General properties of the spectral form factor in open quantum systems

Abstract

The spectral form factor (SFF) can probe the eigenvalue statistic at different energy scales as its time variable varies. In closed quantum chaotic systems, the SFF exhibits a universal dip-ramp-plateau behavior, which reflects the spectrum rigidity of the Hamiltonian. In this work, we explore the general properties of SFF in open quantum systems. We find that in open systems the SFF first decays exponentially, followed by a linear increase at some intermediate time scale, and finally decreases to a saturated plateau value. We derive general relations between (i) the early-time decay exponent and Lindblad operators; (ii) the long-time plateau value and the number of steady states. We also explain the effective field theory perspective of general behaviors. We verify our theoretical predictions by numerically simulating the Sachdev−Ye−Kitaev (SYK) model, random matrix theory (RMT), and the Bose−Hubbard model.

Graphical abstract

Keywords

spectral form factor / open quantum system / quantum chaos

References

[1] E. Brézin, A. Zee. Universality of the correlations between eigenvalues of large random matrices. Nucl. Phys. B, 1993, 402(3): 613

[2] E. Brézin, S. Hikami. Correlations of nearby levels induced by a random potential. Nucl. Phys. B, 1996, 479(3): 697

[3] S. Müller, S. Heusler, P. Braun, F. Haake, A. Altland. Periodic-orbit theory of universality in quantum chaos. Phys. Rev. E, 2005, 72(4): 046207

[4] J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, M. Tezuka. Black holes and random matrices. J. High Energy Phys., 2017, 2017(5): 118

[5] J. Liu. Spectral form factors and late time quantum chaos. Phys. Rev. D, 2018, 98(8): 086026

[6] P. Kos, M. Ljubotina, T. Prosen. Many-body quantum chaos: Analytic connection to random matrix theory. Phys. Rev. X, 2018, 8(2): 021062

[7] B. Bertini, P. Kos, T. Prosen. Exact spectral form factor in a minimal model of many-body quantum chaos. Phys. Rev. Lett., 2018, 121(26): 264101

[8] A. Chan, A. De Luca, J. T. Chalker. Spectral statistics in spatially extended chaotic quantum many-body systems. Phys. Rev. Lett., 2018, 121(6): 060601

[9] J. Kudler-Flam, L. Nie, S. Ryu. Conformal field theory and the web of quantum chaos diagnostics. J. High Energy Phys., 2020, 2020(1): 175

[10] D. Roy, T. Prosen. Random matrix spectral form factor in kicked interacting fermionic chains. Phys. Rev. E, 2020, 102(6): 060202

[11] M. Winer, B. Swingle. Hydrodynamic theory of the connected spectral form factor. Phys. Rev. X, 2022, 12(2): 021009

[12] D. Roy, D. Mishra, T. Prosen. Spectral form factor in a minimal bosonic model of many-body quantum chaos. Phys. Rev. E, 2022, 106(2): 024208

[13] R. Barney, M. Winer, C. L. Baldwin, B. Swingle, V. Galitski. Spectral statistics of a minimal quantum glass model. SciPost Phys., 2023, 15: 084

[14] P.SaadS. H. ShenkerD.Stanford, A semi-classical ramp in SYK and in gravity, arXiv: 1806.06840 (2018)

[15] H. Gharibyan, M. Hanada, S. H. Shenker, M. Tezuka. Onset of random matrix behavior in scrambling systems. J. High Energy Phys., 2018, 2018(7): 124

[16] M. Winer, S. K. Jian, B. Swingle. Exponential ramp in the quadratic Sachdev‒Ye‒Kitaev model. Phys. Rev. Lett., 2020, 125(25): 250602

[17] R.ShirP. Martinez-AzconaA.Chenu, Full range spectral correlations and their spectral form factors in chaotic and integrable models, arXiv: 2311.09292 (2023)

[18] Y.N. ZhouL. MaoH.Zhai, Rényi entropy dynamics and Lindblad spectrum for open quantum systems, Phys. Rev. Res. 3(4), 043060 (2021)

[19] G. Mazzucchi, W. Kozlowski, S. F. Caballero-Benitez, T. J. Elliott, I. B. Mekhov. Quantum measurement-induced dynamics of many-body ultracold bosonic and fermionic systems in optical lattices. Phys. Rev. A, 2016, 93(2): 023632

[20] Y. Li, X. Chen, M. P. A. Fisher. Quantum Zeno effect and the many-body entanglement transition. Phys. Rev. B, 2018, 98(20): 205136

[21] B. Skinner, J. Ruhman, A. Nahum. Measurement-induced phase transitions in the dynamics of entanglement. Phys. Rev. X, 2019, 9(3): 031009

[22] Y. Li, X. Chen, M. P. A. Fisher. Measurement-driven entanglement transition in hybrid quantum circuits. Phys. Rev. B, 2019, 100(13): 134306

