Environmentally Induced Entanglement – Anomalous Behavior in the Adiabatic Regime

Richard Hartmann and Walter T. Strunz

Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany

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Considering two non-interacting qubits in the context of open quantum systems, it is well known that their common environment may act as an entangling agent. In a perturbative regime the influence of the environment on the system dynamics can effectively be described by a unitary and a dissipative contribution. For the two-spin Boson model with (sub-) Ohmic spectral density considered here, the particular unitary contribution (Lamb shift) easily explains the buildup of entanglement between the two qubits. Furthermore it has been argued that in the adiabatic limit, adding the so-called counterterm to the microscopic model compensates the unitary influence of the environment and, thus, inhibits the generation of entanglement. Investigating this assertion is one of the main objectives of the work presented here. Using the hierarchy of pure states (HOPS) method to numerically calculate the exact reduced dynamics, we find and explain that the degree of inhibition crucially depends on the parameter $s$ determining the low frequency power law behavior of the spectral density $J(\omega) \sim \omega^s e^{-\omega/\omega_c}$. Remarkably, we find that for resonant qubits, even in the adiabatic regime (arbitrarily large $\omega_c$), the entanglement dynamics is still influenced by an environmentally induced Hamiltonian interaction. Further, we study the model in detail and present the exact entanglement dynamics for a wide range of coupling strengths, distinguish between resonant and detuned qubits, as well as Ohmic and deep sub-Ohmic environments. Notably, we find that in all cases the asymptotic entanglement does not vanish and conjecture a linear relation between the coupling strength and the asymptotic entanglement measured by means of concurrence. Further we discuss the suitability of various perturbative master equations for obtaining approximate entanglement dynamics.

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[1] Wojciech Hubert Zurek. Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys., 75 (3): 715–775, May 2003. 10.1103/​RevModPhys.75.715.

[2] W. Dür and H.-J. Briegel. Stability of Macroscopic Entanglement under Decoherence. Phys. Rev. Lett., 92 (18): 180403, May 2004. 10.1103/​PhysRevLett.92.180403.

[3] M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich. Environment-Induced Sudden Death of Entanglement. Science, 316 (5824): 579–582, April 2007. ISSN 0036-8075, 1095-9203. 10.1126/​science.1139892.

[4] Ting Yu and J. H. Eberly. Sudden Death of Entanglement. Science, 323 (5914): 598–601, January 2009. ISSN 0036-8075, 1095-9203. 10.1126/​science.1167343.

[5] Daniel Braun. Creation of Entanglement by Interaction with a Common Heat Bath. Phys. Rev. Lett., 89 (27): 277901, December 2002. 10.1103/​PhysRevLett.89.277901.

[6] M. S. Kim, Jinhyoung Lee, D. Ahn, and P. L. Knight. Entanglement induced by a single-mode heat environment. Phys. Rev. A, 65 (4): 040101, April 2002. 10.1103/​PhysRevA.65.040101.

[7] Aurelian Isar. Entanglement Generation and Evolution in Open Quantum Systems. Open Systems & Information Dynamics, 16 (02n03): 205–219, September 2009. ISSN 1230-1612, 1793-7191. 10.1142/​S1230161209000153.

[8] L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, and B. M. Garraway. Sudden death and sudden birth of entanglement in common structured reservoirs. Phys. Rev. A, 79 (4): 042302, April 2009. 10.1103/​PhysRevA.79.042302.

[9] F. Benatti and R. Floreanini. Entangling oscillators through environment noise. J. Phys. A: Math. Gen., 39 (11): 2689–2699, March 2006. ISSN 0305-4470. 10.1088/​0305-4470/​39/​11/​009.

[10] Thomas Zell, Friedemann Queisser, and Rochus Klesse. Distance Dependence of Entanglement Generation via a Bosonic Heat Bath. Phys. Rev. Lett., 102 (16): 160501, April 2009. 10.1103/​PhysRevLett.102.160501.

