Abstract
The construction of exactly-solvable models has recently been advanced by considering integrable \( T\overline{T} \) deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse \( T\overline{T} \) deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.
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References
M. Mariño, Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory, Cambridge University Press (2015) [https://doi.org/10.1017/CBO9781107705968].
L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer (2007) [https://doi.org/10.1007/978-3-540-69969-9].
M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press (1999) [https://doi.org/10.1017/CBO9780511524332].
B. Sutherland, Beautiful Models, World Scientific (2004) [https://doi.org/10.1142/5552].
M. Gaudin, The Bethe Wavefunction, Cambridge University Press (2014) [https://doi.org/10.1017/CBO9781107053885].
M. Jimbo, Yang-Baxter Equation in Integrable Systems, World Scientific (1990) [https://doi.org/10.1142/1021].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1993) [https://doi.org/10.1017/CBO9780511628832].
G. Lusztig, Introduction to Quantum Groups, Birkhäuser (2010) [https://doi.org/10.1007/978-0-8176-4717-9].
M.L. Mehta, Random Matrices, 3rd edition, Academic Press (2004).
P. Forrester, Log-Gases and Random Matrices (LMS-34), London Mathematical Society Monographs, Princeton University Press (2010).
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rept. 251 (1995) 267 [hep-th/9405029] [INSPIRE].
M. Ammon and J. Erdmenger, Gauge/Gravity Duality, Cambridge University Press (2015) [https://doi.org/10.1017/CBO9780511846373].
A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \)-deformed 2D Quantum Field Theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
Y. Jiang, A pedagogical review on solvable irrelevant deformations of 2d quantum field theory, Commun. Theor. Phys. 73 (2021) 057201.
D.J. Gross, J. Kruthoff, A. Rolph and E. Shaghoulian, \( T\overline{T} \) in AdS2 and Quantum Mechanics, Phys. Rev. D 101 (2020) 026011 [arXiv:1907.04873] [INSPIRE].
D.J. Gross, J. Kruthoff, A. Rolph and E. Shaghoulian, Hamiltonian deformations in quantum mechanics, \( T\overline{T} \), and the SYK model, Phys. Rev. D 102 (2020) 046019 [arXiv:1912.06132] [INSPIRE].
J. Kruthoff and O. Parrikar, On the flow of states under \( T\overline{T} \), SciPost Phys. 9 (2020) 078.
F. Rosso, \( T\overline{T} \) deformation of random matrices, Phys. Rev. D 103 (2021) 126017 [arXiv:2012.11714] [INSPIRE].
Y. Jiang, \( T\overline{T} \)-deformed 1d Bose gas, SciPost Phys. 12 (2022) 191 [arXiv:2011.00637] [INSPIRE].
S. Ebert, C. Ferko, H.-Y. Sun and Z. Sun, \( T\overline{T} \) deformations of supersymmetric quantum mechanics, JHEP 08 (2022) 121 [arXiv:2204.05897] [INSPIRE].
S. He and Z.-Y. Xian, \( T\overline{T} \) deformation on multiquantum mechanics and regenesis, Phys. Rev. D 106 (2022) 046002 [arXiv:2104.03852] [INSPIRE].
W.H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75 (2003) 715 [quant-ph/0105127] [INSPIRE].
H.P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002) [https://doi.org/10.1093/acprof:oso/9780199213900.001.0001].
T. Prosen, Third quantization: a general method to solve master equations for quadratic open fermi systems, New J. Phys. 10 (2008) 043026.
M. Beau, J. Kiukas, I.L. Egusquiza and A. del Campo, Nonexponential Quantum Decay under Environmental Decoherence, Phys. Rev. Lett. 119 (2017) 130401 [arXiv:1706.06943] [INSPIRE].
F. Haake, Quantum Signatures of Chaos, Springer (2010) [https://doi.org/10.1007/978-3-642-05428-0].
Z. Xu, L.P. García-Pintos, A. Chenu and A. del Campo, Extreme Decoherence and Quantum Chaos, Phys. Rev. Lett. 122 (2019) 014103 [arXiv:1810.02319] [INSPIRE].
A. Del Campo and T. Takayanagi, Decoherence in Conformal Field Theory, JHEP 02 (2020) 170 [arXiv:1911.07861] [INSPIRE].
Z. Xu, A. Chenu, T. Prosen and A. del Campo, Thermofield dynamics: Quantum Chaos versus Decoherence, Phys. Rev. B 103 (2021) 064309 [arXiv:2008.06444] [INSPIRE].
A.M. García-García, L. Sá and J.J.M. Verbaarschot, Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model, Phys. Rev. X 12 (2022) 021040 [arXiv:2110.03444] [INSPIRE].
T. Can, Random Lindblad Dynamics, J. Phys. A 52 (2019) 485302 [arXiv:1902.01442] [INSPIRE].
L. Sá, P. Ribeiro and T. Prosen, Spectral and steady-state properties of random Liouvillians, J. Phys. A 53 (2020) 305303 [arXiv:1905.02155] [INSPIRE].
