Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

Fabrizio Minganti1,2 and Dolf Huybrechts3,4

1Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
2Center for Quantum Science and Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium
4Univ Lyon, Ens de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is $\textit{general and model-independent}$, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization.

Characterizing the dynamical properties of open quantum systems is a major theoretical challenge. For instance, determining how long quantum properties survive the action of dissipation is pivotal for the development of quantum technologies. Such a task can be accomplished by the diagonalization of the Liouvillian superoperator, i.e., the mathematical object encoding both the coherent and dissipative evolution of a system. The Liouvillian eigendecomposition, however, is numerically challenging, and it is common practice to investigate dynamical properties by simply integrating the corresponding equations of motion. These types of extrapolations, however, discard much of the information about the system’s evolution, and are inefficient and imprecise in many cases.

Here, we present a new algorithm that combines the efficiency of the time evolution with the precision and effectiveness of the Liouvillian diagonalization. We perform a series of consecutive short evolutions on appropriately renormalized density matrices, consistent with an Arnoldi iteration algorithm performed on the system’s dynamical map. Accordingly, we construct a reduced Liouvillian incorporating all the dynamical properties otherwise “hidden” by the complexity of the dissipative dynamics. The diagonalization of this much smaller reduced Liouvillian retrieves the long-living dynamical properties up to numerical precision and, moreover, manages to determine the long-time dynamics from a much shorter one, making it a faster-than-the-clock algorithm. Our algorithm is general and can be applied to any system described by a Lindblad master equation, including Floquet (i.e., periodically driven) ones.

► BibTeX data

► References

[1] J. Q. You and F. Nori, Atomic physics and quantum optics using superconducting circuits, Nature (London) 474, 589 (2011).
https:/​/​doi.org/​10.1038/​nature10122

[2] R. J. Schoelkopf and S. M. Girvin, Wiring up quantum systems, Nature (London) 451, 664 (2008).
https:/​/​doi.org/​10.1038/​451664a

[3] H. J. Carmichael, Breakdown of Photon Blockade: A Dissipative Quantum Phase Transition in Zero Dimensions, Phys. Rev. X 5, 031028 (2015).
https:/​/​doi.org/​10.1103/​PhysRevX.5.031028

[4] J. M. Fink, A. Dombi, A. Vukics, A. Wallraff and P. Domokos, Observation of the Photon-Blockade Breakdown Phase Transition, Phys. Rev. X 7, 011012 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.011012

[5] M. Fitzpatrick, N. M. Sundaresan, A. C. Y. Li, J. Koch and A. A. Houck, Observation of a Dissipative Phase Transition in a One-Dimensional Circuit QED Lattice, Phys. Rev. X 7, 011016 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.011016

[6] M. Müller, S. Diehl, G. Pupillo and P. Zoller, Engineered Open Systems and Quantum Simulations with Atoms and Ions, Adv. At. Mol. Opt. Phys. 61, 1 (2012).
https:/​/​doi.org/​10.1016/​B978-0-12-396482-3.00001-6

[7] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
https:/​/​doi.org/​10.1038/​nature24622

[8] M. Aspelmeyer, T. J. Kippenberg and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
https:/​/​doi.org/​10.1103/​RevModPhys.86.1391

[9] 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 475, 359 (2011).
https:/​/​doi.org/​10.1038/​nature10261

[10] S. Kolkowitz, A. C. Bleszynski Jayich, Q. P. Unterreithmeier, S. D. Bennett, P. Rabl, J. G. E. Harris and M. D. Lukin, Coherent Sensing of a Mechanical Resonator with a Single-Spin Qubit, Science 335, 1603 (2012).
https:/​/​doi.org/​10.1126/​science.1216821

[11] E. Gil-Santos, M. Labousse, C. Baker, A. Goetschy, W. Hease, C. Gomez, A. Lemaı̂tre, G. Leo, C. Ciuti and I. Favero, Light-Mediated Cascaded Locking of Multiple Nano-Optomechanical Oscillators, Phys. Rev. Lett. 118, 063605 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.118.063605

[12] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
https:/​/​doi.org/​10.1038/​s41586-019-1666-5

