Quantum control of continuous systems via nonharmonic potential modulation

Piotr T. Grochowski; Hannes Pichler; Cindy A. Regal; Oriol Romero-Isart

doi:10.22331/q-2025-08-08-1824

Quantum control of continuous systems via nonharmonic potential modulation

Piotr T. Grochowski^1,2,3, Hannes Pichler^1,2, Cindy A. Regal^4,5, and Oriol Romero-Isart^1,2,6,7

¹Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
²Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
³Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic
⁴JILA, National Institute of Standards and Technology and University of Colorado, Boulder, Colorado 80309, USA
⁵Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
⁶ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
⁷ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain

Published:	2025-08-08, volume 9, page 1824
Editor:	Christos Gagatsos
Eprint:	arXiv:2311.16819v4
Doi:	https://doi.org/10.22331/q-2025-08-08-1824
Citation:	Quantum 9, 1824 (2025).

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Abstract

We present a theoretical proposal for preparing and manipulating a state of a single continuous-variable degree of freedom confined to a nonharmonic potential. By utilizing optimally controlled modulation of the potential's position and depth, we demonstrate the generation of non-Gaussian states, including Fock, Gottesman-Kitaev-Preskill, multi-legged-cat, and cubic-phase states, as well as the implementation of arbitrary unitaries within a selected two-level subspace. Additionally, we propose protocols for single-shot orthogonal state discrimination, algorithmic cooling, and correcting for nonlinear evolution. We analyze the robustness of this control scheme against noise. Since all the presented protocols rely solely on the precise modulation of the effective nonharmonic potential landscape, they are relevant to several experiments with continuous-variable systems, including the motion of a single particle in an optical tweezer or lattice, or current in circuit quantum electrodynamics.

Featured image: (Top) Examples of continuous-variable nonlinear systems that can be optimally controlled without the need for auxiliary systems—single atoms in optical tweezers and flux-tunable transmons.
(Bottom) Time-evolved probability density $P(x,\tau) = |\psi (x,\tau)|^2$, during the state preparation protocol with the snapshots of Wigner functions $W(x,p)$ at the beginning and the end of the protocol.

Popular summary

No extra qubits? No problem.
Many quantum technologies rely on auxiliary systems—like spins or qubits—to control continuous-variable quantum states. But what if we could do it all with just a simple trap?
We show how to prepare and manipulate nonclassical states—like Fock, cat, cubic-phase, and GKP states—by optimally modulating nonharmonic potential wells, as found in optical tweezers or superconducting circuits. No extra nonlinear elements are needed.
Using optimal control theory, we compute how best to "shake" the trap to implement quantum gates, perform cooling, and distinguish quantum states in a single shot.
This minimal approach enables precise quantum control with reduced hardware, opening new directions for experiments in neutral atoms, circuit QED, and beyond.

► BibTeX data

@article{Grochowski2025quantumcontrolof,
  doi = {10.22331/q-2025-08-08-1824},
  url = {https://doi.org/10.22331/q-2025-08-08-1824},
  title = {Quantum control of continuous systems via nonharmonic potential modulation},
  author = {Grochowski, Piotr T. and Pichler, Hannes and Regal, Cindy A. and Romero-Isart, Oriol},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1824},
  month = aug,
  year = {2025}
}

► References

[1] W. H. Zurek, ``Sub-Planck structure in phase space and its relevance for quantum decoherence,'' Nature 412, 712–717 (2001).
https://doi.org/10.1038/35089017

[2] S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, ``Reconstruction of non-classical cavity field states with snapshots of their decoherence,'' Nature 455, 510–514 (2008).
https://doi.org/10.1038/nature07288

[3] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, ``Wave–particle duality of C60 molecules,'' Nature 401, 680–682 (1999).
https://doi.org/10.1038/44348

[4] Y. Y. Fein, P. Geyer, P. Zwick, F. Kiałka, S. Pedalino, M. Mayor, S. Gerlich, and M. Arndt, ``Quantum superposition of molecules beyond 25 kDa,'' Nat. Phys. 15, 1242–1245 (2019).
https://doi.org/10.1038/s41567-019-0663-9

[5] 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, 274–278 (2023).
https://doi.org/10.1126/science.adf7553

[6] A. Peters, K. Y. Chung, and S. Chu, ``High-precision gravity measurements using atom interferometry,'' Metrologia 38, 25 (2001).
https://doi.org/10.1088/0026-1394/38/1/4

[7] P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran, ``New Method for Gravitational Wave Detection with Atomic Sensors,'' Phys. Rev. Lett. 110, 171102 (2013).
https://doi.org/10.1103/PhysRevLett.110.171102

[8] R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller, ``Measurement of the fine-structure constant as a test of the Standard Model,'' Science 360, 191–195 (2018).
https://doi.org/10.1126/science.aap7706

