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

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.

Authors:
ORCiD logo [1]
  1. Georgia Inst. of Technology, Atlanta, GA (United States)
Publication Date:
Research Org.:
Georgia Inst. of Technology, Atlanta, GA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division; Georgia Tech Quantum Alliance (GTQA)
OSTI Identifier:
1666137
Grant/Contract Number:  
FG02-06ER46281
Resource Type:
Accepted Manuscript
Journal Name:
Quantum
Additional Journal Information:
Journal Volume: 4; Journal ID: ISSN 2521-327X
Publisher:
Quantum Science Open Community
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Quantum Dynamics; Energy Transfer

Citation Formats

Davidović, Dragomir. Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation. United States: N. p., 2020. Web. doi:10.22331/q-2020-09-21-326.
Davidović, Dragomir. Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation. United States. doi:10.22331/q-2020-09-21-326.
Davidović, Dragomir. Mon . "Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation". United States. doi:10.22331/q-2020-09-21-326. https://www.osti.gov/servlets/purl/1666137.
@article{osti_1666137,
title = {Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation},
author = {Davidović, Dragomir},
abstractNote = {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.},
doi = {10.22331/q-2020-09-21-326},
journal = {Quantum},
number = ,
volume = 4,
place = {United States},
year = {2020},
month = {9}
}

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Works referenced in this record:

Completely positive dynamical semigroups of N-level systems
journal, January 1976

  • Gorini, Vittorio
  • Journal of Mathematical Physics, Vol. 17, Issue 5
  • DOI: 10.1063/1.522979

From completely positive maps to the quantum Markovian semigroup master equation
journal, June 2001


Phase space approach to theories of quantum dissipation
journal, October 1997

  • Kohen, D.; Marston, C. C.; Tannor, D. J.
  • The Journal of Chemical Physics, Vol. 107, Issue 13
  • DOI: 10.1063/1.474887

Ensemble Method in the Theory of Irreversibility
journal, November 1960

  • Zwanzig, Robert
  • The Journal of Chemical Physics, Vol. 33, Issue 5
  • DOI: 10.1063/1.1731409

On Quantum Theory of Transport Phenomena: Steady Diffusion
journal, December 1958

  • Nakajima, Sadao
  • Progress of Theoretical Physics, Vol. 20, Issue 6
  • DOI: 10.1143/PTP.20.948

On the Theory of Relaxation Processes
journal, January 1957

  • Redfield, A. G.
  • IBM Journal of Research and Development, Vol. 1, Issue 1
  • DOI: 10.1147/rd.11.0019

Computable measure of entanglement
journal, February 2002


Efficiency of different numerical methods for solving Redfield equations
journal, January 2001

  • Kondov, Ivan; Kleinekathöfer, Ulrich; Schreiber, Michael
  • The Journal of Chemical Physics, Vol. 114, Issue 4
  • DOI: 10.1063/1.1335656

Modeling of ultrafast electron-transfer processes: Validity of multilevel Redfield theory
journal, August 2003

  • Egorova, Dassia; Thoss, Michael; Domcke, Wolfgang
  • The Journal of Chemical Physics, Vol. 119, Issue 5
  • DOI: 10.1063/1.1587121

A time-dependent modified Redfield theory for absorption spectra applied to light-harvesting systems
journal, July 2007

  • Schröder, Markus; Schreiber, Michael; Kleinekathöfer, Ulrich
  • Journal of Luminescence, Vol. 125, Issue 1-2
  • DOI: 10.1016/j.jlumin.2006.08.086

Extending the applicability of Redfield theories into highly non-Markovian regimes
journal, November 2015

  • Montoya-Castillo, