{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T23:01:13Z","timestamp":1775257273103,"version":"3.50.1"},"reference-count":48,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T00:00:00Z","timestamp":1589500800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>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 <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>N<\/mml:mi><\/mml:math> 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.<\/jats:p>","DOI":"10.22331\/q-2020-05-15-268","type":"journal-article","created":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T09:26:13Z","timestamp":1589534773000},"page":"268","source":"Crossref","is-referenced-by-count":42,"title":["Ramsey interferometry with generalized one-axis twisting echoes"],"prefix":"10.22331","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0389-0988","authenticated-orcid":false,"given":"Marius","family":"Schulte","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik und Institut f\u00fcr Gravitationsphysik (Albert-Einstein-Institut), Leibniz Universit\u00e4t Hannover, Appelstra\u00dfe 2, 30167 Hannover, Germany"}]},{"given":"Victor J.","family":"Mart\u00ednez-Lahuerta","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik und Institut f\u00fcr Gravitationsphysik (Albert-Einstein-Institut), Leibniz Universit\u00e4t Hannover, Appelstra\u00dfe 2, 30167 Hannover, Germany"}]},{"given":"Maja S.","family":"Scharnagl","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik und Institut f\u00fcr Gravitationsphysik (Albert-Einstein-Institut), Leibniz Universit\u00e4t Hannover, Appelstra\u00dfe 2, 30167 Hannover, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7179-0666","authenticated-orcid":false,"given":"Klemens","family":"Hammerer","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik und Institut f\u00fcr Gravitationsphysik (Albert-Einstein-Institut), Leibniz Universit\u00e4t Hannover, Appelstra\u00dfe 2, 30167 Hannover, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2020,5,15]]},"reference":[{"key":"0","unstructured":"A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt. Optical atomic clocks. Rev. Mod. Phys., 87: 637\u2013701, Jun 2015. https:\/\/doi.org\/10.1103\/RevModPhys.87.637."},{"key":"1","unstructured":"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."},{"key":"2","unstructured":"V. Giovannetti, S. Lloyd, and L. Maccone. Quantum metrology. Phys. Rev. Lett., 96: 010401, Jan 2006. https:\/\/doi.org\/10.1103\/PhysRevLett.96.010401."},{"key":"3","unstructured":"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."},{"key":"4","unstructured":"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."},{"key":"5","unstructured":"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."},{"key":"6","unstructured":"L. Pezz\u00e8, 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."},{"key":"7","unstructured":"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\u20133868, Nov 1997. https:\/\/doi.org\/10.1103\/PhysRevLett.79.3865."},{"key":"8","unstructured":"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\u2013411, Mar 2011. https:\/\/doi.org\/10.1038\/nphys1958."},{"key":"9","unstructured":"R. Demkowicz-Dobrza\u0144ski, J. Ko\u0142ody\u0144ski, and M. Gu\u0163\u0103. The elusive Heisenberg limit in quantum-enhanced metrology. Nature Communications, 3 (1): 1063, Jan 2012. https:\/\/doi.org\/10.1038\/ncomms2067."},{"key":"10","unstructured":"O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich. Quantum phase magnification. Science, 352 (6293): 1552\u20131555, Jun 2016. https:\/\/doi.org\/10.1126\/science.aaf3397."},{"key":"11","unstructured":"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\u20131478, Jun 2004. https:\/\/doi.org\/10.1126\/science.1097576."},{"key":"12","unstructured":"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."},{"key":"13","unstructured":"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\u20131165, Jun 2019. https:\/\/doi.org\/10.1126\/science.aaw2884."},{"key":"14","unstructured":"M. Kitagawa and M. Ueda. Squeezed spin states. Phys. Rev. A, 47: 5138\u20135143, Jun 1993. https:\/\/doi.org\/10.1103\/PhysRevA.47.5138."},{"key":"15","unstructured":"M. H. Schleier-Smith, I. D. Leroux, and V. Vuleti\u0107. 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."},{"key":"16","unstructured":"R. Blatt and D. Wineland. Entangled states of trapped atomic ions. Nature, 453 (7198): 1008\u20131015, Jun 2008. https:\/\/doi.org\/10.1038\/nature07125."},{"key":"17","unstructured":"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."},{"key":"18","unstructured":"T. Macr\u00ec, A. Smerzi, and L. Pezz\u00e8. Loschmidt echo for quantum metrology. Phys. Rev. A, 94: 010102, Jul 2016. https:\/\/doi.org\/10.1103\/PhysRevA.94.010102."