[9] Y. Margalit, O. Dobkowski, Z. Zhou, O. Amit, Y. Japha, S. Moukouri, D. Rohrlich, A. Mazumdar, S. Bose, C. Henkel, and R. Folman, ``Realization of a complete Stern-Gerlach interferometer: Toward a test of quantum gravity,'' Sci. Adv. 7, eabg2879 (2021).
https://doi.org/10.1126/sciadv.abg2879

[10] D. Gottesman, A. Kitaev, and J. Preskill, ``Encoding a qubit in an oscillator,'' Phys. Rev. A 64, 012310 (2001).
https://doi.org/10.1103/PhysRevA.64.012310

[11] E. Knill, R. Laflamme, and G. J. Milburn, ``A scheme for efficient quantum computation with linear optics,'' Nature 409, 46–52 (2001).
https://doi.org/10.1038/35051009

[12] R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, ``Implementing a universal gate set on a logical qubit encoded in an oscillator,'' Nat. Commun. 8, 94 (2017).
https://doi.org/10.1038/s41467-017-00045-1

[13] P. Reinhold, S. Rosenblum, W.-L. Ma, L. Frunzio, L. Jiang, and R. J. Schoelkopf, ``Error-corrected gates on an encoded qubit,'' Nat. Phys. 16, 822–826 (2020).
https://doi.org/10.1038/s41567-020-0931-8

[14] Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan, H. Wang, Y. P. Song, C.-L. Zou, and L. Sun, ``Error-transparent operations on a logical qubit protected by quantum error correction,'' Nat. Phys. 16, 827–831 (2020).
https://doi.org/10.1038/s41567-020-0893-x

[15] 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 584, 205–209 (2020).
https://doi.org/10.1038/s41586-020-2587-z

[16] J. M. Gertler, B. Baker, J. Li, S. Shirol, J. Koch, and C. Wang, ``Protecting a bosonic qubit with autonomous quantum error correction,'' Nature 590, 243–248 (2021).
https://doi.org/10.1038/s41586-021-03257-0

[17] B. de Neeve, T.-L. Nguyen, T. Behrle, and J. P. Home, ``Error correction of a logical grid state qubit by dissipative pumping,'' Nat. Phys. 18, 296–300 (2022).
https://doi.org/10.1038/s41567-021-01487-7

[18] D. González-Cuadra, D. Bluvstein, M. Kalinowski, R. Kaubruegger, N. Maskara, P. Naldesi, T. V. Zache, A. M. Kaufman, M. D. Lukin, H. Pichler, B. Vermersch, J. Ye, and P. Zoller, ``Fermionic quantum processing with programmable neutral atom arrays,'' Proc. Natl. Acad. Sci. U.S.A. 120, e2304294120 (2023).
https://doi.org/10.1073/pnas.2304294120

[19] J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, ``Nonclassical States of Motion in Ion Traps,'' in Advances In Atomic, Molecular, and Optical Physics, Vol. 37, edited by B. Bederson and H. Walther (Academic Press, 1996) pp. 237–296.
https://doi.org/10.1016/S1049-250X(08)60102-0

[20] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, ``Generation of Nonclassical Motional States of a Trapped Atom,'' Phys. Rev. Lett. 76, 1796–1799 (1996).
https://doi.org/10.1103/PhysRevLett.76.1796

[21] J. C. Loredo, C. Antón, B. Reznychenko, P. Hilaire, A. Harouri, C. Millet, H. Ollivier, N. Somaschi, L. De Santis, A. Lemaı̂tre, I. Sagnes, L. Lanco, A. Auffèves, O. Krebs, and P. Senellart, ``Generation of non-classical light in a photon-number superposition,'' Nat. Photonics 13, 803–808 (2019).
https://doi.org/10.1038/s41566-019-0506-3

[22] R. Trivedi, K. A. Fischer, J. Vučković, and K. Müller, ``Generation of Non-Classical Light Using Semiconductor Quantum Dots,'' Adv. Quantum Technol. 3, 1900007 (2020).
https://doi.org/10.1002/qute.201900007

[23] R. Jáuregui, N. Poli, G. Roati, and G. Modugno, ``Anharmonic parametric excitation in optical lattices,'' Phys. Rev. A 64, 033403 (2001).
https://doi.org/10.1103/PhysRevA.64.033403

[24] J. P. Home, D. Hanneke, J. D. Jost, D. Leibfried, and D. J. Wineland, ``Normal modes of trapped ions in the presence of anharmonic trap potentials,'' New J. Phys. 13, 073026 (2011).
https://doi.org/10.1088/1367-2630/13/7/073026

[25] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, ``Circuit quantum electrodynamics,'' Rev. Mod. Phys. 93, 025005 (2021).
https://doi.org/10.1103/RevModPhys.93.025005

[26] J. Gorman, D. G. Hasko, and D. A. Williams, ``Charge-Qubit Operation of an Isolated Double Quantum Dot,'' Phys. Rev. Lett. 95, 090502 (2005).
https://doi.org/10.1103/PhysRevLett.95.090502

