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arXiv:1006.3767 · Physical Review A

Atomic population distribution in the output ports of cold-atom interferometers with optical splitting and recombination

Ebubechukwu O. Ilo-Okeke, Alex A. Zozulya

Published Nov 3, 2010• 18 citations
arXivHTMLPDF

Abstract

Cold-atom interferometers with optical splitting and recombination use off-resonant laser beams to split a cloud of Bose-Einstein condensate (BEC) into two clouds that travel along different paths and are then recombined again using optical beams. After the recombination, the BEC in general populates both the cloud at rest and the moving clouds. Measuring a relative number of atoms in each of these clouds yields information about the relative phase shift accumulated by the atoms in the two moving clouds during the interferometric cycle. We derive the expression for the probability of finding any given number of atoms in each of the clouds, discuss features of the probability density distribution, analyze its dependence on the relative accumulated phase shift as a function of the strength of the interatomic interactions, and compare our results with experiment.

DOI
10.1103/physreva.82.053603

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Cold-atom interferometers with optical splitting and recombination use off-resonant laser beams to split a cloud of Bose-Einstein condensate (BEC) into two clouds that travel along different paths and are then recombined again using optical beams. After the recombination, the BEC in general populates both the cloud at rest and the moving clouds. Measuring a relative number of atoms in each of these clouds yields information about the relative phase shift accumulated by the atoms in the two moving clouds during the interferometric cycle. We derive the expression for the probability of finding any given number of atoms in each of the clouds, discuss features of the probability density distribution, analyze its dependence on the relative accumulated phase shift as a function of the strength of the interatomic interactions, and compare our results with experiment.
This chapter describes the Bose--Einstein condensate (BEC) interferometry. We first introduce the optical interferometer, briefly discussing the role of fringe contrast in observing interference patterns. Next, we examine a BEC interferometer in a Michelson geometry that consists of a BEC in a trapping potential well. The BEC cloud in the potential well is put into a linear superposition of two clouds that travel along different paths in the trap and are recombined using the same light beams. By studying the population of atoms in the different motional states of the BEC clouds that emerge after recombination, we obtain information about the relative phase shift accumulated by the two traveling superposed condensates during the interferometry. We then characterize the atom distribution that is found in the output of the interferometer by finding the probability density distribution and calculating the features of the probability density such as the mean, the variance, and the interference fringe contrast. Finally, we parameterize the effect of two-body collisional interactions on the observed interference fringes in a way that can be directly controlled in an experiment.
An atom interferometer is a sensitive device that has potential for many useful applications. Atoms are sensitive to electromagnetic fields due to their electric and magnetic moments and their mass allows them to be deflected in a gravitational field, thereby making them attractive for measuring inertial forces. The narrow momentum distribution of Bose-Einstein condensate (BEC) is a great asset in realizing portable atom interferometers. An example is a guided-wave atom interferometer that uses a confining potential to guide the motion of the condensate. Despite the promise of guided-wave atom interferometry with BEC, spatial phase and phase diffusion limit the contrast of the interference fringes. The control of these phases is required for successful development of a BEC-based guided-wave atom interferometer. This thesis analyses the guided-wave atom interferometer, where an atomic BEC cloud at the center of a confining potential is split into two clouds that move along different arms of the interferometer. The clouds accumulate relative phase due to the environment, spatially inhomogeneous trapping potential and atom-atom interactions within the condensate. At the end of the interferometric cycle, the clouds are recombined producing a cloud at rest and moving clouds. The number of atoms in the clouds that emerge depends on the relative phase accumulated by the clouds during propagation. This is investigated by deriving an expression for the probability of finding any given number of atoms in the clouds that emerge after recombination. Characteristic features like mean, standard deviation and cross-correlation function of the probability density distribution are calculated and the contrast of the interference fringes is optimized. This thesis found that optimum contrast is achieved through the control of total population of atoms in the condensate, trap frequencies, s-wave scattering length, and the duration of the interferometric cycle.
Atom interferometers and gyroscopes are highly sensitive atom-optical devices which are capable to measure inertial, gravitational, electric, and magnetic fields and to sense rotations. Theoretically, the signal-to-noise ratio of atomic gyroscopes is about a hundred billion times more than that of their optical counterparts for the same particle flux and the enclosed area. Ultra cold atoms from a Bose-Einstein condensate (BEC) can easily be controlled and coherently manipulated on small chips by laser pulses. Atom-optical devices will therefore play a significant role in fundamental research, precision measurements, and navigation systems. In BEC-based atom interferometers, a BEC in a trap is split by using laser pulses, the split clouds are allowed to evolve, they are reflected, and then recombined by laser pulses to observe interference. The split clouds accumulate spatial phase because of the trap and the nonlinearity caused by atom-atom interactions. A velocity mismatch due to reflection laser pulses also introduces a phase gradient across each cloud. These factors contribute to spatial relative phase between the clouds at recombination, causing the loss of contrast of the interference fringes. The main objective of this dissertation is to study the dynamics of a split condensate in atom Michelson interferometers, investigate the effect of trap frequencies, nonlinearity, and the velocity mismatch on the contrast, and to obtain the best theoretical limit of performance in terms of the experimental parameters: trap frequencies, number of atoms, and the velocity imparted to the clouds by the splitting laser pulses.
Cold-atom interferometers use guiding potentials that split the wave function of the Bose-Einstein condensate and then recombine it. We present a theoretical analysis of the wave-function recombination instability that is due to the weak nonlinearity of the condensate. It is most pronounced when the accumulated phase difference between the arms of the interferometer is close to an odd multiple of $\ensuremath{\pi}$ and consists in exponential amplification of the weak ground state mode by the strong first excited mode. The instability exists for both trapped-atom and beam interferometers.
An atom interferometer using Bose-Einstein condensates has been implemented and is used to measure the dynamic polarizability of 87Rb. An off-resonant standing-wave laser beam is used to manipulate the motion of atomic wavepackets in order to implement splitting and reflection operations. It is demonstrated that these operations can be performed with near unity fidelity. Interferometer times of up to 72 ms as well as arm separations of up to 0.42 mm are achieved experimentally. The primary limitation for guided-wave atom interferometers with a large arm separation is found to be the effect of the confining potential along the waveguide axis. Such a potential causes a spatially varying phase shift on the wavepackets as they propagate in the interferometer. Different parts of the cloud therefore interfere with different phases, leading to a reduction of the interferometer contrast. A model of the phase shifts on the wavepackets agrees well with the experimental data. In a different scheme in which the wavepackets oscillate freely in the confining potential, these phase shifts completely cancel. Interference is observed at times up to 0.9 s in this case, but with a fluctuating overall phase of the output, which is attributed to mechanical instabilities. Finally, the dynamic polarizability of 87Rb is measured. A well-calibrated laser beam is applied to one atomic packet and not the other, inducing a differential phase shift. This technique requires relatively low laser intensity and works for arbitrary optical frequencies. For offresonant light, the ac polarizability is obtained with a statistical accuracy of 3% and a calibration uncertainty of 6%. On resonance, the dispersion-shaped behavior of the Stark shift is observed, but with a broadened linewidth that is attributed to multiple and collective light scattering effects. The resulting nonlinearity may prove useful for the production and control of squeezed quantum states. In general, this work also demonstrates the applicability of condensate interferometry to practical measurements.