Bending the rules of low-temperature thermometry with periodic driving

Jonas Glatthard and Luis A. Correa

Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom

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Abstract

There exist severe limitations on the accuracy of low-temperature thermometry, which poses a major challenge for future quantum-technological applications. Low-temperature sensitivity might be manipulated by tailoring the interactions between probe and sample. Unfortunately, the tunability of these interactions is usually very restricted. Here, we focus on a more practical solution to boost thermometric precision – driving the probe. Specifically, we solve for the limit cycle of a periodically modulated linear probe in an equilibrium sample. We treat the probe-sample interactions $exactly$ and hence, our results are valid for arbitrarily low temperatures $ T $ and any spectral density. We find that weak near-resonant modulation strongly enhances the signal-to-noise ratio of low-temperature measurements, while causing minimal back action on the sample. Furthermore, we show that near-resonant driving changes the power law that governs thermal sensitivity over a broad range of temperatures, thus `bending' the fundamental precision limits and enabling more sensitive low-temperature thermometry. We then focus on a concrete example – impurity thermometry in an atomic condensate. We demonstrate that periodic driving allows for a sensitivity improvement of several orders of magnitude in sub-nanokelvin temperature estimates drawn from the density profile of the impurity atoms. We thus provide a feasible upgrade that can be easily integrated into low-$T$ thermometry experiments.

Precisely measuring ultracold temperatures is a challenging task not only in practice, but also in principle. Yet it is a crucial challenge to be met in order to facilitate the progress of quantum technologies. While the fragility of quantum systems is often seen as a nuisance, it can also be used to construct more sensitive instruments — quantum sensors. In this article we study a way of boosting the sensitivity of a quantum Brownian thermometer by periodically modulating its energy. Importantly, we analyse the problem avoiding commonly used approximations which break down at low temperatures. We find that periodic modulation can substantially boost thermal sensitivity with respect to undriven probes, and also make it drop off much more slowly as the absolute zero is approached. We illustrate the method in an experimentally realisable situation — an impurity thermometer measuring the temperature of an ultracold atomic gas. We further show that the protocol is robust when it comes to lack of synchronisation between measurements and drive. Finally, we show that the negative side-effects of pumping energy into a temperature probe immersed in an ultracold sample, i.e., spurious heating, are controllably small, thus rending our proposal a practical solution for sub-nanokelvin impurity thermometry in cold atomic gases.

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[1] Aron E. Leanhardt, Thomas A. Pasquini, Michele Saba, André Schirotzek, Yong-il Shin, Dave Kielpinski, David E. Pritchard, and Wolfgang Ketterle. Cooling bose-einstein condensates below 500 picokelvin. Science, 301 (5639): 1513–1515, 2003. 10.1126/​science.1088827.
https:/​/​doi.org/​10.1126/​science.1088827

[2] Immanuel Bloch, Jean Dalibard, and Sylvain Nascimbene. Quantum simulations with ultracold quantum gases. Nat. Phys., 8 (4): 267–276, 2012. 10.1038/​nphys2259.
https:/​/​doi.org/​10.1038/​nphys2259

[3] Ryan Olf, Fang Fang, G. Edward Marti, Andrew MacRae, and Dan M. Stamper-Kurn. Thermometry and cooling of a bose gas to 0.02 times the condensation temperature. Nat. Phys., 11 (9): 720–723, 2015. 10.1038/​nphys3408.
https:/​/​doi.org/​10.1038/​nphys3408

[4] Christian Deppner, Waldemar Herr, Merle Cornelius, Peter Stromberger, Tammo Sternke, Christoph Grzeschik, Alexander Grote, Jan Rudolph, Sven Herrmann, Markus Krutzik, André Wenzlawski, Robin Corgier, Eric Charron, David Guéry-Odelin, Naceur Gaaloul, Claus Lämmerzahl, Achim Peters, Patrick Windpassinger, and Ernst M. Rasel. Collective-mode enhanced matter-wave optics. Phys. Rev. Lett., 127: 100401, 2021. 10.1103/​PhysRevLett.127.100401.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.100401

