- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Diffusion in quantum state preparation: From passive cooling to system-bath engineering

Tim Pokart1, Lukas König1, Sebastian Diehl2,*, and Jan Carl Budich1,3,4

  • *Contact author: diehl@thp.uni-koeln.de
PDF

Phys. Rev. Research 8, 023336 – Published 25 June, 2026

DOI: https://doi.org/10.1103/4q72-7gg6

Abstract

We investigate and compare two particle number-conserving protocols for the preparation of a topologically nontrivial state. The first is derived from thermally coupling the system to a cold bath, while the second is based on engineered dissipation. We numerically study the time required to reach the target state as well as its robustness against physically important perturbations. Crucially, in both protocols, the cooling capability is limited by dissipatively induced diffusion processes. The resulting quadratic scaling of the cooling time with system size is also corroborated analytically using mean-field approximations and a purely classical random-walk model. Furthermore, we find that the engineered protocol admits a unique and stable dark state, which contributes to an ongoing discussion regarding the applicability of dissipative state preparation to many-body systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (102)

  1. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  2. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  3. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, Nature (London) 464, 45 (2010).
  4. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  5. T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018).
  6. V. Danos, E. Kashefi, and P. Panangaden, The measurement calculus, J. ACM 54, 8 (2007).
  7. D. P. DiVincenzo, The physical implementation of quantum computation, Fortschr. Phys. 48, 771 (2000).
  8. J. I. Cirac and P. Zoller, Goals and opportunities in quantum simulation, Nat. Phys. 8, 264 (2012).
  9. A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature (London) 607, 667 (2022).
  10. J. Kempe, A. Kitaev, and O. Regev, The complexity of the local Hamiltonian problem, SIAM J. Comput. 35, 1070 (2006).
  11. D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, The power of quantum systems on a line, Commun. Math. Phys. 287, 41 (2009).
  12. J. Preskill, Quantum computing 40 years later, Feynman Lectures on Computation, edited by T. Hey (CRC Press, Boca Raton, 2023), pp. 193–244.
  13. J. Watrous, Quantum computational complexity, in Encyclopedia of Complexity and Systems Science, edited by R. A. Meyers (Springer, New York, NY, 2009), pp. 7174–7201.
  14. I. Buluta and F. Nori, Quantum simulators, Science 326, 108 (2009).
  15. I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  16. E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Demler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriksson, K.-M. C. Fu, et al., Quantum simulators: Architectures and opportunities, PRX Quantum 2, 017003 (2021).
  17. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  18. Y. Guo, R. M. Kroeze, B. P. Marsh, S. Gopalakrishnan, J. Keeling, and B. L. Lev, An optical lattice with sound, Nature (London) 599, 211 (2021).
  19. V. D. Vaidya, Y. Guo, R. M. Kroeze, K. E. Ballantine, A. J. Kollár, J. Keeling, and B. L. Lev, Tunable-range, photon-mediated atomic interactions in multimode cavity QED, Phys. Rev. X 8, 011002 (2018).
  20. D. Dreon, A. Baumgärtner, X. Li, S. Hertlein, T. Esslinger, and T. Donner, Self-oscillating pump in a topological dissipative atom–cavity system, Nature (London) 608, 494 (2022).
  21. P. Kongkhambut, J. Skulte, L. Mathey, J. G. Cosme, A. Hemmerich, and H. Keßler, Observation of a continuous time crystal, Science 377, 670 (2022).
  22. R. Landig, L. Hruby, N. Dogra, M. Landini, R. Mottl, T. Donner, and T. Esslinger, Quantum phases from competing short- and long-range interactions in an optical lattice, Nature (London) 532, 476 (2016).