[27] C. Gollub, U. Troppmann, and R. de Vivie-Riedle, ``The role of anharmonicity and coupling in quantum computing based on vibrational qubits,'' New J. Phys. 8, 48 (2006).
https://doi.org/10.1088/1367-2630/8/4/048

[28] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, ``Charge-insensitive qubit design derived from the Cooper pair box,'' Phys. Rev. A 76, 042319 (2007).
https://doi.org/10.1103/PhysRevA.76.042319

[29] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, ``Suppressing charge noise decoherence in superconducting charge qubits,'' Phys. Rev. B 77, 180502 (2008).
https://doi.org/10.1103/PhysRevB.77.180502

[30] F. Pistolesi, A. N. Cleland, and A. Bachtold, ``Proposal for a Nanomechanical Qubit,'' Phys. Rev. X 11, 031027 (2021).
https://doi.org/10.1103/PhysRevX.11.031027

[31] Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi, ``Hardware-Efficient Autonomous Quantum Memory Protection,'' Phys. Rev. Lett. 111, 120501 (2013a).
https://doi.org/10.1103/PhysRevLett.111.120501

[32] Y. Zheng, O. Hahn, P. Stadler, P. Holmvall, F. Quijandría, A. Ferraro, and G. Ferrini, ``Gaussian Conversion Protocols for Cubic Phase State Generation,'' PRX Quantum 2, 010327 (2021).
https://doi.org/10.1103/PRXQuantum.2.010327

[33] S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, ``Training Schrödinger's cat: Quantum optimal control,'' Eur. Phys. J. D 69, 279 (2015).
https://doi.org/10.1140/epjd/e2015-60464-1

[34] C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, ``Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe,'' EPJ Quantum Technol. 9, 1–60 (2022).
https://doi.org/10.1140/epjqt/s40507-022-00138-x

[35] M. Rossignolo, T. Reisser, A. Marshall, P. Rembold, A. Pagano, P. J. Vetter, R. S. Said, M. M. Müller, F. Motzoi, T. Calarco, F. Jelezko, and S. Montangero, ``QuOCS: The quantum optimal control suite,'' Comput. Phys. Commun. 291, 108782 (2023).
https://doi.org/10.1016/j.cpc.2023.108782

[36] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, ``Resolving photon number states in a superconducting circuit,'' Nature 445, 515–518 (2007).
https://doi.org/10.1038/nature05461

[37] M. Mallweger, M. H. de Oliveira, R. Thomm, H. Parke, N. Kuk, G. Higgins, R. Bachelard, C. J. Villas-Boas, and M. Hennrich, ``Single-Shot Measurements of Phonon Number States Using the Autler-Townes Effect,'' Phys. Rev. Lett. 131, 223603 (2023).
https://doi.org/10.1103/PhysRevLett.131.223603

[38] M. Popp, J.-J. Garcia-Ripoll, K. G. H. Vollbrecht, and J. I. Cirac, ``Cooling toolbox for atoms in optical lattices,'' New J. Phys. 8, 164 (2006).
https://doi.org/10.1088/1367-2630/8/8/164

[39] A. L. Shaw, P. Scholl, R. Finkelstein, R. B.-S. Tsai, J. Choi, and M. Endres, ``Erasure cooling, control, and hyperentanglement of motion in optical tweezers,'' Science 388, 845–849 (2025).
https://doi.org/10.1126/science.adn2618

[40] F. Serwane, G. Zürn, T. Lompe, T. B. Ottenstein, A. N. Wenz, and S. Jochim, ``Deterministic Preparation of a Tunable Few-Fermion System,'' Science 332, 336–338 (2011).
https://doi.org/10.1126/science.1201351

[41] F. Serwane, Deterministic Preparation of a Tunable Few-Fermion System, Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (2011).
https://doi.org/10.11588/heidok.00012961

[42] A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig, K. R. A. Hazzard, A. M. Rey, and C. A. Regal, ``Two-particle quantum interference in tunnel-coupled optical tweezers,'' Science 345, 306–309 (2014).
https://doi.org/10.1126/science.1250057

[43] A. M. Kaufman, B. J. Lester, M. Foss-Feig, M. L. Wall, A. M. Rey, and C. A. Regal, ``Entangling two transportable neutral atoms via local spin exchange,'' Nature 527, 208–211 (2015).
https://doi.org/10.1038/nature16073

[44] M. O. Brown, S. R. Muleady, W. J. Dworschack, R. J. Lewis-Swan, A. M. Rey, O. Romero-Isart, and C. A. Regal, ``Time-of-flight quantum tomography of an atom in an optical tweezer,'' Nat. Phys. 19, 569–573 (2023).
https://doi.org/10.1038/s41567-022-01890-8

[45] M. R. Lam, N. Peter, T. Groh, W. Alt, C. Robens, D. Meschede, A. Negretti, S. Montangero, T. Calarco, and A. Alberti, ``Demonstration of Quantum Brachistochrones between Distant States of an Atom,'' Phys. Rev. X 11, 011035 (2021).
https://doi.org/10.1103/PhysRevX.11.011035

