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  • Featured in Physics
  • Open Access

Observation of Pairwise Level Degeneracies and the Quantum Regime of the Arrhenius Law in a Double-Well Parametric Oscillator

Nicholas E. Frattini*,†, Rodrigo G. Cortiñas*,‡, Jayameenakshi Venkatraman, Xu Xiao, Qile Su, Chan U. Lei, Benjamin J. Chapman, Vidul R. Joshi, S. M. Girvin et al.

Robert J. Schoelkopf, Shruti Puri, and Michel H. Devoret§

  • Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA

  • *These authors contributed equally to this work.
  • †Contact author: nick@nordquantique.ca
  • ‡Contact author: rodrigo.cortinas@yale.edu
  • §Contact author: michel.devoret@yale.edu
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Phys. Rev. X 14, 031040 – Published 3 September, 2024

DOI: https://doi.org/10.1103/PhysRevX.14.031040

Abstract

By applying a microwave drive to a specially designed Josephson circuit, we have realized a double-well model system: a Kerr oscillator submitted to a squeezing force. We have observed, for the first time, the spectroscopic fingerprint of a quantum double-well Hamiltonian when its barrier height is increased: a pairwise level kissing (coalescence) corresponding to the exponential reduction of tunnel splitting in the excited states as they sink under the barrier. The discrete levels in the wells also manifest themselves in the activation time across the barrier which, instead of increasing smoothly as a function of the barrier height, presents steps each time a pair of excited states is captured by the wells. This experiment illustrates the quantum regime of Arrhenius’s law, whose observation is made possible here by the unprecedented combination of low dissipation, time-resolved state control, 98.5% quantum nondemolition single shot measurement fidelity, and complete microwave control over all Hamiltonian parameters in the quantum regime. Direct applications to quantum computation and simulation are discussed.

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New Quantum Effect in Textbook Chemistry Law

Published 3 September, 2024

The observation of quantum modifications to a well-known chemical law could lead to performance improvements for quantum information storage.

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Popular Summary

A prototypical model in quantum mechanics is that of a massive particle in a symmetric double-well potential. The rate at which the particle moves from one well to the other is important for multiple applications, including predicting the rate of nuclear reactions as well as the decoherence rate of quantum information stored in such a system for quantum computation. However, experimentally tuning the double-well potential and preparing excited quantum states in it is difficult. Here, we realize an experimental platform that does just that.

Our prototypical double-well system consists of a quantum electromagnetic oscillator in which photons attract each other while subjected to a squeezing interaction (generating pairs of photons). By measuring the spectrum of the system up to ten excited states, we find that, as the squeezing strength is increased, the spectrum of the anharmonic oscillator morphs into pairs of degenerate states—a signature of tunneling in a double-well potential. We also observe that the coherence lifetime of the ground-state manifold increases in steps as the squeezing strength increases. This brings to light a new effect of general interest: the quantum regime of Arrhenius’s law, which describes thermal activation across a barrier.

Our results reveal how out-of-equilibrium quantum circuits can create new control mechanisms in quantum computation and simulation, showcasing an experimental test bed. Furthermore, the strongly enhanced coherence times along with the fast quantum control and high-fidelity readout leave our system poised for implementation in proposed quantum computing architectures.

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References (140)

