Squeezing Enhancement in Lossy Multi-Path Atom Interferometers
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AbstractThis 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.Featured image: Space-time diagram of a MZI with two input and two output ports, each described by field operators $\hat{\psi}_i^{\mathrm{in(out)}}(p)$ in momentum space for port $i=1,2$. Light-pulse beam splitter and mirror operations based on Bragg diffraction are inherently lossy due to scattering into undesired diffraction orders (dashed lines)Popular summaryQuantum entanglement between particles can drastically enhance the sensitivity of gravitational acceleration measurements in atom interferometers. However, entangled states are more susceptible to losses and noise than unentangled states which reduces the obtained sensitivity. In this paper, we introduce a mathematical framework that describes the sensitivity for a lossy atom interferometer when using entangled input states generated via one-axis twisting. We apply this formalism to simulate a realistic Bragg Mach-Zehnder interferometer to find the optimal degree of entanglement and the expected sensitivity while taking into account the multi-path nature of the Bragg diffraction process.► BibTeX data@article{Gunther2026squeezing, doi = {10.22331/q-2026-06-01-2122}, url = {https://doi.org/10.22331/q-2026-06-01-2122}, title = {Squeezing {E}nhancement in {L}ossy {M}ulti-{P}ath {A}tom {I}nterferometers}, author = {G{\"{u}}nther, Julian and Kirsten-Siem{\ss{}}, Jan-Niclas and Gaaloul, Naceur and Hammerer, Klemens}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2122}, month = jun, year = {2026} }► References [1] D. Ganapathy, W. Jia, M. 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Thompson, Entanglement-enhanced matter-wave interferometry in a high-finesse cavity, Nature 610, 472–477 (2022). https://doi.org/10.1038/s41586-022-05197-9 [11] C. Cassens, B. Meyer-Hoppe, E. Rasel, and C. Klempt, Entanglement-Enhanced Atomic Gravimeter, Phys. Rev. X 15, 011029 (2025). https://doi.org/10.1103/PhysRevX.15.011029 [12] L. Morel, Z. Yao, P. Cladé, and S. Guellati-Khélifa, Determination of the fine-structure constant with an accuracy of 81 parts per trillion, Nature 588, 61 (2020). https://doi.org/10.1038/s41586-020-2964-7 [13] R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller, Measurement of the fine-structure constant as a test of the standard model, Science 360, 191–195 (2018). https://doi.org/10.1126/science.aap7706 [14] P. Asenbaum, C. Overstreet, T. Kovachy, D. D. Brown, J. M. Hogan, and M. A. Kasevich, Phase shift in an atom interferometer due to spacetime curvature across its wave function, Phys. Rev. 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Hammerer, Ramsey interferometry with generalized one-axis twisting echoes, Quantum 4, 268 (2020). https://doi.org/10.22331/q-2020-05-15-268 [48] D. Pfeiffer, M. Dietrich, P. Schach, G. Birkl, and E. Giese, Dichroic mirror pulses for optimized higher-order atomic Bragg diffraction Phys. Rev. Res. 7, L012028 (2025). https://doi.org/10.1103/PhysRevResearch.7.L012028 [49] G. Louie, Z. Chen, T. Deshpande, and T. Kovachy, Robust atom optics for Bragg atom interferometry, New Journal of Physics 25, 083017 (2023). https://doi.org/10.1088/1367-2630/aceb15Cited byCould not fetch Crossref cited-by data during last attempt 2026-06-01 07:52:45: Could not fetch cited-by data for 10.22331/q-2026-06-01-2122 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-06-01 07:52:45: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThis 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.Featured