Tomographically reconstructed master equations for any open quantum dynamics

Felix A. Pollock and Kavan Modi

School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia

Memory effects in open quantum dynamics are often incorporated in the equation of motion through a superoperator known as the memory kernel, which encodes how past states affect future dynamics. However, the usual prescription for determining the memory kernel requires information about the underlying system-environment dynamics. Here, by deriving the transfer tensor method from first principles, we show how a memory kernel master equation, for any quantum process, can be entirely expressed in terms of a family of completely positive dynamical maps. These can be reconstructed through quantum process tomography on the system alone, either experimentally or numerically, and the resulting equation of motion is equivalent to a generalised Nakajima-Zwanzig equation. For experimental settings, we give a full prescription for the reconstruction procedure, rendering the memory kernel operational. When simulation of an open system is the goal, we show how our procedure yields a considerable advantage for numerically calculating dynamics, even when the system is arbitrarily periodically (or transiently) driven or initially correlated with its environment. Namely, we show that the long time dynamics can be efficiently obtained from a set of reconstructed maps over a much shorter time.

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[1] I. L. Chuang and M. A. Nielsen, J. Mod. Opt. 44, 2455 (1997).

[2] K. Modi, Sci. Rep. 2, 581 (2012).

[3] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. A 97, 012127 (2018a).

[4] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. Lett. 120, 040405 (2018b).

[5] S. Milz, F. A. Pollock, and K. Modi, accepted in Phys. Rev. A (2018).

[6] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002).

[7] A. Fruchtman, N. Lambert, and E. M. Gauger, Sci. Rep. 6, 28204 (2016).

[8] J. Iles-Smith, A. G. Dijkstra, N. Lambert, and A. Nazir, J. Chem. Phys. 144, 044110 (2016).

[9] Y. Tanimura, J. Phys. Soc. Jpn. 75, 082001 (2006).

[10] J. Strümpfer and K. Schulten, J. Chem. Theory Comput. 8, 2808 (2012).

[11] A. W. Chin, S. F. Huelga, and M. B. Plenio, in Semiconductors and Semimetals, Vol. 85, edited by U. Würfel, M. Thorwart, and E. R. Weber (Elsevier, 2011) pp. 115 - 143.

[12] N. Makri and D. E. Makarov, J. Chem. Phys. 102, 4611 (1995).

[13] I. de Vega and D. Alonso, Rev. Mod. Phys. 89, 015001 (2017).

[14] J. Cerrillo and J. Cao, Phys. Rev. Lett. 112, 110401 (2014).

[15] P. Nalbach, A. Ishizaki, G. R. Fleming, and M. Thorwart, New J. Phys. 13, 063040 (2011).

[16] A. Strathearn, B. W. Lovett, and P. Kirton, New J. Phys. 19, 093009 (2017a).

[17] A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, arXiv:1711.09641 (2017b).

[18] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).

[19] V. Gorini, A. Kossakokowski, and E. C. G. Sudarshan, J. Math. Phys 17, 821 (1976).

[20] K. Modi, Open Systems & Information Dynamics 18, 253 (2011).

[21] M. Ringbauer, C. J. Wood, K. Modi, A. Gilchrist, A. G. White, and A. Fedrizzi, Phys. Rev. Lett. 114, 090402 (2015).

[22] D. Kretschmann, D. Schlingemann, and R. F. Werner, J. Funct. Anal. 255, 1889 (2008).

[23] B. Dive, F. Mintert, and D. Burgarth, Phys. Rev. A 92, 032111 (2015).

[24] L. M. Norris, G. A. Paz-Silva, and L. Viola, Phys. Rev. Lett. 116, 150503 (2016).

[25] F. A. Pollock, A. Chęcińska, S. Pascazio, and K. Modi, Phys. Rev. A 94, 032112 (2016).

[26] J. Jeske, J. H. Cole, C. Müller, M. Marthaler, and G. Schön, New J. Phys. 14, 023013 (2012).

[27] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, Phys. Rev. A 80, 052108 (2009).

[28] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, Phys. Rev. A 82, 062104 (2010a).

[29] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, J. Phys. A 43, 395303 (2010b).