[11] Mohammad M. Sahrapour and Nancy Makri. Tunneling, decoherence, and entanglement of two spins interacting with a dissipative bath. The Journal of Chemical Physics, 138 (11): 114109, March 2013. ISSN 0021-9606, 1089-7690. 10.1063/​1.4795159.

[12] M. Dubé and P. C. E. Stamp. Dynamics of a Pair of Interacting Spins Coupled to an Environmental Sea. Int. J. Mod. Phys. B, 12 (11): 1191–1245, May 1998. ISSN 0217-9792. 10.1142/​S0217979298000661.

[13] Karol Życzkowski, Paweł Horodecki, Michał Horodecki, and Ryszard Horodecki. Dynamics of quantum entanglement. Phys. Rev. A, 65 (1): 012101, December 2001. 10.1103/​PhysRevA.65.012101.

[14] A. M. Basharov. Decoherence and entanglement in radiative decay of a diatomic system. J. Exp. Theor. Phys., 94 (6): 1070–1079, June 2002. ISSN 1063-7761, 1090-6509. 10.1134/​1.1493157.

[15] L. Jakóbczyk. Entangling two qubits by dissipation. J. Phys. A: Math. Gen., 35 (30): 6383, 2002. ISSN 0305-4470. 10.1088/​0305-4470/​35/​30/​313.

[16] S. Schneider and G. J. Milburn. Entanglement in the steady state of a collective-angular-momentum (Dicke) model. Phys. Rev. A, 65 (4): 042107, March 2002. 10.1103/​PhysRevA.65.042107.

[17] K. Lendi and A. J. van Wonderen. Davies theory for reservoir-induced entanglement in a bipartite system. J. Phys. A: Math. Theor., 40 (2): 279–288, December 2006. ISSN 1751-8121. 10.1088/​1751-8113/​40/​2/​007.

[18] Juan Pablo Paz and Augusto J. Roncaglia. Dynamics of the Entanglement between Two Oscillators in the Same Environment. Phys. Rev. Lett., 100 (22): 220401, June 2008. 10.1103/​PhysRevLett.100.220401.

[19] Denis Kast and Joachim Ankerhold. Bipartite entanglement dynamics of two-level systems in sub-Ohmic reservoirs. Phys. Rev. B, 90 (10): 100301, September 2014. 10.1103/​PhysRevB.90.100301.

[20] Leandro Aolita, Fernando de Melo, and Luiz Davidovich. Open-system dynamics of entanglement:a key issues review. Rep. Prog. Phys., 78 (4): 042001, March 2015. ISSN 0034-4885. 10.1088/​0034-4885/​78/​4/​042001.

[21] Tianrui Deng, Yiying Yan, Lipeng Chen, and Yang Zhao. Dynamics of the two-spin spin-boson model with a common bath. J. Chem. Phys., 144 (14): 144102, April 2016. ISSN 0021-9606. 10.1063/​1.4945390.

[22] P. R. Eastham, P. Kirton, H. M. Cammack, B. W. Lovett, and J. Keeling. Bath-induced coherence and the secular approximation. Phys. Rev. A, 94 (1): 012110, July 2016. 10.1103/​PhysRevA.94.012110.

[23] D. Suess, A. Eisfeld, and W. T. Strunz. Hierarchy of Stochastic Pure States for Open Quantum System Dynamics. Phys. Rev. Lett., 113 (15): 150403, October 2014. 10.1103/​PhysRevLett.113.150403.

[24] Pan-Pan Zhang and Alexander Eisfeld. Non-Perturbative Calculation of Two-Dimensional Spectra Using the Stochastic Hierarchy of Pure States. J. Phys. Chem. Lett., 7 (22): 4488–4494, November 2016. ISSN 1948-7185. 10.1021/​acs.jpclett.6b02111.