[23] M. Szyniszewski, A. Romito, H. Schomerus. Entanglement transition from variable-strength weak measurements. Phys. Rev. B, 2019, 100(6): 064204

[24] A. Chan, R. M. Nandkishore, M. Pretko, G. Smith. Unitary-projective entanglement dynamics. Phys. Rev. B, 2019, 99(22): 224307

[25] R. Vasseur, A. C. Potter, Y. Z. You, A. W. W. Ludwig. Entanglement transitions from holographic random tensor networks. Phys. Rev. B, 2019, 100(13): 134203

[26] T. Zhou, A. Nahum. Emergent statistical mechanics of entanglement in random unitary circuits. Phys. Rev. B, 2019, 99(17): 174205

[27] M. J. Gullans, D. A. Huse. Scalable probes of measurement-induced criticality. Phys. Rev. Lett., 2020, 125(7): 070606

[28] C. M. Jian, Y. Z. You, R. Vasseur, A. W. W. Ludwig. Measurement-induced criticality in random quantum circuits. Phys. Rev. B, 2020, 101(10): 104302

[29] Y. Fuji, Y. Ashida. Measurement-induced quantum criticality under continuous monitoring. Phys. Rev. B, 2020, 102(5): 054302

[30] A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrishnan, D. A. Huse, J. H. Pixley. Critical properties of the measurement-induced transition in random quantum circuits. Phys. Rev. B, 2020, 101(6): 060301

[31] M. J. Gullans, D. A. Huse. Dynamical purification phase transition induced by quantum measurements. Phys. Rev. X, 2020, 10(4): 041020

[32] S. Choi, Y. Bao, X. L. Qi, E. Altman. Quantum error correction in scrambling dynamics and measurement-induced phase transition. Phys. Rev. Lett., 2020, 125(3): 030505

[33] Y. Bao, S. Choi, E. Altman. Theory of the phase transition in random unitary circuits with measurements. Phys. Rev. B, 2020, 101(10): 104301

[34] A. Nahum, S. Roy, B. Skinner, J. Ruhman. Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau‒Ginsburg theory. PRX Quantum, 2021, 2(1): 010352

[35] R. Fan, S. Vijay, A. Vishwanath, Y. Z. You. Self-organized error correction in random unitary circuits with measurement. Phys. Rev. B, 2021, 103(17): 174309

[36] S. Sang, T. H. Hsieh. Measurement-protected quantum phases. Phys. Rev. Res., 2021, 3(2): 023200

[37] O. Alberton, M. Buchhold, S. Diehl. Entanglement transition in a monitored free-fermion chain: From extended criticality to area law. Phys. Rev. Lett., 2021, 126(17): 170602

[38] A. Lavasani, Y. Alavirad, M. Barkeshli. Measurement-induced topological entanglement transitions in symmetric random quantum circuits. Nat. Phys., 2021, 17(3): 342

[39] X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, M. Schiró. Measurement-induced entanglement transitions in the quantum Ising chain: From infinite to zero clicks. Phys. Rev. B, 2021, 103(22): 224210

[40] Y. Le Gal, X. Turkeshi, M. Schiró. Volume-to-area law entanglement transition in a non-Hermitian free fermionic Chain. SciPost Phys., 2023, 14: 138

[41] S. K. Jian, C. Liu, X. Chen, B. Swingle, P. Zhang. Measurement-induced phase transition in the monitored Sachdev‒Ye‒Kitaev model. Phys. Rev. Lett., 2021, 127(14): 140601

[42] P. Zhang, C. Liu, S. K. Jian, X. Chen. Universal entanglement transitions of free fermions with long-range non-unitary dynamics. Quantum, 2022, 6: 723

[43] C. Liu, P. Zhang, X. Chen. Non-unitary dynamics of Sachdev‒Ye‒Kitaev chain. SciPost Phys., 2021, 10: 048

[44] P. Zhang, S. K. Jian, C. Liu, X. Chen. Emergent replica conformal symmetry in non-Hermitian SYK2 chains. Quantum, 2021, 5: 579

[45] P. Zhang. Quantum entanglement in the Sachdev‒Ye‒Kitaev model and its generalizations. Front. Phys., 2022, 17(4): 43201

[46] S. Sahu, S. K. Jian, G. Bentsen, B. Swingle. Entanglement phases in large-n hybrid Brownian circuits with long-range couplings. Phys. Rev. B, 2022, 106(22): 224305

[47] C. Liu, H. Tang, H. Zhai. Krylov complexity in open quantum systems. Phys. Rev. Res., 2023, 5: 033085

[48] A. Bhattacharya, P. Nandy, P. P. Nath, H. Sahu. Operator growth and Krylov construction in dissipative open quantum systems. J. High Energy Phys., 2022, 2022(12): 81

