Abstract

We propose how to achieve quantum nonreciprocity via unconventional photon blockade (UPB) in a compound device consisting of an optical harmonic resonator and a spinning optomechanical resonator. We show that, even with very weak single-photon nonlinearity, nonreciprocal UPB can emerge in this system, i.e., strong photon antibunching can emerge only by driving the device from one side but not from the other side. This nonreciprocity results from the Fizeau drag, leading to different splitting of the resonance frequencies for the optical counter-circulating modes. Such quantum nonreciprocal devices can be particularly useful in achieving back-action-free quantum sensing or chiral photonic communications.

© 2019 Chinese Laser Press

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2019 (2)

A. Y. Song, Y. Shi, Q. Lin, and S. Fan, “Direction-dependent parity-time phase transition and non-reciprocal directional amplification with dynamic gain–loss modulation,” Phys. Rev. A 99, 013824 (2019).
[Crossref]

G. Enzian, M. Szczykulska, J. Silver, L. Del Bino, S. Zhang, I. A. Walmsley, P. Del’Haye, and M. R. Vanner, “Observation of Brillouin optomechanical strong coupling with an 11  GHz mechanical mode,” Optica 6, 7–14 (2019).
[Crossref]

2018 (19)

Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref]

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X. B. Zou, G. C. Guo, C.-L. Zou, and C. H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. M. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
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Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
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H. Zhang, F. Salf, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26, 25199–25210 (2018).
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L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect,” Optica 5, 279–282 (2018).
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S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018).
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K. Y. Xia, F. Nori, and M. Xiao, “Cavity-free optical isolators and circulators using a chiral cross-Kerr nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018).
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C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569–572 (2018).
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H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
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H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the unconventional photon blockade,” Phys. Rev. Lett. 121, 043601 (2018).
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C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
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2017 (10)

H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A 96, 053810 (2017).
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H. Flayac and V. Savona, “Nonclassical statistics from a polaritonic Josephson junction,” Phys. Rev. A 95, 043838 (2017).
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H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photon. Res. 5, 367–371 (2017).
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D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
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N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
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Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033907 (2017).
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H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional points in random-defect phonon lasers,” Phys. Rev. Appl. 8, 044020 (2017).
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2016 (7)

H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
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V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
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V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
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M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
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H. Xie, G.-W. Lin, X. Chen, Z.-H. Chen, and X.-M. Lin, “Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling,” Phys. Rev. A 93, 063860 (2016).
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2015 (4)

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
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K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. 114, 233601 (2015).
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A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).
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Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
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2014 (8)

K. Y. Xia, G. W. Lu, G. W. Lin, Y. Q. Cheng, Y. P. Niu, S. Q. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90, 043802 (2014).
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B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
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X.-W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
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W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
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H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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2013 (8)

J. R. Johansson, P. D. Nation, and F. Nori, “Qutip 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
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J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
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S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
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X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B 46, 035502 (2013).
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N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
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I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
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2012 (4)

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
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T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012).
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A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
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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 (London) 482, 63–67 (2012).
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2011 (8)

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 (London) 475, 359–363 (2011).
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L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
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M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
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C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
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A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
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A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
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I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
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2010 (4)

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
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J.-Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
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T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
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I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
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2009 (3)

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009).
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V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
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Z. R. Gong, H. Ian, Y.-X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
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2008 (1)

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
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2006 (1)

A. Miranowicz and W. Leoński, “Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers,” J. Phys. B 39, 1683–1700 (2006).
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2005 (1)

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
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2004 (1)

W. Leoński and A. Miranowicz, “Kerr nonlinear coupler and entanglement,” J. Opt. B 6, S37–S42 (2004).
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2003 (3)

W. Leoński and A. Miranowicz, “Quantum-optical states in finite-dimensional Hilbert space. II. State generation,” Adv. Chem. Phys. 119, 155–193 (2003).
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S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
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K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
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2000 (1)

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1998 (1)

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1992 (1)

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1988 (1)

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B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. Schoelkopf, and M. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Phys. Rev. Lett. 112, 167701 (2014).
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Abdumalikov, A. A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
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Achouri, K.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
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J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
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Aiello, G.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
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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 (London) 475, 359–363 (2011).
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C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
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D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
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Andreani, L. C.

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
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Aprili, M.

C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the unconventional photon blockade in the microwave domain,” Phys. Rev. Lett. 121, 043602 (2018).
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I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
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A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 163601 (2012).
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A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
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Baker, C.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
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H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Purification of a single-photon nonlinearity,” Nat. Commun. 7, 12578 (2016).
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M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
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Bang, J.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Phys. Rev. Lett. 121, 033603 (2018).
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S. Barzanjeh, M. Aquilina, and A. Xuereb, “Manipulating the flow of thermal noise in quantum devices,” Phys. Rev. Lett. 120, 060601 (2018).
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C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
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V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
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V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
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B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
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P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
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Bino, L. D.

