Quantum manipulation of a two-level mechanical system

Salvatore Chiavazzo1, Anders Søndberg Sørensen2, Oleksandr Kyriienko1, and Luca Dellantonio1,3,4

1Department of Physics and Astronomy, University of Exeter, Exeter, Devon EX4 4QL, UK
2Center for Hybrid Quantum Networks (Hy-Q), Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
3Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
4Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We consider a nonlinearly coupled electromechanical system, and develop a quantitative theory for two-phonon cooling. In the presence of two-phonon cooling, the mechanical Hilbert space is effectively reduced to its ground and first excited states, allowing for quantum operations at the level of individual phonons and preparing nonclassical mechanical states with negative Wigner functions. We propose a scheme for performing arbitrary Bloch sphere rotations, and derive the fidelity in the specific case of a $\pi$-pulse. We characterise detrimental processes that reduce the coherence in the system, and demonstrate that our scheme can be implemented in state-of-the-art electromechanical devices.

In this work we show how to perform a two-phonon cooling with micro-mechanical membranes. This cooling is a second-order process acting on two excitation-quanta per time, ultimately leading to protected 2-level state. Finally, we introduce possible procedures to correctly manipulate the state, demonstrating its use as a 2-level system.

► BibTeX data

► References

[1] Markus Aspelmeyer, Tobias J. Kippenberg, and Florian Marquardt. Cavity optomechanics. Rev. Mod. Phys., 86: 1391–1452, Dec 2014. URL https:/​/​doi.org/​10.1103/​RevModPhys.86.1391.
https:/​/​doi.org/​10.1103/​RevModPhys.86.1391

[2] Scott B Papp, Katja Beha, Pascal Del’Haye, Franklyn Quinlan, Hansuek Lee, Kerry J Vahala, and Scott A Diddams. Microresonator frequency comb optical clock. Optica, 1 (1): 10–14, 2014. URL https:/​/​doi.org/​10.1364/​OPTICA.1.000010.
https:/​/​doi.org/​10.1364/​OPTICA.1.000010

[3] Lue Wu, Heming Wang, Qi-Fan Yang, Maodong Gao, Qing-Xin Ji, Boqiang Shen, Chengying Bao, and Kerry Vahala. On-chip q-factor greater than 1 billion. In 2020 Conference on Lasers and Electro-Optics (CLEO), pages 1–2. IEEE, 2020. URL https:/​/​doi.org/​10.1364/​CLEO_SI.2020.SW3J.7.
https:/​/​doi.org/​10.1364/​CLEO_SI.2020.SW3J.7

[4] Y Lai, S Pirotta, G Urbinati, D Gerace, M Minkov, V Savona, A Badolato, and M Galli. Genetically designed l3 photonic crystal nanocavities with measured quality factor exceeding one million. Applied Physics Letters, 104 (24): 241101, 2014. URL https:/​/​doi.org/​10.1063/​1.4882860.
https:/​/​doi.org/​10.1063/​1.4882860

[5] Matthew Reagor, Hanhee Paik, Gianluigi Catelani, Luyan Sun, Christopher Axline, Eric Holland, Ioan M Pop, Nicholas A Masluk, Teresa Brecht, Luigi Frunzio, et al. Reaching 10 ms single photon lifetimes for superconducting aluminum cavities. Applied Physics Letters, 102 (19): 192604, 2013. URL https:/​/​doi.org/​10.1063/​1.4807015.
https:/​/​doi.org/​10.1063/​1.4807015

[6] M Kudra, J Biznárová, A Fadavi Roudsari, JJ Burnett, D Niepce, S Gasparinetti, B Wickman, and P Delsing. High quality three-dimensional aluminum microwave cavities. Applied Physics Letters, 117 (7): 070601, 2020. URL https:/​/​doi.org/​10.1063/​5.0016463.
https:/​/​doi.org/​10.1063/​5.0016463