[49] B. Bhattacharjee, X. Cao, P. Nandy, T. Pathak. Operator growth in open quantum systems: Lessons from the dissipative SYK. J. High Energy Phys., 2023, 2023(3): 54

[50] A. Bhattacharya, P. Nandy, P. P. Nath, H. Sahu. On Krylov complexity in open systems: An approach via bi-Lanczos algorithm. J. High Energy Phys., 2023, 2023: 66

[51] T. Can. Random Lindblad dynamics. J. Phys. A Math. Theor., 2019, 52(48): 485302

[52] J. Li, T. Prosen, A. Chan. Spectral statistics of non-Hermitian matrices and dissipative quantum chaos. Phys. Rev. Lett., 2021, 127(17): 170602

[53] P. Kos, B. Bertini, T. Prosen. Chaos and ergodicity in extended quantum systems with noisy driving. Phys. Rev. Lett., 2021, 126(19): 190601

[54] K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, S. Ryu. Dynamical quantum phase transitions in SYK Lindbladians. Phys. Rev. B, 2023, 108: 075110

[55] Z. Xu, A. Chenu, T. Prosen, A. del Campo. Thermofield dynamics: Quantum chaos versus decoherence. Phys. Rev. B, 2021, 103(6): 064309

[56] J. Cornelius, Z. Xu, A. Saxena, A. Chenu, A. del Campo. Spectral filtering induced by non-Hermitian evolution with balanced gain and loss: Enhancing quantum chaos. Phys. Rev. Lett., 2022, 128(19): 190402

[57] A. S. Matsoukas-Roubeas, F. Roccati, J. Cornelius, Z. Xu, A. Chenu, A. del Campo. Non-Hermitian Hamiltonian deformations in quantum mechanics. J. High Energy Phys., 2023, 2023(1): 60

[58] F.RoccatiF. BalducciR.ShirA.Chenu, Diagnosing non-Hermitian many-body localization and quantum chaos via singular value decomposition, arXiv: 2311.16229 (2023)

[59] If we simply generalize the definition of the SFF for non-Hermitian systems as follows: Fγ(t)=1[ Z(0)]2∑ m,n e− i( ϵm− ϵn)t. where {ϵn} is the set of eigenvalues of the non-Hermitian system, and we denote the real and imaginary parts of the eigenvalues as α_n and β_n respectively. Since the energy eigenvalues of a general non-Hermitian system are complex, implying that the imaginary part β_n is generally nonzero, from the definition we observe that Fγ(t)=1[ Z(0)]2∑ m,n e− i( αm− αn)te(βm−βn) t. Hence, for the set of m, n that satisfies βm −βn > 0, there will be an exponential growth term e(βm−βn) t in the above definition, resulting in the exponential growth of the SFF as time increases.

[60] P. Martinez-Azcona, A. Chenu. Analyticity constraints bound the decay of the spectral form factor. Quantum, 2022, 6: 852

[61] O. Agam, B. L. Altshuler, A. V. Andreev. Spectral statistics: From disordered to chaotic systems. Phys. Rev. Lett., 1995, 75(24): 4389

[62] E. B. Bogomolny, J. P. Keating. Gutzwiller’s trace formula and spectral statistics: Beyond the diagonal approximation. Phys. Rev. Lett., 1996, 77(8): 1472

[63] J. E. Tyson. Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries. J. Phys. Math. Gen., 2003, 36(39): 10101

[64] M. Zwolak, G. Vidal. Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm. Phys. Rev. Lett., 2004, 93(20): 207205

[65] In this supplementary, we show (A) alternative definitions of SFF; (B) the derivation of the pre-factor α in early decay region; (C, D, E) detailed calculation of SFF in three examples; (F) possible experimental realization of SFF.

[66] In general, we think the types of different Lindblad operators will not change the general properties of the normalized SFF regarding its short-time exponential decay and long-time plateau behavior. Since the argument we provide just below Eq. (9) does not resume some specific form of the Lindblad operators. Nevertheless, different Lindblad operators may lead to a different number of steady states, thereby altering the value of θ. For example, let us consider a Hamiltonian H with charge conservation, such as our Bose‒Hubbard model. In the main text, we focus on Lindblad operators that preserve the particle number, ensuring that charge conservation is a strong U(1) symmetry of the open system. In this scenario, there is at least one steady state in each charge sector, resulting in at least N + 1 steady states in the full Fock space with arbitrary particle numbers. (Note that our discussions in the main text focus on a single charge sector.) In contrast, when some Lindblad operators couple different charge sectors, the system exhibits only a weak U(1) symmetry. Consequently, there may be only one steady state even in the full Fock space.

