Generalized Adiabatic Theorem and Strong-Coupling Limits

Daniel Burgarth1, Paolo Facchi2,3, Hiromichi Nakazato4, Saverio Pascazio2,3,5, and Kazuya Yuasa4

1Center for Engineered Quantum Systems, Dept. of Physics & Astronomy, Macquarie University, 2109 NSW, Australia
2Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy
3INFN, Sezione di Bari, I-70126 Bari, Italy
4Department of Physics, Waseda University, Tokyo 169-8555, Japan
5Istituto Nazionale di Ottica (INO-CNR), I-50125 Firenze, Italy

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Abstract

We generalize Kato's adiabatic theorem to nonunitary dynamics with an isospectral generator. This enables us to unify two strong-coupling limits: one driven by fast oscillations under a Hamiltonian, and the other driven by strong damping under a Lindbladian. We discuss the case where both mechanisms are present and provide nonperturbative error bounds. We also analyze the links with the quantum Zeno effect and dynamics.

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► References

[1] Quantum Error Correction, edited by D. A. Lidar and T. A. Brun (Cambridge University Press, New York, 2013).
https:/​/​doi.org/​10.1017/​CBO9781139034807

[2] B. Misra and E. C. G. Sudarshan, The Zeno's Paradox in Quantum Theory, J. Math. Phys. 18, 756 (1977).
https:/​/​doi.org/​10.1063/​1.523304

[3] P. Facchi and S. Pascazio, Quantum Zeno Subspaces, Phys. Rev. Lett. 89, 080401 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.89.080401

[4] P. Facchi, Quantum Zeno Effect, Adiabaticity and Dynamical Superselection Rules, in Fundamental Aspects of Quantum Physics, Vol. 17 of QP-PQ: Quantum Probability and White Noise Analysis, edited by L. Accardi and S. Tasaki (World Scientific, Singapore, 2003), pp. 197–221.
https:/​/​doi.org/​10.1142/​5213

[5] E. B. Davies, Markovian Master Equations, Commun. Math. Phys. 39, 91 (1974).
https:/​/​doi.org/​10.1007/​BF01608389

[6] P. Zanardi and L. Campos Venuti, Coherent Quantum Dynamics in Steady-State Manifolds of Strongly Dissipative Systems, Phys. Rev. Lett. 113, 240406 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.240406

[7] P. Zanardi and L. Campos Venuti, Geometry, Robustness, and Emerging Unitarity in Dissipation-Projected Dynamics, Phys. Rev. A 91, 052324 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.052324

[8] T. Kato, On the Adiabatic Theorem of Quantum Mechanics, J. Phys. Soc. Jpn. 5, 435 (1950).
https:/​/​doi.org/​10.1143/​JPSJ.5.435

[9] W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Quantum Zeno Effect, Phys. Rev. A 41, 2295 (1990).
https:/​/​doi.org/​10.1103/​PhysRevA.41.2295

[10] E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, Continuous and Pulsed Quantum Zeno Effect, Phys. Rev. Lett. 97, 260402 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.97.260402

[11] F. Schäfer, I. Herrera, S. Cherukattil, C. Lovecchio, F. S. Cataliotti, F. Caruso, and A. Smerzi, Experimental Realization of Quantum Zeno Dynamics, Nat. Commun. 5, 3194 (2014).
https:/​/​doi.org/​10.1038/​ncomms4194

[12] A. Signoles, A. Facon, D. Grosso, I. Dotsenko, S. Haroche, J.-M. Raimond, M. Brune, and S. Gleyzes, Confined Quantum Zeno Dynamics of a Watched Atomic Arrow, Nat. Phys. 10, 715 (2014).
https:/​/​doi.org/​10.1038/​nphys3076

[13] L. Bretheau, P. Campagne-Ibarcq, E. Flurin, F. Mallet, and B. Huard, Quantum Dynamics of an Electromagnetic Mode that Cannot Contain $N$ Photons, Science 348, 776 (2015).
https:/​/​doi.org/​10.1126/​science.1259345

[14] G. Barontini, L. Hohmann, F. Haas, J. Estève, and J. Reichel, Deterministic Generation of Multiparticle Entanglement by Quantum Zeno Dynamics, Science 349, 1317 (2015).
https:/​/​doi.org/​10.1126/​science.aaa0754

[15] N. Kalb, J. Cramer, D. J. Twitchen, M. Markham, R. Hanson, and T. H. Taminiau, Experimental Creation of Quantum Zeno Subspaces by Repeated Multi-Spin Projections in Diamond, Nat. Commun. 7, 13111 (2016).
https:/​/​doi.org/​10.1038/​ncomms13111

[16] P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, and D. A. Lidar, Control of Decoherence: Analysis and Comparison of Three Different Strategies, Phys. Rev. A 71, 022302 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.022302

[17] P. Facchi and S. Pascazio, Quantum Zeno Dynamics: Mathematical and Physical Aspects, J. Phys. A: Math. Theor. 41, 493001 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​49/​493001

[18] T. Petrosky, S. Tasaki, and I. Prigogine, Quantum Zeno Effect, Phys. Lett. A 151, 109 (1990).
https:/​/​doi.org/​10.1016/​0375-9601(90)90173-L

[19] L. S. Schulman, Continuous and Pulsed Observations in the Quantum Zeno Effect, Phys. Rev. A 57, 1509 (1998).
https:/​/​doi.org/​10.1103/​PhysRevA.57.1509