[7] Yeghishe Tsaturyan, Andreas Barg, Anders Simonsen, Luis Guillermo Villanueva, Silvan Schmid, Albert Schliesser, and Eugene S Polzik. Demonstration of suppressed phonon tunneling losses in phononic bandgap shielded membrane resonators for high-q optomechanics. Optics express, 22 (6): 6810–6821, 2014. URL https:/​/​doi.org/​10.1364/​OE.22.006810.
https:/​/​doi.org/​10.1364/​OE.22.006810

[8] Amir H Safavi-Naeini, Jeff T Hill, Seán Meenehan, Jasper Chan, Simon Gröblacher, and Oskar Painter. Two-dimensional phononic-photonic band gap optomechanical crystal cavity. Physical Review Letters, 112 (15): 153603, 2014. URL https:/​/​doi.org/​10.1103/​PhysRevLett.112.153603.
https:/​/​doi.org/​10.1103/​PhysRevLett.112.153603

[9] B Vogell, T Kampschulte, MT Rakher, A Faber, P Treutlein, Klemens Hammerer, and P Zoller. Long distance coupling of a quantum mechanical oscillator to the internal states of an atomic ensemble. New Journal of Physics, 17 (4): 043044, 2015. URL https:/​/​doi.org/​10.1088/​1367-2630/​17/​4/​043044.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​4/​043044

[10] Yeghishe Tsaturyan, Andreas Barg, Eugene S Polzik, and Albert Schliesser. Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution. Nature nanotechnology, 12 (8): 776, 2017. URL https:/​/​doi.org/​10.1038/​nnano.2017.101.
https:/​/​doi.org/​10.1038/​nnano.2017.101

[11] Gregory S MacCabe, Hengjiang Ren, Jie Luo, Justin D Cohen, Hengyun Zhou, Alp Sipahigil, Mohammad Mirhosseini, and Oskar Painter. Phononic bandgap nano-acoustic cavity with ultralong phonon lifetime. arXiv preprint arXiv:1901.04129, 2019. URL https:/​/​doi.org/​10.48550/​arXiv.1901.04129.
https:/​/​doi.org/​10.48550/​arXiv.1901.04129
arXiv:1901.04129

[12] Ewold Verhagen, Samuel Deléglise, Stefan Weis, Albert Schliesser, and Tobias J Kippenberg. Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode. Nature, 482 (7383): 63, 2012. URL https:/​/​doi.org/​10.1038/​nature10787.
https:/​/​doi.org/​10.1038/​nature10787

[13] Jasper Chan, T. P. Mayer Alegre, Amir H. Safavi-Naeini, Jeff T. Hill, Alex Krause, Simon Gröblacher, Markus Aspelmeyer, and Oskar Painter. Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature, 478: 89–92, 2011. URL https:/​/​doi.org/​10.1038/​nature10461.
https:/​/​doi.org/​10.1038/​nature10461

[14] J. D. Teufel, T. Donner, Dale 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: 359–363, 2011a. URL https:/​/​doi.org/​10.1038/​nature10261.
https:/​/​doi.org/​10.1038/​nature10261

[15] R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P.-L. Yu, K. W. Lehnert, and C. A. Regal. Laser cooling of a micromechanical membrane to the quantum backaction limit. Phys. Rev. Lett., 116: 063601, Feb 2016. URL https:/​/​doi.org/​10.1103/​PhysRevLett.116.063601.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.063601

[16] A. D. O’Connell, M. Hofheinz, M. Ansmann, Radoslaw C. Bialczak, M. Lenander, Erik Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, John M. Martinis, and A. N. Cleland. Quantum ground state and single-phonon control of a mechanical resonator. Nature, 464: 697–703, 2010. URL https:/​/​doi.org/​10.1038/​nature08967.
https:/​/​doi.org/​10.1038/​nature08967

