We consider a large class of Ramsey interferometry protocols which are enhanced by squeezing and un-squeezing operations before and after a phase signal is imprinted on the collective spin of $N$ particles. We report an analytical optimization for any given particle number and strengths of (un-)squeezing. These results can be applied even when experimentally relevant decoherence processes during the squeezing and un-squeezing interactions are included. Noise between the two interactions is however not considered in this work. This provides a generalized characterization of squeezing echo protocols, recovering a number of known quantum metrological protocols as local sensitivity maxima, thereby proving their optimality. We discover a single new protocol. Its sensitivity enhancement relies on a double inversion of squeezing. In the general class of echo protocols, the newly found over-un-twisting protocol is singled out due to its Heisenberg scaling even at strong collective dephasing.
Many theoretical proposals were developed, which suggest the use of entangled probes in order to improve upon the quantum projection noise limit e.g. by squeezing quantum fluctuations. However, the experimental implementation of these schemes faces major hurdles due to the often extremely demanding requirements on the manipulation and measurement of the particles and due to increased susceptibility to noise and decoherence of highly entangled states. Notably, some of the best metrological measurements have been achieved in schemes where well-controlled squeezing interactions were used several times, before and after imprinting a signal. The application of such an ‘echo’ was essential to encode the signal in a robust and readily measurable quantity and thus allow an improvement already at small particle numbers and with imperfect control capabilities.
In our work we give for the first time a theory of generalized squeezing echoes in Ramsey interferometry. We show how to perform an analytical optimization of the geometrical parameters involved in the problem. Our methods allow us to attain a complete overview of all possible echo protocols within this large variational class, and we recover many prominent known proposals as local sensitivity maxima. In some cases our treatment proves optimality of these protocols which could not be claimed so far. Most importantly, we are able to show that there is exactly one more, so-far unknown protocol. Remarkably, this new protocol achieves a sensitivity at the Heisenberg limit and turns out to be very robust with respect to the experimentally most relevant decoherence processes.
 A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt. Optical atomic clocks. Rev. Mod. Phys., 87: 637–701, Jun 2015. https://doi.org/10.1103/RevModPhys.87.637.
 M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark. Search for new physics with atoms and molecules. Rev. Mod. Phys., 90: 025008, Jun 2018. https://doi.org/10.1103/RevModPhys.90.025008.
 V. Giovannetti, S. Lloyd, and L. Maccone. Quantum metrology. Phys. Rev. Lett., 96: 010401, Jan 2006. https://doi.org/10.1103/PhysRevLett.96.010401.
 L. Barsotti, J. Harms, and R. Schnabel. Squeezed vacuum states of light for gravitational wave detectors. Reports on Progress in Physics, 82 (1): 016905, Dec 2018. https://doi.org/10.1088/1361-6633/aab906.
 M. Tse et al. Quantum-enhanced advanced LIGO detectors in the era of gravitational-wave astronomy. Phys. Rev. Lett., 123: 231107, Dec 2019. https://doi.org/10.1103/PhysRevLett.123.231107.
 F. Acernese et al. Increasing the astrophysical reach of the advanced Virgo detector via the application of squeezed vacuum states of light. Phys. Rev. Lett., 123: 231108, Dec 2019. https://doi.org/10.1103/PhysRevLett.123.231108.
 L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys., 90: 035005, Sep 2018. https://doi.org/10.1103/RevModPhys.90.035005.
 S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac. Improvement of frequency standards with quantum entanglement. Phys. Rev. Lett., 79: 3865–3868, Nov 1997. https://doi.org/10.1103/PhysRevLett.79.3865.
 B. M. Escher, R. L. de Matos Filho, and L. Davidovich. General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nature Physics, 7 (5): 406–411, Mar 2011. https://doi.org/10.1038/nphys1958.
 R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă. The elusive Heisenberg limit in quantum-enhanced metrology. Nature Communications, 3 (1): 1063, Jan 2012. https://doi.org/10.1038/ncomms2067.
 O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich. Quantum phase magnification. Science, 352 (6293): 1552–1555, Jun 2016. https://doi.org/10.1126/science.aaf3397.
 D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland. Toward Heisenberg-limited spectroscopy with multiparticle entangled states. Science, 304 (5676): 1476–1478, Jun 2004. https://doi.org/10.1126/science.1097576.
 D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler. Quantum-enhanced sensing based on time reversal of nonlinear dynamics. Phys. Rev. Lett., 117: 013001, Jun 2016. https://doi.org/10.1103/PhysRevLett.117.013001.