L. Sá, P. Ribeiro, T. Can and T. Prosen, Spectral transitions and universal steady states in random Kraus maps and circuits, Phys. Rev. B 102 (2020) 134310 [arXiv:2007.04326] [INSPIRE].
A. Chenu, M. Beau, J. Cao and A. del Campo, Quantum simulation of generic many-body open system dynamics using classical noise, Phys. Rev. Lett. 118 (2017) 140403.
Á. Rubio-García, R.A. Molina and J. Dukelsky, From integrability to chaos in quantum Liouvillians, SciPost Phys. Core 5 (2022) 026 [arXiv:2102.13452] [INSPIRE].
Y. Li, X. Chen and M.P.A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98 (2018) 205136 [arXiv:1808.06134] [INSPIRE].
B. Skinner, J. Ruhman and A. Nahum, Measurement-Induced Phase Transitions in the Dynamics of Entanglement, Phys. Rev. X 9 (2019) 031009 [arXiv:1808.05953] [INSPIRE].
M.J. Gullans and D.A. Huse, Dynamical Purification Phase Transition Induced by Quantum Measurements, Phys. Rev. X 10 (2020) 041020 [arXiv:1905.05195] [INSPIRE].
M. Ippoliti, M.J. Gullans, S. Gopalakrishnan, D.A. Huse and V. Khemani, Entanglement Phase Transitions in Measurement-Only Dynamics, Phys. Rev. X 11 (2021) 011030 [INSPIRE].
L. Sá, P. Ribeiro and T. Prosen, Integrable nonunitary open quantum circuits, Phys. Rev. B 103 (2021) 115132 [arXiv:2011.06565] [INSPIRE].
Y. Ashida, Z. Gong and M. Ueda, Non-hermitian physics, Adv. Phys. 69 (2020) 249.
H. Verlinde, ER = EPR revisited: On the Entropy of an Einstein-Rosen Bridge, arXiv:2003.13117 [INSPIRE].
T. Anegawa, N. Iizuka, K. Tamaoka and T. Ugajin, Wormholes and holographic decoherence, JHEP 03 (2021) 214 [arXiv:2012.03514] [INSPIRE].
H. Verlinde, Deconstructing the Wormhole: Factorization, Entanglement and Decoherence, arXiv:2105.02142 [INSPIRE].
K. Goto, Y. Kusuki, K. Tamaoka and T. Ugajin, Product of random states and spatial (half-)wormholes, JHEP 10 (2021) 205 [arXiv:2108.08308] [INSPIRE].
A.M. García-García, L. Sá, J.J.M. Verbaarschot and J.P. Zheng, Keldysh Wormholes and Anomalous Relaxation in the Dissipative Sachdev-Ye-Kitaev Model, arXiv:2210.01695 [INSPIRE].
A. Bhattacharya, P. Nandy, P.P. Nath and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, JHEP 12 (2022) 081 [arXiv:2207.05347] [INSPIRE].
C. Liu, H. Tang and H. Zhai, Krylov Complexity in Open Quantum Systems, arXiv:2207.13603 [INSPIRE].
J. Cornelius, Z. Xu, A. Saxena, A. Chenu and A. del Campo, Spectral Filtering Induced by Non-Hermitian Evolution with Balanced Gain and Loss: Enhancing Quantum Chaos, Phys. Rev. Lett. 128 (2022) 190402 [arXiv:2108.06784] [INSPIRE].
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 8th edition, Academic Press (2014).
D.C. Brody, Biorthogonal quantum mechanics, J. Phys. A 47 (2013) 035305.
G. Gamow, Zur Quantentheorie des Atomkernes, Z. Phys. 51 (1928) 204 [INSPIRE].
E. Majorana, Scattering of an α Particle by a Radioactive Nucleus, Electron. J. Theor. Phys. 3 (2006) 293.
N. Moiseyev, Non-Hermitian Quantum Mechanics, Cambridge University Press (2011) [https://doi.org/10.1017/CBO9780511976186].
C.M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243 [physics/9712001] [INSPIRE].
C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70 (2007) 947 [hep-th/0703096] [INSPIRE].
D.C. Brody and E.-M. Graefe, Mixed-state evolution in the presence of gain and loss, Phys. Rev. Lett. 109 (2012) 230405.
H. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields, Theoretical and Mathematical Physics, Springer (2009) [https://doi.org/10.1007/978-3-540-71320-3].
S. Alipour, A. Chenu, A.T. Rezakhani and A. del Campo, Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution, Quantum 4 (2020) 336.
V. Gorini, A. Kossakowski and E.C.G. Sudarshan, Completely Positive Dynamical Semigroups of N Level Systems, J. Math. Phys. 17 (1976) 821 [INSPIRE].
G. Lindblad, On the Generators of Quantum Dynamical Semigroups, Commun. Math. Phys. 48 (1976) 119 [INSPIRE].
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 (2019) 062131.
F. Roccati, G.M. Palma, F. Ciccarello and F. Bagarello, Non-hermitian physics and master equations, Open Syst. Info. Dyn. 29 (2022) 2250004.