[46] M. D. Hutchings, J. B. Hertzberg, Y. Liu, N. T. Bronn, G. A. Keefe, M. Brink, J. M. Chow, and B. L. T. Plourde, ``Tunable Superconducting Qubits with Flux-Independent Coherence,'' Phys. Rev. Appl. 8, 044003 (2017).
https://doi.org/10.1103/PhysRevApplied.8.044003

[47] U. Hohenester, P. K. Rekdal, A. Borzì, and J. Schmiedmayer, ``Optimal quantum control of Bose-Einstein condensates in magnetic microtraps,'' Phys. Rev. A 75, 023602 (2007).
https://doi.org/10.1103/PhysRevA.75.023602

[48] R. Bücker, J. Grond, S. Manz, T. Berrada, T. Betz, C. Koller, U. Hohenester, T. Schumm, A. Perrin, and J. Schmiedmayer, ``Twin-atom beams,'' Nat. Phys. 7, 608–611 (2011).
https://doi.org/10.1038/nphys1992

[49] R. Bücker, T. Berrada, S. van Frank, J.-F. Schaff, T. Schumm, J. Schmiedmayer, G. Jäger, J. Grond, and U. Hohenester, ``Vibrational state inversion of a Bose–Einstein condensate: Optimal control and state tomography,'' J. Phys. B: At. Mol. Opt. Phys. 46, 104012 (2013a).
https://doi.org/10.1088/0953-4075/46/10/104012

[50] G. Jäger, D. M. Reich, M. H. Goerz, C. P. Koch, and U. Hohenester, ``Optimal quantum control of Bose-Einstein condensates in magnetic microtraps: Comparison of gradient-ascent-pulse-engineering and Krotov optimization schemes,'' Phys. Rev. A 90, 033628 (2014).
https://doi.org/10.1103/PhysRevA.90.033628

[51] S. van Frank, M. Bonneau, J. Schmiedmayer, S. Hild, C. Gross, M. Cheneau, I. Bloch, T. Pichler, A. Negretti, T. Calarco, and S. Montangero, ``Optimal control of complex atomic quantum systems,'' Sci. Rep. 6, 34187 (2016).
https://doi.org/10.1038/srep34187

[52] D. Hocker, J. Yan, and H. Rabitz, ``Optimal nonlinear coherent mode transitions in Bose-Einstein condensates utilizing spatiotemporal controls,'' Phys. Rev. A 93, 053612 (2016).
https://doi.org/10.1103/PhysRevA.93.053612

[53] J. J. W. H. Sørensen, M. O. Aranburu, T. Heinzel, and J. F. Sherson, ``Quantum optimal control in a chopped basis: Applications in control of Bose-Einstein condensates,'' Phys. Rev. A 98, 022119 (2018).
https://doi.org/10.1103/PhysRevA.98.022119

[54] J. Schmidt, D. Hönig, P. Weckesser, F. Thielemann, T. Schaetz, and L. Karpa, ``Mass-selective removal of ions from Paul traps using parametric excitation,'' Appl. Phys. B 126, 176 (2020).
https://doi.org/10.1007/s00340-020-07491-8

[55] N. Dupont, G. Chatelain, L. Gabardos, M. Arnal, J. Billy, B. Peaudecerf, D. Sugny, and D. Guéry-Odelin, ``Quantum State Control of a Bose-Einstein Condensate in an Optical Lattice,'' PRX Quantum 2, 040303 (2021).
https://doi.org/10.1103/PRXQuantum.2.040303

[56] S. Xu, J. Schmiedmayer, and B. C. Sanders, ``Nonlinear quantum gates for a Bose-Einstein condensate,'' Phys. Rev. Res. 4, 023071 (2022).
https://doi.org/10.1103/PhysRevResearch.4.023071

[57] T. Calarco, U. Dorner, P. S. Julienne, C. J. Williams, and P. Zoller, ``Quantum computations with atoms in optical lattices: Marker qubits and molecular interactions,'' Phys. Rev. A 70, 012306 (2004).
https://doi.org/10.1103/PhysRevA.70.012306

[58] U. Dorner, T. Calarco, P. Zoller, A. Browaeys, and P. Grangier, ``Quantum logic via optimal control in holographic dipole traps,'' J. Opt. B: Quantum Semiclass. Opt. 7, S341 (2005).
https://doi.org/10.1088/1464-4266/7/10/020

[59] J. D. Sterk, H. Coakley, J. Goldberg, V. Hietala, J. Lechtenberg, H. McGuinness, D. McMurtrey, L. P. Parazzoli, J. Van Der Wall, and D. Stick, ``Closed-loop optimization of fast trapped-ion shuttling with sub-quanta excitation,'' npj Quantum Inf. 8, 1–6 (2022).
https://doi.org/10.1038/s41534-022-00579-3

[60] 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, 1–7 (2017).
https://doi.org/10.1038/s41534-017-0019-1