[25] Richard Hartmann and Walter T. Strunz. Exact Open Quantum System Dynamics Using the Hierarchy of Pure States (HOPS). Journal of Chemical Theory and Computation, 13 (12): 5834–5845, December 2017. ISSN 1549-9618, 1549-9626. 10.1021/​acs.jctc.7b00751.

[26] P.-P. Zhang, C. D. B. Bentley, and A. Eisfeld. Flexible scheme to truncate the hierarchy of pure states. The Journal of Chemical Physics, 148 (13): 134103, April 2018. ISSN 0021-9606, 1089-7690. 10.1063/​1.5022225.

[27] Richard Hartmann, Michael Werther, Frank Grossmann, and Walter T. Strunz. Exact open quantum system dynamics: Optimal frequency vs time representation of bath correlations. J. Chem. Phys., 150 (23): 234105, June 2019. ISSN 0021-9606. 10.1063/​1.5097158.

[28] Richard Hartmann and Walter T. Strunz. Accuracy assessment of perturbative master equations: Embracing nonpositivity. Phys. Rev. A, 101 (1): 012103, January 2020. 10.1103/​PhysRevA.101.012103.

[29] Ulrich Weiss. Quantum Dissipative Systems. World Scientific, Singapore, March 2008. ISBN 978-981-279-162-7.

[30] Fabio Benatti, Roberto Floreanini, and Marco Piani. Environment Induced Entanglement in Markovian Dissipative Dynamics. Phys. Rev. Lett., 91 (7): 070402, August 2003. 10.1103/​PhysRevLett.91.070402.

[31] A. J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P. A. Fisher, Anupam Garg, and W. Zwerger. Dynamics of the dissipative two-state system. Rev. Mod. Phys., 59 (1): 1–85, January 1987. 10.1103/​RevModPhys.59.1.

[32] E. B. Davies. Markovian master equations. Communications in Mathematical Physics, 39 (2): 91–110, June 1974. ISSN 0010-3616, 1432-0916. 10.1007/​BF01608389.

[33] Heinz-Peter Breuer and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford University Press, Oxford, New York, January 2007. ISBN 978-0-19-921390-0.

[34] Dariusz Chruściński and Saverio Pascazio. A Brief History of the GKLS Equation. Open Syst. Inf. Dyn., 24 (03): 1740001, September 2017. ISSN 1230-1612, 1793-7191. 10.1142/​S1230161217400017.

[35] Stanislaw Kryszewski and Justyna Czechowska-Kryszk. Master equation - tutorial approach. arXiv:0801.1757 [quant-ph], January 2008.

[36] Robert S. Whitney. Staying positive: Going beyond Lindblad with perturbative master equations. J. Phys. A: Math. Theor., 41 (17): 175304, 2008. ISSN 1751-8121. 10.1088/​1751-8113/​41/​17/​175304.

[37] D. P. S. McCutcheon, A. Nazir, S. Bose, and A. J. Fisher. Long-lived spin entanglement induced by a spatially correlated thermal bath. Physical Review A, 80 (2): 022337, August 2009. ISSN 1050-2947, 1094-1622. 10.1103/​PhysRevA.80.022337.

[38] Dmitry Solenov, Denis Tolkunov, and Vladimir Privman. Exchange interaction, entanglement, and quantum noise due to a thermal bosonic field. Physical Review B, 75 (3): 035134, January 2007. ISSN 1098-0121, 1550-235X. 10.1103/​PhysRevB.75.035134.

[39] F. Benatti, R. Floreanini, and U. Marzolino. Entangling two unequal atoms through a common bath. Phys. Rev. A, 81 (1): 012105, January 2010. 10.1103/​PhysRevA.81.012105.

[40] Christian Majenz, Tameem Albash, Heinz-Peter Breuer, and Daniel A. Lidar. Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations. Phys. Rev. A, 88 (1): 012103, July 2013. 10.1103/​PhysRevA.88.012103.