  1. W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715 (2003).
  2. E. Merzbacher, The early history of quantum tunneling, Phys. Today 55, No. 8, 44 (2002).
  3. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Measurement of the single-photon tunneling time, Phys. Rev. Lett. 71, 708 (1993).
  4. J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo, Macroscopic measurement of resonant magnetization tunneling in high-spin molecules, Phys. Rev. Lett. 76, 3830 (1996).
  5. A. I. Kolesnikov, G. F. Reiter, N. Choudhury, T. R. Prisk, E. Mamontov, A. Podlesnyak, G. Ehlers, A. G. Seel, D. J. Wesolowski, and L. M. Anovitz, Quantum tunneling of water in beryl: A new state of the water molecule, Phys. Rev. Lett. 116, 167802 (2016).
  6. D. DeVault, J. H. Parkes, and B. Chance, Electron tunnelling in cytochromes, Nature (London) 215, 642 (1967).
  7. M. H. Devoret, J. M. Martinis, and J. Clarke, Measurements of macroscopic quantum tunneling out of the zero-voltage state of a current-biased Josephson junction, Phys. Rev. Lett. 55, 1908 (1985).
  8. C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, Tunneling of optical pulses through photonic band gaps, Phys. Rev. Lett. 73, 2308 (1994).
  9. P. Joyez, V. Bouchiat, D. Esteve, C. Urbina, and M. H. Devoret, Strong tunneling in the single-electron transistor, Phys. Rev. Lett. 79, 1349 (1997).
  10. A. Slobodskyy, C. Gould, T. Slobodskyy, C. R. Becker, G. Schmidt, and L. W. Molenkamp, Voltage-controlled spin selection in a magnetic resonant tunneling diode, Phys. Rev. Lett. 90, 246601 (2003).
  11. S. Fölling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera, T. Müller, and I. Bloch, Direct observation of second-order atom tunnelling, Nature (London) 448, 1029 (2007).
  12. E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic, and M. K. Oberthaler, Single-particle tunneling in strongly driven double-well potentials, Phys. Rev. Lett. 100, 190405 (2008).
  13. W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice, Nature (London) 462, 74 (2009).
  14. R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, and M. Greiner, Photon-assisted tunneling in a biased strongly correlated Bose gas, Phys. Rev. Lett. 107, 095301 (2011).
  15. I. Bloch, J. Dalibard, and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012).
  16. M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices, Phys. Rev. Lett. 111, 185301 (2013).
  17. A. Kaufman, B. Lester, C. Reynolds, M. Wall, M. Foss-Feig, K. Hazzard, A. Rey, and C. Regal, Two-particle quantum interference in tunnel-coupled optical tweezers, Science 345, 306 (2014).
  18. A. Noguchi, Y. Shikano, K. Toyoda, and S. Urabe, Aharonov–Bohm effect in the tunnelling of a quantum rotor in a linear Paul trap, Nat. Commun. 5, 1 (2014).
  19. A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom Fermi–Hubbard antiferromagnet, Nature (London) 545, 462 (2017).
  20. H. Zheng, J. Zhang, and R. Berndt, A minimal double quantum dot, Sci. Rep. 7, 10764 (2017).
  21. F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultracold atoms in optical lattices, Nat. Rev. Phys. 2, 411 (2020).