[13] K. Schneider, Y. Baumgartner, S. Hönl, P. Welter, H. Hahn, D. J. Wilson, L. Czornomaz and P. Seidler, Optomechanics with one-dimensional gallium phosphide photonic crystal cavities, Optica 6, 577 (2019).
https:/​/​doi.org/​10.1364/​OPTICA.6.000577

[14] C. K. Andersen, A. Remm, S. Lazar, S. Krinner, N. Lacroix, G. J. Norris, M. Gabureac, C. Eichler and A. Wallraff, Repeated quantum error detection in a surface code, Nat. Phys. 16, 875 (2020).
https:/​/​doi.org/​10.1038/​s41567-020-0920-y

[15] I. Carusotto and C. Ciuti, Quantum fluids of light, Rev. Mod. Phys. 85, 299.
https:/​/​doi.org/​10.1103/​RevModPhys.85.299

[16] A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta and F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
https:/​/​doi.org/​10.1038/​s42254-018-0006-2

[17] F. Verstraete, M. M. Wolf and J. I. Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009).
https:/​/​doi.org/​10.1038/​nphys1342

[18] M. Soriente, T. L. Heugel, K. Arimitsu, R. Chitra and O. Zilberberg, Distinctive class of dissipation-induced phase transitions and their universal characteristics, Phys. Rev. Res. 3, 023100 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.023100

[19] R. D. Candia, F. Minganti, K. V. Petrovnin, G. S. Paraoanu and S. Felicetti, Critical parametric quantum sensing, (2021), arXiv:2107.04503.
arXiv:arXiv:2107.04503

[20] E. M. Kessler, G. Giedke, A. Imamoğlu, S. F. Yelin, M. D. Lukin and J. I. Cirac, Dissipative phase transition in a central spin system, Phys. Rev. A 86, 012116 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.012116

[21] B. Buča and T. Prosen, A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains, New J. Phys. 14, 073007 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​7/​073007

[22] V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.022118

[23] F. Minganti, A. Biella, N. Bartolo and C. Ciuti, Spectral theory of Liouvillians for dissipative phase transitions, Phys. Rev. A 98, 042118 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.042118

[24] D. Nigro, On the uniqueness of the steady-state solution of the Lindblad–Gorini–Kossakowski–Sudarshan equation, J. Stat. Mech. Theory Exp. 2019, 043202 (2019).
https:/​/​doi.org/​10.1088/​1742-5468/​ab0c1c

[25] C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, Berlin, 2004).

[26] S. Haroche and J. M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University Press, Oxford, 2006).
https:/​/​doi.org/​10.1093/​acprof:oso/​9780198509141.001.0001

[27] A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
https:/​/​doi.org/​10.1080/​00018732.2014.933502

[28] B. Cao, K. W. Mahmud and M. Hafezi, Two coupled nonlinear cavities in a driven-dissipative environment, Phys. Rev. A 94, 063805 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.063805

[29] A. Le Boité, G. Orso and C. Ciuti, Steady-State Phases and Tunneling-Induced Instabilities in the Driven Dissipative Bose-Hubbard Model, Phys. Rev. Lett. 110, 233601 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.233601

[30] H. Weimer, Variational Principle for Steady States of Dissipative Quantum Many-Body Systems, Phys. Rev. Lett. 114, 040402 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.040402

[31] J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio and D. Rossini, Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems, Phys. Rev. X 6, 031011 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.031011

[32] M. Biondi, G. Blatter, H. E. Türeci and S. Schmidt, Nonequilibrium gas-liquid transition in the driven-dissipative photonic lattice, Phys. Rev. A 96, 043809 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.043809

[33] W. Casteels, R. M. Wilson and M. Wouters, Gutzwiller Monte Carlo approach for a critical dissipative spin model, Phys. Rev. A 97, 062107 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.062107

[34] D. Huybrechts and M. Wouters, Cluster methods for the description of a driven-dissipative spin model, Phys. Rev. A 99, 043841 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.043841

