- Letter
- Open Access
Preparation of conditionally squeezed states in qubit-oscillator systems
Phys. Rev. Research 8, L012046 – Published 26 February, 2026
DOI: https://doi.org/10.1103/jrrb-hymw
Abstract
Inspired by recent advances in the manipulation of superconducting circuits coupled to mechanical modes in the quantum regime, we propose a protocol for generating superpositions of orthogonally squeezed states in a quantum harmonic oscillator. The protocol relies on a quadratic coupling between the oscillator and a qubit, and is conceptually similar to methods used for preparing cat states in qubit-oscillator systems. We numerically evaluate the robustness of the state-preparation scheme in the presence of decoherence, considering environmental coupling for both the harmonic oscillator and the qubit. As a potential application, we consider a quantum error-correcting code based on conditionally squeezed states and analyze its error correction properties.
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References (52)
- J. P. Dowling and G. J. Milburn, Quantum technology: The second quantum revolution, Philos. Trans. R. Soc. London Ser. A 361, 1655 (2003).
- C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, 2010).
- T. H. Johnson, S. R. Clark, and D. Jaksch, What is a quantum simulator? EPJ Quantum Technol. 1, 10 (2014).
- D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin, 2008).
- H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, Cavity quantum electrodynamics, Rep. Prog. Phys. 69, 1325 (2006).
- R. Blatt, J. I. Cirac, A. S. Parkins, and P. Zoller, Quantum motion of trapped ions, Phys. Scr. 1995, 294 (1995).
- A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Phys. Rev. A 69, 062320 (2004).
- W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, Bosonic quantum error correction codes in superconducting quantum circuits, Fundam. Res. 1, 50 (2021).
- V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, et al., Real-time quantum error correction beyond break-even, Nature (London) 616, 50 (2023).
- A. D. O'Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, et al., Quantum ground state and single-phonon control of a mechanical resonator, Nature (London) 464, 697 (2010).
- 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 (2011).
- E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, Quantum squeezing of motion in a mechanical resonator, Science 349, 952 (2015).
- J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, Squeezing of quantum noise of motion in a micromechanical resonator, Phys. Rev. Lett. 115, 243601 (2015).
- R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, Remote quantum entanglement between two micromechanical oscillators, Nature (London) 556, 473 (2018).
- C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanpää, Stabilized entanglement of massive mechanical oscillators, Nature (London) 556, 478 (2018).
- Y. Chu, P. Kharel, T. Yoon, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Creation and control of multi-phonon Fock states in a bulk acoustic-wave resonator, Nature (London) 563, 666 (2018).
- M. Bild, M. Fadel, Y. Yang, U. von Lüpke, P. Martin, A. Bruno, and Y. Chu, Schrödinger cat states of a 16-microgram mechanical oscillator, Science 380, 274 (2023).
- M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
- Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Quantum acoustics with superconducting qubits, Science 358, 199 (2017).
- C. K. Law and J. H. Eberly, Arbitrary control of a quantum electromagnetic field, Phys. Rev. Lett. 76, 1055 (1996).
- S. Krastanov, V. V. Albert, C. Shen, C.-L. Zou, R. W. Heeres, B. Vlastakis, R. J. Schoelkopf, and L. Jiang, Universal control of an oscillator with dispersive coupling to a qubit, Phys. Rev. A 92, 040303(R) (2015).
- M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, Synthesizing arbitrary quantum states in a superconducting resonator, Nature (London) 459, 546 (2009).
- A. Eickbusch, V. Sivak, A. Z. Ding, S. S. Elder, S. R. Jha, J. Venkatraman, B. Royer, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Fast universal control of an oscillator with weak dispersive coupling to a qubit, Nat. Phys. 18, 1464 (2022).
- B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Deterministically encoding quantum information using 100-photon Schrödinger cat states, Science 342, 607 (2013).
- J.-Q. Liao, J.-F. Huang, and L. Tian, Generation of macroscopic Schrödinger-cat states in qubit-oscillator systems, Phys. Rev. A 93, 033853 (2016).
- V. Bužek, H. Moya-Cessa, P. L. Knight, and S. J. D. Phoenix, Schrödinger-cat states in the resonant Jaynes-Cummings model: Collapse and revival of oscillations of the photon-number distribution, Phys. Rev. A 45, 8190 (1992).
- M. Krenn, J. Landgraf, T. Foesel, and F. Marquardt, Artificial intelligence and machine learning for quantum technologies, Phys. Rev. A 107, 010101 (2023).
