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arXiv:1007.0703 · Physical Review A

Phase estimation with interfering Bose-Einstein-condensed atomic clouds

Jan Chwedeńczuk, Francesco Piazza, Augusto Smerzi

Published Nov 4, 2010• 11 citations
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Abstract

We investigate how to estimate from atom-position measurements the relative phase of two Bose-Einstein condensates released from a double-well potential. We demonstrate that the phase-estimation sensitivity via the fit of the average density to the interference pattern is fundamentally bounded by shot noise. The shot noise can be overcome by estimating the phase from the measurement of order $\sqrt{N}$ (or higher) correlation functions. The optimal estimation strategy, saturating the quantum Fisher information, requires the measurement of the $N$th-order correlation function. We demonstrate that a different estimation method---based on the detection of the center of mass of the interference pattern---also provides sub-shot-noise sensitivity. The implementation of both protocols, however, might be experimentally challenging.

DOI
10.1103/physreva.82.051601

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We investigate how to estimate from atom-position measurements the relative phase of two Bose-Einstein condensates released from a double-well potential. We demonstrate that the phase-estimation sensitivity via the fit of the average density to the interference pattern is fundamentally bounded by shot noise. The shot noise can be overcome by estimating the phase from the measurement of order N\sqrt{N}N​ (or higher) correlation functions. The optimal estimation strategy, saturating the quantum Fisher information, requires the measurement of the NNNth-order correlation function. We demonstrate that a different estimation method---based on the detection of the center of mass of the interference pattern---also provides sub-shot-noise sensitivity. The implementation of both protocols, however, might be experimentally challenging.
We study the measurement of the positions of atoms as a means of estimating the relative phase between two Bose–Einstein condensates. We consider N bosonic atoms released from a double-well trap, which form an interference pattern; we show that the measurement of the position of N atoms has a sensitivity that saturates the bound set by the quantum Fisher information, and allows for estimation at the Heisenberg limit of precision. Phase estimation through the measurement of the center of mass of the interference pattern can also provide sub-shot-noise sensitivity. Finally, we study the effect of an overlap of the two clouds on the estimation precision when Mach–Zehnder interferometry is performed in a double well. We find that a nonzero overlap of the clouds strongly reduces the phase sensitivity.
We study the measurement of the positions of atoms as a means of estimating the relative phase between two Bose–Einstein condensates. We consider N bosonic atoms released from a double-well trap, which form an interference pattern; we show that the measurement of the position of N atoms has a sensitivity that saturates the bound set by the quantum Fisher information, and allows for estimation at the Heisenberg limit of precision. Phase estimation through the measurement of the center of mass of the interference pattern can also provide sub-shot-noise sensitivity. Finally, we study the effect of an overlap of the two clouds on the estimation precision when Mach–Zehnder interferometry is performed in a double well. We find that a nonzero overlap of the clouds strongly reduces the phase sensitivity.
We conduct a detailed analysis of a phase estimation experiment with a Bose-Einstein condensate in a double potential well. Our goal is to confront the theoretical considerations with the conditions of a real experiment. We present a typical experimental procedure and the theoretical model. We simulate the estimation of phase between two modes in two cases: based on the full correlation function measurement for two atoms and on the center of mass position measurement for 20 atoms. We show that a lossless atom detection lets us exceed the shot noise (that is purely classical) limit of phase estimation precision with the currently available spatial resolution, while a few times higher resolution will make it possible to approach the highest precision allowed by quantum physics.
We derive the asymptotic maximum-likelihood phase estimation uncertainty for any interferometric protocol where the positions of the probe particles are measured to infer the phase, but where correlations between the particles are not accessible. First, we apply our formula to the estimation of the phase acquired in the Mach–Zehnder interferometer and recover the well-known momentum formula for the phase sensitivity. Then, we apply our results to interferometers with two spatially separated modes, which could be implemented with a Bose–Einstein condensate trapped in a double-well potential. We show that in a simple protocol which estimates the phase from an interference pattern, a sub-shot-noise phase uncertainty of up to Δ θ ∝ N −2/3 can be achieved. One important property of this estimation protocol is that its sensitivity does not depend on the value of the phase θ , contrary to the sensitivity given by the momentum formula for the Mach–Zehnder transformation. Finally, we study the experimental implementation of the above protocol in detail, by numerically simulating the full statistics as well as by considering the main sources of detection noise, and argue that the shot-noise limit could be surpassed with current technology.
A method is presented that is able to predict the probability of outcomes of snapshot measurements, such as the images of the instantaneous particle density distribution in a quantum many-body system. It is shown that a gauge-like transformation of the phase of the many-body wave function allows one to construct a probability generating functional, the Fourier transform of which with respect to the “gauge” field returns the joint probability distribution to detect any given number of particles at various locations. The method is applied to the problem of interference of two independent clouds of Bose-Einstein condensates, where the initially separated clouds with fixed boson numbers expand and the density profile image of the overlapping clouds is registered. In the limit of large particle numbers, the probability to observe a particular image of the density profile is shown to be given by a sum of partial probability distributions, each of which corresponds to a noisy image of interference of two matter waves with definite phase difference. In agreement with earlier theoretical arguments, interference fringes are, therefore, expected in any single shot measurement, the fringe pattern randomly varying from run to run. These results conform to the physical picture where the Bose-Einstein clouds are in spontaneously symmetry broken states, the hidden phases of which are revealed by the density profile measurement via the position of the interference fringes.
We present a detailed study of an atom interferometer which can be realized in a double-well potential. We assume that the interferometric phase is imprinted in the presence of coherent tunneling between the wells. We calculate the ultimate bounds for the estimation sensitivity and show how they relate to the precision of the Mach-Zehnder interferometer. The interferometer presented here allows for sub-shot-noise sensitivity when fed with spin-squeezed states with either a reduced relative population imbalance or a reduced relative phase. We also calculate the precision of the estimation from the population imbalance and show that it overcomes the shot-noise limit when the entangled squeezed states are used at the input.