  22. K. Wintersperger, C. Braun, F. N. Ünal, A. Eckardt, M. Di Liberto, N. Goldman, I. Bloch, and M. Aidelsburger, Realization of an anomalous Floquet topological system with ultracold atoms, Nat. Phys. 16, 1058 (2020).
  23. R. Ramos, D. Spierings, I. Racicot, and A. M. Steinberg, Measurement of the time spent by a tunnelling atom within the barrier region, Nature (London) 583, 529 (2020).
  24. T.-K. Hsiao, C. J. van Diepen, U. Mukhopadhyay, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, Efficient orthogonal control of tunnel couplings in a quantum dot array, Phys. Rev. Appl. 13, 054018 (2020).
  25. N. Bohr and J. A. Wheeler, The mechanism of nuclear fission, Phys. Rev. 56, 426 (1939).
  26. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932).
  27. H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica (Amsterdam) 7A, 284 (1940).
  28. A. Larkin and Y. N. Ovchinnikov, Quantum-mechanical tunneling sith dissipation. the pre-exponential factor, Zh. Eksp. Teor. Fiz. 86, 726 (1984).
  29. M. Marthaler and M. I. Dykman, Switching via quantum activation: A parametrically modulated oscillator, Phys. Rev. A 73, 042108 (2006).
  30. M. I. Dykman, M. Marthaler, and V. Peano, Quantum heating of a parametrically modulated oscillator: Spectral signatures, Phys. Rev. A 83, 052115 (2011).
  31. J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. v. d. Wal, and S. Lloyd, Josephson Persistent-current qubit, Science 285, 1036 (1999).
  32. C. H. v. d. Wal, A. C. J. t. Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, Quantum superposition of macroscopic persistent-current states, Science 290, 773 (2000).
  33. R. Rouse, S. Han, and J. E. Lukens, Observation of resonant tunneling between macroscopically distinct quantum levels, Phys. Rev. Lett. 75, 1614 (1995).
  34. S. Han, R. Rouse, and J. E. Lukens, Generation of a population inversion between quantum states of a macroscopic variable, Phys. Rev. Lett. 76, 3404 (1996).
  35. J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E. Lukens, Quantum superposition of distinct macroscopic states, Nature (London) 406, 43 (2000).
  36. D. M. Berns, M. S. Rudner, S. O. Valenzuela, K. K. Berggren, W. D. Oliver, L. S. Levitov, and T. P. Orlando, Amplitude spectroscopy of a solid-state artificial atom, Nature (London) 455, 51 (2008).
  37. A. J. Berkley, A. J. Przybysz, T. Lanting, R. Harris, N. Dickson, F. Altomare, M. H. Amin, P. Bunyk, C. Enderud, E. Hoskinson, M. W. Johnson, E. Ladizinsky, R. Neufeld, C. Rich, A. Y. Smirnov, E. Tolkacheva, S. Uchaikin, and A. B. Wilson, Tunneling spectroscopy using a probe qubit, Phys. Rev. B 87, 020502(R) (2013).
  38. F. G. Paauw, A. Fedorov, C. J. P. M. Harmans, and J. E. Mooij, Tuning the gap of a superconducting flux qubit, Phys. Rev. Lett. 102, 090501 (2009).
  39. S. Gustavsson, J. Bylander, F. Yan, W. D. Oliver, F. Yoshihara, and Y. Nakamura, Noise correlations in a flux qubit with tunable tunnel coupling, Phys. Rev. B 84, 014525 (2011).
  40. M. J. Schwarz, J. Goetz, Z. Jiang, T. Niemczyk, F. Deppe, A. Marx, and R. Gross, Gradiometric flux qubits with a tunable gap, New J. Phys. 15, 045001 (2013).