image: Space-time diagram of a MZI with two input and two output ports, each described by field operators $\hat{\psi}_i^{\mathrm{in(out)}}(p)$ in momentum space for port $i=1,2$. Light-pulse beam splitter and mirror operations based on Bragg diffraction are inherently lossy due to scattering into undesired diffraction orders (dashed lines)Popular summaryQuantum entanglement between particles can drastically enhance the sensitivity of gravitational acceleration measurements in atom interferometers. However, entangled states are more susceptible to losses and noise than unentangled states which reduces the obtained sensitivity. In this paper, we introduce a mathematical framework that describes the sensitivity for a lossy atom interferometer when using entangled input states generated via one-axis twisting. We apply this formalism to simulate a realistic Bragg Mach-Zehnder interferometer to find the optimal degree of entanglement and the expected sensitivity while taking into account the multi-path nature of the Bragg diffraction process.► BibTeX data@article{Gunther2026squeezing, doi = {10.22331/q-2026-06-01-2122}, url = {https://doi.org/10.22331/q-2026-06-01-2122}, title = {Squeezing {E}nhancement in {L}ossy {M}ulti-{P}ath {A}tom {I}nterferometers}, author = {G{\"{u}}nther, Julian and Kirsten-Siem{\ss{}}, Jan-Niclas and Gaaloul, Naceur and Hammerer, Klemens}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2122}, month = jun, year = {2026} }► References [1] D. Ganapathy, W. Jia, M. Nakano, et al. (LIGO O4 Detector Collaboration), Broadband quantum enhancement of the ligo detectors with frequency-dependent squeezing, Phys. Rev. X 13, 041021 (2023). https://doi.org/10.1103/PhysRevX.13.041021 [2] S. Colombo, E. Pedrozo-Peñafiel, and V. Vuletić, Entanglement-enhanced optical atomic clocks, Applied Physics Letters 121, 210502 (2022). https://doi.org/10.1063/5.0121372 [3] S. S. Szigeti, O. Hosten, and S. A. Haine, Improving cold-atom sensors with quantum entanglement: Prospects and challenges, Applied Physics Letters 118, 140501 (2021). https://doi.org/10.1063/5.0050235 [4] S. S. Szigeti, S. P. Nolan, J. D. Close, and S. A. Haine, High-precision quantum-enhanced gravimetry with a Bose-Einstein Condensate, Phys. Rev. Lett. 125, 100402 (2020). https://doi.org/10.1103/PhysRevLett.125.100402 [5] R. Corgier, N. Gaaloul, A. Smerzi, and L. Pezzè, Delta-kick squeezing, Phys. Rev. Lett. 127, 183401 (2021a). https://doi.org/10.1103/PhysRevLett.127.183401 [6] R. Corgier, L. Pezzè, and A. Smerzi, Nonlinear Bragg interferometer with a trapped bose-einstein condensate, Phys. Rev. A 103, L061301 (2021b). https://doi.org/10.1103/PhysRevA.103.L061301 [7] L. Salvi, N. Poli, V. Vuletić, and G. M. Tino, Squeezing on momentum states for atom interferometry, Phys. Rev. Lett. 120, 033601 (2018). https://doi.org/10.1103/PhysRevLett.120.033601 [8] F. Anders, A. Idel, P. Feldmann, D. Bondarenko, S. Loriani, K. Lange, J. Peise, M. Gersemann, B. Meyer-Hoppe, S. Abend, N. Gaaloul, C. Schubert, D. Schlippert, L. Santos, E. Rasel, and C. Klempt, Momentum entanglement for atom interferometry, Phys. Rev. Lett. 127, 140402 (2021). https://doi.org/10.1103/PhysRevLett.127.140402 [9] B. K. Malia, Y. Wu, J. Martínez-Rincón, and M. A. Kasevich, Distributed quantum sensing with mode-entangled spin-squeezed atomic states, Nature 612, 661–665 (2022). https://doi.org/10.1038/s41586-022-05363-z [10] G. P. Greve, C. Luo, B. Wu, and J. K. Thompson, Entanglement-enhanced matter-wave interferometry in a high-finesse cavity, Nature 610, 472–477 (2022). https://doi.org/10.1038/s41586-022-05197-9 [11] C. Cassens, B. Meyer-Hoppe, E. Rasel, and C. Klempt, Entanglement-Enhanced Atomic Gravimeter, Phys. Rev. X 15, 011029 (2025). https://doi.org/10.1103/PhysRevX.15.011029 [12] L. Morel, Z. Yao, P. Cladé, and S. Guellati-Khélifa, Determination of the fine-structure constant with an accuracy of 81 parts per trillion, Nature 588, 61 (2020). https://doi.org/10.1038/s41586-020-2964-7 [13] R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller, Measurement of the fine-structure constant as a test of the standard model, Science 360, 191–195 (2018). https://doi.org/10.1126/science.aap7706 [14] P. Asenbaum, C. Overstreet, T. Kovachy, D. D. Brown, J. M. Hogan, and M. A. Kasevich, Phase shift in an atom interferometer due to spacetime curvature across its wave function, Phys. Rev. Lett. 118, 183602 (2017). https://doi.org/10.1103/PhysRevLett.118.183602 [15] S.-w. Chiow, J. Williams, N. Yu, and H. Müller, Gravity-gradient suppression in spaceborne atomic tests of the equivalence principle, Phys. Rev. A 95, 021603 (2017). https://doi.org/10.1103/PhysRevA.95.021603 [16] G. W. Biedermann, X. Wu, L. Deslauriers, S. Roy, C. Mahadeswaraswamy, and M. A. Kasevich, Testing gravity with cold-atom interferometers, Phys. Rev. A 91, 033629 (2015). https://doi.org/10.1103/PhysRevA.91.033629 [17] J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, Sensitive absolute-gravity gradiometry using atom interferometry, Phys. Rev. A 65, 033608 (2002). https://doi.org/10.1103/PhysRevA.65.033608 [18] S. Dimopoulos, P. W. Graham, J. M. Hogan, and M. A. Kasevich, Testing general relativity with atom interferometry, Phys. Rev. Lett. 98, 111102 (2007). https://doi.org/10.1103/PhysRevLett.98.111102 [19] S. Dimopoulos, P. W. Graham, J. M. Hogan, and M. A. Kasevich, General relativistic effects in atom interferometry, Phys. Rev. D 78, 042003 (2008a). https://doi.org/10.1103/PhysRevD.78.042003 [20] C. Ufrecht, F. Di Pumpo, A. Friedrich, A. Roura, C. Schubert, D. Schlippert, E. M. Rasel, W. P. Schleich, and E. Giese, Atom-interferometric test of the universality of gravitational redshift and free fall, Phys. Rev. Res. 2, 043240 (2020). https://doi.org/10.1103/PhysRevResearch.2.043240 [21] P. Asenbaum, C. Overstreet, M. Kim, J. Curti, and M. A. Kasevich, Atom-Interferometric Test of the Equivalence Principle at the $10^{-12}$ Level, Physical Review Letters 125, 191101 (2020). https://doi.org/10.1103/PhysRevLett.125.191101 [22] M. Werner, P. K. Schwartz, J.-N. Kirsten-Siemß, N. Gaaloul, D. Giulini, and K. Hammerer, Atom interferometers in weakly curved spacetimes using Bragg diffraction and bloch oscillations, Phys. Rev. D 109, 022008 (2024). https://doi.org/10.1103/PhysRevD.109.022008 [23] S. Abend, B. Allard, I. Alonso, et al., Terrestrial very-long-baseline atom interferometry: Workshop summary, AVS Quantum Science 6, 10.1116/5.0185291 (2024). https://doi.org/10.1116/5.0185291 [24] M. Abe, P. Adamson, M. Borcean, et al., Matter-wave atomic gradiometer interferometric sensor (magis-100), Quantum Science and Technology 6, 044003 (2021). https://doi.org/10.1088/2058-9565/abf719 [25] L. Badurina, E. Bentine, D. Blas, et al., Aion: an atom interferometer observatory and network, Journal of Cosmology and Astroparticle Physics 2020 (05), 011–011. https://doi.org/10.1088/1475-7516/2020/05/011 [26] B. Canuel, S. Abend, P. Amaro-Seoane, et al., Elgar—a european laboratory for gravitation and atom-interferometric research, Classical and Quantum Gravity 37, 225017 (2020). https://doi.org/10.1088/1361-6382/aba80e [27] G. M. Tino, A. Bassi, G. 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