[5] Karen V. Hovhannisyan and Luis A. Correa. Measuring the temperature of cold many-body quantum systems. Phys. Rev. B, 98: 045101, 2018. 10.1103/​PhysRevB.98.045101.
https:/​/​doi.org/​10.1103/​PhysRevB.98.045101

[6] Antonella De Pasquale, Davide Rossini, Rosario Fazio, and Vittorio Giovannetti. Local quantum thermal susceptibility. Nat. Commun., 7 (1): 1–8, 2016. 10.1038/​ncomms12782.
https:/​/​doi.org/​10.1038/​ncomms12782

[7] Luis A. Correa, Martí Perarnau-Llobet, Karen V. Hovhannisyan, Senaida Hernández-Santana, Mohammad Mehboudi, and Anna Sanpera. Enhancement of low-temperature thermometry by strong coupling. Phys. Rev. A, 96: 062103, 2017. 10.1103/​PhysRevA.96.062103.
https:/​/​doi.org/​10.1103/​PhysRevA.96.062103

[8] Harry J. D. Miller and Janet Anders. Energy-temperature uncertainty relation in quantum thermodynamics. Nat. Commun., 9 (1): 1–8, 2018. 10.1038/​s41467-018-04536-7.
https:/​/​doi.org/​10.1038/​s41467-018-04536-7

[9] Antonella de Pascuale and Tom Stace. Thermodynamics in the Quantum Regime. Fundamental Theories of Physics, vol. 195, chapter Quantum thermometry. Springer, Cham, 2019. ISBN 978-3-319-99046-0. 10.1007/​978-3-319-99046-0_21.
https:/​/​doi.org/​10.1007/​978-3-319-99046-0_21

[10] Mohammad Mehboudi, Anna Sanpera, and Luis A. Correa. Thermometry in the quantum regime: recent theoretical progress. J. Phys. A Math. Theor., 52: 011611, 2019a. 10.1088/​1751-8121/​ab2828.
https:/​/​doi.org/​10.1088/​1751-8121/​ab2828

[11] Patrick P. Potts, Jonatan Bohr Brask, and Nicolas Brunner. Fundamental limits on low-temperature quantum thermometry with finite resolution. Quantum, 3: 161, 2019. ISSN 2521-327X. 10.22331/​q-2019-07-09-161.
https:/​/​doi.org/​10.22331/​q-2019-07-09-161

[12] Mathias R. Jørgensen, Patrick P. Potts, Matteo G. A. Paris, and Jonatan B. Brask. Tight bound on finite-resolution quantum thermometry at low temperatures. Phys. Rev. Res., 2: 033394, 2020. 10.1103/​PhysRevResearch.2.033394.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.033394

[13] Victor Mukherjee, Analia Zwick, Arnab Ghosh, Xi Chen, and Gershon Kurizki. Enhanced precision bound of low-temperature quantum thermometry via dynamical control. Commun. Phys., 2: 162, 2019. 10.1038/​s42005-019-0265-y.
https:/​/​doi.org/​10.1038/​s42005-019-0265-y

[14] Ivan Henao, Karen V Hovhannisyan, and Raam Uzdin. Thermometric machine for ultraprecise thermometry of low temperatures. arXiv preprint arXiv:2108.10469, 2021. 10.48550/​arXiv.2108.10469.
https:/​/​doi.org/​10.48550/​arXiv.2108.10469
arXiv:2108.10469

[15] Mohammad Mehboudi, Aniello Lampo, Christos Charalambous, Luis A. Correa, Miguel Ángel García-March, and Maciej Lewenstein. Using polarons for sub-nk quantum nondemolition thermometry in a bose-einstein condensate. Phys. Rev. Lett., 122: 030403, 2019b. 10.1103/​PhysRevLett.122.030403.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.030403

[16] Utkarsh Mishra and Abolfazl Bayat. Integrable quantum many-body sensors for ac field sensing. arXiv preprint arXiv:2105.13507, 2021a. 10.48550/​arXiv.2105.13507.
https:/​/​doi.org/​10.48550/​arXiv.2105.13507
arXiv:2105.13507