[35] D. Huybrechts, F. Minganti, F. Nori, M. Wouters and N. Shammah, Validity of mean-field theory in a dissipative critical system: Liouvillian gap, $\mathbb{PT}$-symmetric antigap, and permutational symmetry in the $XYZ$ model, Phys. Rev. B 101, 214302 (2020).
https:/​/​doi.org/​10.1103/​PhysRevB.101.214302

[36] A. Nagy and V. Savona, Driven-dissipative quantum Monte Carlo method for open quantum systems, Phys. Rev. A 97, 052129 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.052129

[37] J. Cui, J. I. Cirac and M. C. Bañuls, Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems, Phys. Rev. Lett. 114, 220601 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.220601

[38] 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).
https:/​/​doi.org/​10.1103/​PhysRevA.92.022116

[39] A. Kshetrimayum, H. Weimer and R. Orús, A simple tensor network algorithm for two-dimensional steady states, Nat. Commun. 8, 1291 (2017).
https:/​/​doi.org/​10.1038/​s41467-017-01511-6

[40] A. Nagy and V. Savona, Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems, Phys. Rev. Lett. 122, 250501 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.250501

[41] F. Vicentini, A. Biella, N. Regnault and C. Ciuti, Variational Neural-Network Ansatz for Steady States in Open Quantum Systems, Phys. Rev. Lett. 122, 250503 (2019a).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.250503

[42] M. J. Hartmann and G. Carleo, Neural-Network Approach to Dissipative Quantum Many-Body Dynamics, Phys. Rev. Lett. 122, 250502 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.250502

[43] M. Gegg and M. Richter, PsiQuaSP–A library for efficient computation of symmetric open quantum systems, Sci. Rep. 7, 16304 (2017).
https:/​/​doi.org/​10.1038/​s41598-017-16178-8

[44] N. Shammah, S. Ahmed, N. Lambert, S. De Liberato and F. Nori, Open quantum systems with local and collective incoherent processes: Efficient numerical simulations using permutational invariance, Phys. Rev. A 98, 063815 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.063815

[45] 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).
https:/​/​doi.org/​10.1126/​science.aaa2085

[46] 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, 18 (2017).
https:/​/​doi.org/​10.1038/​s41534-017-0019-1

[47] S. Finazzi, A. Le Boité, F. Storme, A. Baksic and C. Ciuti, Corner-Space Renormalization Method for Driven-Dissipative Two-Dimensional Correlated Systems, Phys. Rev. Lett. 115, 080604 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.080604

[48] R. Rota, F. Storme, N. Bartolo, R. Fazio and C. Ciuti, Critical behavior of dissipative two-dimensional spin lattices, Phys. Rev. B 95, 134431 (2017).
https:/​/​doi.org/​10.1103/​PhysRevB.95.134431

[49] M. Foss-Feig, P. Niroula, J. T. Young, M. Hafezi, A. V. Gorshkov, R. M. Wilson and M. F. Maghrebi, Emergent equilibrium in many-body optical bistability, Phys. Rev. A 95, 043826 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.043826

[50] F. Vicentini, F. Minganti, R. Rota, G. Orso and C. Ciuti, Critical slowing down in driven-dissipative Bose-Hubbard lattices, Phys. Rev. A 97, 013853 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.013853

[51] W. Casteels, F. Storme, A. Le Boité and C. Ciuti, Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators, Phys. Rev. A 93, 033824 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.033824

[52] W. Casteels, R. Fazio and C. Ciuti, Critical dynamical properties of a first-order dissipative phase transition, Phys. Rev. A 95, 012128 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.012128

[53] W. Verstraelen and M. Wouters, Classical critical dynamics in quadratically driven Kerr resonators, Phys. Rev. A 101, 043826 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.043826

[54] D. Huybrechts and M. Wouters, Dynamical hysteresis properties of the driven-dissipative Bose-Hubbard model with a Gutzwiller Monte Carlo approach, Phys. Rev. A 102, 053706 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.053706

[55] N. Bartolo, F. Minganti, J. Lolli and C. Ciuti, Homodyne versus photon-counting quantum trajectories for dissipative Kerr resonators with two-photon driving, Eur. Phys. J. Spec. Top. 226, 2705 (2017).
https:/​/​doi.org/​10.1140/​epjst/​e2016-60385-8

