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.

Share

► BibTeX data

► References

[1] I. L. Chuang and M. A. Nielsen, J. Mod. Opt. 44, 2455 (1997).
https://doi.org/10.1080/09500349708231894

[2] K. Modi, Sci. Rep. 2, 581 (2012).
https://doi.org/10.1038/srep00581

[3] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. A 97, 012127 (2018a).
https://doi.org/10.1103/PhysRevA.97.012127

[4] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. Lett. 120, 040405 (2018b).
https://doi.org/10.1103/PhysRevLett.120.040405

[5] S. Milz, F. A. Pollock, and K. Modi, accepted in Phys. Rev. A (2018).
arXiv:1610.02152

[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).
https://doi.org/10.1038/srep28204

[8] J. Iles-Smith, A. G. Dijkstra, N. Lambert, and A. Nazir, J. Chem. Phys. 144, 044110 (2016).
https://doi.org/10.1063/1.4940218

[9] Y. Tanimura, J. Phys. Soc. Jpn. 75, 082001 (2006).
https://doi.org/10.1143/JPSJ.75.082001

[10] J. Strümpfer and K. Schulten, J. Chem. Theory Comput. 8, 2808 (2012).
https://doi.org/10.1021/ct3003833

[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.
https://doi.org/10.1016/B978-0-12-391060-8.00004-6

[12] N. Makri and D. E. Makarov, J. Chem. Phys. 102, 4611 (1995).
https://doi.org/10.1063/1.469509

[13] I. de Vega and D. Alonso, Rev. Mod. Phys. 89, 015001 (2017).
https://doi.org/10.1103/RevModPhys.89.015001

[14] J. Cerrillo and J. Cao, Phys. Rev. Lett. 112, 110401 (2014).
https://doi.org/10.1103/PhysRevLett.112.110401

[15] P. Nalbach, A. Ishizaki, G. R. Fleming, and M. Thorwart, New J. Phys. 13, 063040 (2011).
https://doi.org/10.1088/1367-2630/13/6/063040

[16] A. Strathearn, B. W. Lovett, and P. Kirton, New J. Phys. 19, 093009 (2017a).
https://doi.org/10.1088/1367-2630/aa8744

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

[18] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).
https:/​/​projecteuclid.org/​euclid.cmp/​1103899849

[19] V. Gorini, A. Kossakokowski, and E. C. G. Sudarshan, J. Math. Phys 17, 821 (1976).
https://doi.org/10.1063/1.522979

[20] K. Modi, Open Systems & Information Dynamics 18, 253 (2011).
https://doi.org/10.1142/S1230161211000170

[21] M. Ringbauer, C. J. Wood, K. Modi, A. Gilchrist, A. G. White, and A. Fedrizzi, Phys. Rev. Lett. 114, 090402 (2015).
https://doi.org/10.1103/PhysRevLett.114.090402

[22] D. Kretschmann, D. Schlingemann, and R. F. Werner, J. Funct. Anal. 255, 1889 (2008).
https://doi.org/10.1016/j.jfa.2008.07.023

[23] B. Dive, F. Mintert, and D. Burgarth, Phys. Rev. A 92, 032111 (2015).
https://doi.org/10.1103/PhysRevA.92.032111

[24] L. M. Norris, G. A. Paz-Silva, and L. Viola, Phys. Rev. Lett. 116, 150503 (2016).
https://doi.org/10.1103/PhysRevLett.116.150503

[25] F. A. Pollock, A. Chęcińska, S. Pascazio, and K. Modi, Phys. Rev. A 94, 032112 (2016).
https://doi.org/10.1103/PhysRevA.94.032112

[26] J. Jeske, J. H. Cole, C. Müller, M. Marthaler, and G. Schön, New J. Phys. 14, 023013 (2012).
https://doi.org/10.1088/1367-2630/14/2/023013

[27] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, Phys. Rev. A 80, 052108 (2009).
https://doi.org/10.1103/PhysRevA.80.052108

[28] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, Phys. Rev. A 82, 062104 (2010a).
https://doi.org/10.1103/PhysRevA.82.062104

