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arXiv:quant-ph/0507061 · Physical Review A

Differential atom interferometry beyond the standard quantum limit

K. Eckert, P. Hyllus, Dagmar Bruß, Uffe V. Poulsen, Maciej Lewenstein, C. Jentsch, T. Müller, Ernst Maria Rasel, Wolfgang A. Ertmer

Published Jan 18, 2006• 34 citations
arXivHTMLPDF

Abstract

We analyze methods designed to go beyond the standard quantum limit for a class of atomic interferometers, where the quantity of interest is the difference of phase shifts obtained by two independent atomic ensembles. An example is given by an atomic Sagnac interferometer, where for two ensembles propagating in opposite directions in the interferometer this phase difference encodes the angular velocity of the experimental setup. We discuss methods of separately or jointly squeezing observables of the two atomic ensembles, and compare in detail the advantages and drawbacks of such schemes. In particular, we show that the method of joint squeezing may improve the variance by up to a factor of 2. We take into account fluctuations of the number of atoms in both the preparation and the measurement stage, and obtain bounds on the difference between the numbers of atoms in the two ensembles, as well as on the detection efficiency, which have to be fulfilled in order to surpass the standard quantum limit. Under realistic conditions, the performance of both schemes can be improved significantly by reading out the phase difference via a quantum nondemolition measurement. Finally, we discuss a scheme using macroscopically entangled ensembles.