[61] C. A. Weidner and D. Z. Anderson, ``Experimental Demonstration of Shaken-Lattice Interferometry,'' Phys. Rev. Lett. 120, 263201 (2018).
https://doi.org/10.1103/PhysRevLett.120.263201

[62] C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home, ``Encoding a qubit in a trapped-ion mechanical oscillator,'' Nature 566, 513–517 (2019).
https://doi.org/10.1038/s41586-019-0960-6

[63] K. C. McCormick, J. Keller, S. C. Burd, D. J. Wineland, A. C. Wilson, and D. Leibfried, ``Quantum-enhanced sensing of a single-ion mechanical oscillator,'' Nature 572, 86–90 (2019).
https://doi.org/10.1038/s41586-019-1421-y

[64] L. Podhora, L. Lachman, T. Pham, A. Lešundák, O. Číp, L. Slodička, and R. Filip, ``Quantum Non-Gaussianity of Multiphonon States of a Single Atom,'' Phys. Rev. Lett. 129, 013602 (2022).
https://doi.org/10.1103/PhysRevLett.129.013602

[65] V. G. Matsos, C. H. Valahu, T. Navickas, A. D. Rao, M. J. Millican, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, ``Robust and Deterministic Preparation of Bosonic Logical States in a Trapped Ion,'' Phys. Rev. Lett. 133, 050602 (2024).
https://doi.org/10.1103/PhysRevLett.133.050602

[66] W. Wang, Y. Wu, Y. Ma, W. Cai, L. Hu, X. Mu, Y. Xu, Z.-J. Chen, H. Wang, Y. P. Song, H. Yuan, C.-L. Zou, L.-M. Duan, and L. Sun, ``Heisenberg-limited single-mode quantum metrology in a superconducting circuit,'' Nat. Commun. 10, 4382 (2019).
https://doi.org/10.1038/s41467-019-12290-7

[67] 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, 1464–1469 (2022).
https://doi.org/10.1038/s41567-022-01776-9

[68] M. Kudra, M. Kervinen, I. Strandberg, S. Ahmed, M. Scigliuzzo, A. Osman, D. P. Lozano, M. O. Tholén, R. Borgani, D. B. Haviland, G. Ferrini, J. Bylander, A. F. Kockum, F. Quijandría, P. Delsing, and S. Gasparinetti, ``Robust Preparation of Wigner-Negative States with Optimized SNAP-Displacement Sequences,'' PRX Quantum 3, 030301 (2022).
https://doi.org/10.1103/PRXQuantum.3.030301

[69] M. Endo, R. He, T. Sonoyama, K. Takahashi, T. Kashiwazaki, T. Umeki, S. Takasu, K. Hattori, D. Fukuda, K. Fukui, K. Takase, W. Asavanant, P. Marek, R. Filip, and A. Furusawa, ``Non-Gaussian quantum state generation by multi-photon subtraction at the telecommunication wavelength,'' Opt. Express 31, 12865–12879 (2023).
https://doi.org/10.1364/OE.486270

[70] S. Konno, W. Asavanant, F. Hanamura, H. Nagayoshi, K. Fukui, A. Sakaguchi, R. Ide, F. China, M. Yabuno, S. Miki, H. Terai, K. Takase, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, ``Logical states for fault-tolerant quantum computation with propagating light,'' Science 383, 289–293 (2024).
https://doi.org/10.1126/science.adk7560

[71] M. Gutierrez Latorre, G. Higgins, A. Paradkar, T. Bauch, and W. Wieczorek, ``Superconducting Microsphere Magnetically Levitated in an Anharmonic Potential with Integrated Magnetic Readout,'' Phys. Rev. Appl. 19, 054047 (2023).
https://doi.org/10.1103/PhysRevApplied.19.054047

[72] J. Hofer, R. Gross, G. Higgins, H. Huebl, O. F. Kieler, R. Kleiner, D. Koelle, P. Schmidt, J. A. Slater, M. Trupke, K. Uhl, T. Weimann, W. Wieczorek, and M. Aspelmeyer, ``High-$Q$ Magnetic Levitation and Control of Superconducting Microspheres at Millikelvin Temperatures,'' Phys. Rev. Lett. 131, 043603 (2023).
https://doi.org/10.1103/PhysRevLett.131.043603

[73] M. Fuwa, R. Sakagami, and T. Tamegai, ``Ferromagnetic levitation and harmonic trapping of a milligram-scale yttrium iron garnet sphere,'' Phys. Rev. A 108, 063511 (2023).
https://doi.org/10.1103/PhysRevA.108.063511

[74] E. Bonvin, L. Devaud, M. Rossi, A. Militaru, L. Dania, D. S. Bykov, M. Teller, T. E. Northup, L. Novotny, and M. Frimmer, ``Hybrid Paul-optical trap with large optical access for levitated optomechanics,'' Phys. Rev. Res. 6, 043129 (2024).
https://doi.org/10.1103/PhysRevResearch.6.043129