- H. Hutin, P. Bilous, C. Ye, S. Abdollahi, L. Cros, T. Dvir, T. Shah, Y. Cohen, A. Bienfait, F. Marquardt, and B. Huard, Preparing Schrödinger cat states in a microwave cavity using a neural network, PRX Quantum 6, 010321 (2025).
- B. C. Sanders, Superposition of two squeezed vacuum states and interference effects, Phys. Rev. A 39, 4284 (1989).
- M. Drechsler, M. Belén Farías, N. Freitas, C. T. Schmiegelow, and J. P. Paz, State-dependent motional squeezing of a trapped ion: Proposed method and applications, Phys. Rev. A 101, 052331 (2020).
- Y. Liu, S. Singh, K. C. Smith, E. Crane, J. M. Martyn, A. Eickbusch, A. Schuckert, R. D. Li, J. Sinanan-Singh, M. B. Soley, et al., Hybrid oscillator-qubit quantum processors: Instruction set architectures, abstract machine models, and applications, PRX Quantum 7, 010201 (2026).
- M. Ayyash, X. Xu, S. Ashhab, and M. Mariantoni, Driven multiphoton qubit-resonator interactions, Phys. Rev. A 110, 053711 (2024).
- N. F. Del Grosso, R. G. Cortiñas, P. I. Villar, F. C. Lombardo, and J. P. Paz, Controlled-squeeze gate in superconducting quantum circuits, Phys. Rev. A 111, 042606 (2025).
- S.-B. Zheng, Preparation of motional macroscopic quantum-interference states of a trapped ion, Phys. Rev. A 58, 761 (1998).
- S. Saner, O. Băzăvan, D. J. Webb, G. Araneda, D. M. Lucas, C. J. Ballance, and R. Srinivas, Generating arbitrary superpositions of nonclassical quantum harmonic oscillator states, Phys. Rev. X (to be published).
- F. Beaudoin, J. M. Gambetta, and A. Blais, Dissipation and ultrastrong coupling in circuit QED, Phys. Rev. A 84, 043832 (2011).
- X. Ma, J. J. Viennot, S. Kotler, J. D. Teufel, and K. W. Lehnert, Non-classical energy squeezing of a macroscopic mechanical oscillator, Nat. Phys. 17, 322 (2021).
- J. Manninen, R. H. Blick, and F. Massel, Hybrid optomechanical superconducting qubit system, Phys. Rev. Res. 6, 023029 (2024).
- E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum computing with rotation-symmetric bosonic codes, Phys. Rev. X 10, 011058 (2020).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/jrrb-hymw for a detailed derivation of the RWA Hamiltonian, the logical states in the Fock basis, a comparison with other rotation codes, and the open system dynamics, including improved stability with qubit measurement.
- 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).
- D. S. Schlegel, F. Minganti, and V. Savona, Quantum error correction using squeezed Schrödinger cat states, Phys. Rev. A 106, 022431 (2022).
- J. A. Vaccaro, S. M. Barnett, and D. Pegg, Phase fluctuations and squeezing, J. Mod. Opt. 39, 603 (1992).
- 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).
- M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quantum error-correcting codes for a bosonic mode, Phys. Rev. X 6, 031006 (2016).
- J. J. Burnett, A. Bengtsson, M. Scigliuzzo, D. Niepce, M. Kudra, P. Delsing, and J. Bylander, Decoherence benchmarking of superconducting qubits, npj Quantum Inf. 5, 54 (2019).
- C. Wang, X. Li, H. Xu, Z. Li, J. Wang, Z. Yang, Z. Mi, X. Liang, T. Su, C. Yang, et al., Towards practical quantum computers: Transmon qubit with a lifetime approaching 0.5 milliseconds, npj Quantum Inf. 8, 3 (2022).
- S. Marti, U. von Lüpke, O. Joshi, Y. Yang, M. Bild, A. Omahen, Y. Chu, and M. Fadel, Quantum squeezing in a nonlinear mechanical oscillator, Nat. Phys. 20, 1448 (2024).
- S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch, QuantumOptics.jl: A Julia framework for simulating open quantum systems, Comput. Phys. Commun. 227, 109 (2018).
- M. K. Hope, J. Lidal, and F. Massel, Preparation of conditionally squeezed states in qubit-oscillator systems, Zenodo (2026), https://doi.org/10.5281/zenodo.18608746.