 S. C. Burd, R. Srinivas, J. J. Bollinger, A. C. Wilson, D. J. Wineland, D. Leibfried, D. H. Slichter, and D. T. C. Allcock. Quantum amplification of mechanical oscillator motion. Science, 364 (6446): 1163–1165, Jun 2019. https://doi.org/10.1126/science.aaw2884.
 M. H. Schleier-Smith, I. D. Leroux, and V. Vuletić. States of an ensemble of two-level atoms with reduced quantum uncertainty. Phys. Rev. Lett., 104: 073604, Feb 2010. https://doi.org/10.1103/PhysRevLett.104.073604.
 R. Kaubruegger, P. Silvi, C. Kokail, R. van Bijnen, A. M. Rey, J. Ye, A. M. Kaufman, and P. Zoller. Variational spin-squeezing algorithms on programmable quantum sensors. Phys. Rev. Lett., 123: 260505, Dec 2019. https://doi.org/10.1103/PhysRevLett.123.260505.
 T. Macrì, A. Smerzi, and L. Pezzè. Loschmidt echo for quantum metrology. Phys. Rev. A, 94: 010102, Jul 2016. https://doi.org/10.1103/PhysRevA.94.010102.
 S. A. Haine. Using interaction-based readouts to approach the ultimate limit of detection-noise robustness for quantum-enhanced metrology in collective spin systems. Phys. Rev. A, 98: 030303, Sep 2018. https://doi.org/10.1103/PhysRevA.98.030303.
 S. S. Mirkhalaf, S. P. Nolan, and S. A. Haine. Robustifying twist-and-turn entanglement with interaction-based readout. Phys. Rev. A, 97: 053618, May 2018. https://doi.org/10.1103/PhysRevA.97.053618.
 F. Anders, L. Pezzè, A. Smerzi, and C. Klempt. Phase magnification by two-axis countertwisting for detection-noise robust interferometry. Phys. Rev. A, 97: 043813, Apr 2018. https://doi.org/10.1103/PhysRevA.97.043813.
 J. Huang, M. Zhuang, B. Lu, Y. Ke, and C. Lee. Achieving Heisenberg-limited metrology with spin cat states via interaction-based readout. Phys. Rev. A, 98: 012129, Jul 2018. https://doi.org/10.1103/PhysRevA.98.012129.
 A. Niezgoda, D. Kajtoch, J. Dziekańska, and E. Witkowska. Optimal quantum interferometry robust to detection noise using spin-1 atomic condensates. New Journal of Physics, 21 (9): 093037, Sep 2019. https://doi.org/10.1088/1367-2630/ab4099.
 N. F. Ramsey. A molecular beam resonance method with separated oscillating fields. Phys. Rev., 78: 695–699, Jun 1950. https://doi.org/10.1103/PhysRev.78.695.
 M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey. Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet. Nature Physics, 13 (8): 781–786, May 2017. https://doi.org/10.1038/nphys4119.
 A. André, A. S. Sørensen, and M. D. Lukin. Stability of atomic clocks based on entangled atoms. Phys. Rev. Lett., 92: 230801, Jun 2004. https://doi.org/10.1103/PhysRevLett.92.230801.
 E. Davis, G. Bentsen, and M. Schleier-Smith. Approaching the Heisenberg limit without single-particle detection. Phys. Rev. Lett., 116: 053601, Feb 2016. https://doi.org/10.1103/PhysRevLett.116.053601.
 F. Fröwis, P. Sekatski, and W. Dür. Detecting large quantum Fisher information with finite measurement precision. Phys. Rev. Lett., 116: 090801, Mar 2016. https://doi.org/10.1103/PhysRevLett.116.090801.
 S. P. Nolan, S. S. Szigeti, and S. A. Haine. Optimal and robust quantum metrology using interaction-based readouts. Phys. Rev. Lett., 119: 193601, Nov 2017. https://doi.org/10.1103/PhysRevLett.119.193601.
 M. Gessner, A. Smerzi, and L. Pezzè. Metrological nonlinear squeezing parameter. Phys. Rev. Lett., 122: 090503, Mar 2019. https://doi.org/10.1103/PhysRevLett.122.090503.
 D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen. Spin squeezing and reduced quantum noise in spectroscopy. Phys. Rev. A, 46: R6797–R6800, Dec 1992. https://doi.org/10.1103/PhysRevA.46.R6797.