[29] B. Bellomo, A. De Pasquale, G. Gualdi, and U. Marzolino, J. Phys. A 43, 395303 (2010b).
https://doi.org/10.1088/1751-8113/43/39/395303

[30] R. Rosenbach, J. Cerrillo, S. F. Huelga, J. Cao, and M. B. Plenio, New J. Phys. 18, 023035 (2016).
https://doi.org/10.1088/1367-2630/18/2/023035

[31] A. A. Kananenka, C.-Y. Hsieh, J. Cao, and E. Geva, J. Phys. Chem. Lett. 7, 4809 (2016).
https://doi.org/10.1021/acs.jpclett.6b02389

[32] S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).
https://doi.org/10.1103/PhysRevA.64.033808

[33] S. Daffer, K. Wódkiewicz, J. D. Cresser, and J. K. McIver, Phys. Rev. A 70, 010304 (2004).
https://doi.org/10.1103/PhysRevA.70.010304

[34] H.-P. Breuer and B. Vacchini, Phys. Rev. Lett. 101, 140402 (2008).
https://doi.org/10.1103/PhysRevLett.101.140402

[35] D. Chruściński and A. Kossakowski, Phys. Rev. A 94, 020103 (2016).
https://doi.org/10.1103/PhysRevA.94.020103

[36] B. Vacchini, Phys. Rev. Lett. 117, 230401 (2016).
https://doi.org/10.1103/PhysRevLett.117.230401

[37] Q. Shi and E. Geva, J. Chem. Phys. 119, 12063 (2003).
https://doi.org/10.1063/1.1624830

[38] G. Cohen and E. Rabani, Phys. Rev. B 84, 075150 (2011).
https://doi.org/10.1103/PhysRevB.84.075150

[39] G. Cohen, E. Gull, D. R. Reichman, A. J. Millis, and E. Rabani, Phys. Rev. B 87, 195108 (2013).
https://doi.org/10.1103/PhysRevB.87.195108

[40] M. Buser, J. Cerrillo, G. Schaller, and J. Cao, Phys. Rev. A 96, 062122 (2017).
https://doi.org/10.1103/PhysRevA.96.062122

[41] D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 120, 030402 (2018).
https://doi.org/10.1103/PhysRevLett.120.030402

[42] C. R. Willis and R. H. Picard, Phys. Rev. A 9, 1343 (1974).
https://doi.org/10.1103/PhysRevA.9.1343

[43] R. H. Picard and C. R. Willis, Phys. Rev. A 16, 1625 (1977).
https://doi.org/10.1103/PhysRevA.16.1625

[44] P. Degenfeld-Schonburg and M. J. Hartmann, Phys. Rev. B 89, 245108 (2014).
https://doi.org/10.1103/PhysRevB.89.245108

[45] P. Degenfeld-Schonburg, C. Navarrete-Benlloch, and M. J. Hartmann, Phys. Rev. A 91, 053850 (2015).
https://doi.org/10.1103/PhysRevA.91.053850

[46] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).
https://doi.org/10.1103/PhysRevLett.82.2417

[47] P. Facchi and S. Pascazio, J. Phys. A 41, 493001 (2008).
https://doi.org/10.1088/1751-8113/41/49/493001

[48] T. M. Stace, A. C. Doherty, and D. J. Reilly, Phys. Rev. Lett. 111, 180602 (2013).
https://doi.org/10.1103/PhysRevLett.111.180602

[49] R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
https://doi.org/10.1146/annurev-conmatphys-031214-014726

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

[51] S. Kitajima, M. Ban, and F. Shibata, J. Phys. A 50, 125303 (2017).
https://doi.org/10.1088/1751-8121/aa5d85

[52] V. Prepeliţă, M. Doroftei, and T. Vasilache, Balkan J. Geom. Appl. 3, 111 (1998).
http:/​/​www.mathem.pub.ro/​bjga/​v03n1/​B03-1-PREPE.pdf

► Cited by (beta)

Crossref's cited-by service has no data on citing works. Unfortunately not all publishers provide suitable citation data.