency vibrations and of the Coriolis force are expected. The compensation is numerically confirmed and its effectiveness is compared to existing techniques. The second main result is the demonstration of Bragg diffraction of complex molecules on optical gratings, to which the author contributed the basic experimental design, simulations, and the interpretation of the data. This demonstration is a first step towards applying large momentum transfer coherent manipulation techniques to the interference of heavy molecules and clusters. The third main result is a new interference scheme for matter waves in toroidal confinement. The beam splitting and recombination in this scheme rely on the free evolution in the waveguide and require no interaction with the matter waves. The scheme is within reach of state-of-the-art experiments with Bose-Einstein condensates, where it could be used to measure weak atom-atom interactions or magnetic field gradients.
We propose and demonstrate a momentum filter for atomic gas-based on a designed Talbot–Lau interferometer. It consists of two identical optical standing-wave pulses separated by a delay equal to odd multiples of the half Talbot time. The one-dimensional momentum width along the long direction of a cigar-shaped condensate is rapidly and greatly purified to a minimum, which corresponds to the ground state energy of the confining trap in our experiment. We find good agreement between theoretical analysis and experimental results. The filter is also effective for non-condensed cold atoms and could be applied widely.
We have used an echo-type atom interferometer that manipulates laser-cooled atoms in a single ground state to investigate the effect of light scattering from pulsed and continuous-wave light. The interferometer uses two off-resonant standing-wave pulses applied at times t=0t=0t=0 and t=Tt=Tt=T to diffract and recombine momentum states separated by $2\ensuremath{\hbar}k$ at t=2Tt=2Tt=2T. Matter wave interference is associated with the formation of a density grating with period $\ensuremath{\lambda}/2$ in the vicinity of this echo time. The grating contrast is measured by recording the intensity of coherently backscattered light. The interferometer is perturbed by an additional pulse applied at $t=2T\ensuremath{-}\ensuremath{\delta}T$ or by continuous-wave background light. If the additional pulse is a standing wave, the momentum states interfering at t=2Tt=2Tt=2T are displaced and the grating contrast can be completely recovered due to constructive interference. In this case, the contrast shows a periodic modulation at the atomic recoil frequency as a function of $\ensuremath{\delta}T$. In a recent work, it was shown that the atomic recoil frequency can be measured easily and precisely when using coherence functions to model the signal shape. This paper provides an alternative description of the signal shape through an analytical calculation of echo formation in the presence of an additional standing-wave pulse. Using this treatment, it is possible to model the effects of spontaneous emission and spatial profile of the laser beam on the signal shape. Additionally, the theory predicts scaling laws as a function of the pulse area and the number of additional standing-wave pulses. These scaling laws are investigated experimentally and can be exploited to improve precision measurements of the atomic recoil frequency. We also show that coherence functions can be used to make a direct measurement of the populations of momentum states associated with the ground state under conditions where the Doppler-broadened velocity distribution of the sample is much larger than the recoil velocity. These measurements are consistent with Monte Carlo wave-function simulations. If the additional pulse is a traveling wave, we find that the grating contrast measured as a function of $\ensuremath{\delta}T$ can be modeled by a quasiperiodic coherence function as in previous experiments that utilized atomic beams. In this work, we investigate the dependence of the photon scattering rate on the intensity and detuning of the traveling wave. We also study the effects of perturbing the interferometer with continuous-wave light and find that the dependence of the photon scattering rate on the intensity and detuning of the perturbing field is consistent with expectations.