[17] A. Schliesser, P. Del'Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg. Radiation pressure cooling of a micromechanical oscillator using dynamical backaction. Phys. Rev. Lett., 97: 243905, Dec 2006. URL https:/​/​doi.org/​10.1103/​PhysRevLett.97.243905.
https:/​/​doi.org/​10.1103/​PhysRevLett.97.243905

[18] Žarko Zobenica, Rob W Heijden, Maurangelo Petruzzella, Francesco Pagliano, Rick Leijssen, Tian Xia, Leonardo Midolo, Michele Cotrufo, YongJin Cho, Frank WM Otten, et al. Integrated nano-opto-electro-mechanical sensor for spectrometry and nanometrology. Nature communications, 8 (1): 2216, 2017. URL https:/​/​doi.org/​10.1038/​s41467-017-02392-5.
https:/​/​doi.org/​10.1038/​s41467-017-02392-5

[19] JD Teufel, T Donner, MA Castellanos-Beltran, JW Harlow, and KW Lehnert. Nanomechanical motion measured with an imprecision below that at the standard quantum limit. Nature nanotechnology, 4 (12): 820–823, 2009. URL https:/​/​doi.org/​10.1038/​nnano.2009.343.
https:/​/​doi.org/​10.1038/​nnano.2009.343

[20] Eduardo Gil-Santos, Daniel Ramos, Javier Martinez, Marta Fernandez-Regulez, Ricardo Garcia, Alvaro San Paulo, Montserrat Calleja, and Javier Tamayo. Nanomechanical mass sensing and stiffness spectrometry based on two-dimensional vibrations of resonant nanowires. Nat Nano, 5: 641–645, 2010. URL https:/​/​doi.org/​10.1038/​nnano.2010.151.
https:/​/​doi.org/​10.1038/​nnano.2010.151

[21] MS Hanay, S Kelber, AK Naik, D Chi, S Hentz, EC Bullard, E Colinet, L Duraffourg, and ML Roukes. Single-protein nanomechanical mass spectrometry in real time. Nature nanotechnology, 7 (9): 602–608, 2012. URL https:/​/​doi.org/​10.1038/​nnano.2012.119.
https:/​/​doi.org/​10.1038/​nnano.2012.119

[22] P. Weber, J. Güttinger, A. Noury, J. Vergara-Cruz, and A. Bachtold. Force sensitivity of multilayer graphene optomechanical devices. Nature Communications, 7: 12496, 2016. URL https:/​/​doi.org/​10.1038/​ncomms12496.
https:/​/​doi.org/​10.1038/​ncomms12496

[23] Daniel WC Brooks, Thierry Botter, Sydney Schreppler, Thomas P Purdy, Nathan Brahms, and Dan M Stamper-Kurn. Non-classical light generated by quantum-noise-driven cavity optomechanics. Nature, 488 (7412): 476–480, 2012. URL https:/​/​doi.org/​10.1038/​nnano.2012.119.
https:/​/​doi.org/​10.1038/​nnano.2012.119

[24] Amir H Safavi-Naeini, Simon Gröblacher, Jeff T Hill, Jasper Chan, Markus Aspelmeyer, and Oskar Painter. Squeezed light from a silicon micromechanical resonator. Nature, 500 (7461): 185–189, 2013. URL https:/​/​doi.org/​10.1038/​nature12307.
https:/​/​doi.org/​10.1038/​nature12307

[25] T. P. Purdy, P.-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal. Strong optomechanical squeezing of light. Phys. Rev. X, 3: 031012, Sep 2013. URL https:/​/​doi.org/​10.1103/​PhysRevX.3.031012.
https:/​/​doi.org/​10.1103/​PhysRevX.3.031012

[26] Emma Edwina Wollman, CU Lei, AJ Weinstein, J Suh, A Kronwald, F Marquardt, AA Clerk, and KC Schwab. Quantum squeezing of motion in a mechanical resonator. Science, 349 (6251): 952–955, 2015. URL https:/​/​doi.org/​10.1126/​science.aac5138.
https:/​/​doi.org/​10.1126/​science.aac5138