  23. J. Klinder, H. Keßler, M. R. Bakhtiari, M. Thorwart, and A. Hemmerich, Observation of a superradiant Mott insulator in the Dicke-Hubbard model, Phys. Rev. Lett. 115, 230403 (2015).
  24. Z. Wu, J. Fan, X. Zhang, J. Qi, and H. Wu, Signatures of prethermalization in a quenched cavity-mediated long-range interacting Fermi gas, Phys. Rev. Lett. 131, 243401 (2023).
  25. T. Zwettler, G. Del Pace, F. Marijanovic, S. Chattopadhyay, T. Bühler, C.-M. Halati, L. Skolc, L. Tolle, V. Helson, G. Bolognini, A. Fabre, S. Uchino, T. Giamarchi, E. Demler, and J. P. Brantut, Nonequilibrium dynamics of long-range interacting fermions, Phys. Rev. X 15, 021089 (2025).
  26. A. Periwal, E. S. Cooper, P. Kunkel, J. F. Wienand, E. J. Davis, and M. Schleier-Smith, Programmable interactions and emergent geometry in an array of atom clouds, Nature (London) 600, 630 (2021).
  27. D. J. Young, A. Chu, E. Y. Song, D. Barberena, D. Wellnitz, Z. Niu, V. M. Schäfer, R. J. Lewis-Swan, A. M. Rey, and J. K. Thompson, Observing dynamical phases of BCS superconductors in a cavity QED simulator, Nature (London) 625, 679 (2024).
  28. N. Sauerwein, F. Orsi, P. Uhrich, S. Bandyopadhyay, F. Mattiotti, T. Cantat-Moltrecht, G. Pupillo, P. Hauke, and J.-P. Brantut, Engineering random spin models with atoms in a high-finesse cavity, Nat. Phys. 19, 1128 (2023).
  29. A. A. Houck, H. E. Türeci, and J. Koch, On-chip quantum simulation with superconducting circuits, Nat. Phys. 8, 292 (2012).
  30. 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).
  31. T. Müller, S. Fölling, A. Widera, and I. Bloch, State preparation and dynamics of ultracold atoms in higher lattice orbitals, Phys. Rev. Lett. 99, 200405 (2007).
  32. A. Schindewolf, R. Bause, X.-Y. Chen, M. Duda, T. Karman, I. Bloch, and X.-Y. Luo, Evaporation of microwave-shielded polar molecules to quantum degeneracy, Nature (London) 607, 677 (2022).
  33. G. Pagano, P. W. Hess, H. B. Kaplan, W. L. Tan, P. Richerme, P. Becker, A. Kyprianidis, J. Zhang, E. Birckelbaw, M. R. Hernandez, Y. Wu, and C. Monroe, Cryogenic trapped-ion system for large scale quantum simulation, Quantum Sci. Technol. 4, 014004 (2018).
  34. P. Magnard, S. Storz, P. Kurpiers, J. Schär, F. Marxer, J. Lütolf, T. Walter, J.-C. Besse, M. Gabureac, K. Reuer, A. Akin, B. Royer, A. Blais, and A. Wallraff, Microwave quantum link between superconducting circuits housed in spatially separated cryogenic systems, Phys. Rev. Lett. 125, 260502 (2020).
  35. C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021).
  36. M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet, Nat. Phys. 13, 781 (2017).
  37. H. Chang and J. Zhang, Refrigeration technologies of cryogenic chips, Chip 2, 100054 (2023).
  38. C.-F. Chen, H.-Y. Huang, J. Preskill, and L. Zhou, Local minima in quantum systems, Nat. Phys. 21, 654 (2025).
  39. S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4, 878 (2008).
  40. F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009).
  41. B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller, Preparation of entangled states by quantum Markov processes, Phys. Rev. A 78, 042307 (2008).
  42. M. Müller, S. Diehl, G. Pupillo, and P. Zoller, Engineered open systems and quantum simulations with atoms and ions, in Advances in Atomic, Molecular, and Optical Physics, edited by P. Berman, E. Arimondo, and C. Lin (Academic Press, Oxford, UK, 2012), Vol. 61, pp. 1–80.
  43. P. M. Harrington, E. J. Mueller, and K. W. Murch, Engineered dissipation for quantum information science, Nat. Rev. Phys. 4, 660 (2022).
  44. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Great Clarendon Street, Oxford, 2007).
  45. A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
  46. L.-M. Duan and G.-C. Guo, Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment, Phys. Rev. A 57, 737 (1998).
  47. D. DiVincenzo and B. Terhal, Decoherence: The obstacle to quantum computation, Phys. World 11, 53 (1998).
  48. L. Gyongyosi and S. Imre, A survey on quantum computing technology, Comput. Sci. Rev. 31, 51 (2019).
  49. S. Daffer, K. Wódkiewicz, and J. K. McIver, Quantum Markov channels for qubits, Phys. Rev. A 67, 062312 (2003).
