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.
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.
 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).
 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).
 R. J. Glauber, Quantum theory of optical coherence : selected papers and lectures (Wiley-VCH, 2007) p. 639.
 W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).
 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).
 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).
 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).
 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).
 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.
 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).
 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).
 T. Kiesel, W. Vogel, B. Hage, and R. Schnabel, Direct Sampling of Negative Quasiprobabilities of a Squeezed State, Phys. Rev. Lett. 107, 113604 (2011).
 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).
 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).
 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).
 K. R. Parthasarathy and R. Sengupta, From particle counting to gaussian tomography, Infinite Dimensional Analysis, Quantum Probability and Related Topics 18, 1550023 (2015).
 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).
 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).
 R. A. Norte, J. P. Moura, and S. Gröblacher, Mechanical Resonators for Quantum Optomechanics Experiments at Room Temperature, Phys. Rev. Lett. 116, 147202 (2016).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 K. Husimi, Some Formal Properties of the Density Matrix, Proc. Phys. Math. Soc. Japan 3rd Series 22, 264 (1940).
 E. Wigner, On the Quantum Correction For Thermodynamic Equil, Phys. Rev. 40, 749 (1932).
 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).
 W. P. Schleich, Quantum optics in phase space (Wiley-VCH, 2015).
 F. Shahandeh and M. R. Bazrafkan, The general boson ordering problem and its combinatorial roots, Phys. Scr. T153, 014056 (2013).
 L. A. Howard, T. J. Weinhold, F. Shahandeh, J. Combes, M. R. Vanner, A. G. White, and M. Ringbauer, "Quantum Hypercube States", Physical Review Letters 123 2, 020402 (2019).
 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).
 Martin Koppenhöfer, Christoph Bruder, and Niels Lörch, "Unraveling nonclassicality in the optomechanical instability", Physical Review A 97 6, 063812 (2018).
 P. Warszawski, A. Szorkovszky, W. P. Bowen, and A. C. Doherty, "Tomography of an optomechanical oscillator via parametrically amplified position measurement", New Journal of Physics 21 2, 023020 (2019).
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