[27] C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää. Noiseless quantum measurement and squeezing of microwave fields utilizing mechanical vibrations. Phys. Rev. Lett., 118: 103601, Mar 2017. URL https:/​/​doi.org/​10.1103/​PhysRevLett.118.103601.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.103601

[28] Mauro Paternostro. Engineering nonclassicality in a mechanical system through photon subtraction. Phys. Rev. Lett., 106: 183601, May 2011. URL https:/​/​doi.org/​10.1103/​PhysRevLett.106.183601.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.183601

[29] M. R. Vanner, M. Aspelmeyer, and M. S. Kim. Quantum state orthogonalization and a toolset for quantum optomechanical phonon control. Phys. Rev. Lett., 110: 010504, Jan 2013. URL https:/​/​doi.org/​10.1103/​PhysRevLett.110.010504.
https:/​/​doi.org/​10.1103/​PhysRevLett.110.010504

[30] Antonio S. Coelho, Luca S. Costanzo, Alessandro Zavatta, Catherine Hughes, M. S. Kim, and Marco Bellini. Universal continuous-variable state orthogonalizer and qubit generator. Phys. Rev. Lett., 116: 110501, Mar 2016. URL https:/​/​doi.org/​10.1103/​PhysRevLett.116.110501.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.110501

[31] T. J. Milburn, M. S. Kim, and M. R. Vanner. Nonclassical-state generation in macroscopic systems via hybrid discrete-continuous quantum measurements. Phys. Rev. A, 93: 053818, May 2016. URL https:/​/​doi.org/​10.1103/​PhysRevA.93.053818.
https:/​/​doi.org/​10.1103/​PhysRevA.93.053818

[32] Lydia A. Kanari-Naish, Jack Clarke, Sofia Qvarfort, and Michael R. Vanner. Non-gaussian mechanical entanglement with nonlinear optomechanics: generation and verification, 2021. URL https:/​/​doi.org/​10.1088/​2058-9565/​ac6dfd.
https:/​/​doi.org/​10.1088/​2058-9565/​ac6dfd

[33] Jack Clarke and Michael R Vanner. Growing macroscopic superposition states via cavity quantum optomechanics. Quantum Science and Technology, 4 (1): 014003, sep 2018. URL https:/​/​doi.org/​10.1088/​2058-9565/​aada1d.
https:/​/​doi.org/​10.1088/​2058-9565/​aada1d

[34] Christoffer B Møller, Rodrigo A Thomas, Georgios Vasilakis, Emil Zeuthen, Yeghishe Tsaturyan, Kasper Jensen, Albert Schliesser, Klemens Hammerer, and Eugene S Polzik. Back action evading quantum measurement of motion in a negative mass reference frame. Nature, 547 (2): 191, 2017. URL https:/​/​doi.org/​http:/​/​doi.org/​10.1038/​nature22980.
https:/​/​doi.org/​10.1038/​nature22980

[35] C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, A. A. Clerk, M. J. Woolley, and M. A. Sillanpää. Quantum backaction evading measurement of collective mechanical modes. Phys. Rev. Lett., 117: 140401, Sep 2016. URL https:/​/​doi.org/​10.1103/​PhysRevLett.117.140401.
https:/​/​doi.org/​10.1103/​PhysRevLett.117.140401

[36] Massimiliano Rossi, David Mason, Junxin Chen, Yeghishe Tsaturyan, and Albert Schliesser. Measurement-based quantum control of mechanical motion. Nature, 563 (7729): 53–58, 2018. URL https:/​/​doi.org/​10.1038/​s41586-018-0643-8.
https:/​/​doi.org/​10.1038/​s41586-018-0643-8