[75] F.-G. Deng, B.-C. Ren, and X.-H. Li, ``Quantum hyperentanglement and its applications in quantum information processing,'' Sci. Bull. 62, 46 (2017).
https://doi.org/10.1016/j.scib.2016.11.007

[76] M. O. Brown, T. Thiele, C. Kiehl, T.-W. Hsu, and C. A. Regal, ``Gray-Molasses Optical-Tweezer Loading: Controlling Collisions for Scaling Atom-Array Assembly,'' Phys. Rev. X 9, 011057 (2019).
https://doi.org/10.1103/PhysRevX.9.011057

[77] A. M. Kaufman and K.-K. Ni, ``Quantum science with optical tweezer arrays of ultracold atoms and molecules,'' Nat. Phys. 17, 1324–1333 (2021).
https://doi.org/10.1038/s41567-021-01357-2

[78] C. Robens, S. Brakhane, W. Alt, D. Meschede, J. Zopes, and A. Alberti, ``Fast, High-Precision Optical Polarization Synthesizer for Ultracold-Atom Experiments,'' Phys. Rev. Appl. 9, 034016 (2018).
https://doi.org/10.1103/PhysRevApplied.9.034016

[79] M. A. Rol, F. Battistel, F. K. Malinowski, C. C. Bultink, B. M. Tarasinski, R. Vollmer, N. Haider, N. Muthusubramanian, A. Bruno, B. M. Terhal, and L. DiCarlo, ``Fast, High-Fidelity Conditional-Phase Gate Exploiting Leakage Interference in Weakly Anharmonic Superconducting Qubits,'' Phys. Rev. Lett. 123, 120502 (2019).
https://doi.org/10.1103/PhysRevLett.123.120502

[80] M. A. Rol, L. Ciorciaro, F. K. Malinowski, B. M. Tarasinski, R. E. Sagastizabal, C. C. Bultink, Y. Salathe, N. Haandbaek, J. Sedivy, and L. DiCarlo, ``Time-domain characterization and correction of on-chip distortion of control pulses in a quantum processor,'' Appl. Phys. Lett. 116, 054001 (2020).
https://doi.org/10.1063/1.5133894

[81] R. Bücker, T. Berrada, S. van Frank, J.-F. Schaff, T. Schumm, J. Schmiedmayer, G. Jäger, J. Grond, and U. Hohenester, ``Vibrational state inversion of a Bose–Einstein condensate: Optimal control and state tomography,'' J. Phys. B: At. Mol. Opt. Phys. 46, 104012 (2013b).
https://doi.org/10.1088/0953-4075/46/10/104012

[82] M. Dykman, ed., Fluctuating Nonlinear Oscillators: From Nanomechanics to Quantum Superconducting Circuits (Oxford University Press, Oxford, United Kingdom, 2012).
https://academic.oup.com/book/7274

[83] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, ``Quantum dynamics of single trapped ions,'' Rev. Mod. Phys. 75, 281–324 (2003).
https://doi.org/10.1103/RevModPhys.75.281

[84] C. Gonzalez-Ballestero, M. Aspelmeyer, L. Novotny, R. Quidant, and O. Romero-Isart, ``Levitodynamics: Levitation and control of microscopic objects in vacuum,'' Science 374, eabg3027 (2021).
https://doi.org/10.1126/science.abg3027

[85] M. Roda-Llordes, A. Riera-Campeny, D. Candoli, P. T. Grochowski, and O. Romero-Isart, ``Macroscopic Quantum Superpositions via Dynamics in a Wide Double-Well Potential,'' Phys. Rev. Lett. 132, 023601 (2024a).
https://doi.org/10.1103/PhysRevLett.132.023601

[86] A. Lingenfelter, D. Roberts, and A. A. Clerk, ``Unconditional Fock state generation using arbitrarily weak photonic nonlinearities,'' Sci. Adv. 7, eabj1916 (2021).
https://doi.org/10.1126/sciadv.abj1916

[87] M. Yuan, A. Seif, A. Lingenfelter, D. I. Schuster, A. A. Clerk, and L. Jiang, ``Universal control in bosonic systems with weak Kerr nonlinearities,'' Phys. Rev. A 111, 032606 (2025).
https://doi.org/10.1103/PhysRevA.111.032606

[88] J. Werschnik and E. K. U. Gross, ``Quantum optimal control theory,'' J. Phys. B: At. Mol. Opt. Phys. 40, R175 (2007).
https://doi.org/10.1088/0953-4075/40/18/R01

[89] N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, ``Stimulated Raman adiabatic passage in physics, chemistry, and beyond,'' Rev. Mod. Phys. 89, 015006 (2017).
https://doi.org/10.1103/RevModPhys.89.015006

[90] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, ``Shortcuts to adiabaticity: Concepts, methods, and applications,'' Rev. Mod. Phys. 91, 045001 (2019).
https://doi.org/10.1103/RevModPhys.91.045001