  41. S. Novikov, R. Hinkey, S. Disseler, J. I. Basham, T. Albash, A. Risinger, D. Ferguson, D. A. Lidar, and K. M. Zick, Exploring more-coherent quantum annealing, in IEEE International Conference on Rebooting Computing (ICRC), McLean, VA, USA, 2018 (2018) pp. 1–7, 10.1109/ICRC.2018.8638625.
  42. W. D. Oliver, Y. Yu, J. C. Lee, K. K. Berggren, L. S. Levitov, and T. P. Orlando, Mach-Zehnder interferometry in a strongly driven superconducting qubit, Science 310, 1653 (2005).
  43. J. O. Hirschfelder and E. Wigner, Some quantum-mechanical considerations in the theory of reactions involving an activation energy, J. Chem. Phys. 7, 616 (1939).
  44. J. H. Weiner, Quantum rate theory for a symmetric double-well potential, J. Chem. Phys. 68, 2492 (1978).
  45. P. Hänggi, P. Talkner, and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62, 251 (1990).
  46. D. A. Garanin and E. M. Chudnovsky, Thermally activated resonant magnetization tunneling in molecular magnets: Mn12Ac and others, Phys. Rev. B 56, 11102 (1997).
  47. J. R. Friedman, Quantum tunneling and classical barrier reduction for a mesoscopic spin, Phys. Rev. B 57, 10291 (1998).
  48. A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, Stabilization and operation of a Kerr-cat qubit, Nature (London) 584, 205 (2020).
  49. S. Puri, A. Grimm, P. Campagne-Ibarcq, A. Eickbusch, K. Noh, G. Roberts, L. Jiang, M. Mirrahimi, M. H. Devoret, and S. M. Girvin, Stabilized cat in a driven nonlinear cavity: A fault-tolerant error syndrome detector, Phys. Rev. X 9, 041009 (2019).
  50. J. Guillaud and M. Mirrahimi, Repetition cat qubits for fault-tolerant quantum computation, Phys. Rev. X 9, 041053 (2019).
  51. S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frattini, P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais, S. T. Flammia, and S. M. Girvin, Bias-preserving gates with stabilized cat qubits, Sci. Adv. 6, eaay5901 (2020).
  52. J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, The XZZX surface code, Nat. Commun. 12, 2172 (2021).
  53. A. S. Darmawan, B. J. Brown, A. L. Grimsmo, D. K. Tuckett, and S. Puri, Practical quantum error correction with the XZZX code and Kerr-cat qubits, PRX Quantum 2, 030345 (2021).
  54. C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, A. Keller, G. Refael, J. Preskill, L. Jiang, A. H. Safavi-Naeini, O. Painter, and F. G. S. L. Brandão, Building a fault-tolerant quantum computer using concatenated cat codes, PRX Quantum 3, 010329 (2022).
  55. W. Wustmann and V. Shumeiko, Parametric effects in circuit quantum electrodynamics, Low Temp. Phys. 45, 848 (2019).
  56. S. M. Girvin, Circuit QED: Superconducting qubits coupled to microwave photons, Quantum machines: measurement and control of engineered quantum systems, in Proceedings of the 2011 Les Houches Summer School on Quantum Machines, edited by M. H. Devoret, R. J. Schoelkopf, B. Huard, and L. F. Cugliandolo (Oxford University Press, New York, 2014).
  57. N. Frattini, U. Vool, S. Shankar, A. Narla, K. Sliwa, and M. Devoret, 3-wave mixing Josephson dipole element, Appl. Phys. Lett. 110, 222603 (2017).