[37] Abbott et al. Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett., 116: 061102, Feb 2016. URL https:/​/​doi.org/​10.1103/​PhysRevLett.116.061102.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.061102

[38] Abbott et al. Gw190521: A binary black hole merger with a total mass of $150\text{ }\text{ }{M}_{{\bigodot}}$. Phys. Rev. Lett., 125: 101102, Sep 2020. URL https:/​/​doi.org/​10.1103/​PhysRevLett.125.101102.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.101102

[39] Samuel L. Braunstein and Peter van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77: 513–577, Jun 2005. URL https:/​/​doi.org/​10.1103/​RevModPhys.77.513.
https:/​/​doi.org/​10.1103/​RevModPhys.77.513

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

[41] J. J. Viennot, X. Ma, and K. W. Lehnert. Phonon-number-sensitive electromechanics. Phys. Rev. Lett., 121: 183601, Oct 2018. URL https:/​/​doi.org/​10.1103/​PhysRevLett.121.183601.
https:/​/​doi.org/​10.1103/​PhysRevLett.121.183601

[42] Xizheng Ma, Jeremie J. Viennot, Shlomi Kotler, John D. Teufel, and Konrad W. Lehnert. Nonclassical energy squeezing of a macroscopic mechanical oscillator, 2020. URL https:/​/​doi.org/​10.1038/​s41567-020-01102-1.
https:/​/​doi.org/​10.1038/​s41567-020-01102-1

[43] L. R. Sletten, B. A. Moores, J. J. Viennot, and K. W. Lehnert. Resolving phonon fock states in a multimode cavity with a double-slit qubit. Phys. Rev. X, 9: 021056, Jun 2019. URL https:/​/​doi.org/​10.1103/​PhysRevX.9.021056.
https:/​/​doi.org/​10.1103/​PhysRevX.9.021056

[44] GA Brawley, MR Vanner, Peter Emil Larsen, Silvan Schmid, Anja Boisen, and WP Bowen. Nonlinear optomechanical measurement of mechanical motion. Nature communications, 7 (1): 1–7, 2016. URL https:/​/​doi.org/​10.1038/​ncomms10988.
https:/​/​doi.org/​10.1038/​ncomms10988

[45] Rick Leijssen, Giada R La Gala, Lars Freisem, Juha T Muhonen, and Ewold Verhagen. Nonlinear cavity optomechanics with nanomechanical thermal fluctuations. Nature communications, 8: ncomms16024, 2017. URL https:/​/​doi.org/​10.1038/​ncomms16024.
https:/​/​doi.org/​10.1038/​ncomms16024

[46] D. Cattiaux, X. Zhou, S. Kumar, I. Golokolenov, R. R. Gazizulin, A. Luck, L. Mercier de Lépinay, M. Sillanpää, A. D. Armour, A. Fefferman, and E. Collin. Beyond linear coupling in microwave optomechanics. Phys. Rev. Research, 2: 033480, Sep 2020. URL https:/​/​doi.org/​10.1103/​PhysRevResearch.2.033480.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.033480

[47] Alpo Välimaa, Wayne Crump, Mikael Kervinen, and Mika A. Sillanpää. Multiphonon transitions in a quantum electromechanical system, 2021. URL https:/​/​doi.org/​10.1103/​PhysRevApplied.17.064003.
https:/​/​doi.org/​10.1103/​PhysRevApplied.17.064003

[48] Luca Dellantonio, Oleksandr Kyriienko, Florian Marquardt, and Anders S Sørensen. Quantum nondemolition measurement of mechanical motion quanta. Nature communications, 9 (1): 1–8, 2018. URL https:/​/​doi.org/​10.1038/​s41467-018-06070-y.
https:/​/​doi.org/​10.1038/​s41467-018-06070-y

[49] A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin. Cooling and squeezing via quadratic optomechanical coupling. Phys. Rev. A, 82: 021806, Aug 2010. URL https:/​/​doi.org/​10.1103/​PhysRevA.82.021806.
https:/​/​doi.org/​10.1103/​PhysRevA.82.021806