[56] R. Rota, F. Minganti, A. Biella and C. Ciuti, Dynamical properties of dissipative XYZ Heisenberg lattices, New J. Phys. 20, 045003 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aab703

[57] C. Sánchez Muñoz, B. Buča, J. Tindall, A. González-Tudela, D. Jaksch and D. Porras, Symmetries and conservation laws in quantum trajectories: Dissipative freezing, Phys. Rev. A 100, 042113 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.042113

[58] M. J. Hartmann, Polariton Crystallization in Driven Arrays of Lossy Nonlinear Resonators, Phys. Rev. Lett. 104, 113601 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.113601

[59] D. A. Lidar, Lecture Notes on the Theory of Open Quantum Systems, arXiv:1902.00967.
arXiv:arXiv:1902.00967
https:/​/​arxiv.org/​abs/​1902.00967

[60] P. D. Nation, Steady-state solution methods for open quantum optical systems, (2015), arXiv:1504.06768.
arXiv:arXiv:1504.06768

[61] F. Iemini, A. Russomanno, J. Keeling, M. Schirò, M. Dalmonte and R. Fazio, Boundary Time Crystals, Phys. Rev. Lett. 121, 035301 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.035301

[62] K. Seibold, R. Rota and V. Savona, Dissipative time crystal in an asymmetric nonlinear photonic dimer, Phys. Rev. A 101, 033839 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.033839

[63] F. Minganti, I. I. Arkhipov, A. Miranowicz and F. Nori, Correspondence between dissipative phase transitions of light and time crystals, arXiv:2008.08075 (2020a).
arXiv:2008.08075

[64] J. Tindall, C. S. Muñoz, B. Buča and D. Jaksch, Quantum synchronisation enabled by dynamical symmetries and dissipation, New J. Phys. 22, 013026 (2020).
https:/​/​doi.org/​10.1088/​1367-2630/​ab60f5

[65] K. Macieszczak, M. Guţă, I. Lesanovsky and J. P. Garrahan, Dynamical phase transitions as a resource for quantum enhanced metrology, Phys. Rev. A 93, 022103 (2016a).
https:/​/​doi.org/​10.1103/​PhysRevA.93.022103

[66] R. Rota, F. Minganti, C. Ciuti and V. Savona, Quantum Critical Regime in a Quadratically Driven Nonlinear Photonic Lattice, Phys. Rev. Lett. 122, 110405 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.110405

[67] C. Sánchez Muñoz and D. Jaksch, Squeezed Lasing, Phys. Rev. Lett. 127, 183603 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.127.183603

[68] V. V. Albert, K. Noh, K. Duivenvoorden, D. J. Young, R. T. Brierley, P. Reinhold, C. Vuillot, L. Li, C. Shen, S. M. Girvin et al., Performance and structure of single-mode bosonic codes, Phys. Rev. A 97, 032346 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.032346

[69] S. Lieu, R. Belyansky, J. T. Young, R. Lundgren, V. V. Albert and A. V. Gorshkov, Symmetry Breaking and Error Correction in Open Quantum Systems, Phys. Rev. Lett. 125, 240405 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.240405

[70] F. Minganti, A. Miranowicz, R. W. Chhajlany and F. Nori, Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps, Phys. Rev. A 100, 062131 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.062131

[71] N. Hatano, Exceptional points of the Lindblad operator of a two-level system, Mol. Phys. 117, 2121 (2019).
https:/​/​doi.org/​10.1080/​00268976.2019.1593535

[72] F. Minganti, A. Miranowicz, R. W. Chhajlany, I. I. Arkhipov and F. Nori, Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories, Phys. Rev. A 101, 062112 (2020b).
https:/​/​doi.org/​10.1103/​PhysRevA.101.062112

[73] I. I. Arkhipov, A. Miranowicz, F. Minganti and F. Nori, Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory, Phys. Rev. A 101, 013812 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.013812

[74] I. I. Arkhipov, F. Minganti, A. Miranowicz and F. Nori, Generating high-order quantum exceptional points in synthetic dimensions, Phys. Rev. A 104, 012205 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.104.012205