Birnbaum, K. M.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
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C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
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K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005).
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Figures (6)

Fig. 1.
Fig. 1. Nonreciprocal UPB in a coupled-resonator system. Spinning the OM (Kerr-type) resonator results in different Fizeau drag ΔF for the counter-circulating whispering-gallery modes of the resonator. (a) By driving the system from the left-hand side, the direct excitation from state |1,0 to state |2,0 (red dotted arrow) will be forbidden by destructive quantum interference with the other paths drawn by green arrows, leading to photon antibunching. (b) Photon bunching occurs when the system is driven from the right side, due to lack of complete destructive quantum interference between the indicated levels (drawn by crossed green dotted arrows). Here, δ=g2/ωm is the energy shift induced by the OM nonlinearity.
Fig. 2.
Fig. 2. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) with (a) Ω=0 and (b) Ω=12  kHz, which is found numerically (solid curves) and analytically (dotted curve). The PB can be generated (red curves) or suppressed (blue curves) for different driving directions, which can be seen more clearly in panel (c). The other parameters are g/κ=0.63, ωm/κ=10 [91], J/κ=3, T=0.1  mK (case 1), and g/κ=0.1 [28], ωm/κ=30 [92], J/κ=20, T=1  mK (case 2).
Fig. 3.
Fig. 3. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) at various angular velocities Ω upon driving the device from (a) the right-hand side or (b) the left-hand side. The dashed curves show our approximate analytical results, given in Eq. (12), whereas the solid curves are our numerical solutions. The other parameters are the same as those in Fig. 2 (case 1).
Fig. 4.
Fig. 4. Correlation function gL(2)(0) in logarithmic scale [i.e., log10gL(2)(0)] versus (a) radiation-pressure coupling g/κ (in units of cavity loss rate κ=κL=κR) and optical detuning Δ/κ, and (b) coupling strength of the resonators J/κ and radiation-pressure coupling g/κ for optical detuning of Δ/κ=0.05. The angular velocity is Ω=12  kHz and the white dashed curve corresponds to gL(2)(0)=1. The other parameters are the same as those in Fig. 3.
Fig. 5.
Fig. 5. Correlation function gL(2)(0) versus optical detuning Δ/κ (in units of cavity loss rate κL=κR=κ) with varied mean thermal phonon numbers nth for various angular velocities Ω, and the resulting Fizeau shifts ΔF. The other parameters are the same as those in Fig. 4.
Fig. 6.
Fig. 6. (a) Correlation function gL(2)(0) versus effective temperature T of the environment of the mechanical resonator for three values of Fizeau shift ΔF (ΔF>0, ΔF=0, and ΔF<0) for optimal values of Δopt and gopt. The other parameters are set the same as in case 2 in Fig. 2. Also shown is the correlation function gL(2)(0) versus T for various values of (b) spinning frequency, (c) mechanical decay, and (d) cavity decay, assuming the device is driven from the left-hand side and optical detuning is fixed at the optimal values.

Equations (55)