  58. N. E. Frattini, V. V. Sivak, A. Lingenfelter, S. Shankar, and M. H. Devoret, Optimizing the nonlinearity and dissipation of a SNAIL parametric amplifier for dynamic range, Phys. Rev. Appl. 10, 054020 (2018).
  59. V. V. Sivak, N. E. Frattini, V. R. Joshi, A. Lingenfelter, S. Shankar, and M. H. Devoret, Kerr-free three-wave mixing in superconducting quantum circuits, Phys. Rev. Appl. 11, 054060 (2019).
  60. M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, Dynamically protected cat-qubits: A new paradigm for universal quantum computation, New J. Phys. 16, 045014 (2014).
  61. Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J. Hatridge et al., Confining the state of light to a quantum manifold by engineered two-photon loss, Science 347, 853 (2015).
  62. S. Touzard, A. Grimm, Z. Leghtas, S. O. Mundhada, P. Reinhold, C. Axline, M. Reagor, K. Chou, J. Blumoff, K. M. Sliwa, S. Shankar, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, Coherent oscillations inside a quantum manifold stabilized by dissipation, Phys. Rev. X 8, 021005 (2018).
  63. R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Exponential suppression of bit-flips in a qubit encoded in an oscillator, Nat. Phys. 16, 509 (2020).
  64. C. Berdou, A. Murani, U. Réglade, W. Smith, M. Villiers, J. Palomo, M. Rosticher, A. Denis, P. Morfin, M. Delbecq et al., One hundred second bit-flip time in a two-photon dissipative oscillator, PRX Quantum 4, 020350 (2023).
  65. J. Venkatraman, X. Xiao, R. G. Cortiñas, A. Eickbusch, and M. H. Devoret, Static effective Hamiltonian of a rapidly driven nonlinear system, Phys. Rev. Lett. 129, 100601 (2022).
  66. S. Harris, Tunable optical parametric oscillators, Proc. IEEE 57, 2096 (1969).
  67. G. J. Milburn and C. A. Holmes, Quantum coherence and classical chaos in a pulsed parametric oscillator with a Kerr nonlinearity, Phys. Rev. A 44, 4704 (1991).
  68. G. Y. Kryuchkyan and K. Kheruntsyan, Exact quantum theory of a parametrically driven dissipative anharmonic oscillator, Opt. Commun. 127, 230 (1996).
  69. P. T. Cochrane, G. J. Milburn, and W. J. Munro, Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping, Phys. Rev. A 59, 2631 (1999).
  70. B. Wielinga and G. J. Milburn, Quantum tunneling in a Kerr medium with parametric pumping, Phys. Rev. A 48, 2494 (1993).
  71. C. M. Wilson, T. Duty, M. Sandberg, F. Persson, V. Shumeiko, and P. Delsing, Photon generation in an electromagnetic cavity with a time-dependent boundary, Phys. Rev. Lett. 105, 233907 (2010).
  72. V. Peano, M. Marthaler, and M. I. Dykman, Sharp tunneling peaks in a parametric oscillator: Quantum resonances missing in the rotating wave approximation, Phys. Rev. Lett. 109, 090401 (2012).
  73. H. Goto, Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network, Sci. Rep. 6, 21686 (2016).
  74. Y. Zhang and M. I. Dykman, Preparing quasienergy states on demand: A parametric oscillator, Phys. Rev. A 95, 053841 (2017).
  75. S. Puri, S. Boutin, and A. Blais, Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving, npj Quantum Inf. 3, 1 (2017).