[75] P. Kumar, K. Snizhko, Y. Gefen and B. Rosenow, Optimized steering: Quantum state engineering and exceptional points, Phys. Rev. A 105, L010203 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.105.L010203

[76] P. Kumar, K. Snizhko and Y. Gefen, Near-unit efficiency of chiral state conversion via hybrid-Liouvillian dynamics, Phys. Rev. A 104, L050405 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.104.L050405

[77] M. Naghiloo, M. Abbasi, Y. N. Joglekar and K. W. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit, Nat. Phys. 15, 1232 (2019).
https:/​/​doi.org/​10.1038/​s41567-019-0652-z

[78] W. Chen, M. Abbasi, Y. N. Joglekar and K. W. Murch, Quantum Jumps in the Non-Hermitian Dynamics of a Superconducting Qubit, Phys. Rev. Lett. 127, 140504 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.127.140504

[79] Z. Gong, R. Hamazaki and M. Ueda, Discrete Time-Crystalline Order in Cavity and Circuit QED Systems, Phys. Rev. Lett. 120, 040404 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.040404

[80] A. Lazarides, S. Roy, F. Piazza and R. Moessner, Time crystallinity in dissipative Floquet systems, Phys. Rev. Res. 2, 022002 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.022002

[81] S. A. Sato, U. D. Giovannini, S. Aeschlimann, I. Gierz, H. Hübener and A. Rubio, Floquet states in dissipative open quantum systems, J. Phys. B At. Mol. Opt. Phys. 53, 225601 (2020).
https:/​/​doi.org/​10.1088/​1361-6455/​abb127

[82] T. N. Ikeda and M. Sato, General description for nonequilibrium steady states in periodically driven dissipative quantum systems, Sci. Adv. 6 (2020).
https:/​/​doi.org/​10.1126/​sciadv.abb4019

[83] T. N. Ikeda, K. Chinzei and M. Sato, Nonequilibrium steady states in the Floquet-Lindblad systems: van Vleck's high-frequency expansion approach, (2021), arXiv:2107.07911 [cond-mat.mes-hall].
arXiv:2107.07911

[84] S. Restrepo, J. Cerrillo, V. M. Bastidas, D. G. Angelakis and T. Brandes, Driven Open Quantum Systems and Floquet Stroboscopic Dynamics, Phys. Rev. Lett. 117, 250401 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.250401

[85] M. Hartmann, D. Poletti, M. Ivanchenko, S. Denisov and P. Hänggi, Asymptotic Floquet states of open quantum systems: the role of interaction, New J. Phys. 19, 083011 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa7ceb

[86] F. Minganti and D. Huybrechts, GitHub repository: Arnoldi-Lindblad time evolution (2022).
https:/​/​github.com/​DHuybrechts/​Arnoldi-Lindblad-time-evolution

[87] H. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[88] Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction (Springer, Berlin, 2011).
https:/​/​doi.org/​10.1007/​978-3-642-23354-8

[89] K. Macieszczak, M. Gută, I. Lesanovsky and J. P. Garrahan, Towards a Theory of Metastability in Open Quantum Dynamics, Phys. Rev. Lett. 116, 240404 (2016b).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.240404

[90] D. O. Krimer and M. Pletyukhov, Few-Mode Geometric Description of a Driven-Dissipative Phase Transition in an Open Quantum System, Phys. Rev. Lett. 123, 110604 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.110604

[91] F. Minganti, Out-of-Equilibrium Phase Transitions in Nonlinear Optical Systems, Phd thesis, Université Sorbonne Paris Cité (2018).
https:/​/​tel.archives-ouvertes.fr/​tel-02003919

[92] M. Nakagawa, N. Kawakami and M. Ueda, Exact Liouvillian Spectrum of a One-Dimensional Dissipative Hubbard Model, Phys. Rev. Lett. 126, 110404 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.110404

[93] V. Popkov and C. Presilla, Full Spectrum of the Liouvillian of Open Dissipative Quantum Systems in the Zeno Limit, Phys. Rev. Lett. 126, 190402 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.190402

[94] L. Trefethen and D. Bau, Numerical Linear Algebra (Society for Industrial and Applied Mathematics, Philadelphia, 1997).