[30] R. Rosenbach, J. Cerrillo, S. F. Huelga, J. Cao, and M. B. Plenio, New J. Phys. 18, 023035 (2016).

[31] A. A. Kananenka, C.-Y. Hsieh, J. Cao, and E. Geva, J. Phys. Chem. Lett. 7, 4809 (2016).

[32] S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).

[33] S. Daffer, K. Wódkiewicz, J. D. Cresser, and J. K. McIver, Phys. Rev. A 70, 010304 (2004).

[34] H.-P. Breuer and B. Vacchini, Phys. Rev. Lett. 101, 140402 (2008).

[35] D. Chruściński and A. Kossakowski, Phys. Rev. A 94, 020103 (2016).

[36] B. Vacchini, Phys. Rev. Lett. 117, 230401 (2016).

[37] Q. Shi and E. Geva, J. Chem. Phys. 119, 12063 (2003).

[38] G. Cohen and E. Rabani, Phys. Rev. B 84, 075150 (2011).

[39] G. Cohen, E. Gull, D. R. Reichman, A. J. Millis, and E. Rabani, Phys. Rev. B 87, 195108 (2013).

[40] M. Buser, J. Cerrillo, G. Schaller, and J. Cao, Phys. Rev. A 96, 062122 (2017).

[41] D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 120, 030402 (2018).

[42] C. R. Willis and R. H. Picard, Phys. Rev. A 9, 1343 (1974).

[43] R. H. Picard and C. R. Willis, Phys. Rev. A 16, 1625 (1977).

[44] P. Degenfeld-Schonburg and M. J. Hartmann, Phys. Rev. B 89, 245108 (2014).

[45] P. Degenfeld-Schonburg, C. Navarrete-Benlloch, and M. J. Hartmann, Phys. Rev. A 91, 053850 (2015).

[46] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).

[47] P. Facchi and S. Pascazio, J. Phys. A 41, 493001 (2008).

[48] T. M. Stace, A. C. Doherty, and D. J. Reilly, Phys. Rev. Lett. 111, 180602 (2013).

[49] R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).

[50] P.-Y. Yang and W.-M. Zhang, arXiv:1605.08521 (2016).

[51] S. Kitajima, M. Ban, and F. Shibata, J. Phys. A 50, 125303 (2017).

[52] V. Prepeliţă, M. Doroftei, and T. Vasilache, Balkan J. Geom. Appl. 3, 111 (1998).

Cited by

[1] Michael te Vrugt and Raphael Wittkowski, "Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians", Physical Review E 99 6, 062118 (2019).

[2] Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock, "Almost Markovian processes from closed dynamics", Quantum 3, 136 (2019).

[3] Leonardo Banchi, Edward Grant, Andrea Rocchetto, and Simone Severini, "Modelling non-markovian quantum processes with recurrent neural networks", arXiv:1808.01374, New Journal of Physics 20 12, 123030 (2018).

[4] Philip Taranto, Felix A. Pollock, Simon Milz, Marco Tomamichel, and Kavan Modi, "Quantum Markov Order", Physical Review Letters 122 14, 140401 (2019).

[5] Philip Taranto, Simon Milz, Felix A. Pollock, and Kavan Modi, "Structure of quantum stochastic processes with finite Markov order", Physical Review A 99 4, 042108 (2019).

[6] Shlok Nahar and Sai Vinjanampathy, "Preparations and Weak Quantum Control can Witness non-Markovianity", arXiv:1803.08443.

[7] Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternostro, and Kavan Modi, "Non-Markovian quantum processes: Complete framework and efficient characterization", arXiv:1512.00589, Physical Review A 97 1, 012127 (2018).

[8] Maximilian Buser, Javier Cerrillo, Gernot Schaller, and Jianshu Cao, "Initial system-environment correlations via the transfer-tensor method", Physical Review A 96 6, 062122 (2017).

[9] Simon Milz, Felix A. Pollock, and Kavan Modi, "An Introduction to Operational Quantum Dynamics", Open Systems and Information Dynamics 24 4, 1740016 (2017).

The above citations are from Crossref's cited-by service (last updated 2019-06-18 12:52:53) and SAO/NASA ADS (last updated 2019-06-18 12:52:54). The list may be incomplete as not all publishers provide suitable and complete citation data.