[20] P. Facchi, D. A. Lidar, and S. Pascazio, Unification of Dynamical Decoupling and the Quantum Zeno Effect, Phys. Rev. A 69, 032314 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.032314

[21] K. Koshino and A. Shimizu, Quantum Zeno Effect by General Measurements, Phys. Rep. 412, 191 (2005).
https:/​/​doi.org/​10.1016/​j.physrep.2005.03.001

[22] P. Facchi, H. Nakazato, and S. Pascazio, From the Quantum Zeno to the Inverse Quantum Zeno Effect, Phys. Rev. Lett. 86, 2699 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.86.2699

[23] P. Facchi and M. Ligabò, Quantum Zeno Effect and Dynamics, J. Math. Phys. 51, 022103 (2010).
https:/​/​doi.org/​10.1063/​1.3290971

[24] J. Schwinger, The Algebra of Microscopic Measurement, Proc. Natl. Acad. Sci. USA 45, 1542 (1959).
https:/​/​doi.org/​10.1073/​pnas.45.10.1542

[25] A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, New York, 2002).
https:/​/​doi.org/​10.1007/​0-306-47120-5

[26] S. Pascazio, On Noise-Induced Superselection Rules, J. Mod. Opt. 51, 925 (2004).
https:/​/​doi.org/​10.1080/​09500340408233606

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

[28] V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, Geometry and Response of Lindbladians, Phys. Rev. X 6, 041031 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.041031

[29] J. Marshall, L. Campos Venuti, and P. Zanardi, Noise Suppression via Generalized-Markovian Processes, Phys. Rev. A 96, 052113 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.052113

[30] R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, 2nd ed. (Springer, Berlin, 2007).
https:/​/​doi.org/​10.1007/​3-540-70861-8

[31] D. Chruściński and S. Pascazio, A Brief History of the GKLS Equation, Open Sys. Inf. Dyn. 24, 1740001 (2017).
https:/​/​doi.org/​10.1142/​S1230161217400017

[32] A. Messiah, Quantum Mechanics (Dover, New York, 2017).
http:/​/​store.doverpublications.com/​048678455x.html

[33] J. E. Avron, M. Fraas, and G. M. Graf, Adiabatic Response for Lindblad Dynamics, J. Stat. Phys. 148, 800 (2012).
https:/​/​doi.org/​10.1007/​s10955-012-0550-6

[34] J. Schmid, Adiabatic Theorems for General Linear Operators with Time-Independent Domains, Rev. Math. Phys. 31, 1950014 (2019).
https:/​/​doi.org/​10.1142/​S0129055X19500144

[35] E. B. Davies, One-Parameter Semigroups (Academic Press, San Diego, 1980).
https:/​/​books.google.it/​books?id=IQOFAAAAIAAJ

[36] C. Cohen‐Tannoudji, J. Dupont‐Roc, and G. Grynberg, Atom-Photon Interactions: Basic Process and Appilcations (Wiley, Weinheim, 1998).
https:/​/​doi.org/​10.1002/​9783527617197

[37] N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, 2nd ed. (Elsevier, Amsterdam, 1992).
https:/​/​books.google.it/​books?isbn=0080571387

[38] R. Azouit, A. Sarlette, and P. Rouchon, Adiabatic Elimination for Open Quantum Systems with Effective Lindblad Master Equations, in 2016 IEEE 55th Conference on Decision and Control (CDC), Dec. 2016, pp. 4559–4565.
https:/​/​doi.org/​10.1109/​CDC.2016.7798963
https:/​/​ieeexplore.ieee.org/​xpl/​mostRecentIssue.jsp?punumber=7786694

[39] G. Dirr and U. Helmke, Lie Theory for Quantum Control, GAMM-Mitt. 31, 59 (2008).
https:/​/​doi.org/​10.1002/​gamm.200890003

[40] Z. K. Minev, S. O. Mundhada, S. Shankar, P. Reinhold, R. Gutiérrez-Jáuregui, R. J. Schoelkopf, M. Mirrahimi, H. J. Carmichael, and M. H. Devoret, To Catch and Reverse a Quantum Jump Mid-Flight, arXiv:1803.00545 [quant-ph] (2018).
arXiv:1803.00545

[41] T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer, Berlin, 1980).
https:/​/​doi.org/​10.1007/​978-3-642-66282-9

[42] M. M. Wolf, ``Quantum Channels & Operations: Guided Tour,'' URL: https:/​/​www-m5.ma.tum.de/​foswiki/​pub/​M5/​Allgemeines/​MichaelWolf/​QChannelLecture.pdf.
https:/​/​www-m5.ma.tum.de/​foswiki/​pub/​M5/​Allgemeines/​MichaelWolf/​QChannelLecture.pdf

[43] B. Baumgartner, H. Narnhofer, and W. Thirring, Analysis of Quantum Semigroups with GKS-Lindblad Generators: I. Simple Generators, J. Phys. A: Math. Theor. 41, 065201 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​6/​065201

[44] B. Baumgartner and H. Narnhofer, 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

[45] B. Baumgartner and H. Narnhofer, The Structures of State Space Concerning Quantum Dynamical Semigroups, Rev. Math. Phys. 24, 1250001 (2012).
https:/​/​doi.org/​10.1142/​S0129055X12500018

[46] V. V. Albert, Lindbladians with Multiple Steady States: Theory and Applications, Ph.D. Thesis, Yale University, Connecticut, 2017, available at arXiv:1802.00010 [quant-ph].
arXiv:1802.00010

[47] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. (Cambridge University Press, Cambridge, 2012).
http:/​/​www.cambridge.org/​9780521548236

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