[41] Peter P. Orth, David Roosen, Walter Hofstetter, and Karyn Le Hur. Dynamics, synchronization, and quantum phase transitions of two dissipative spins. Phys. Rev. B, 82 (14): 144423, October 2010. 10.1103/​PhysRevB.82.144423.

[42] A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett. Efficient non-Markovian quantum dynamics using time-evolving matrix product operators. Nature Communications, 9 (1): 3322, August 2018. ISSN 2041-1723. 10.1038/​s41467-018-05617-3.

[43] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81 (2): 865–942, June 2009. 10.1103/​RevModPhys.81.865.

[44] L. Diósi, N. Gisin, and Walter T. Strunz. Non-Markovian quantum state diffusion. Phys. Rev. A, 58 (3): 1699–1712, September 1998. 10.1103/​PhysRevA.58.1699.

[45] Walter T. Strunz, Lajos Diósi, and Nicolas Gisin. Open System Dynamics with Non-Markovian Quantum Trajectories. Phys. Rev. Lett., 82 (9): 1801–1805, March 1999. 10.1103/​PhysRevLett.82.1801.

[46] Scott Hill and William K. Wootters. Entanglement of a Pair of Quantum Bits. Phys. Rev. Lett., 78 (26): 5022–5025, June 1997. 10.1103/​PhysRevLett.78.5022.

[47] A. R.R. Carvalho, F. Mintert, S. Palzer, and A. Buchleitner. Entanglement dynamics under decoherence: From qubits to qudits. The European Physical Journal D, 41 (2): 425–432, February 2007. ISSN 1434-6060, 1434-6079. 10.1140/​epjd/​e2006-00246-4.

[48] Nicolas Vogt, Jan Jeske, and Jared H. Cole. Stochastic Bloch-Redfield theory: Quantum jumps in a solid-state environment. Physical Review B, 88 (17): 174514, November 2013. ISSN 1098-0121, 1550-235X. 10.1103/​PhysRevB.88.174514.

[49] Jan Jeske, David J. Ing, Martin B. Plenio, Susana F. Huelga, and Jared H. Cole. Bloch-Redfield equations for modeling light-harvesting complexes. The Journal of Chemical Physics, 142 (6): 064104, February 2015. ISSN 0021-9606. 10.1063/​1.4907370.

[50] Timur V. Tscherbul and Paul Brumer. Partial secular Bloch-Redfield master equation for incoherent excitation of multilevel quantum systems. J. Chem. Phys., 142 (10): 104107, March 2015. ISSN 0021-9606. 10.1063/​1.4908130.

[51] A. G. Redfield. On the Theory of Relaxation Processes. IBM J. Res. Dev., 1 (1): 19–31, January 1957. ISSN 0018-8646. 10.1147/​rd.11.0019.

[52] Dragomir Davidović. Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation. Quantum, 4: 326, September 2020. 10.22331/​q-2020-09-21-326.

[53] Gernot Schaller and Tobias Brandes. Preservation of positivity by dynamical coarse graining. Phys. Rev. A, 78 (2): 022106, August 2008. 10.1103/​PhysRevA.78.022106.

[54] Chris Fleming, N. I. Cummings, Charis Anastopoulos, and B. L. Hu. The rotating-wave approximation: Consistency and applicability from an open quantum system analysis. J. Phys. A: Math. Theor., 43 (40): 405304, 2010. ISSN 1751-8121. 10.1088/​1751-8113/​43/​40/​405304.

[55] Jian Ma, Zhe Sun, Xiaoguang Wang, and Franco Nori. Entanglement dynamics of two qubits in a common bath. Phys. Rev. A, 85 (6): 062323, June 2012. 10.1103/​PhysRevA.85.062323.

[56] E. B. Davies. Markovian master equations. II. Mathematische Annalen, 219 (2): 147–158, June 1976. ISSN 0025-5831, 1432-1807. 10.1007/​BF01351898.

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