[95] L. Rosso, A. Biella and L. Mazza, The one-dimensional Bose gas with strong two-body losses: dissipative fermionisation and the harmonic confinement, (2021a), arXiv:2106.08092 [cond-mat.quant-gas].
https:/​/​doi.org/​10.21468/​SciPostPhys.12.1.044
arXiv:2106.08092

[96] D. Rossini, A. Ghermaoui, M. B. Aguilera, R. Vatré, R. Bouganne, J. Beugnon, F. Gerbier and L. Mazza, Strong correlations in lossy one-dimensional quantum gases: From the quantum Zeno effect to the generalized Gibbs ensemble, Phys. Rev. A 103, L060201 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.L060201

[97] L. Rosso, D. Rossini, A. Biella and L. Mazza, One-dimensional spin-1/​2 fermionic gases with two-body losses: Weak dissipation and spin conservation, Phys. Rev. A 104, 053305 (2021b).
https:/​/​doi.org/​10.1103/​PhysRevA.104.053305

[98] F. Minganti, I. I. Arkhipov, A. Miranowicz and F. Nori, Liouvillian spectral collapse in the Scully-Lamb laser model, Phys. Rev. Res. 3, 043197 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.043197

[99] X. Gu, A. F. Kockum, A. Miranowicz, Y. X. Liu and F. Nori, Microwave photonics with superconducting quantum circuits, Phys. Rep. 718-719, 1 (2017).
https:/​/​doi.org/​10.1016/​j.physrep.2017.10.002

[100] P. D. Drummond and D. F. Walls, Quantum theory of optical bistability. I. Nonlinear polarisability model, J. Phys. A: Math. Gen. 13, 725 (1980).
https:/​/​doi.org/​10.1088/​0305-4470/​13/​2/​034

[101] K. Stannigel, P. Rabl and P. Zoller, Driven-dissipative preparation of entangled states in cascaded quantum-optical networks, New J. Phys. 14, 063014 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​6/​063014

[102] 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).
https:/​/​doi.org/​10.1103/​PhysRevX.10.021022

[103] W. Casteels and C. Ciuti, Quantum entanglement in the spatial-symmetry-breaking phase transition of a driven-dissipative Bose-Hubbard dimer, Phys. Rev. A 95, 013812 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.013812

[104] S. R. K. Rodriguez, W. Casteels, F. Storme, N. Carlon Zambon, I. Sagnes, L. Le Gratiet, E. Galopin, A. Lemaı̂tre, A. Amo, C. Ciuti et al., Probing a Dissipative Phase Transition via Dynamical Optical Hysteresis, Phys. Rev. Lett. 118, 247402 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.118.247402

[105] N. Bartolo, F. Minganti, W. Casteels and C. Ciuti, Exact steady state of a Kerr resonator with one- and two-photon driving and dissipation: Controllable Wigner-function multimodality and dissipative phase transitions, Phys. Rev. A 94, 033841 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.033841

[106] W. Verstraelen, R. Rota, V. Savona and M. Wouters, Gaussian trajectory approach to dissipative phase transitions: The case of quadratically driven photonic lattices, Phys. Rev. Res. 2, 022037 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.022037

[107] C. Lledó and M. H. Szymańska, A dissipative time crystal with or without Z2 symmetry breaking, New J. Phys. 22, 075002 (2020).
https:/​/​doi.org/​10.1088/​1367-2630/​ab9ae3

[108] J. Johansson, P. Nation and F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
https:/​/​doi.org/​10.1016/​j.cpc.2012.02.021

[109] J. Johansson, P. Nation and F. Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 184, 1234 (2013).
https:/​/​doi.org/​10.1016/​j.cpc.2012.11.019

[110] E. Jones, T. Oliphant, P. Peterson et al., SciPy: Open source scientific tools for Python, http:/​/​www.scipy.org/​, (2001–).
http:/​/​www.scipy.org/​

[111] S. Blanes, F. Casas, J. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470, 151 (2009).
https:/​/​doi.org/​10.1016/​j.physrep.2008.11.001