M.V. Berry, M. Tabor and J.M. Ziman, Level clustering in the regular spectrum, Proc. Roy. Soc. Lond. A 356 (1977) 375.
O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984) 1 [INSPIRE].
J. Feinberg and A. Zee, NonHermitian random matrix theory: Method of Hermitean reduction, Nucl. Phys. B 504 (1997) 579 [cond-mat/9703087] [INSPIRE].
K. Wang, F. Piazza and D.J. Luitz, Hierarchy of Relaxation Timescales in Local Random Liouvillians, Phys. Rev. Lett. 124 (2020) 100604.
G. Marinello and M.P. Pato, Random non-hermitian tight-binding models, J. Phys. Conf. Ser. 738 (2016) 012040.
K. Mochizuki, N. Hatano, J. Feinberg and H. Obuse, Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms, Phys. Rev. E 102 (2020) 012101 [arXiv:2005.02705] [INSPIRE].
C. Wang and X.R. Wang, Level statistics of extended states in random non-hermitian hamiltonians, Phys. Rev. B 101 (2020) 165114.
T. Can, V. Oganesyan, D. Orgad and S. Gopalakrishnan, Spectral Gaps and Midgap States in Random Quantum Master Equations, Phys. Rev. Lett. 123 (2019) 234103 [arXiv:1902.01414] [INSPIRE].
S. Denisov, T. Laptyeva, W. Tarnowski, D. Chruściński and K. Życzkowski, Universal spectra of random Lindblad operators, Phys. Rev. Lett. 123 (2019) 140403 [arXiv:1811.12282] [INSPIRE].
J.A. Gyamfi, Fundamentals of quantum mechanics in liouville space, Eur. J. Phys. 41 (2020) 063002.
E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Annals Math. 62 (1955) 548.
L. García-Álvarez, I.L. Egusquiza, L. Lamata, A. del Campo, J. Sonner and E. Solano, Digital Quantum Simulation of Minimal AdS/CFT, Phys. Rev. Lett. 119 (2017) 040501 [arXiv:1607.08560] [INSPIRE].
A.M. García-García and J.J.M. Verbaarschot, Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N, Phys. Rev. D 96 (2017) 066012 [arXiv:1701.06593] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
A. Jevicki, K. Suzuki and J. Yoon, Bi-Local Holography in the SYK Model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
Y. Takahashi and H. Umezawa, Thermo field dynamics, Int. J. Mod. Phys. B 10 (1996) 1755 [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
A. del Campo, J. Molina-Vilaplana and J. Sonner, Scrambling the spectral form factor: unitarity constraints and exact results, Phys. Rev. D 95 (2017) 126008 [arXiv:1702.04350] [INSPIRE].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
M.B. Plenio, Logarithmic Negativity: A Full Entanglement Monotone That is not Convex, Phys. Rev. Lett. 95 (2005) 090503 [quant-ph/0505071] [INSPIRE].
L. Leviandier, M. Lombardi, R. Jost and J.P. Pique, Fourier transform: A tool to measure statistical level properties in very complex spectra, Phys. Rev. Lett. 56 (1986) 2449.
J. Wilkie and P. Brumer, Time-dependent manifestations of quantum chaos, Phys. Rev. Lett. 67 (1991) 1185 [INSPIRE].
Y. Alhassid and N. Whelan, Onset of chaos and its signature in the spectral autocorrelation function, Phys. Rev. Lett. 70 (1993) 572.
J.-Z. Ma, Correlation hole of survival probability and level statistics, J. Phys. Soc. Jap. 64 (1995) 4059.
E. Brézin and S. Hikami, Spectral form factor in a random matrix theory, Phys. Rev. E 55 (1997) 4067.
T. Gorin, T. Prosen, T.H. Seligman and M. Žnidarič, Dynamics of loschmidt echoes and fidelity decay, Phys. Rept. 435 (2006) 33.
E. Dyer and G. Gur-Ari, 2D CFT Partition Functions at Late Times, JHEP 08 (2017) 075 [arXiv:1611.04592] [INSPIRE].
S. He, P.H.C. Lau, Z.-Y. Xian and L. Zhao, Quantum chaos, scrambling and operator growth in \( T\overline{T} \) deformed SYK models, JHEP 12 (2022) 070 [arXiv:2209.14936] [INSPIRE].
P. Jacquod and C. Petitjean, Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom, Adv. Phys. 58 (2009) 67.
J. Li, T. Prosen and A. Chan, Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos, Phys. Rev. Lett. 127 (2021) 170602 [arXiv:2103.05001] [INSPIRE].
R.E. Prange, The spectral form factor is not self-averaging, Phys. Rev. Lett. 78 (1997) 2280.
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Matsoukas-Roubeas, A.S., Roccati, F., Cornelius, J. et al. Non-Hermitian Hamiltonian deformations in quantum mechanics. J. High Energ. Phys. 2023, 60 (2023). https://doi.org/10.1007/JHEP01(2023)060
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DOI: https://doi.org/10.1007/JHEP01(2023)060