  50. P. Zanardi, Dissipation and decoherence in a quantum register, Phys. Rev. A 57, 3276 (1998).
  51. H. Pichler, A. J. Daley, and P. Zoller, Nonequilibrium dynamics of bosonic atoms in optical lattices: Decoherence of many-body states due to spontaneous emission, Phys. Rev. A 82, 063605 (2010).
  52. A. Griessner, A. J. Daley, S. R. Clark, D. Jaksch, and P. Zoller, Dark-state cooling of atoms by superfluid immersion, Phys. Rev. Lett. 97, 220403 (2006).
  53. A. Griessner, A. J. Daley, S. R. Clark, D. Jaksch, and P. Zoller, Dissipative dynamics of atomic Hubbard models coupled to a phonon bath: Dark state cooling of atoms within a Bloch band of an optical lattice, New J. Phys. 9, 44 (2007).
  54. T. Albash and D. A. Lidar, Decoherence in adiabatic quantum computation, Phys. Rev. A 91, 062320 (2015).
  55. J. F. Poyatos, J. I. Cirac, and P. Zoller, Quantum reservoir engineering with laser cooled trapped ions, Phys. Rev. Lett. 77, 4728 (1996).
  56. D. Molpeceres, S. Lu, J. I. Cirac, and B. Kraus, Quantum algorithms for cooling: A simple case study, Phys. Rev. Res. 7, 033162 (2025).
  57. J. Lloyd, A. A. Michailidis, X. Mi, V. Smelyanskiy, and D. A. Abanin, Quasiparticle cooling algorithms for quantum many-body state preparation, PRX Quantum 6, 010361 (2025).
  58. X. Mi, A. A. Michailidis, S. Shabani, K. C. Miao, P. V. Klimov, J. Lloyd, E. Rosenberg, R. Acharya, I. Aleiner, T. I. Andersen, et al., Stable quantum-correlated many-body states through engineered dissipation, Science 383, 1332 (2024).
  59. F. Tonielli, J. C. Budich, A. Altland, and S. Diehl, Topological field theory far from equilibrium, Phys. Rev. Lett. 124, 240404 (2020).
  60. Z. Liu, E. J. Bergholtz, and J. C. Budich, Dissipative preparation of fractional Chern insulators, Phys. Rev. Res. 3, 043119 (2021).
  61. P. A. Nosov, D. S. Shapiro, M. Goldstein, and I. S. Burmistrov, Reaction-diffusive dynamics of number-conserving dissipative quantum state preparation, Phys. Rev. B 107, 174312 (2023).
  62. A. A. Lyublinskaya and I. S. Burmistrov, Diffusive modes of two-band fermions under number-conserving dissipative dynamics, JETP Lett. 118, 524 (2023).
  63. A. A. Lyublinskaya, P. A. Nosov, and I. S. Burmistrov, Instability of the engineered dark state in two-band fermions under number-conserving dissipative dynamics, Phys. Rev. B 111, 094302 (2025).
  64. O. I. Baburin and I. S. Burmistrov, Minimal velocity of the travelling wave solutions in two coupled FKPP equations with the global conservation law, Phys. Rev. E 113, L032202 (2026).
  65. M. S. Shustin, S. V. Aksenov, and I. S. Burmistrov, Dissipation-induced steady states in topological superconductors: Mechanisms and design principles, SciPost Phys. 20, 138 (2026).
  66. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  67. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976).
  68. K. Shiraishi, M. Nakagawa, T. Mori, and M. Ueda, Quantum master equation for many-body systems based on the Lieb-Robinson bound, Phys. Rev. B 111, 184311 (2025).
  69. E. B. Davies, Markovian master equations, Commun. Math. Phys. 39, 91 (1974).
  70. F. Nathan and M. S. Rudner, Universal Lindblad equation for open quantum systems, Phys. Rev. B 102, 115109 (2020).
  71. In the case of the Davies equation, this will render the power spectrum diagonal as γμν(ω)∝δμν. For the ULE, we then assume a diagonal spectral density function Jμν∝δμν.
  72. E. Mozgunov and D. Lidar, Completely positive master equation for arbitrary driving and small level spacing, Quantum 4, 227 (2020).
  73. D. Davidović, Completely positive, simple, and possibly highly accurate approximation of the Redfield equation, Quantum 4, 326 (2020).
  74. D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu, and J. R. Petta, Quantum spintronics: Engineering and manipulating atom-like spins in semiconductors, Science 339, 1174 (2013).
  75. A. Altland, M. Fleischhauer, and S. Diehl, Symmetry classes of open fermionic quantum matter, Phys. Rev. X 11, 021037 (2021).