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ΔF=±nrΩωRc(11n2λndndλ)=±ηΩ,
H=ΔLaLaL+(ΔR+ΔF)aRaR+ωmbb+J(aLaR+aRaL)+gaRaR(b+b)+iϵd(aLaL),
ddtq=ωmp,ddtp=ωmqgbaRaRγm2p+ξ,ddtaL=(κL2+iΔL)aLiJaR+ϵd+κLaL,in,ddtaR=(κR2+iΔR)aRiJaLigbqaR+κLaR,in,
ξ(t)ξ(t)=12πdωeiω(tt)Γm(ω),
Γm(ω)=ωγm2ωm[1+coth(ω2kBT)],
aK,in(t)aK,in(t)=0,aK,in(t)aK,in(t)=δ(tt),
ddtδq=ωmδp,ddtδp=ωmδqgb(β*δaR+βδaR)γm2δp+ξ,ddtδaL=(κL2+iΔL)δaLiJδaR+κLaL,in,ddtδaR=(κR2+iΔR)δaRiJδaLigbqsδaRigbβδq+κRaR,in.
δaL(ω)=E(ω)aL,in(ω)+F(ω)aL,in(ω)+G(ω)aR,in(ω)+H(ω)aR,in(ω)+Q(ω)ξ(ω),
E(ω)=κLA1(ω)A5(ω),F(ω)=κLA2(ω)A5(ω),G(ω)=κRA3(ω)A5(ω),H(ω)=κRA4(ω)A5(ω),Q(ω)=igbχ(ω)ωmA5(ω)[βA3(ω)+β*A4(ω)],
A1(ω)=[(κR2+iω)2+ΔR2]V1(ω)gb4|β|4[χ(ω)ωm]2V1(ω)+J2V2+,A2(ω)=iJ2gb2β2χ(ω)ωm,A3(ω)=iJV1(ω)V2iJ3,A4(ω)=Jgb2β2χ(ω)ωmV1(ω),A5(ω)=V1+A1(ω)+iJA3(ω),
ΔR=ΔR+gbqsgb2|β|2χ(ω),χ(ω)=ωm2/(ωm2ω2+iωγm2),V1±(ω)=κL2±i(ΔLω),V2±(ω)=κR2±i(ΔRω).
gL(2)(0)=|α|4+4|α|2R1+2Re[α*2R2]+R3(|α|2+R1)2,
δaL±(t)δaL(t)=12π+XaL±aLdω,
δaL+(t)=δaL(t),δaL(t)=δaL(t),andXaLaL=|Q(ω)|2Γm(ω)+|F(ω)|2+|H(ω)|2,XaLaL=Q(ω)Q(ω)Γm(ω)+E(ω)F(ω)+G(ω)H(ω).
ρ˙=1i[H,ρ]+κL2L[aL](ρ)+κR2L[aR](ρ)+γm2(n¯m+1)L[b](ρ)+γm2n¯mL[b](ρ),
|1,02ϵd|2,0,|1,0J|0,1ϵd|1,12J|2,0.
|φ=C00|0,0+C10|1,0+C01|0,1+C20|2,0+C11|1,1+C02|0,2,
Δopta3+sgn(E)λ1λ24a4,gopt=ωm[Δopt(4Δopt2+5κ2)+ΔFλ3]2(2J2κ2)+2ΔFλ4,
H=H0+Hin+Hdr,H0=ωLaLaL+(ωR+ΔF)aRaR+ωmbb,Hin=J(aLaR+aRaL)+gaRaR(b+b),Hdr=iϵd(aLeiωdtaLeiωdt),
Heff=UHU=ωLaLaL+(ωR+ΔF)aRaRδ(aRaR)2+J[aLaReδ(bb)+aLaReδ(bb)]+iϵd(aLeiωdtaLeiωdt),
Heff=ωLaLaL+(ωR+ΔF)aRaRδ(aRaR)2+J(aLaR+aLaR)+iϵd(aLeiωdtaLeiωdt).
0=(κL2+iΔL)α+iJβϵd,0=[κR2+i(ΔR+gbqs)]βiJα,0=ωmqsgb|β|2.
b3qs3+b2qs2+b1qs+b0=0,
b0=gbJ2ϵd2,b1=ωm(κLκR4+J2)2+ωm(κLΔR2+κRΔL2)2ωmΔLΔR(κLκR2+2J2ΔLΔR),b2=2ωmgb[κL2ΔR4+ΔL(ΔLΔRJ2)],b3=ωmgb2(κL24+ΔL2).
ddtδq=ωmδp,ddtδp=ωmδqgb(β*δaR+βδaR)γm2δp+ξ,ddtδaL=(κL2+iΔL)δaLiJδaR+κLaL,in,ddtδaR=(κR2+iΔR)δaRiJδaLigbqsδaRigbβδq+κRaR,in,
iωδaL(ω)=(κL2+iΔL)δaL(ω)iJδaR(ω)+κLaL,in(ω),iωδaR(ω)=(κR2+iΔR)δaL(ω)iJδaR(ω)igbβδq(ω)+κRaR,in(ω),iωδq(ω)=ωmδp(ω),iωδp(ω)=ωmδq(ω)gb[β*δaR(ω)+βδaR(ω)]γm2δp(ω)+ξ(ω),
δq(ω)=gbβ*χ(ω)δaR(ω)gbβχ(ω)δaR(ω)+χ(ω)ξ(ω),
χ(ω)=ωmωm2ω2+iωγm/2.
M(ω)δaR(ω)=igb2β2χ(ω)δaR(ω)igbβχ(ω)ξ(ω)iJδL(ω)+κRaR,in(ω),