[17] Utkarsh Mishra and Abolfazl Bayat. Driving enhanced quantum sensing in partially accessible many-body systems. Phys. Rev. Lett., 127: 080504, 2021b. 10.1103/​PhysRevLett.127.080504.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.080504

[18] E. Brian Davies. Markovian master equations. Commun. Math. Phys., 39 (2): 91 – 110, 1974. 10.1007/​BF01608389.
https:/​/​doi.org/​10.1007/​BF01608389

[19] Vittorio Gorini, Andrzej Kossakowski, and Ennackal Chandy George Sudarshan. Completely positive dynamical semigroups of n-level systems. J. Math. Phys., 17 (5): 821–825, 1976. 10.1063/​1.522979.
https:/​/​doi.org/​10.1063/​1.522979

[20] Goran Lindblad. On the generators of quantum dynamical semigroups. Commun. Math. Phys., 48 (2): 119–130, 1976. 10.1007/​BF01608499.
https:/​/​doi.org/​10.1007/​BF01608499

[21] Robert Alicki, Daniel A. Lidar, and Paolo Zanardi. Internal consistency of fault-tolerant quantum error correction in light of rigorous derivations of the quantum markovian limit. Phys. Rev. A, 73: 052311, 2006. 10.1103/​PhysRevA.73.052311.
https:/​/​doi.org/​10.1103/​PhysRevA.73.052311

[22] Robert Alicki, David Gelbwaser-Klimovsky, and Gershon Kurizki. Periodically driven quantum open systems: Tutorial. arXiv preprint arXiv:1205.4552, 2012. 10.48550/​arXiv.1205.4552.
https:/​/​doi.org/​10.48550/​arXiv.1205.4552
arXiv:1205.4552

[23] Heinz-Peter Breuer, Francesco Petruccione, et al. The theory of open quantum systems. Oxford University Press on Demand, 2002. ISBN 9780199213900. 10.1093/​acprof:oso/​9780199213900.001.0001.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[24] Christine Zerbe and Peter Hänggi. Brownian parametric quantum oscillator with dissipation. Phys. Rev. E, 52: 1533–1543, 1995. 10.1103/​PhysRevE.52.1533.
https:/​/​doi.org/​10.1103/​PhysRevE.52.1533

[25] Nahuel Freitas and Juan Pablo Paz. Analytic solution for heat flow through a general harmonic network. Phys. Rev. E, 90: 042128, 2014. 10.1103/​PhysRevE.90.042128.
https:/​/​doi.org/​10.1103/​PhysRevE.90.042128

[26] Nahuel Freitas and Juan Pablo Paz. Fundamental limits for cooling of linear quantum refrigerators. Phys. Rev. E, 95: 012146, 2017. 10.1103/​PhysRevE.95.012146.
https:/​/​doi.org/​10.1103/​PhysRevE.95.012146

[27] Aniello Lampo, Soon Hoe Lim, Miguel Ángel García-March, and Maciej Lewenstein. Bose polaron as an instance of quantum Brownian motion. Quantum, 1: 30, 2017. ISSN 2521-327X. 10.22331/​q-2017-09-27-30.
https:/​/​doi.org/​10.22331/​q-2017-09-27-30

[28] Aniello Lampo, Christos Charalambous, Miguel Ángel García-March, and Maciej Lewenstein. Non-markovian polaron dynamics in a trapped bose-einstein condensate. Phys. Rev. A, 98: 063630, 2018. 10.1103/​PhysRevA.98.063630.
https:/​/​doi.org/​10.1103/​PhysRevA.98.063630

[29] Ulrich Weiss. Quantum dissipative systems, volume 13. World Scientific, 2nd edition, 1999. ISBN 978-981-4374-91-0. 10.1142/​8334.
https:/​/​doi.org/​10.1142/​8334

[30] Alessandro Ferraro, Stefano Olivares, and Matteo G. A. Paris. Gaussian States in Quantum Information. Napoli Series on physics and Astrophysics. Bibliopolis, 2005. ISBN 88-7088-483-X. 10.48550/​arXiv.quant-ph/​0503237.
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0503237
arXiv:quant-ph/0503237

