Optomechanical state reconstruction and nonclassicality verification beyond the resolved-sideband regime

Farid Shahandeh1,2 and Martin Ringbauer3,4

1Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, St Lucia, Queensland 4072, Australia
2Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
3Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, St Lucia, Queensland 4072, Australia
4Institute for Experimental Physics, University of Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria

Quantum optomechanics uses optical means to generate and manipulate quantum states of motion of mechanical resonators. This provides an intriguing platform for the study of fundamental physics and the development of novel quantum devices. Yet, the challenge of reconstructing and verifying the quantum state of mechanical systems has remained a major roadblock in the field. Here, we present a novel approach that allows for tomographic reconstruction of the quantum state of a mechanical system without the need for extremely high quality optical cavities. We show that, without relying on the usual state transfer presumption between light an mechanics, the full optomechanical Hamiltonian can be exploited to imprint mechanical tomograms on a strong optical coherent pulse, which can then be read out using well-established techniques. Furthermore, with only a small number of measurements, our method can be used to witness nonclassical features of mechanical systems without requiring full tomography. By relaxing the experimental requirements, our technique thus opens a feasible route towards verifying the quantum state of mechanical resonators and their nonclassical behaviour in a wide range of optomechanical systems.

Quantum optomechanics exploits the interaction between light and massive mechanical resonators to engineer and control their quantum state of motion. Such systems hold the promise to ultra-sensitive force sensors and tests of fundamental physics. Over recent years, the field has made great progress—from the preparation of nonclassical states of mechanical motion to the violation of Bell’s inequality using mechanical systems. Despite all these achievements, the reconstruction of the state of motion of a mechanical system has remained an outstanding challenge.

Here, we introduce a new technique for optomechanical quantum state tomography and nonclassicality verification with the potential to avoid much of the experimental challenges typically associated with this task. While previous approaches mostly aimed at completely transferring the mechanical state of motion onto a single photon, our method works more like taking two-dimensional snapshots of the three-dimensional phase-space distribution of the mechanical system, using strong pulses of light. From these snapshots the state of motion can be reconstructed using standard procedures. Due to the significantly reduced experimental requirements, this method can be applied to a wide range of state-of-the-art optomechanical experiments. Furthermore, it makes it possible to certify that a certain state of motion is nonclassical without having to reconstruct the state and using only a small number of noisy measurements.

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[1] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014a).
https:/​/​doi.org/​10.1103/​RevModPhys.86.1391

[2] S. Bose, K. Jacobs, and P. L. Knight, Scheme to probe the decoherence of a macroscopic object, Phys. Rev. A 59, 3204 (1999).
https:/​/​doi.org/​10.1109/​EQEC.1998.714780

[3] W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Towards Quantum Superpositions of a Mirror, Phys. Rev. Lett. 91, 130401 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.91.130401

[4] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Č. Brukner, Probing Planck-scale physics with quantum optics, Nat. Phys. 8, 393 (2012).
https:/​/​doi.org/​10.1038/​nphys2262

[5] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, Single spin detection by magnetic resonance force microscopy, Nature 430, 329 (2004).
https:/​/​doi.org/​10.1038/​nature02658

[6] M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, Cooling-by-measurement and mechanical state tomography via pulsed optomechanics. Nat. Commun. 4, 2295 (2013).
https:/​/​doi.org/​10.1038/​ncomms3295

[7] M. Ringbauer, T. J. Weinhold, L. A. Howard, A. G. White, and M. R. Vanner, Generation of mechanical interference fringes by multi-photon counting, New Journal of Physics, 20, 053042 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aabb8d

[8] R. J. Glauber, Quantum theory of optical coherence : selected papers and lectures (Wiley-VCH, 2007) p. 639.

[9] W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).

[10] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, Quantum State Reconstruction of the Single-Photon Fock State, Phys. Rev. Lett. 87, 050402 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.87.050402

[11] A. Zavatta, S. Viciani, and M. Bellini, Tomographic reconstruction of the single-photon Fock state by high-frequency homodyne detection, Phys. Rev. A 70, 053821 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.70.053821

[12] U. Leonhardt, Cambridge Studies in Modern Optics, edited by P. Knight and A. Miller (Cambridge University Press, 1997).
https:/​/​doi.org/​10.1016/​0079-6727(94)00007-L

[13] H. Hansen, T. Aichele, C. Hettich, P. Lodahl, A. I. Lvovsky, J. Mlynek, and S. Schiller, Ultrasensitive pulsed, balanced homodyne detector: application to time-domain quantum measurements. Opt. Lett. 26, 1714 (2001).
https:/​/​doi.org/​10.1364/​OL.26.001714

[14] M. R. Vanner, I. Pikovski, and M. S. Kim, Towards optomechanical quantum state reconstruction of mechanical motion, Ann. Phys. 527, 15 (2015).
https:/​/​doi.org/​10.1002/​andp.201400124

[15] M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, Pulsed quantum optomechanics, Proc. Natl. Acad. Sci. 108, 16182 (2011).
https:/​/​doi.org/​10.1073/​pnas.1105098108

