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arXiv:2007.03306 · PRX Quantum

Characterization of Errors in Interferometry with Entangled Atoms

Constantin Brif, Brandon Ruzic, Grant W. Biedermann

Published Sep 3, 2020• 11 citations
arXivHTMLPDF

Abstract

Recent progress in generating entangled spin states of neutral atoms provides opportunities to advance quantum sensing technology.In particular, entanglement can enhance the performance of accelerometers and gravimeters based on light-pulse atom interferometry.We study the effects of error sources that may limit the sensitivity of such devices, including errors in the preparation of the initial entangled state, imperfections in the laser pulses, momentum spread of the initial atomic wave packet, measurement errors, spontaneous emission, and atom loss.We determine that, for each of these errors, the expectation value of the parity operator has the general form = 0 cos(N φ), where φ is the interferometer phase and N is the number of atoms prepared in the maximally entangled Greenberger-Horne-Zeilinger state.Correspondingly, the minimum phase uncertainty has the general form φ = ( 0 N ) -1 .Each error manifests itself through a reduction of the amplitude of the parity oscillations, 0 , below the ideal value of 0 = 1.For each of the errors, we derive an analytic result that expresses the dependence of 0 on error parameter(s) and N , and also obtain a simplified approximate expression valid when the error is small.Based on the performed analysis, entanglement-enhanced atom interferometry appears to be feasible with existing experimental capabilities.

DOI
10.1103/prxquantum.1.010306

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Recent progress in generating entangled spin states of neutral atoms provides opportunities to advance quantum sensing technology.In particular, entanglement can enhance the performance of accelerometers and gravimeters based on light-pulse atom interferometry.We study the effects of error sources that may limit the sensitivity of such devices, including errors in the preparation of the initial entangled state, imperfections in the laser pulses, momentum spread of the initial atomic wave packet, measurement errors, spontaneous emission, and atom loss.We determine that, for each of these errors, the expectation value of the parity operator has the general form = 0 cos(N φ), where φ is the interferometer phase and N is the number of atoms prepared in the maximally entangled Greenberger-Horne-Zeilinger state.Correspondingly, the minimum phase uncertainty has the general form φ = ( 0 N ) -1 .Each error manifests itself through a reduction of the amplitude of the parity oscillations, 0 , below the ideal value of 0 = 1.For each of the errors, we derive an analytic result that expresses the dependence of 0 on error parameter(s) and N , and also obtain a simplified approximate expression valid when the error is small.Based on the performed analysis, entanglement-enhanced atom interferometry appears to be feasible with existing experimental capabilities.
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.
Compared to light interferometers, the flux in cold-atom interferometers is low and the associated shot noise large. Sensitivities beyond these limitations require the preparation of entangled atoms in different momentum modes. Here, we demonstrate a source of entangled atoms that is compatible with state-of-the-art interferometers. Entanglement is transferred from the spin degree of freedom of a Bose-Einstein condensate to well-separated momentum modes, witnessed by a squeezing parameter of -3.1(8) dB. Entanglement-enhanced atom interferometers open up unprecedented sensitivities for quantum gradiometers or gravitational wave detectors.
Compared to light interferometers, the flux in cold-atom interferometers is low and the associated shot noise is large. Sensitivities beyond these limitations require the preparation of entangled atoms in different momentum modes. Here, we demonstrate a source of entangled atoms that is compatible with state-of-the-art interferometers. Entanglement is transferred from the spin degree of freedom of a Bose-Einstein condensate to well-separated momentum modes, witnessed by a squeezing parameter of -3.1(8) dB. Entanglement-enhanced atom interferometers promise unprecedented sensitivities for quantum gradiometers or gravitational wave detectors.
We develop a framework for describing the effects of systematic state-preparation error in quantum-enhanced atom interferometry on sensing performance. We do this in the context of both spin-squeezed and non-Gaussian states for the two-axis-twisting (TAT), one-axis-twisting (OAT), and twist-and-turn (TNT) state-preparation schemes, and derive general conditions for robustness and susceptibility of quantum states to state-preparation error. In the spin-squeezing regime, we find that OAT is more susceptible to state-preparation error than TAT due to its parameter-dependent phase-space rotation. In the non-Gaussian regime, we find that OAT is robust to state-preparation errors, which can be explained by a small ratio of off-diagonal to diagonal elements in its Fisher-covariance matrix. In contrast, TNT does not exhibit this robustness. We find that the single-parameter unbiased estimators that are habitually used in quantum-enhanced atom interferometry are not always optimal, and that there may be occasions where biased estimators, or two-parameter unbiased estimators, lead to lower net error.
Interferometers based on ultracold atoms enable an absolute measurement of inertial forces with unprecedented precision. However, their resolution is fundamentally restricted by quantum fluctuations. Improved resolutions with entangled or squeezed atoms were demonstrated in internal-state measurements for thermal and quantum-degenerate atoms and, recently, for momentum-state interferometers with laser-cooled atoms. Here, we present a gravimeter based on Bose-Einstein condensates with a sensitivity of − 1.7 − 0.5 + 0.4 dB beyond the standard quantum limit. Interferometry with Bose-Einstein condensates combined with delta-kick collimation minimizes atom loss in and improves scalability of the interferometer to very-long-baseline atom interferometers.
This paper explores the sensitivity gains afforded by spin-squeezed states in atom interferometry, in particular using Bragg diffraction. We introduce a generalised input-output formalism that accurately describes realistic, non-unitary interferometers, including losses due to velocity selectivity and scattering into undesired momentum states. This formalism is applied to evaluate the performance of one-axis twisted spin-squeezed states in improving phase sensitivity. Our results show that by carefully optimising the parameters of the Bragg beam splitters and controlling the degree of squeezing, it is possible to improve the sensitivity of the interferometer by several dB with respect to the standard quantum limit despite realistic levels of losses in light pulse operations. However, the analysis also highlights the challenges associated with achieving these improvements in practice, most notably the impact of finite temperature on the benefits of entanglement. The results suggest ways of optimising interferometric setups to exploit quantum entanglement under realistic conditions, thereby contributing to advances in precision metrology with atom interferometers.