 D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen. Squeezed atomic states and projection noise in spectroscopy. Phys. Rev. A, 50: 67–88, Jul 1994. https://doi.org/10.1103/PhysRevA.50.67.
 D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland. Creation of a six-atom `Schrödinger cat' state. Nature, 438 (7068): 639–642, Dec 2005. https://doi.org/10.1038/nature04251.
 H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezze, A. Smerzi, and M. K. Oberthaler. Fisher information and entanglement of non-Gaussian spin states. Science, 345 (6195): 424–427, Jul 2014. https://doi.org/10.1126/science.1250147.
 L. Pezzé and A. Smerzi. Entanglement, nonlinear dynamics, and the Heisenberg limit. Phys. Rev. Lett., 102: 100401, Mar 2009. https://doi.org/10.1103/PhysRevLett.102.100401.
 D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland. A decoherence-free quantum memory using trapped ions. Science, 291 (5506): 1013–1015, Jan 2001. https://doi.org/10.1126/science.1057357.
 C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt. `Designer atoms' for quantum metrology. Nature, 443 (7109): 316–319, Sep 2006. https://doi.org/10.1038/nature05101.
 G. S. Agarwal, R. R. Puri, and R. P. Singh. Atomic Schrödinger cat states. Phys. Rev. A, 56: 2249–2254, Sep 1997. https://doi.org/10.1103/PhysRevA.56.2249.
 J. P. Dowling, G. S. Agarwal, and W. P. Schleich. Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms. Phys. Rev. A, 49: 4101–4109, May 1994. https://doi.org/10.1103/PhysRevA.49.4101.
 D. Gottesman, A. Kitaev, and J. Preskill. Encoding a qubit in an oscillator. Phys. Rev. A, 64: 012310, Jun 2001. https://doi.org/10.1103/PhysRevA.64.012310.
 K. Duivenvoorden, B. M. Terhal, and D. Weigand. Single-mode displacement sensor. Phys. Rev. A, 95: 012305, Jan 2017. https://doi.org/10.1103/PhysRevA.95.012305.
 G. S. Agarwal. Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions. Phys. Rev. A, 24: 2889–2896, Dec 1981. https://doi.org/10.1103/PhysRevA.24.2889.
 M. J. W. Hall and H. M. Wiseman. Does nonlinear metrology offer improved resolution? Answers from quantum information theory. Phys. Rev. X, 2: 041006, Oct 2012. https://doi.org/10.1103/PhysRevX.2.041006.
 M. Schulte, C. Lisdat, P. O. Schmidt, U. Sterr, and K. Hammerer. Prospects and challenges for squeezing-enhanced optical atomic clocks. arXiv e-prints, art. arXiv:1911.00882, Nov 2019.
 F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas. Atomic coherent states in quantum optics. Phys. Rev. A, 6: 2211–2237, Dec 1972. https://doi.org/10.1103/PhysRevA.6.2211.
 S. L. Braunstein and C. M. Caves. Statistical distance and the geometry of quantum states. Phys. Rev. Lett., 72: 3439–3443, May 1994. https://doi.org/10.1103/PhysRevLett.72.3439.
 Artur Niezgoda, Emilia Witkowska, and Safoura Sadat Mirkhalaf, "Twist-and-store entanglement in bimodal and spin-1 Bose-Einstein condensates", Physical Review A 102 5, 053315 (2020).
 Daniel Basilewitsch, Haidong Yuan, and Christiane P. Koch, "Optimally controlled quantum discrimination and estimation", Physical Review Research 2 3, 033396 (2020).
 Thomas Bilitewski, Luigi De Marco, Jun-Ru Li, Kyle Matsuda, William G. Tobias, Giacomo Valtolina, Jun Ye, and Ana Maria Rey, "Dynamical Generation of Spin Squeezing in Ultracold Dipolar Molecules", Physical Review Letters 126 11, 113401 (2021).
 Stuart S. Szigeti, Onur Hosten, and Simon A. Haine, "Improving cold-atom sensors with quantum entanglement: Prospects and challenges", Applied Physics Letters 118 14, 140501 (2021).
 Luca Pezzè, "Twisting the noise away", Quantum Views 4, 36 (2020).
 A. Hüper, C. Pür, M. Hetzel, J. Geng, J. Peise, I. Kruse, M. Kristensen, W. Ertmer, J. Arlt, and C. Klempt, "Preparation of mesoscopic atomic ensembles with single-particle resolution", arXiv:1912.05689.
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