[16] M. Aspelmeyer, T. J. Kippenberg, and F. Marquard, Cavity Optomechanics, edited by M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Vol. 86 (Springer Berlin Heidelberg, 2014) pp. 1391-1452.
https:/​/​doi.org/​10.1007/​978-3-642-55312-7

[17] X.-B. Wang, S.-X. Yu, and Y.-D. Zhang, Linear quantum transformation and normal product calculation of boson exponential quadratic operators, J. Phys. A. 27, 6563 (1994).
https:/​/​doi.org/​10.1088/​0305-4470/​27/​19/​026

[18] Y.-d. Zhang and Z. Tang, General theory of linear quantum transformation of Bargmann-Fock space, Nuovo Cim. B Ser. 11 109, 387 (1994).
https:/​/​doi.org/​10.1007/​BF02722519

[19] S. Wallentowitz and W. Vogel, Unbalanced homodyning for quantum state measurements, Phys. Rev. A 53, 4528 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.53.4528

[20] T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, and M. Bellini, Experimental determination of a nonclassical Glauber-Sudarshan P-function, Phys. Rev. A 78, 021804 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.021804

[21] T. Kiesel, W. Vogel, B. Hage, and R. Schnabel, Direct Sampling of Negative Quasiprobabilities of a Squeezed State, Phys. Rev. Lett. 107, 113604 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.113604

[22] E. Agudelo, J. Sperling, W. Vogel, S. Köhnke, M. Mraz, and B. Hage, Continuous sampling of the squeezed-state nonclassicality, Phys. Rev. A 92, 033837 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.033837

[23] E. Shchukin, T. Richter, and W. Vogel, Nonclassicality criteria in terms of moments, Phys. Rev. A 71, 011802 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.011802

[24] E. V. Shchukin and W. Vogel, Nonclassical moments and their measurement, Phys. Rev. A 72, 043808 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.043808

[25] T. Kiesel, W. Vogel, B. Hage, J. DiGuglielmo, A. Samblowski, and R. Schnabel, Experimental test of nonclassicality criteria for phase-diffused squeezed states, Phys. Rev. A 79, 022122 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.022122

[26] T. Kiesel and W. Vogel, Complete nonclassicality test with a photon-number-resolving detector, Phys. Rev. A 86, 032119 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.032119

[27] S. Ryl, J. Sperling, E. Agudelo, M. Mraz, S. Köhnke, B. Hage, and W. Vogel, Unified nonclassicality criteria, Phys. Rev. A 92, 011801 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.011801

[28] J. Park, Y. Lu, J. Lee, Y. Shen, K. Zhang, S. Zhang, M. S. Zubairy, K. Kim, and H. Nha, Revealing nonclassicality beyond Gaussian states via a single marginal distribution, Proc. Natl. Acad. Sci. 114, 891 (2017).
https:/​/​doi.org/​10.1073/​pnas.1617621114

[29] J. Eisert and M. M. Wolf, in Quantum Inf. with Contin. Var. Atoms Light (Imperial College Press, London, 2007) pp. 23-42.
https:/​/​doi.org/​10.1142/​9781860948169_0002

[30] K. R. Parthasarathy and R. Sengupta, From particle counting to gaussian tomography, Infinite Dimensional Analysis, Quantum Probability and Related Topics 18, 1550023 (2015).
https:/​/​doi.org/​10.1142/​S021902571550023X

[31] K. S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, Quantum nondemolition measurements of harmonic oscillators, Phys. Rev. Lett. 40, 667 (1978).
https:/​/​doi.org/​10.1103/​PhysRevLett.40.667

[32] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, Quantum nondemolition measurements, Science 209, 547 (1980).
https:/​/​doi.org/​10.1126/​science.209.4456.547

[33] A. Ferraro and M. G. a. Paris, Nonclassicality Criteria from Phase-Space Representations and Information-Theoretical Constraints Are Maximally Inequivalent, Phys. Rev. Lett. 108, 260403 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.260403

[34] D. Kleckner, B. Pepper, E. Jeffrey, P. Sonin, S. M. Thon, and D. Bouwmeester, Optomechanical trampoline resonators. Opt. Expr. 19, 19708 (2011).
https:/​/​doi.org/​10.1364/​OE.19.019708

[35] R. A. Norte, J. P. Moura, and S. Gröblacher, Mechanical Resonators for Quantum Optomechanics Experiments at Room Temperature, Phys. Rev. Lett. 116, 147202 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.147202

[36] S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, Observation of strong coupling between a micromechanical resonator and an optical cavity field, Nature 460, 724 (2009).
https:/​/​doi.org/​10.1038/​nature08171

[37] F. E. Becerra, J. Fan, and A. Migdall, Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states, Nat. Commun. 4, 2028 (2013).
https:/​/​doi.org/​10.1038/​ncomms3028

[38] A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.299

[39] M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, Non-Gaussianity of quantum states: An experimental test on single-photon-added coherent states, Phys. Rev. A 82, 063833 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.82.063833

[40] K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, Observation of quantum-measurement backaction with an ultracold atomic gas, Nat. Phys. 4, 561 (2008).
https:/​/​doi.org/​10.1038/​nphys965

