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.
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.
 C. W.Gardiner, ``Quantum noise,'' (1991).
 H.-P. Breuer and F. Petruccione, ``The theory of open quantum systems,'' (2007).
 G. Lindblad, Comm. Math. Phys. 48, 119 (1976).
 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.
 D. Kohen, C. C. Marston, and D. J. Tannor, The Journal of Chemical Physics 107, 5236 (1997), https://doi.org/10.1063/1.474887.
 S. Nakajima, Progress of Theoretical Physics 20, 948 (1958), https://academic.oup.com/ptp/article-pdf/20/6/948/5440766/20-6-948.pdf.
 K. Ryogo, T. Morikazu, and H. Natsuki, ``Statistical physics ii,'' (1998).
 A. G. Redfield, IBM Journal of Research and Development 1, 19 (1957).
 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.
 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.
 I. Kondov, U. Kleinekathöfer, and M. Schreiber, The Journal of Chemical Physics 114, 1497 (2001), https://doi.org/10.1063/1.1335656.
 D. Egorova, M. Thoss, W. Domcke, and H. Wang, The Journal of Chemical Physics 119, 2761 (2003), https://doi.org/10.1063/1.1587121.
 M. Schröder, M. Schreiber, and U. Kleinekathöfer, Journal of Luminescence 125, 126 (2007), festschrift in Honor of Academician Alexander A. Kaplyanskii.
 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.
 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.
 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.
 R. S. Whitney, Journal of Physics A: Mathematical and Theoretical 41, 175304 (2008).
 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.
 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.
 Y. Tanimura and R. Kubo, Journal of the Physical Society of Japan 58, 1199 (1989), https://doi.org/10.1143/JPSJ.58.1199.
 Y. Tanimura, Journal of the Physical Society of Japan 75, 082001 (2006), https://doi.org/10.1143/JPSJ.75.082001.
 Z. Li, N. Tong, X. Zheng, D. Hou, J. Wei, J. Hu, and Y. Yan, Phys. Rev. Lett. 109, 266403 (2012).
 Y. Cheng, W. Hou, Y. Wang, Z. Li, J. Wei, and Y. Yan, New Journal of Physics 17, 033009 (2015).
 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.
 D. Suess, A. Eisfeld, and W. T. Strunz, Phys. Rev. Lett. 113, 150403 (2014).
 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.
 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.
 E. B. Davies, Comm. Math. Phys. 39, 91 (1974).
 T. V. Tscherbul and P. Brumer, The Journal of Chemical Physics 142, 104107 (2015), https://doi.org/10.1063/1.4908130.
 B. Palmieri, D. Abramavicius, and S. Mukamel, The Journal of Chemical Physics 130, 204512 (2009), https://doi.org/10.1063/1.3142485.
 T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, New Journal of Physics 14, 123016 (2012).
 Y. C. Cheng and R. J. Silbey, The Journal of Physical Chemistry B 109, 21399 (2005), pMID: 16853776, https://doi.org/10.1021/jp051303o.
 A. Suarez, R. Silbey, and I. Oppenheim, The Journal of Chemical Physics 97, 5101 (1992), https://doi.org/10.1063/1.463831.
 G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015).
 T. Shirai, J. Thingna, T. Mori, S. Denisov, P. Hänggi, and S. Miyashita, New Journal of Physics 18, 053008 (2016).
 Archak Purkayastha, Madhumita Saha, and Bijay Kumar Agarwalla, "Subdiffusive Phases in Open Clean Long-Range Systems", Physical Review Letters 127 24, 240601 (2021).
 Richard Hartmann and Walter T. Strunz, "Environmentally Induced Entanglement – Anomalous Behavior in the Adiabatic Regime", Quantum 4, 347 (2020).
 Lorenzo Campos Venuti, Domenico D’Alessandro, and Daniel A. Lidar, "Optimal Control for Quantum Optimization of Closed and Open Systems", Physical Review Applied 16 5, 054023 (2021).
 Brecht Donvil and Paolo Muratore-Ginanneschi, "Quantum trajectory framework for general time-local master equations", Nature Communications 13 1, 4140 (2022).
 F. Benatti, D. Chruściński, and R. Floreanini, "Local Generation of Entanglement with Redfield Dynamics", Open Systems & Information Dynamics 29 01, 2250001 (2022).
 Patrick P Potts, Alex Arash Sand Kalaee, and Andreas Wacker, "A thermodynamically consistent Markovian master equation beyond the secular approximation", New Journal of Physics 23 12, 123013 (2021).
 Archak Purkayastha, "Lyapunov equation in open quantum systems and non-Hermitian physics", Physical Review A 105 6, 062204 (2022).
 Massimo Borrelli and Hans Christian Öttinger, "Dissipation in spin chains using quantized nonequilibrium thermodynamics", Physical Review A 106 2, 022220 (2022).
 Devashish Tupkary, Abhishek Dhar, Manas Kulkarni, and Archak Purkayastha, "Fundamental limitations in Lindblad descriptions of systems weakly coupled to baths", Physical Review A 105 3, 032208 (2022).
 Huo Chen and Daniel A. Lidar, "Hamiltonian open quantum system toolkit", Communications Physics 5 1, 112 (2022).
 Marek Winczewski, Antonio Mandarino, Michał Horodecki, and Robert Alicki, "Bypassing the Intermediate Times Dilemma for Open Quantum System", arXiv:2106.05776.
 Marek Winczewski and Robert Alicki, "Renormalization in the Theory of Open Quantum Systems via the Self-Consistency Condition", arXiv:2112.11962.
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