[50] Haixing Miao, Stefan Danilishin, Thomas Corbitt, and Yanbei Chen. Standard quantum limit for probing mechanical energy quantization. Phys. Rev. Lett., 103: 100402, Sep 2009. URL https:/​/​doi.org/​10.1103/​PhysRevLett.103.100402.
https:/​/​doi.org/​10.1103/​PhysRevLett.103.100402

[51] Yariv Yanay and Aashish A Clerk. Shelving-style qnd phonon-number detection in quantum optomechanics. New Journal of Physics, 19 (3): 033014, 2017. URL https:/​/​doi.org/​10.1088/​1367-2630/​aa6206.
https:/​/​doi.org/​10.1088/​1367-2630/​aa6206

[52] B. D. Hauer, A. Metelmann, and J. P. Davis. Phonon quantum nondemolition measurements in nonlinearly coupled optomechanical cavities. Phys. Rev. A, 98: 043804, Oct 2018. URL https:/​/​doi.org/​10.1103/​PhysRevA.98.043804.
https:/​/​doi.org/​10.1103/​PhysRevA.98.043804

[53] A M Jayich, J C Sankey, B M Zwickl, C Yang, J D Thompson, S M Girvin, A A Clerk, F Marquardt, and J G E Harris. Dispersive optomechanics: a membrane inside a cavity. New Journal of Physics, 10 (9): 095008, 2008. URL https:/​/​doi.org/​10.1088/​1367-2630/​10/​9/​095008.
https:/​/​doi.org/​10.1088/​1367-2630/​10/​9/​095008

[54] Max Ludwig, Amir H. Safavi-Naeini, Oskar Painter, and Florian Marquardt. Enhanced quantum nonlinearities in a two-mode optomechanical system. Phys. Rev. Lett., 109: 063601, Aug 2012. URL https:/​/​doi.org/​10.1103/​PhysRevLett.109.063601.
https:/​/​doi.org/​10.1103/​PhysRevLett.109.063601

[55] Shiqian Ding, Gleb Maslennikov, Roland Hablützel, and Dzmitry Matsukevich. Cross-kerr nonlinearity for phonon counting. Phys. Rev. Lett., 119: 193602, Nov 2017. URL https:/​/​doi.org/​10.1103/​PhysRevLett.119.193602.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.193602

[56] F. Pistolesi, A. N. Cleland, and A. Bachtold. Proposal for a nanomechanical qubit, 2020. URL https:/​/​doi.org/​10.1103/​PhysRevX.11.031027.
https:/​/​doi.org/​10.1103/​PhysRevX.11.031027

[57] Jun-Ya Yang, Dong-Yang Wang, Cheng-Hua Bai, Si-Yu Guan, Xiao-Yuan Gao, Ai-Dong Zhu, and Hong-Fu Wang. Ground-state cooling of mechanical oscillator via quadratic optomechanical coupling with two coupled optical cavities. Optics express, 27 (16): 22855–22867, 2019. URL https:/​/​doi.org/​10.1364/​OE.27.022855.
https:/​/​doi.org/​10.1364/​OE.27.022855

[58] JD Teufel, Dale Li, MS Allman, K Cicak, AJ Sirois, JD Whittaker, and RW Simmonds. Circuit cavity electromechanics in the strong-coupling regime. Nature, 471 (7337): 204–208, 2011b. URL https:/​/​doi.org/​10.1038/​nature0989.
https:/​/​doi.org/​10.1038/​nature0989

[59] Georg Heinrich and Florian Marquardt. Coupled multimode optomechanics in the microwave regime. EPL (Europhysics Letters), 93 (1): 18003, 2011. URL https:/​/​doi.org/​10.1209/​0295-5075/​93/​18003.
https:/​/​doi.org/​10.1209/​0295-5075/​93/​18003