[91] M. H. Levitt, ``Composite pulses,'' Prog. Nucl. Magn. Reson. Spectrosc. 18, 61–122 (1986).
https://doi.org/10.1016/0079-6565(86)80005-X

[92] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, ``Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms,'' J. Magn. Reson. 172, 296–305 (2005).
https://doi.org/10.1016/j.jmr.2004.11.004

[93] D. M. Reich, M. Ndong, and C. P. Koch, ``Monotonically convergent optimization in quantum control using Krotov's method,'' J. Chem. Phys. 136, 104103 (2012).
https://doi.org/10.1063/1.3691827

[94] M. Goerz, D. Basilewitsch, F. Gago-Encinas, M. G. Krauss, K. P. Horn, D. M. Reich, and C. Koch, ``Krotov: A Python implementation of Krotov's method for quantum optimal control,'' SciPost Phys. 7, 080 (2019).
https://doi.org/10.21468/SciPostPhys.7.6.080

[95] J. A. Nelder and R. Mead, ``A Simplex Method for Function Minimization,'' Comput. J. 7, 308–313 (1965).
https://doi.org/10.1093/comjnl/7.4.308

[96] D. Goldberg, K. Sastry, and D. Goldberg, Genetic Algorithms: The Design of Innovation (Springer, 2007).

[97] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, ``Optimization by simulated annealing,'' Science 220, 671–680 (1983).
https://doi.org/10.1126/science.220.4598.671

[98] L. Giannelli, S. Sgroi, J. Brown, G. S. Paraoanu, M. Paternostro, E. Paladino, and G. Falci, ``A tutorial on optimal control and reinforcement learning methods for quantum technologies,'' Phys. Lett. A 434, 128054 (2022).
https://doi.org/10.1016/j.physleta.2022.128054

[99] M. M. Müller, R. S. Said, F. Jelezko, T. Calarco, and S. Montangero, ``One decade of quantum optimal control in the chopped random basis,'' Rep. Prog. Phys. 85, 076001 (2022).
https://doi.org/10.1088/1361-6633/ac723c

[100] T. Caneva, T. Calarco, and S. Montangero, ``Chopped random-basis quantum optimization,'' Phys. Rev. A 84, 022326 (2011).
https://doi.org/10.1103/PhysRevA.84.022326

[101] S. Zhao, M. G. Krauss, T. Bienaime, S. Whitlock, C. P. Koch, S. Qvarfort, and A. Metelmann, ``Fast and robust cat state preparation utilizing higher order nonlinearities,'' arXiv:2312.05218 (2023).
arXiv:2312.05218

[102] C. Leforestier, R. H. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, ``A comparison of different propagation schemes for the time dependent Schrödinger equation,'' J. Comput. Phys. 94, 59–80 (1991).
https://doi.org/10.1016/0021-9991(91)90137-A

[103] W.-L. Ma, S. Puri, R. J. Schoelkopf, M. H. Devoret, S. M. Girvin, and L. Jiang, ``Quantum control of bosonic modes with superconducting circuits,'' Sci. Bull. 66, 1789–1805 (2021).
https://doi.org/10.1016/j.scib.2021.05.024

[104] S. Rosenblum, P. Reinhold, M. Mirrahimi, L. Jiang, L. Frunzio, and R. J. Schoelkopf, ``Fault-tolerant detection of a quantum error,'' Science 361, 266–270 (2018).
https://doi.org/10.1126/science.aat3996

[105] L. H. Pedersen, N. M. Møller, and K. Mølmer, ``Fidelity of quantum operations,'' Phys. Lett. A 367, 47–51 (2007).
https://doi.org/10.1016/j.physleta.2007.02.069

[106] Z. Leghtas, G. Kirchmair, B. Vlastakis, M. H. Devoret, R. J. Schoelkopf, and M. Mirrahimi, ``Deterministic protocol for mapping a qubit to coherent state superpositions in a cavity,'' Phys. Rev. A 87, 042315 (2013b).
https://doi.org/10.1103/PhysRevA.87.042315

[107] 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).
https://doi.org/10.1088/1367-2630/16/4/045014

[108] S. Murmann, A. Bergschneider, V. M. Klinkhamer, G. Zürn, T. Lompe, and S. Jochim, ``Two Fermions in a Double Well: Exploring a Fundamental Building Block of the Hubbard Model,'' Phys. Rev. Lett. 114, 080402 (2015).
https://doi.org/10.1103/PhysRevLett.114.080402

[109] R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, ``Measuring entanglement entropy in a quantum many-body system,'' Nature 528, 77–83 (2015).
https://doi.org/10.1038/nature15750

[110] Z. Z. Yan, B. M. Spar, M. L. Prichard, S. Chi, H.-T. Wei, E. Ibarra-García-Padilla, K. R. A. Hazzard, and W. S. Bakr, ``Two-Dimensional Programmable Tweezer Arrays of Fermions,'' Phys. Rev. Lett. 129, 123201 (2022).
https://doi.org/10.1103/PhysRevLett.129.123201