DOI
10.1103/physreva.73.013814

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We analyze methods designed to go beyond the standard quantum limit for a class of atomic interferometers, where the quantity of interest is the difference of phase shifts obtained by two independent atomic ensembles. An example is given by an atomic Sagnac interferometer, where for two ensembles propagating in opposite directions in the interferometer this phase difference encodes the angular velocity of the experimental setup. We discuss methods of separately or jointly squeezing observables of the two atomic ensembles, and compare in detail the advantages and drawbacks of such schemes. In particular, we show that the method of joint squeezing may improve the variance by up to a factor of 2. We take into account fluctuations of the number of atoms in both the preparation and the measurement stage, and obtain bounds on the difference between the numbers of atoms in the two ensembles, as well as on the detection efficiency, which have to be fulfilled in order to surpass the standard quantum limit. Under realistic conditions, the performance of both schemes can be improved significantly by reading out the phase difference via a quantum nondemolition measurement. Finally, we discuss a scheme using macroscopically entangled ensembles.
techniques from quantum optics, which are realized here through the tunneling interaction among three trapped states.The final part of this thesis focuses on a different regime of the manipulation of neutral atoms, namely on the use of ensembles containing a large number of atoms for the purpose of atom interferometry.More precisely, differential interferometers are studied, where the quantity of interest is encoded in the difference of two phases obtained by two independent atomic ensembles.Methods using separately or jointly squeezed states of the two atomic ensembles are discussed, which allow to surpass the standard quantum limit to the resolution of the interferometer.It is shown that non-local squeezing gives an optimal method, provided that certain conditions on the preparation and read-out of the atomic ensembles are met.Under realistic conditions the performance of both schemes can be further improved by reading out the phase difference via a quantum non-demolition measurement.
Quantum Sensors, like atom interferometers (AI), can be employed for high-precision measurements of inertial forces, including their application as gravimeters, gradiometers, accelerometers, and gyroscopes. Their measurement principle relies on ultracold atoms that are prepared in quantum-mechanical superposition states in external degrees of freedom. These states can be prepared by a momentum transfer of a Raman laser. Then the superposition state senses the effect of an inertial force, which induce a corresponding relative phase. The phase is read out by a final coupling which converts the interferometric phase into a atom number difference between the two states. The difference provides an estimate of the interferometric phase and the corresponding quantity of interest. The quantum mechanical noise of the atomic ensemble cause a fundamental uncertainty of this estimation, which I analyze for generic AIs. For small atomic densities, the quantum phase noise of the ensemble limits the interferometric sensitivity. For large densities, quantum number fluctuations generate density fluctuations, which generates phase noise. I show that these two competing effects result in an optimal atom number with a maximal interferometer resolution. Squeezed atomic samples allow for a reduction of the quantum noise of one quantity at the expense of an increased noise along of a conjugate quantity. Phase and number are such quantities which obey to a variant of Heisenberg’s uncertainty principle. Neither phase nor number squeezing can improve the maximal interferometer resolution. As one main result of this thesis, I show how an optimal squeezing in between number and phase squeezing, allows for a fundamental improvement. I evaluate possible experimental paths to implement the proposed protocol. Concepts for a squeezing-enhanced operation of external-degree AIs have not yet been demonstrated. I propose and implement an atomic gravimeter, which is designed to accept spin-squeezed atomic states as input states. The interferometer is designed such that the interferometer couplings are performed in spin space, while the phase accumulation is performed in momentum states. For this interferometer, the squeezed input can be directly obtained from spin dynamics in spinor Bose-Einstein condensates. The main noise contributions in the experiment are analyzed, which results in a factor of 84 above the relevant quantum limit, preventing a squeezing enhancement so far. I outline a suppression of the main noise source, uncontrolled AC Stark shift on the squeezed mode and propose future important applications, including test of spontaneous collapse theories and an improvement of large-scale, high-precision gradiometers.
We show that the accuracy of atomic interferometry can be improved by using QND measurements of the atomic populations at the inputs to the interferometer. The accuracy of such a scheme surpasses the standard quantum limit of phase measurement δϕ SQL = 1/√ N and could reach the Heisenberg limit δϕ ∼ 1/ N . We propose to perform QND measurements of atomic populations with an off-resonant laser field. The conditions necessary for this kind of QND measurement could be fulfilled in a variety of ways with current experimental techniques, including magneto-optical traps and atomic cells.
This chapter describes the use of squeezed states to improve the sensitivity in the estimation of the relative phase accumulated in a Bose--Einstein condensate (BEC) beyond the standard quantum limit. We begin by examining the specific example of a two-component BEC, where atoms in two different hyperfine levels interact with each other. By evolving an initial superposition of states under an effective Hamiltonian, a squeezed state is realized. Next, the squeezing is visualized using the Q-function, giving the mean spin direction and shape of the BEC distribution. We discuss the basic operation of Ramsey interferometry, which is a classic example of two-slit atom diffraction. We then calculate the error in estimating the relative phase shift using the error propagation formula that allows us to define the squeezing parameter. This measures the amount of metrological gain using squeezed states over unsqueezed states. We give an example of the metrological gain in the estimation of the relative phase shift using a squeezed state of the two-component BEC as an input state to an atom Mach--Zehnder interferometer. The Fisher information is used to find which rotation axis gives the optimal information from a given initial state. Finally, we parameterize the nonlinear phase per atom responsible for squeezing and discuss how this parameter is controlled in different experiments.
Atom interferometers are reaching sensitivities fundamentally constrained by quantum fluctuations. A main challenge is to integrate entanglement into quantum sensing protocols to enhance precision while ensuring robustness against noise and systematics. Here, we theoretically investigate differential phase measurements with two atom interferometers using spin-squeezed states, accounting for common-mode phase noise spanning the full 2π2π2π range. We estimate the differential signal using model-free ellipse fitting, a robust method requiring no device calibration and resilient to additional noise sources. Our results show that spin-squeezing enables sensitivities below the standard quantum limit. Specifically, we identify optimal squeezed states that minimize the differential phase variance, scaling as N−2/3N^{-2/3}N−2/3, while eliminating bias inherent in ellipse fitting methods. We benchmark our protocol against the Cramér-Rao bound and compare it with hybrid methods that incorporate auxiliary classical sensors. Our findings provide a pathway to robust and high-precision atom interferometry, in realistic noisy environments and using readily available states and estimation methods.
Every measurement of the population in an uncorrelated ensemble of two-level systems is limited by what is known as the quantum projection noise limit. Here, we present quantum-projection-noise-limited performance of a Ramsey-type interferometer using freely propagating coherent atoms. The experimental setup is based on an electro-optic modulator in an inherently stable Sagnac interferometer, optically coupling the two interfering atomic states via a two-photon Raman transition. Going beyond the quantum projection noise limit requires the use of reduced quantum uncertainty (squeezed) states. The experiment described demonstrates atom interferometry at the fundamental noise level and allows the observation of possible squeezing effects in an atom laser, potentially leading to improved sensitivity in atom interferometers.