},{"key":"19","unstructured":"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."},{"key":"20","unstructured":"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."},{"key":"21","unstructured":"F. Anders, L. Pezz\u00e8, 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."},{"key":"22","unstructured":"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."},{"key":"23","unstructured":"A. Niezgoda, D. Kajtoch, J. Dzieka\u0144ska, 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."},{"key":"24","unstructured":"N. F. Ramsey. A molecular beam resonance method with separated oscillating fields. Phys. Rev., 78: 695\u2013699, Jun 1950. https:\/\/doi.org\/10.1103\/PhysRev.78.695."},{"key":"25","unstructured":"M. G\u00e4rttner, 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\u2013786, May 2017. https:\/\/doi.org\/10.1038\/nphys4119."},{"key":"26","unstructured":"A. Andr\u00e9, A. S. S\u00f8rensen, 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."},{"key":"27","unstructured":"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."},{"key":"28","unstructured":"F. Fr\u00f6wis, P. Sekatski, and W. D\u00fcr. Detecting large quantum Fisher information with finite measurement precision. Phys. Rev. Lett., 116: 090801, Mar 2016. https:\/\/doi.org\/10.1103\/PhysRevLett.116.090801."},{"key":"29","unstructured":"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."},{"key":"30","unstructured":"M. Gessner, A. Smerzi, and L. Pezz\u00e8. Metrological nonlinear squeezing parameter. Phys. Rev. Lett., 122: 090503, Mar 2019. https:\/\/doi.org\/10.1103\/PhysRevLett.122.090503."},{"key":"31","unstructured":"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\u2013R6800, Dec 1992. https:\/\/doi.org\/10.1103\/PhysRevA.46.R6797."},{"key":"32","unstructured":"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\u201388, Jul 1994. https:\/\/doi.org\/10.1103\/PhysRevA.50.67."},{"key":"33","unstructured":"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 `Schr\u00f6dinger cat' state. Nature, 438 (7068): 639\u2013642, Dec 2005. https:\/\/doi.org\/10.1038\/nature04251."},{"key":"34","unstructured":"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\u2013427, Jul 2014. https:\/\/doi.org\/10.1126\/science.1250147."},{"key":"35","unstructured":"L. Pezz\u00e9 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."},{"key":"36","unstructured":"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\u20131015, Jan 2001. https:\/\/doi.org\/10.1126\/science.1057357."},{"key":"37","unstructured":"C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt. `Designer atoms' for quantum metrology. Nature, 443 (7109): 316\u2013319, Sep 2006. https:\/\/doi.org\/10.1038\/nature05101."},{"key":"38","unstructured":"G. S. Agarwal, R. R. Puri, and R. P. Singh. Atomic Schr\u00f6dinger cat states. Phys. Rev. A, 56: 2249\u20132254, Sep 1997. https:\/\/doi.org\/10.1103\/PhysRevA.56.2249."},{"key":"39","unstructured":"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\u20134109, May 1994. https:\/\/doi.org\/10.1103\/PhysRevA.49.4101."},{"key":"40","unstructured":"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."},{"key":"41","unstructured":"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."},{"key":"42","unstructured":"G. S. Agarwal. Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions. Phys. Rev. A, 24: 2889\u20132896, Dec 1981. https:\/\/doi.org\/10.1103\/PhysRevA.24.2889."},{"key":"43","unstructured":"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."},{"key":"44","unstructured":"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."},{"key":"45","unstructured":"F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas. Atomic coherent states in quantum optics. Phys. Rev. A, 6: 2211\u20132237, Dec 1972. https:\/\/doi.org\/10.1103\/PhysRevA.6.2211."},{"key":"46","unstructured":"C. W. Helstrom. Quantum detection and estimation theory. Journal of Statistical Physics, 1 (2): 231\u2013252, 1969. https:\/\/doi.org\/10.1007\/bf01007479."},{"key":"47","unstructured":"S. L. Braunstein and C. M. Caves. Statistical distance and the geometry of quantum states. Phys. Rev. Lett., 72: 3439\u20133443, May 1994. https:\/\/doi.org\/10.1103\/PhysRevLett.72.3439."}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2020-05-15-268\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T09:26:14Z","timestamp":1589534774000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2020-05-15-268\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,15]]},"references-count":48,"URL":"https:\/\/doi.org\/10.22331\/q-2020-05-15-268","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,15]]},"article-number":"268"}}