  76. Z. Guo, Z.-T. Xu, M. Li, L. You, and S. Yang, Variational matrix product state approach for non-Hermitian system based on a companion Hermitian Hamiltonian, arXiv:2210.14858.
  77. P. Zhong, W. Pan, H. Lin, X. Wang, and S. Hu, Density matrix renormalization group algorithm for non-Hermitian systems, Phys. Rev. Lett. 135, 106502 (2025).
  78. K. Yamamoto, M. Nakagawa, M. Tezuka, M. Ueda, and N. Kawakami, Universal properties of dissipative Tomonaga-Luttinger liquids: case study of a non-Hermitian XXZ spin chain, Phys. Rev. B 105, 205125 (2022).
  79. G. Chen, F. Song, and J. L. Lado, Topological spin excitations in non-Hermitian spin chains with a generalized kernel polynomial algorithm, Phys. Rev. Lett. 130, 100401 (2023).
  80. P. Westhoff, M. Moroder, U. Schollwöck, and S. Paeckel, A tensor network framework for Lindbladian spectra and steady states, arXiv:2509.07709.
  81. G. Chen, J. L. Lado, and F. Song, Many-body Liouvillian dynamics with a non-Hermitian tensor-network kernel polynomial algorithm, Phys. Rev. Res. 6, 043182 (2024).
  82. A. Sander, M. Fröhlich, M. Eigel, J. Eisert, P. Gelß, M. Hintermüller, R. M. Milbradt, R. Wille, and C. B. Mendl, Large-scale stochastic simulation of open quantum systems, Nat. Commun. 16, 11074 (2025).
  83. L. Causer, M. C. Bañuls, and J. P. Garrahan, Dynamical heterogeneity and large deviations in the open quantum East glass model from tensor networks, Phys. Rev. B 111, L060303 (2025).
  84. E. Mascarenhas, H. Flayac, and V. Savona, Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays, Phys. Rev. A 92, 022116 (2015).
  85. P.-O. Löwdin, On the nonorthogonality problem, in Advances in Quantum Chemistry, edited by P.-O. Löwdin (Academic Press, New York, London, 1970), pp. 185–199.
  86. N. Albert, Y. Zhang, and H.-H. Tu, Truncated Gaussian basis approach for simulating many-body dynamics, arXiv:2410.04204.
  87. This follows from a counting argument noticing that ∑m=0LLm2=2LL=NL=dimH.
  88. See Supplemental Material at http://link.aps.org/supplemental/10.1103/4q72-7gg6 for the jump operator matrix elements.
  89. A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, UK, 2011).
  90. U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge University Press, Cambridge, UK, 2014).
  91. P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010.
  92. M. Žnidarič, Entanglement growth in diffusive systems, Commun. Phys. 3, 100 (2020).
  93. The full description of the system operators in this basis is provided in the Supplemental Material [88].
  94. Z.-M. Huang, X.-Q. Sun, and S. Diehl, Topological gauge theory for mixed Dirac stationary states in all dimensions, Phys. Rev. B 106, 245204 (2022).
  95. R. Mittal, T. Zander, J. Lang, and S. Diehl, Fermion quantum criticality far from equilibrium, Phys. Rev. X 16, 011069 (2026).
  96. F. Iemini, D. Rossini, R. Fazio, S. Diehl, and L. Mazza, Dissipative topological superconductors in number-conserving systems, Phys. Rev. B 93, 115113 (2016).
  97. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022); Codebase release 0.3 for ITensor 4-r0.3 (2022).
  98. C. Rackauckas and Q. Nie, DifferentialEquations.jl—A performant and feature-rich ecosystem for solving differential equations in Julia, J. Open Res. Software 5, 15 (2017).
  99. J. Haegeman, KrylovKit, Zenodo, 2025, https://zenodo.org/records/17316953.
  100. J. Verzani, Roots.jl: Root finding functions for Julia, GitHub, 2020, https://github.com/JuliaMath/Roots.jl.
  101. C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, et al., Array programming with NumPy, Nature (London) 585, 357 (2020).
  102. T. Pokart, L. König, S. Diehl, and J. C. Budich, Dissipative diffusion in quantum state preparation: from passive cooling to system-bath engineering [Data set], Zenodo, 2026, https://zenodo.org/records/18379558.

Outline

Information

Phys. Rev. Research 8, 023336– Published 25 June, 2026
  • Received 3 February 2026
  • Accepted 11 May 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation