Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation

Dragomir Davidović

doi:10.22331/q-2020-09-21-326

Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation

Georgia Institute of Technology, Atlanta, Georgia, United States

Published:	2020-09-21, volume 4, page 326
Eprint:	arXiv:2003.09063v4
Doi:	https://doi.org/10.22331/q-2020-09-21-326
Citation:	Quantum 4, 326 (2020).

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Abstract

Here we present a Lindblad master equation that approximates the Redfield equation, a well known master equation derived from first principles, without significantly compromising the range of applicability of the Redfield equation. Instead of full-scale coarse-graining, this approximation only truncates terms in the Redfield equation that average out over a time-scale typical of the quantum system. The first step in this approximation is to properly renormalize the system Hamiltonian, to symmetrize the gains and losses of the state due to the environmental coupling. In the second step, we swap out an arithmetic mean of the spectral density with a geometric one, in these gains and losses, thereby restoring complete positivity. This completely positive approximation, GAME (geometric-arithmetic master equation), is adaptable between its time-independent, time-dependent, and Floquet form. In the exactly solvable, three-level, Jaynes-Cummings model, we find that the error of the approximate state is almost an order of magnitude lower than that obtained by solving the coarse-grained stochastic master equation. As a test-bed, we use a ferromagnetic Heisenberg spin-chain with long-range dipole-dipole coupling between up to 25-spins, and study the differences between various master equations. We find that GAME has the highest accuracy per computational resource.

Featured image: Playing GAME with vacuum energy fluctuations (Geometric-Arithmetic Master Equation)

Popular summary

New Lamb-shift:
H(ω, ω' ) = ½[ S(ω) + S(ω' ) ]+$i$¼[γ(ω) − γ(ω') ]
Quantum systems coupled to an environment are renormalized by the vacuum energy fluctuations or the Lamb-shift. Here I identify the Lamb-shift in a 63-year old perturbative Redfield equation, which enables me to find a highly accurate completely positive (CP) approximation of the equation.

The graph shows the trace error from the Redfield solutions versus time, for various CP-master equations. In contrast to the previous approximations, the GAME error does not grow in time and remains low in perpetuity.

► BibTeX data

@article{Davidovic2020completelypositive,
  doi = {10.22331/q-2020-09-21-326},
  url = {https://doi.org/10.22331/q-2020-09-21-326},
  title = {Completely {P}ositive, {S}imple, and {P}ossibly {H}ighly {A}ccurate {A}pproximation of the {R}edfield {E}quation},
  author = {Davidovi{\'{c}}, Dragomir},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {326},
  month = sep,
  year = {2020}
}

► References

[1] C. W.Gardiner, ``Quantum noise,'' (1991).

[2] H.-P. Breuer and F. Petruccione, ``The theory of open quantum systems,'' (2007).

[3] G. Lindblad, Comm. Math. Phys. 48, 119 (1976).
https://projecteuclid.org:443/euclid.cmp/1103899849

[4] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Journal of Mathematical Physics 17, 821 (1976), https://aip.scitation.org/doi/pdf/10.1063/1.522979.
https://doi.org/10.1063/1.522979
arXiv:https://aip.scitation.org/doi/pdf/10.1063/1.522979

[5] D. A. Lidar, Z. Bihary, and K. Whaley, Chemical Physics 268, 35 (2001).
https://doi.org/10.1016/S0301-0104(01)00330-5

[6] D. Kohen, C. C. Marston, and D. J. Tannor, The Journal of Chemical Physics 107, 5236 (1997), https://doi.org/10.1063/1.474887.
https://doi.org/10.1063/1.474887
arXiv:https://doi.org/10.1063/1.474887

[7] R. Zwanzig, The Journal of Chemical Physics 33, 1338 (1960), https://doi.org/10.1063/1.1731409.
https://doi.org/10.1063/1.1731409
arXiv:https://doi.org/10.1063/1.1731409

[8] S. Nakajima, Progress of Theoretical Physics 20, 948 (1958), https://academic.oup.com/ptp/article-pdf/20/6/948/5440766/20-6-948.pdf.
https://doi.org/10.1143/PTP.20.948
arXiv:https://academic.oup.com/ptp/article-pdf/20/6/948/5440766/20-6-948.pdf

[9] K. Ryogo, T. Morikazu, and H. Natsuki, ``Statistical physics ii,'' (1998).

[10] A. G. Redfield, IBM Journal of Research and Development 1, 19 (1957).

[11] A. REDFIELD, in Advances in Magnetic Resonance, Advances in Magnetic and Optical Resonance, Vol. 1, edited by J. S. Waugh (Academic Press, 1965) pp. 1 – 32.
https://doi.org/10.1016/B978-1-4832-3114-3.50007-6

[12] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).
https://doi.org/10.1103/PhysRevA.65.032314

[13] W. T. Pollard, A. K. Felts, and R. A. Friesner, ``The redfield equation in condensed-phase quantum dynamics,'' in Advances in Chemical Physics (John Wiley & Sons, Ltd, 2007) pp. 77–134, https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470141526.ch3.
https://doi.org/10.1002/9780470141526.ch3
arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470141526.ch3

[14] I. Kondov, U. Kleinekathöfer, and M. Schreiber, The Journal of Chemical Physics 114, 1497 (2001), https://doi.org/10.1063/1.1335656.
https://doi.org/10.1063/1.1335656
arXiv:https://doi.org/10.1063/1.1335656

[15] D. Egorova, M. Thoss, W. Domcke, and H. Wang, The Journal of Chemical Physics 119, 2761 (2003), https://doi.org/10.1063/1.1587121.
https://doi.org/10.1063/1.1587121
arXiv:https://doi.org/10.1063/1.1587121

[16] M. Schröder, M. Schreiber, and U. Kleinekathöfer, Journal of Luminescence 125, 126 (2007), festschrift in Honor of Academician Alexander A. Kaplyanskii.
https://doi.org/10.1016/j.jlumin.2006.08.086

[17] A. Montoya-Castillo, T. C. Berkelbach, and D. R. Reichman, The Journal of Chemical Physics 143, 194108 (2015), https://doi.org/10.1063/1.4935443.
https://doi.org/10.1063/1.4935443
arXiv:https://doi.org/10.1063/1.4935443

[18] C. Timm, Phys. Rev. B 77, 195416 (2008).
https://doi.org/10.1103/PhysRevB.77.195416

[19] J. Jeske, D. J. Ing, M. B. Plenio, S. F. Huelga, and J. H. Cole, The Journal of Chemical Physics 142, 064104 (2015), https://doi.org/10.1063/1.4907370.
https://doi.org/10.1063/1.4907370
arXiv:https://doi.org/10.1063/1.4907370

[20] W. P. Bricker, J. L. Banal, M. B. Stone, and M. Bathe, The Journal of Chemical Physics 149, 024905 (2018), https://doi.org/10.1063/1.5036656.
https://doi.org/10.1063/1.5036656
arXiv:https://doi.org/10.1063/1.5036656

[21] R. S. Whitney, Journal of Physics A: Mathematical and Theoretical 41, 175304 (2008).
https://doi.org/10.1088/1751-8113/41/17/175304

[22] R. Hartmann and W. T. Strunz, Phys. Rev. A 101, 012103 (2020).
https://doi.org/10.1103/PhysRevA.101.012103

[23] T. Yu, L. Diósi, N. Gisin, and W. T. Strunz, Phys. Rev. A 60, 91 (1999).
https://doi.org/10.1103/PhysRevA.60.91

[24] I. de Vega, D. Alonso, P. Gaspard, and W. T. Strunz, The Journal of Chemical Physics 122, 124106 (2005), https://doi.org/10.1063/1.1867377.
https://doi.org/10.1063/1.1867377
arXiv:https://doi.org/10.1063/1.1867377

[25] N. Makri and D. E. Makarov, The Journal of Chemical Physics 102, 4600 (1995), https://doi.org/10.1063/1.469508.
https://doi.org/10.1063/1.469508
arXiv:https://doi.org/10.1063/1.469508

[26] M. Thorwart, E. Paladino, and M. Grifoni, Chemical Physics 296, 333 (2004), the Spin-Boson Problem: From Electron Transfer to Quantum Computing ... to the 60th Birthday of Professor Ulrich Weiss.
https://doi.org/10.1016/j.chemphys.2003.10.007

[27] P. Nalbach and M. Thorwart, Phys. Rev. B 81, 054308 (2010).
https://doi.org/10.1103/PhysRevB.81.054308

[28] Y. Tanimura and R. Kubo, Journal of the Physical Society of Japan 58, 1199 (1989), https://doi.org/10.1143/JPSJ.58.1199.
https://doi.org/10.1143/JPSJ.58.1199
arXiv:https://doi.org/10.1143/JPSJ.58.1199

[29] Y. Tanimura, Journal of the Physical Society of Japan 75, 082001 (2006), https://doi.org/10.1143/JPSJ.75.082001.
https://doi.org/10.1143/JPSJ.75.082001
arXiv:https://doi.org/10.1143/JPSJ.75.082001

[30] Y. Tanimura, The Journal of Chemical Physics 141, 044114 (2014), https://doi.org/10.1063/1.4890441.
https://doi.org/10.1063/1.4890441
arXiv:https://doi.org/10.1063/1.4890441

[31] Z. Li, N. Tong, X. Zheng, D. Hou, J. Wei, J. Hu, and Y. Yan, Phys. Rev. Lett. 109, 266403 (2012).
https://doi.org/10.1103/PhysRevLett.109.266403

[32] Y. Cheng, W. Hou, Y. Wang, Z. Li, J. Wei, and Y. Yan, New Journal of Physics 17, 033009 (2015).
https://doi.org/10.1088/1367-2630/17/3/033009

[33] H.-D. Meyer, U. Manthe, and L. Cederbaum, Chemical Physics Letters 165, 73 (1990).
https://doi.org/10.1016/0009-2614(90)87014-I

[34] M. Beck, A. Jäckle, G. Worth, and H.-D. Meyer, Physics Reports 324, 1 (2000).
https://doi.org/10.1016/S0370-1573(99)00047-2

[35] H. Wang and M. Thoss, The Journal of Chemical Physics 119, 1289 (2003), https://doi.org/10.1063/1.1580111.
https://doi.org/10.1063/1.1580111
arXiv:https://doi.org/10.1063/1.1580111

[36] J. Zheng, Y. Xie, S. Jiang, and Z. Lan, The Journal of Physical Chemistry C 120, 1375 (2016), https://doi.org/10.1021/acs.jpcc.5b09921.
https://doi.org/10.1021/acs.jpcc.5b09921
arXiv:https://doi.org/10.1021/acs.jpcc.5b09921

[37] D. Suess, A. Eisfeld, and W. T. Strunz, Phys. Rev. Lett. 113, 150403 (2014).
https://doi.org/10.1103/PhysRevLett.113.150403

[38] P.-P. Zhang and A. Eisfeld, The Journal of Physical Chemistry Letters 7, 4488 (2016), pMID: 27775345, https://doi.org/10.1021/acs.jpclett.6b02111.
https://doi.org/10.1021/acs.jpclett.6b02111
arXiv:https://doi.org/10.1021/acs.jpclett.6b02111

[39] R. Hartmann and W. T. Strunz, Journal of Chemical Theory and Computation 13, 5834 (2017), pMID: 29016126, https://doi.org/10.1021/acs.jctc.7b00751.
https://doi.org/10.1021/acs.jctc.7b00751
arXiv:https://doi.org/10.1021/acs.jctc.7b00751

[40] A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, Nature Communications 9, 3322 (2018).
https://doi.org/10.1038/s41467-018-05617-3

[41] F. A. Y. N. Schröder, D. H. P. Turban, A. J. Musser, N. D. M. Hine, and A. W. Chin, Nature Communications 10, 1062 (2019).
https://doi.org/10.1038/s41467-019-09039-7

[42] S. Jang, The Journal of Chemical Physics 131, 164101 (2009), https://doi.org/10.1063/1.3247899.
https://doi.org/10.1063/1.3247899
arXiv:https://doi.org/10.1063/1.3247899

[43] D. P. S. McCutcheon, N. S. Dattani, E. M. Gauger, B. W. Lovett, and A. Nazir, Phys. Rev. B 84, 081305 (2011).
https://doi.org/10.1103/PhysRevB.84.081305

[44] E. B. Davies, Comm. Math. Phys. 39, 91 (1974).
https://projecteuclid.org:443/euclid.cmp/1103860160

[45] C. Majenz, T. Albash, H.-P. Breuer, and D. A. Lidar, Phys. Rev. A 88, 012103 (2013).
https://doi.org/10.1103/PhysRevA.88.012103

[46] D. Farina and V. Giovannetti, Phys. Rev. A 100, 012107 (2019).
https://doi.org/10.1103/PhysRevA.100.012107

[47] E. Mozgunov and D. Lidar, 4, 227 (2020), 1908.01095 [Quantum].
https://doi.org/10.22331/q-2020-02-06-227
arXiv:1908.01095

[48] N. Vogt, J. Jeske, and J. H. Cole, Phys. Rev. B 88, 174514 (2013).
https://doi.org/10.1103/PhysRevB.88.174514

[49] T. V. Tscherbul and P. Brumer, The Journal of Chemical Physics 142, 104107 (2015), https://doi.org/10.1063/1.4908130.
https://doi.org/10.1063/1.4908130
arXiv:https://doi.org/10.1063/1.4908130

[50] G. Schaller and T. Brandes, Phys. Rev. A 78, 022106 (2008).
https://doi.org/10.1103/PhysRevA.78.022106

[51] F. Benatti, R. Floreanini, and U. Marzolino, Phys. Rev. A 81, 012105 (2010).
https://doi.org/10.1103/PhysRevA.81.012105

[52] W. J. Munro and C. W. Gardiner, Phys. Rev. A 53, 2633 (1996).
https://doi.org/10.1103/PhysRevA.53.2633

[53] J. Wilkie, Phys. Rev. E 62, 8808 (2000).
https://doi.org/10.1103/PhysRevE.62.8808

[54] B. Palmieri, D. Abramavicius, and S. Mukamel, The Journal of Chemical Physics 130, 204512 (2009), https://doi.org/10.1063/1.3142485.
https://doi.org/10.1063/1.3142485
arXiv:https://doi.org/10.1063/1.3142485

[55] G. Kiršanskas, M. Franckié, and A. Wacker, Phys. Rev. B 97, 035432 (2018).
https://doi.org/10.1103/PhysRevB.97.035432

[56] F. Nathan and M. S. Rudner, Phys. Rev. B 102, 115109 (2020).
https://doi.org/10.1103/PhysRevB.102.115109

[57] S. Kryszewski and J. Czechowska-Kryszk, ``Master equation - tutorial approach,'' (2008), arXiv:0801.1757 [quant-ph].
arXiv:0801.1757

[58] H.-P. Breuer, Phys. Rev. A 70, 012106 (2004).
https://doi.org/10.1103/PhysRevA.70.012106

[59] D. W. Hone, R. Ketzmerick, and W. Kohn, Phys. Rev. E 79, 051129 (2009).
https://doi.org/10.1103/PhysRevE.79.051129

[60] T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, New Journal of Physics 14, 123016 (2012).
https://doi.org/10.1088/1367-2630/14/12/123016

[61] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
https://doi.org/10.1103/RevModPhys.82.1155

[62] A. Daley, Advances in Physics 63 (2014), 10.1080/00018732.2014.933502.
https://doi.org/10.1080/00018732.2014.933502

[63] F. Benatti, R. Floreanini, and U. Marzolino, EPL (Europhysics Letters) 88, 20011 (2009).
https://doi.org/10.1209/0295-5075/88/20011

[64] A. Rivas, Phys. Rev. A 95, 042104 (2017).
https://doi.org/10.1103/PhysRevA.95.042104

[65] F. Benatti, R. Floreanini, and M. Piani, Phys. Rev. Lett. 91, 070402 (2003).
https://doi.org/10.1103/PhysRevLett.91.070402

[66] R. Tana and Z. Ficek, Journal of Optics B: Quantum and Semiclassical Optics 6, S90 (2004).
https://doi.org/10.1088/1464-4266/6/3/015

[67] S. E. Clifton and W. E. P., Philosophical Transactions of the Royal Society of London 240, 599 (1948).
https://doi.org/10.1098/rsta.1948.0007

[68] W. F. Brown, Phys. Rev. 130, 1677 (1963).
https://doi.org/10.1103/PhysRev.130.1677

[69] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007).
https://doi.org/10.1103/PhysRevLett.99.027204

[70] F. Haake and M. Lewenstein, Phys. Rev. A 28, 3606 (1983).
https://doi.org/10.1103/PhysRevA.28.3606

[71] P. Gaspard and M. Nagaoka, The Journal of Chemical Physics 111, 5668 (1999), https://doi.org/10.1063/1.479867.
https://doi.org/10.1063/1.479867
arXiv:https://doi.org/10.1063/1.479867

[72] Y. C. Cheng and R. J. Silbey, The Journal of Physical Chemistry B 109, 21399 (2005), pMID: 16853776, https://doi.org/10.1021/jp051303o.
https://doi.org/10.1021/jp051303o
arXiv:https://doi.org/10.1021/jp051303o

[73] A. Suarez, R. Silbey, and I. Oppenheim, The Journal of Chemical Physics 97, 5101 (1992), https://doi.org/10.1063/1.463831.
https://doi.org/10.1063/1.463831
arXiv:https://doi.org/10.1063/1.463831

[74] T. Yu, L. Diósi, N. Gisin, and W. T. Strunz, Physics Letters A 265, 331 (2000).
https://doi.org/10.1016/S0375-9601(00)00014-1

[75] R. Silbey and R. A. Harris, The Journal of Chemical Physics 80, 2615 (1984), https://doi.org/10.1063/1.447055.
https://doi.org/10.1063/1.447055
arXiv:https://doi.org/10.1063/1.447055

[76] Z.-X. Gong, M. F. Maghrebi, A. Hu, M. Foss-Feig, P. Richerme, C. Monroe, and A. V. Gorshkov, Phys. Rev. B 93, 205115 (2016).
https://doi.org/10.1103/PhysRevB.93.205115

[77] G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015).
https://doi.org/10.1103/PhysRevLett.115.180405

[78] M. Grifoni and P. Hänggi, Physics Reports 304, 229 (1998).
https://doi.org/10.1016/S0370-1573(98)00022-2

[79] T. Shirai, J. Thingna, T. Mori, S. Denisov, P. Hänggi, and S. Miyashita, New Journal of Physics 18, 053008 (2016).
https://doi.org/10.1088/1367-2630/18/5/053008

[80] J. Elzerman, R. Hanson, and L. W. van Beveren et al., Nature 430, 431–435 (2004).
https://doi.org/10.1038/nature02693

[81] A. Morello, J. Pla, and F. Z. et al., Nature 467, 687 (2010).
https://doi.org/10.1038/nature09392

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