[60] Leonardo Midolo, Albert Schliesser, and Andrea Fiore. Nano-opto-electro-mechanical systems. Nature nanotechnology, 13 (1): 11–18, 2018. URL https:/​/​doi.org/​10.1038/​s41565-017-0039-1.
https:/​/​doi.org/​10.1038/​s41565-017-0039-1

[61] P. Weber, J. Güttinger, I. Tsioutsios, D. E. Chang, and A. Bachtold. Coupling graphene mechanical resonators to superconducting microwave cavities. Nano Letters, 14 (5): 2854–2860, 2014. URL https:/​/​doi.org/​10.1021/​nl500879k.
https:/​/​doi.org/​10.1021/​nl500879k

[62] Emil Zeuthen. Electro-Optomechanical Transduction and Quantum Hard-Sphere Model for Dissipative Rydberg-EIT Media. PhD thesis, The Niels Bohr Institute, Faculty of Science, University of Copenhagen, 2015.

[63] Jaroslaw Adam Miszczak, Zbigniew Puchała, Pawel Horodecki, Armin Uhlmann, and Karol Życzkowski. Sub–and super–fidelity as bounds for quantum fidelity. arXiv preprint arXiv:0805.2037, 2008. URL https:/​/​doi.org/​10.48550/​arXiv.0805.2037.
https:/​/​doi.org/​10.48550/​arXiv.0805.2037
arXiv:0805.2037

[64] Richard Jozsa. Fidelity for mixed quantum states. Journal of modern optics, 41 (12): 2315–2323, 1994. URL https:/​/​doi.org/​10.1080/​09500349414552171.
https:/​/​doi.org/​10.1080/​09500349414552171

[65] Xuefeng Song, Mika Oksanen, Mika A. Sillanpää, H. G. Craighead, J. M. Parpia, and Pertti J. Hakonen. Stamp transferred suspended graphene mechanical resonators for radio frequency electrical readout. Nano Letters, 12 (1): 198–202, 2012. URL https:/​/​doi.org/​10.1021/​nl203305q. PMID: 22141577.
https:/​/​doi.org/​10.1021/​nl203305q

[66] Johannes Güttinger, Adrien Noury, Peter Weber, Axel Martin Eriksson, Camille Lagoin, Joel Moser, Christopher Eichler, Andreas Wallraff, Andreas Isacsson, and Adrian Bachtold. Energy-dependent path of dissipation in nanomechanical resonators. Nat Nano, advance online publication, 2017. URL https:/​/​doi.org/​10.1038/​nnano.2017.86.
https:/​/​doi.org/​10.1038/​nnano.2017.86

[67] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold. A nanomechanical mass sensor with yoctogram resolution. Nat Nano, 7: 301–304, 2012. URL https:/​/​doi.org/​10.1038/​nnano.2012.42.
https:/​/​doi.org/​10.1038/​nnano.2012.42

[68] J. Moser, A. Eichler, J. Güttinger, M. I. Dykman, and A. Bachtold. Nanotube mechanical resonators with quality factors of up to 5 million. Nat Nano, 9: 1007–1011, 2014. URL https:/​/​doi.org/​10.1038/​nnano.2014.234.
https:/​/​doi.org/​10.1038/​nnano.2014.234

[69] T Rocheleau, T Ndukum, C Macklin, JB Hertzberg, AA Clerk, and KC Schwab. Preparation and detection of a mechanical resonator near the ground state of motion. Nature, 463 (7277): 72, 2010. URL https:/​/​doi.org/​10.1038/​nature08681.
https:/​/​doi.org/​10.1038/​nature08681