[41] J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane, Nature 452, 72 (2008).
https:/​/​doi.org/​10.1038/​nature06715

[42] S. Anguiano, A. E. Bruchhausen, B. Jusserand, I. Favero, F. R. Lamperti, L. Lanco, I. Sagnes, A. Lemaı̂tre, N. D. Lanzillotti-Kimura, P. Senellart, and A. Fainstein, Time-Resolved Cavity Nano-Optomechanics in the 20-100 GHz range, Preprint at arXiv:1610.04179 (2016).
arXiv:1610.04179

[43] O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Radiation-pressure cooling and optomechanical instability of a micromirror, Nature 444, 71 (2006).
https:/​/​doi.org/​10.1038/​nature05244

[44] B. D. Cuthbertson, M. E. Tobar, E. N. Ivanov, and D. G. Blair, Parametric back-action effects in a high-Q cyrogenic sapphire transducer, Rev. Sci. Instr. 67, 2435 (1996).
https:/​/​doi.org/​10.1063/​1.1147193

[45] J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, Laser cooling of a nanomechanical oscillator into its quantum ground state, Nature 478, 89 (2011).
https:/​/​doi.org/​10.1038/​nature10461

[46] E. Verhagen, S. Deléglise, S. Weis, a. Schliesser, and T. J. Kippenberg, Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode. Nature 482, 63 (2012).
https:/​/​doi.org/​10.1038/​nature10787

[47] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, Sideband cooling of micromechanical motion to the quantum ground state, Nature 475, 359 (2011).
https:/​/​doi.org/​10.1038/​nature10261

[48] K. Husimi, Some Formal Properties of the Density Matrix, Proc. Phys. Math. Soc. Japan 3rd Series 22, 264 (1940).
https:/​/​doi.org/​JST.Journalarchive/​ppmsj1919/​22.264

[49] Y. Kano, A New Phase-Space Distribution Function in the Statistical Theory of the Electromagnetic Field, J. Math. Phys. 6, 1913 (1965).
https:/​/​doi.org/​10.1063/​1.1704739

[50] E. Wigner, On the Quantum Correction For Thermodynamic Equil, Phys. Rev. 40, 749 (1932).

[51] R. J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130, 2529 (1963).
https:/​/​doi.org/​10.1103/​PhysRev.130.2529

[52] E. C. G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams, Phys. Rev. Lett. 10, 277 (1963).
https:/​/​doi.org/​10.1103/​PhysRevLett.10.277

[53] J. Sperling, Characterizing maximally singular phase-space distributions, Phys. Rev. A 94, 013814 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.013814

[54] W. Vogel, Nonclassical States: An Observable Criterion, Phys. Rev. Lett. 84, 1849 (2000).
https:/​/​doi.org/​10.1103/​PhysRevLett.84.1849

[55] J. Sperling, W. R. Clements, A. Eckstein, M. Moore, J. J. Renema, W. S. Kolthammer, S. W. Nam, A. Lita, T. Gerrits, W. Vogel, G. S. Agarwal, and I. A. Walmsley, Detector-independent verification of quantum light, Phys. Rev. Lett. 118, 163602 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.118.163602

[56] W. P. Schleich, Quantum optics in phase space (Wiley-VCH, 2015).

[57] M. R. Bazrafkan and E. Nahvifard, Quantum-state tomogram from s-parameterized quasidistributions, J. Russ. Laser Res. 32, 230 (2011).
https:/​/​doi.org/​10.1007/​s10946-011-9208-0

[58] R. R. Puri, Mathematical Methods of Quantum Optics, edited by W. T. R. Metz (Springer Berlin Heidelberg, 2001).
https:/​/​doi.org/​10.1007/​978-3-540-44953-9

[59] D. Kastler, The C*-algebras of a free boson field, Commun. Math. Phys. 1, 14 (1965).
https:/​/​doi.org/​10.1007/​BF01649588

[60] G. Loupias and S. Miracle-Sole, C*-algèbres des systèmes canoniques. i, Commun. Math. Phys. 2, 31 (1966).
https:/​/​doi.org/​10.1007/​BF01773339

[61] H. Nha, Complete conditions for legitimate wigner distributions, Phys. Rev. A 78, 012103 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.012103

[62] F. Shahandeh and M. R. Bazrafkan, The general boson ordering problem and its combinatorial roots, Phys. Scr. T153, 014056 (2013).
https:/​/​doi.org/​10.1088/​0031-8949/​2013/​T153/​014056

[63] G. Dattoli, Incomplete 2D Hermite polynomials: properties and applications, J. Math. Anal. Appl. 284, 447 (2003).
https:/​/​doi.org/​10.1016/​S0022-247X(03)00259-2

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[4] Matteo Brunelli, Oussama Houhou, Darren W. Moore, Andreas Nunnenkamp, Mauro Paternostro, and Alessandro Ferraro, "Unconditional preparation of nonclassical states via linear-and-quadratic optomechanics", Physical Review A 98 6, 063801 (2018).

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