  76. H. Goto, Quantum computation based on quantum adiabatic bifurcations of Kerr-nonlinear parametric oscillators, J. Phys. Soc. Jpn. 88, 061015 (2019).
  77. Z. Wang, M. Pechal, E. A. Wollack, P. Arrangoiz-Arriola, M. Gao, N. R. Lee, and A. H. Safavi-Naeini, Quantum dynamics of a few-photon parametric oscillator, Phys. Rev. X 9, 021049 (2019).
  78. I. Strandberg, G. Johansson, and F. Quijandría, Wigner negativity in the steady-state output of a Kerr parametric oscillator, Phys. Rev. Res. 3, 023041 (2021).
  79. Q. Xu, J. K. Iverson, F. G. S. L. Brandão, and L. Jiang, Engineering fast bias-preserving gates on stabilized cat qubits, Phys. Rev. Res. 4, 013082 (2022).
  80. G. Margiani, S. Guerrero, T. L. Heugel, C. Marty, R. Pachlatko, T. Gisler, G. D. Vukasin, H.-K. Kwon, J. M. L. Miller, N. E. Bousse, T. W. Kenny, O. Zilberberg, D. Sabonis, and A. Eichler, Extracting the lifetime of a synthetic two-level system, Appl. Phys. Lett. 121, 164101 (2022).
  81. S. Masuda, T. Ishikawa, Y. Matsuzaki, and S. Kawabata, Controls of a superconducting quantum parametron under a strong pump field, Sci. Rep. 11, 11459 (2021).
  82. T. Yamaji, S. Kagami, A. Yamaguchi, T. Satoh, K. Koshino, H. Goto, Z. R. Lin, Y. Nakamura, and T. Yamamoto, Spectroscopic observation of the crossover from a classical Duffing oscillator to a Kerr parametric oscillator, Phys. Rev. A 105, 023519 (2022).
  83. K. Matsumoto, A. Yamaguchi, T. Yamamoto, S. Kawabata, and Y. Matsuzaki, Spectroscopic estimation of the photon number for superconducting Kerr parametric oscillators, Jpn. J. Appl. Phys. 62, SC1097 (2023).
  84. Y. Suzuki, S. Watabe, S. Kawabata, and S. Masuda, Measurement-based state preparation of Kerr parametric oscillators, arXiv:2208.04542.
  85. S. Masuda, T. Kanao, H. Goto, Y. Matsuzaki, T. Ishikawa, and S. Kawabata, Fast tunable coupling scheme of Kerr-nonlinear parametric oscillators based on shortcuts to adiabaticity, Phys. Rev. Appl. 18, 034076 (2022).
  86. R. Miyazaki, Effective spin models of Kerr-nonlinear parametric oscillators for quantum annealing, Phys. Rev. A 105, 062457 (2022).
  87. S. Kwon, S. Watabe, and J.-S. Tsai, Autonomous quantum error correction in a four-photon Kerr parametric oscillator, npj Quantum Inf. 8, 40 (2022).
  88. N. E. Frattini, Three-wave mixing in superconducting circuits: Stabilizing cats with SNAILs, Ph.D thesis, Yale University, 2021.
  89. M. Hillery, R. O’Connell, M. Scully, and E. Wigner, Distribution functions in physics: Fundamentals, Phys. Rep. 106, 121 (1984).
  90. T. L. Curtright, D. B. Fairlie, and C. K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space (World Scientific Publishing Company, Singapore, 2013).
  91. F. Minganti, N. Bartolo, J. Lolli, W. Casteels, and C. Ciuti, Exact results for Schrödinger cats in driven-dissipative systems and their feedback control, Sci. Rep. 6, 1 (2016).
  92. H. Putterman, J. Iverson, Q. Xu, L. Jiang, O. Painter, F. G. S. L. Brandão, and K. Noh, Stabilizing a bosonic qubit using colored cissipation, Phys. Rev. Lett. 128, 110502 (2022).
  93. R. Gautier, A. Sarlette, and M. Mirrahimi, Combined dissipative and Hamiltonian confinement of cat qubits, PRX Quantum 3, 020339 (2022).