M(ω)=κR2+iω+iΔRi|β|2gb2χ(ω).
iωδaL(ω)=(κL2iΔL)δaL(ω)+iJδaR(ω)+κLaL,in(ω),iωδaR(ω)=(κR2iΔR)δaR(ω)+iJδaR(ω)+igbβδq(ω)+κRaR,in(ω),iωδq(ω)=ωmδp(ω),iωδp(ω)=ωmδq(ω)gb[βδaR(ω)+β*δaR(ω)]γm2δp+ξ(ω),
N(ω)δaR(ω)=igb2β*2χ(ω)δaR(ω)+igbβ*χ(ω)ξ(ω)+iJδaL(ω)+κRaR,in(ω),
N(ω)=κR2+iωiΔR+i|β|2gb2χ(ω).
V(ω)δaL(ω)=iJδaR(ω)+κLaL,in(ω),
T(ω)δaR(ω)=iχ(ω)gb2β*2V(ω)δaR(ω)+iχ(ω)gbβ*V(ω)ξ(ω)+iJκLaL,in(ω)+κRV(ω)aR,in(ω),
FR(ω)δaR(ω)=χ2(ω)gb3β|β|2  V(ω)ξ(ω)igbβχ(ω)T(ω)ξ(ω)Jgb2β2χ(ω)κLaL,in+igb2β2χ(ω)κRV(ω)aR,in(ω)iJT(ω)aL,inκRT(ω)aR,in,
FL(ω)δaL(ω)=iJχ2(ω)gb3β|β|2  V(ω)ξ(ω)gbβχ(ω)JT(ω)ξ(ω)+iJ2gb2β2χ(ω)κLaL,in+Jgb2β2χ(ω)κRV(ω)aR,in(ω)iJκRT(ω)aR,inκL[M(ω)T(ω)U(ω)]aL,in,
FL(ω)=[M(ω)T(ω)U(ω)]V1(ω)+J2T(ω),U(ω)=χ2(ω)gb4|β|4(iω+κL2iΔL),V1(ω)=κL2+iω+iΔL.
δaL(ω)=E(ω)aL,in(ω)+F(ω)aL,in(ω)+G(ω)aR,in(ω)+H(ω)aR,in(ω)+Q(ω)ξ(ω).
δaL(ω)=E*(ω)aL,in(ω)+F*(ω)aL,in(ω)+G*(ω)aR,in(ω)+H*(ω)aR,in(ω)+Q*(ω)ξ(ω).
aL,in(ω)aL,in(ω)=12πaL,in(t)eiωtdt×12πaL,in(t)eiωtdt=δ(ω+ω),
aR,in(ω)aR,in(ω)=δ(ω+ω).
H=(ΔLiκL2)aLaL+(ΔRiκR2)aRaR+(ωmiγm2)bb+J(aLaR+aRaL)δ(aRaR)2+iϵd(aLaL),
|φ=C00|0,0+C10|1,0+C01|0,1+C20|2,0+C11|1,1+C02|0,2.
id|φdt=H|φ,
HC00|0,0=iϵdC00|1,0,HC10|1,0=δLC10|1,0+JC10|0,1+iϵdC10(2|2,0|0,0),HC01|0,1=δRC01|0,1+JC01|1,0+iϵdC01|1,1,HC20|2,0=2δLC20|2,0+2JC20|1,1+iϵdC20(3|3,02|1,0),HC11|1,1=δLC11|1,1+δRC11|1,1+2JC11(|2,0+|0,2)+iϵdC11(2|2,1|0,1),HC02|0,2=2δRC02|0,22δC02|0,2+2JC02(|1,1+iϵdC02|1,2,
C00t=ϵdC10,iC10t=δLC10+JC012iϵdC20,iC01t=(δRδ)C01+JC10iϵdC11,iC11t=δLC11+(δRδ)C11+2J(C02+C20)+iϵdC01,iC02t=2(δRδ)C02+2JC112δC02,iC20t=2(δRδ)C20+2JC11+2iϵdC10.
0=δLC10+JC01+iϵdC00,0=δRC01+JC10,
0=2δLC20+2JC11+i2ϵdC10,0=(δL+δR)C11+2JC20+2JC02+iϵdC01,0=2(δRδ)C02+2JC11,
0=κ2(2δ6Δ5ΔF2)+4Δ2(2Δ2δ5δΔF2)+4ΔF(4ΔΔF3δΔδΔF+ΔF2)4J2δ,0=8δΔ12Δ2+κ2+ΔF(6δ20Δ8ΔF).
a4Δ4+a3Δ3+a2Δ2+a1Δ+a0=0,
a0=κ(4J210ΔF2)(κ28ΔF2)2κ(κ444ΔF4),a1=8ΔF(6ΔF2κ+10J2κ+3),a2=8κ(2κ2+6J2+13ΔF2),a3=96ΔFκ,a4=32κ,
Δopta3+sgn(E)λ1λ24a4,gopt=ωm[Δopt(4Δopt2+5κ2)+ΔFλ3]2(2J2κ2)+2ΔFλ4,
λ1=D+z13+z233,λ2=2Dz13z23+z333,λ3=20Δopt28ΔoptΔF4ΔF2+5κ2,λ4=10Δopt2+3Δopt+2ΔF,
sgn(E)={1(E>0),1(E<0),z1,2=AD+3B±B24AC2,z3=D2D(z13+z23)+(z13+z23)23A,A=D23F,B=DF9E2,C=F23DE2,D=3a328a4a2,E=a33+4a4a3a28a42a1,F=3a34+16a42a2216a4a32a2+16a42a3a164a43a0.