[70] Mohammad Mirhosseini, Eunjong Kim, Xueyue Zhang, Alp Sipahigil, Paul B Dieterle, Andrew J Keller, Ana Asenjo-Garcia, Darrick E Chang, and Oskar Painter. Cavity quantum electrodynamics with atom-like mirrors. Nature, 569 (7758): 692–697, 2019. URL https:/​/​doi.org/​10.1038/​s41586-019-1196-1.
https:/​/​doi.org/​10.1038/​s41586-019-1196-1

[71] Dalziel J Wilson, Vivishek Sudhir, Nicolas Piro, Ryan Schilling, Amir Ghadimi, and Tobias J Kippenberg. Measurement-based control of a mechanical oscillator at its thermal decoherence rate. Nature, 524 (7565): 325–329, 2015. URL https:/​/​doi.org/​10.1038/​nature14672.
https:/​/​doi.org/​10.1038/​nature14672

[72] E. L. Hahn. Spin echoes. Phys. Rev., 80: 580–594, Nov 1950. URL https:/​/​doi.org/​10.1103/​PhysRev.80.580.
https:/​/​doi.org/​10.1103/​PhysRev.80.580

[73] Zhen Yi, Gao-xiang Li, Shao-ping Wu, and Ya-ping Yang. Ground-state cooling of an oscillator in a hybrid atom-optomechanical system. Optics express, 22 (17): 20060–20075, 2014. URL https:/​/​doi.org/​10.1364/​OE.22.020060.
https:/​/​doi.org/​10.1364/​OE.22.020060

[74] P. Sekatski, E. Oudot, P. Caspar, R. Thew, and N. Sangouard. Benchmarking single-photon sources from an auto-correlation measurement, 2021. URL https:/​/​doi.org/​10.22331/​q-2022-12-13-875.
https:/​/​doi.org/​10.22331/​q-2022-12-13-875

[75] Seth Lloyd and Samuel L. Braunstein. Quantum computation over continuous variables. Phys. Rev. Lett., 82: 1784–1787, Feb 1999. URL https:/​/​doi.org/​10.1103/​PhysRevLett.82.1784.
https:/​/​doi.org/​10.1103/​PhysRevLett.82.1784

[76] Norbert Schuch and Jens Siewert. Natural two-qubit gate for quantum computation using the $\mathrm{XY}$ interaction. Phys. Rev. A, 67: 032301, Mar 2003. URL https:/​/​doi.org/​10.1103/​PhysRevA.67.032301.
https:/​/​doi.org/​10.1103/​PhysRevA.67.032301

[77] Rolf Landauer. Johnson-nyquist noise derived from quantum mechanical transmission. Physica D: Nonlinear Phenomena, 38 (1-3): 226–229, 1989. URL https:/​/​doi.org/​10.1016/​0167-2789(89)90197-8.
https:/​/​doi.org/​10.1016/​0167-2789(89)90197-8

[78] Daniel F Walls and Gerard J Milburn. Quantum optics. Springer Science & Business Media, 2007.

Cited by

[1] Agostino Migliore and Antonino Messina, "Controlling the charge-transfer dynamics of two-level systems around avoided crossings", The Journal of Chemical Physics 160 8, 084112 (2024).

[2] Jan Wójcik and Grzegorz Chimczak, "Electrically coupled optomechanical cavities as a tool for quantum nondemolition measurement", Physics Letters A 490, 129187 (2023).

[3] Vincent Dumont, Hoi-Kwan Lau, Aashish A. Clerk, and Jack C. Sankey, "Asymmetry-Based Quantum Backaction Suppression in Quadratic Optomechanics", Physical Review Letters 129 6, 063604 (2022).

[4] James M. L. Miller, Ariosto Gomez-Franco, Dongsuk D. Shin, Hyun-Keun Kwon, and Thomas W. Kenny, "Amplitude stabilization of micromechanical oscillators using engineered nonlinearity", Physical Review Research 3 3, 033268 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-19 01:48:02) and SAO/NASA ADS (last updated successfully 2024-04-19 01:48:03). The list may be incomplete as not all publishers provide suitable and complete citation data.