[67] P.SaadS. H. ShenkerD.Stanford, A semiclassical ramp in SYK and in gravity, arXiv: 1806.06840 (2018)

[68] L. Sá, P. Ribeiro, T. Prosen. Lindbladian dissipation of strongly-correlated quantum matter. Phys. Rev. Res., 2022, 4(2): L022068

[69] A. M. García-García, L. Sá, J. J. M. Verbaarschot, J. P. Zheng. Keldysh wormholes and anomalous relaxation in the dissipative Sachdev‒Ye‒Kitaev model. Phys. Rev. D, 2023, 107(10): 106006

[70] K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, S. Ryu. Dynamical quantum phase transitions in Sachdev‒Ye‒Kitaev Lindbladians. Phys. Rev. B, 2023, 108(7): 075110

[71] H. Wang, C. Liu, P. Zhang, A. M. García-García. Entanglement transition and replica wormholes in the dissipative Sachdev‒Ye‒Kitaev model. Phys. Rev. D, 2024, 109(4): 046005

[72] Y. Z. You, A. W. W. Ludwig, C. Xu. Sachdev‒Ye‒Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states. Phys. Rev. B, 2017, 95(11): 115150

[73] I. Danshita, A. Polkovnikov. Superfluid-to-Mott-insulator transition in the one-dimensional Bose‒Hubbard model for arbitrary integer filling factors. Phys. Rev. A, 2011, 84(6): 063637

[74] H. Shen, P. Zhang, R. Fan, H. Zhai. Out-of-time-order correlation at a quantum phase transition. Phys. Rev. B, 2017, 96(5): 054503

[75] I. Boettcher, P. Bienias, R. Belyansky, A. J. Kollár, A. V. Gorshkov. Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry. Phys. Rev. A, 2020, 102(3): 032208

[76] L. Pausch, A. Buchleitner, E. G. Carnio, A. Rodríguez. Optimal route to quantum chaos in the Bose–Hubbard model. J. Phys. A Math. Theor., 2022, 55(32): 324002

[77] S. Denisov, T. Laptyeva, W. Tarnowski, D. Chruscinski, K. Zyczkowski. Universal spectra of random Linblad operators. Phys. Rev. Lett., 2019, 123(14): 140403

[78] D. Poulin, R. Laflamme, G. J. Milburn, J. P. Paz. Testing integrability with a single bit of quantum information. Phys. Rev. A, 2003, 68(2): 022302

[79] D. V. Vasilyev, A. Grankin, M. A. Baranov, L. M. Sieberer, P. Zoller. Monitoring quantum simulators via quantum nondemolition couplings to atomic clock qubits. PRX Quantum, 2020, 1(2): 020302

[80] L. K. Joshi, A. Elben, A. Vikram, B. Vermersch, V. Galitski, P. Zoller. Probing many-body quantum chaos with quantum simulators. Phys. Rev. X, 2022, 12(1): 011018

[81] L.LeviandierM.LombardiR.Jost J.P. Pique, A tool to measure statistical level properties in very complex spectra, Phys. Rev. Lett. 56(23), 2449 (1986)

[82] J. P. Pique, Y. Chen, R. W. Field, J. L. Kinsey. Chaos and dynamics on 0.5–300 ps time scales in vibrationally excited acetylene: Fourier transform of stimulated-emission pumping spectrum. Phys. Rev. Lett., 1987, 58(5): 475

[83] T.GuhrH. A. Weidenmuller, Correlations in anticrossing spectra and scattering theory: Analytical aspects, Chem. Phys. 146(1‒2), 21 (1990)

[84] M. Lombardi, T. H. Seligman. Universal and nonuniversal statistical properties of levels and intensities for chaotic Rydberg molecules. Phys. Rev. A, 1993, 47(5): 3571

[85] E. J. Torres-Herrera, L. F. Santos. Dynamical manifestations of quantum chaos: Correlation hole and bulge. Philos. Trans. Royal Soc. A, 2017, 375(2108): 20160434

[86] H.P. BreuerF. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2007

[87] Y. C. Cheng, R. J. Silbey. Markovian approximation in the relaxation of open quantum systems. J. Phys. Chem. B, 2005, 109: 21399

Declarations

The authors declare that they have no competing interests and there are no conflicts.

Electronic supplementary materials

The online version contains supplementary material available at https://doi.org/10.1007/s11467-024-1406-7 and https://journal.hep.com.cn/fop/EN/10.1007/s11467-024-1406-7.

Acknowledgements

We thank Hui Zhai for the invaluable discussions and for carefully reading the manuscript. We thank Yingfei Gu, Haifeng Tang, Liang Mao, and Hanteng Wang for the helpful discussions. We especially thank Adolfo del Campo for assisting us in rectifying our estimation of the decay exponent in the early-time regime and for bringing to our attention several relevant papers that were overlooked in a previous version.

RIGHTS & PERMISSIONS

2024 Higher Education Press