  94. Note that for the excited states, a photon-loss event a^ creates excitations. This phenomenon is known as quantum heating [29].

  95. Note that for much smaller values of κ1/K, the RWA Lindbladian condition δn<κ1 will appreciably differ from the oversimplified Bohr quantization model.

  96. S. T. Adams, E. H. da Silva Neto, S. Datta, J. F. Ware, C. Lampropoulos, G. Christou, Y. Myaesoedov, E. Zeldov, and J. R. Friedman, Geometric-phase interference in a Mn12 single-molecule magnet with fourfold rotational symmetry, Phys. Rev. Lett. 110, 087205 (2013).
  97. R. W. Alexander, A. E. Hughes, and A. J. Sievers, Far-infrared properties of lattice resonant modes. III. Temperature effects, Phys. Rev. B 1, 1563 (1970).
  98. M. I. Dykman and M. A. Krivoglaz, Spectral distribution of nonlinear oscillators with nonlinear friction due to a medium, Phys. Status Solidi B 68, 111 (1975).
  99. A. Petrescu, M. Malekakhlagh, and H. E. Türeci, Lifetime renormalization of driven weakly anharmonic superconducting qubits. II. The readout problem, Phys. Rev. B 101, 134510 (2020).
  100. M. Marthaler and M. I. Dykman, Quantum interference in the classically forbidden region: A parametric oscillator, Phys. Rev. A 76, 010102(R) (2007).
  101. D. Roberts and A. A. Clerk, Driven-dissipative quantum Kerr resonators: New exact solutions, photon blockade and quantum bistability, Phys. Rev. X 10, 021022 (2020).
  102. J. Venkatraman, R. G. Cortiñas, N. E. Frattini, X. Xiao, and M. H. Devoret, A driven Kerr oscillator with two-fold degeneracies for qubit protection, Proc. Natl. Acad. Sci. U.S.A. 121, e2311241121 (2024).
  103. D. Ruiz, R. Gautier, J. Guillaud, and M. Mirrahimi, Two-photon driven Kerr quantum oscillator with multiple spectral degeneracies, Phys. Rev. A 107, 042407 (2023).
  104. L. Gravina, F. Minganti, and V. Savona, Critical Schrödinger cat qubit, PRX Quantum 4, 020337 (2023).
  105. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995).
  106. Z. Chen et al. (Google Quantum AI), Exponential suppression of bit or phase errors with cyclic error correction, Nature (London) 595, 383 (2021).
  107. P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Quantum error correction of a qubit encoded in grid states of an oscillator, Nature (London) 584, 368 (2020).
  108. A. L. Grimsmo and S. Puri, Quantum error correction with the Gottesman-Kitaev-Preskill code, PRX Quantum 2, 020101 (2021).
  109. Q. Xu, J. K. Iverson, F. G. S. L. Brandão, and L. Jiang, Engineering fast bias-preserving gates on stabilized cat qubits, Phys. Rev. Res. 4, 013082 (2022).
  110. T. Kanao, S. Masuda, S. Kawabata, and H. Goto, Quantum gate for a Kerr nonlinear parametric oscillator using effective excited states, Phys. Rev. Appl. 18, 014019 (2022).
  111. H. Chono, T. Kanao, and H. Goto, Two-qubit gate using conditional driving for highly detuned Kerr nonlinear parametric oscillators, Phys. Rev. Res. 4, 043054 (2022).
  112. C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, Observation of the dynamical Casimir effect in a superconducting circuit, Nature (London) 479, 376 (2011).
  113. M. Dykman, Fluctuating Nonlinear Oscillators: From Nanomechanics to Quantum Superconducting Circuits (Oxford, New York, 2012).
  114. M. Caprio, P. Cejnar, and F. Iachello, Excited state quantum phase transitions in many-body systems, Ann. Phys. (Amsterdam) 323, 1106 (2008).
  115. P. Ribeiro, J. Vidal, and R. Mosseri, Exact spectrum of the Lipkin-Meshkov-Glick model in the thermodynamic limit and finite-size corrections, Phys. Rev. E 78, 021106 (2008).
  116. J. Chávez-Carlos, T. L. M. Lezama, R. G. Cortiñas, J. Venkatraman, M. H. Devoret, V. S. Batista, F. Pé rez-Bernal, and L. F. Santos, Spectral kissing and its dynamical consequences in the squeeze-driven Kerr oscillator, npj Quantum Inf. 9, 1 (2023).
  117. I. Carusotto and C. Ciuti, Quantum fluids of light, Rev. Mod. Phys. 85, 299 (2013).
  118. Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, Coherent Ising machine based on degenerate optical parametric oscillators, Phys. Rev. A 88, 063853 (2013).