[111] N. E. Frattini, R. G. Cortiñas, J. Venkatraman, X. Xiao, Q. Su, C. U. Lei, B. J. Chapman, V. R. Joshi, S. M. Girvin, R. J. Schoelkopf, S. Puri, and M. H. Devoret, ``Observation of Pairwise Level Degeneracies and the Quantum Regime of the Arrhenius Law in a Double-Well Parametric Oscillator,'' Phys. Rev. X 14, 031040 (2024).
https://doi.org/10.1103/PhysRevX.14.031040

[112] S. Martínez-Garaot, E. Torrontegui, X. Chen, M. Modugno, D. Guéry-Odelin, S.-Y. Tseng, and J. G. Muga, ``Vibrational Mode Multiplexing of Ultracold Atoms,'' Phys. Rev. Lett. 111, 213001 (2013).
https://doi.org/10.1103/PhysRevLett.111.213001

[113] J. Beugnon, C. Tuchendler, H. Marion, A. Gaëtan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P. A. Jones, G. Messin, A. Browaeys, and P. Grangier, ``Two-dimensional transport and transfer of a single atomic qubit in optical tweezers,'' Nat. Phys. 3, 696–699 (2007).
https://doi.org/10.1038/nphys698

[114] A. Fuhrmanek, A. M. Lance, C. Tuchendler, P. Grangier, Y. R. P. Sortais, and A. Browaeys, ``Imaging a single atom in a time-of-flight experiment,'' New J. Phys. 12, 053028 (2010).
https://doi.org/10.1088/1367-2630/12/5/053028

[115] S. Deffner and S. Campbell, ``Quantum speed limits: From Heisenberg's uncertainty principle to optimal quantum control,'' J. Phys. A: Math. Theor. 50, 453001 (2017).
https://doi.org/10.1088/1751-8121/aa86c6

[116] L. Mandelstam and I. Tamm, ``The uncertainty relation between energy and time in nonrelativistic quantum mechanics.'' J. Phys. USSR 9, 249 (1945).
https://link.springer.com/chapter/10.1007/978-3-642-74626-0_8

[117] N. Margolus and L. B. Levitin, ``The maximum speed of dynamical evolution,'' Physica D 120, 188–195 (1998).
https://doi.org/10.1016/S0167-2789(98)00054-2

[118] J. Anandan and Y. Aharonov, ``Geometry of quantum evolution,'' Phys. Rev. Lett. 65, 1697–1700 (1990).
https://doi.org/10.1103/PhysRevLett.65.1697

[119] S. Deffner and E. Lutz, ``Energy–time uncertainty relation for driven quantum systems,'' J. Phys. A: Math. Theor. 46, 335302 (2013).
https://doi.org/10.1088/1751-8113/46/33/335302

[120] X. Chen and J. G. Muga, ``Transient energy excitation in shortcuts to adiabaticity for the time-dependent harmonic oscillator,'' Phys. Rev. A 82, 053403 (2010).
https://doi.org/10.1103/PhysRevA.82.053403

[121] G. Ness, M. R. Lam, W. Alt, D. Meschede, Y. Sagi, and A. Alberti, ``Observing crossover between quantum speed limits,'' Sci. Adv. 7, eabj9119 (2021).
https://doi.org/10.1126/sciadv.abj9119

[122] K. Mølmer, Y. Castin, and J. Dalibard, ``Monte Carlo wave-function method in quantum optics,'' J. Opt. Soc. Am. B 10, 524–538 (1993).
https://doi.org/10.1364/JOSAB.10.000524

[123] J.-F. Mennemann, D. Matthes, R.-M. Weishäupl, and T. Langen, ``Optimal control of Bose–Einstein condensates in three dimensions,'' New J. Phys. 17, 113027 (2015).
https://doi.org/10.1088/1367-2630/17/11/113027

[124] M. Roda-Llordes, D. Candoli, P. T. Grochowski, A. Riera-Campeny, T. Agrenius, J. J. García-Ripoll, C. Gonzalez-Ballestero, and O. Romero-Isart, ``Numerical simulation of large-scale nonlinear open quantum mechanics,'' Phys. Rev. Res. 6, 013262 (2024b).
https://doi.org/10.1103/PhysRevResearch.6.013262

[125] A. Riera-Campeny, M. Roda-Llordes, P. T. Grochowski, and O. Romero-Isart, ``Wigner Analysis of Particle Dynamics and Decoherence in Wide Nonharmonic Potentials,'' Quantum 8, 1393 (2024).
https://doi.org/10.22331/q-2024-07-02-1393

[126] C. A. Rosiek, M. Rossi, A. Schliesser, and A. S. Sørensen, ``Quadrature Squeezing Enhances Wigner Negativity in a Mechanical Duffing Oscillator,'' PRX Quantum 5, 030312 (2024).
https://doi.org/10.1103/PRXQuantum.5.030312

[127] All data and simulation codes are available at https://doi.org/10.5281/zenodo.11160534.
https://doi.org/10.5281/zenodo.11160534

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