[112] S. Blanes, F. Casas, J. A. Oteo and J. Ros, A pedagogical approach to the Magnus expansion, Eur. J. Phys. 31, 907.
https:/​/​doi.org/​10.1088/​0143-0807/​31/​4/​020

[113] T. Laptyeva, E. Kozinov, I. Meyerov, M. Ivanchenko, S. Denisov and P. Hänggi, Calculating Floquet states of large quantum systems: A parallelization strategy and its cluster implementation, Comput. Phys. Commun. 201, 85 (2016).
https:/​/​doi.org/​10.1016/​j.cpc.2015.12.024

[114] T. Kuwahara, T. Mori and K. Saito, Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems, Ann. Phys. 367, 96 (2016).
https:/​/​doi.org/​10.1016/​j.aop.2016.01.012

[115] A. Vardi and J. R. Anglin, Bose-Einstein Condensates beyond Mean Field Theory: Quantum Backreaction as Decoherence, Phys. Rev. Lett. 86, 568 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.86.568

[116] F. Trimborn, D. Witthaut and S. Wimberger, Mean-field dynamics of a two-mode Bose–Einstein condensate subject to noise and dissipation, J. Phys. B At. Mol. Opt. 41, 171001 (2008).
https:/​/​doi.org/​10.1088/​0953-4075/​41/​17/​171001

[117] C. Weiss and N. Teichmann, Differences between Mean-Field Dynamics and $N$-Particle Quantum Dynamics as a Signature of Entanglement, Phys. Rev. Lett. 100, 140408 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.100.140408

[118] D. Poletti, J.-S. Bernier, A. Georges and C. Kollath, Interaction-Induced Impeding of Decoherence and Anomalous Diffusion, Phys. Rev. Lett. 109, 045302 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.045302

[119] B. Baumgartner and N. Heide, Analysis of quantum semigroups with GKS-Lindblad generators: II. General, J. Phys. A Math. Theor. 41, 395303 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​39/​395303

[120] D. Huybrechts, M. Wouters and F. Minganti, Mean-field dynamical failure in all-to-all connected $n$-level systems (to appear in 2022, in preparation).

[121] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide (Society for Industrial and Applied Mathematics, 2000).
https:/​/​doi.org/​10.1137/​1.9780898719581

[122] D. Kressner, Numerical Methods for General and Structured Eigenvalue Problems (Springer, Berlin, Heidelberg, 2005).
https:/​/​doi.org/​10.1007/​3-540-28502-4

[123] J. Dalibard, Y. Castin and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992).
https:/​/​doi.org/​10.1103/​PhysRevLett.68.580

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

[125] F. Vicentini, F. Minganti, A. Biella, G. Orso and C. Ciuti, Optimal stochastic unraveling of disordered open quantum systems: Application to driven-dissipative photonic lattices, Phys. Rev. A 99, 032115 (2019b).
https:/​/​doi.org/​10.1103/​PhysRevA.99.032115

[126] F. Pietracaprina, N. Macé, D. J. Luitz and F. Alet, Shift-invert diagonalization of large many-body localizing spin chains, SciPost Phys. 5, 45 (2018).
https:/​/​doi.org/​10.21468/​SciPostPhys.5.5.045

Cited by

[1] Haggai Landa and Grégoire Misguich, "Nonlocal correlations in noisy multiqubit systems simulated using matrix product operators", SciPost Physics Core 6 2, 037 (2023).

[2] Fabrizio Minganti, Vincenzo Savona, and Alberto Biella, "Dissipative phase transitions in n-photon driven quantum nonlinear resonators", Quantum 7, 1170 (2023).

[3] Aranya Bhattacharya, Pratik Nandy, Pingal Pratyush Nath, and Himanshu Sahu, "Operator growth and Krylov construction in dissipative open quantum systems", Journal of High Energy Physics 2022 12, 81 (2022).

[4] Limin Xu, "A Quantum Kinetic Monte Carlo Method For Lindblad Equation", arXiv:2306.05102, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-29 03:58:22) and SAO/NASA ADS (last updated successfully 2024-03-29 03:58:23). The list may be incomplete as not all publishers provide suitable and complete citation data.