  119. S. Puri, C. K. Andersen, A. L. Grimsmo, and A. Blais, Quantum annealing with all-to-all connected nonlinear oscillators, Nat. Commun. 8, 15785 (2017).
  120. H. Goto and T. Kanao, Chaos in coupled Kerr-nonlinear parametric oscillators, Phys. Rev. Res. 3, 043196 (2021).
  121. D. Monteoliva and J. P. Paz, Decoherence and the rate of entropy production in chaotic quantum systems, Phys. Rev. Lett. 85, 3373 (2000).
  122. S. Habib, K. Shizume, and W. H. Zurek, Decoherence, chaos, and the correspondence principle, Phys. Rev. Lett. 80, 4361 (1998).
  123. S. E. Nigg, H. Paik, B. Vlastakis, G. Kirchmair, S. Shankar, L. Frunzio, M. H. Devoret, R. J. Schoelkopf, and S. M. Girvin, Black-box superconducting circuit quantization, Phys. Rev. Lett. 108, 240502 (2012).
  124. Z. K. Minev, Z. Leghtas, S. O. Mundhada, L. Christakis, I. M. Pop, and M. H. Devoret, Energy-participation quantization of Josephson circuits, arXiv:2010.00620.
  125. C. Axline, M. Reagor, R. Heeres, P. Reinhold, C. Wang, K. Shain, W. Pfaff, Y. Chu, L. Frunzio, and R. J. Schoelkopf, An architecture for integrating planar and 3D cQED devices, Appl. Phys. Lett. 109, 042601 (2016).
  126. N. Didier, J. Bourassa, and A. Blais, Fast quantum nondemolition readout by parametric modulation of longitudinal qubit-oscillator interaction, Phys. Rev. Lett. 115, 203601 (2015).
  127. S. Touzard, A. Kou, N. E. Frattini, V. V. Sivak, S. Puri, A. Grimm, L. Frunzio, S. Shankar, and M. H. Devoret, Gated conditional displacement readout of superconducting qubits, Phys. Rev. Lett. 122, 080502 (2019).
  128. P. Campagne-Ibarcq, P. Six, L. Bretheau, A. Sarlette, M. Mirrahimi, P. Rouchon, and B. Huard, Observing quantum state diffusion by heterodyne detection of fluorescence, Phys. Rev. X 6, 011002 (2016).
  129. W. Pfaff, C. J. Axline, L. D. Burkhart, U. Vool, P. Reinhold, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, Controlled release of multiphoton quantum states from a microwave cavity memory, Nat. Phys. 13, 882 (2017).
  130. D. Sank, Z. Chen, M. Khezri, J. Kelly, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. Fowler et al., Measurement-induced state transitions in a superconducting qubit: Beyond the rotating wave approximation, Phys. Rev. Lett. 117, 190503 (2016).
  131. Z. K. Minev, S. O. Mundhada, S. Shankar, P. Reinhold, R. Gutiérrez-Jáuregui, R. J. Schoelkopf, M. Mirrahimi, H. J. Carmichael, and M. H. Devoret, To catch and reverse a quantum jump mid-flight, Nature (London) 570, 200 (2019).
  132. R. Lescanne, L. Verney, Q. Ficheux, M. H. Devoret, B. Huard, M. Mirrahimi, and Z. Leghtas, Escape of a driven quantum Josephson circuit into unconfined states, Phys. Rev. Appl. 11, 014030 (2019).
  133. N. H. McCoy, On the function in quantum mechanics which corresponds to a given function in classical mechanics, Proc. Natl. Acad. Sci. U.S.A. 18, 674 (1932).
  134. H. J. Groenewold, On the principles of elementary quantum mechanics, in On the Principles of Elementary Quantum Mechanics (Springer, New York, 1946), pp. 1–56.
  135. J. E. Moyal, Quantum mechanics as a statistical theory, Math. Proc. Cambridge Philos. Soc. 45, 99 (1949).
  136. P. A. M. Dirac, On the analogy between classical and quantum mechanics, Rev. Mod. Phys. 17, 195 (1945).

  137. Note that the cat coherence times drop as TZ∼1/n¯, the interfringe distance decreases as ∼1/n¯, and the maximal Rabi rate as determined by the gap scales as Δgap∼n¯. Thus, the amount of coherent operation over the Schrödinger cat superposition during the state’s lifetime grows as n¯.

  138. S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University Press, New York, 2006).
  139. B. Yurke and D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett. 57, 13 (1986).
  140. M. Oliva and O. Steuernagel, Quantum Kerr oscillators’ evolution in phase space: Wigner current, symmetries, shear suppression, and special states, Phys. Rev. A 99, 032104 (2019).

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Phys. Rev. X 14, 031040– Published 3 September, 2024
  • Received 28 December 2022
  • Revised 18 May 2024
  • Accepted 9 July 2024
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