[31] Harald Cramér. Mathematical Methods of Statistics (PMS-9). Princeton University Press, 2016. ISBN 9781400883868. 10.1515/​9781400883868.
https:/​/​doi.org/​10.1515/​9781400883868

[32] Calyampudi Radhakrishna Rao. Information and the accuracy attainable in the estimation of statistical parameters. Reson. J. Sci. Educ, 20: 78–90, 1945. 10.1007/​978-1-4612-0919-5_16.
https:/​/​doi.org/​10.1007/​978-1-4612-0919-5_16

[33] Jesús Rubio, Janet Anders, and Luis A. Correa. Global quantum thermometry. Phys. Rev. Lett., 127: 190402, 2021. 10.1103/​PhysRevLett.127.190402.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.190402

[34] Horia Scutaru. Fidelity for displaced squeezed thermal states and the oscillator semigroup. J. Phys. A Math. Gen., 31 (15): 3659–3663, 1998. 10.1088/​0305-4470/​31/​15/​025.
https:/​/​doi.org/​10.1088/​0305-4470/​31/​15/​025

[35] Sahar Alipour and Ali T. Rezakhani. Extended convexity of quantum fisher information in quantum metrology. Phys. Rev. A, 91: 042104, 2015. 10.1103/​PhysRevA.91.042104.
https:/​/​doi.org/​10.1103/​PhysRevA.91.042104

[36] David C. Aveline, Jason R. Williams, Ethan R. Elliott, Chelsea Dutenhoffer, James R. Kellogg, James M. Kohel, Norman E. Lay, Kamal Oudrhiri, Robert F. Shotwell, Nan Yu, and Robert J. Thompson. Observation of Bose–Einstein condensates in an Earth-orbiting research lab. Nature, 582 (7811): 193–197, 2020. 10.1038/​s41586-020-2346-1.
https:/​/​doi.org/​10.1038/​s41586-020-2346-1

[37] Nicolas Spethmann, Farina Kindermann, Shincy John, Claudia Weber, Dieter Meschede, and Artur Widera. Dynamics of single neutral impurity atoms immersed in an ultracold gas. Phys. Rev. Lett., 109: 235301, 2012. 10.1103/​PhysRevLett.109.235301.
https:/​/​doi.org/​10.1103/​PhysRevLett.109.235301

[38] Michael Hohmann, Farina Kindermann, Tobias Lausch, Daniel Mayer, Felix Schmidt, and Artur Widera. Single-atom thermometer for ultracold gases. Phys. Rev. A, 93: 043607, 2016. 10.1103/​PhysRevA.93.043607.
https:/​/​doi.org/​10.1103/​PhysRevA.93.043607

[39] Jacopo Catani, Giacomo Lamporesi, Devang Naik, Michael Gring, Massimo Inguscio, Fransesco Minardi, Aadrian Kantian, and Thierry Giamarchi. Quantum dynamics of impurities in a one-dimensional bose gas. Phys. Rev. A, 85: 023623, 2012. 10.1103/​PhysRevA.85.023623.
https:/​/​doi.org/​10.1103/​PhysRevA.85.023623

[40] Samuel L. Braunstein and Carlton M. Caves. Statistical distance and the geometry of quantum states. Phys. Rev. Lett., 72: 3439–3443, 1994. 10.1103/​PhysRevLett.72.3439.
https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439

[41] Géza Tóth and Iagoba Apellaniz. Quantum metrology from a quantum information science perspective. J. Phys. A Math. Theor., 47: 424006, 2014. 10.1088/​1751-8113/​47/​42/​424006.
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424006

[42] Jonas Zmuidzinas. Superconducting microresonators: Physics and applications. Annu. Rev. Condens. Matter Phys., 3 (1): 169–214, 2012. 10.1146/​annurev-conmatphys-020911-125022.
https:/​/​doi.org/​10.1146/​annurev-conmatphys-020911-125022

[43] Ali Hasan Nayfeh and Dean T. Mook. Nonlinear Oscillations. John Wiley and Sons, Ltd, 1995. ISBN 9780471121428. 10.1002/​9783527617586.
https:/​/​doi.org/​10.1002/​9783527617586

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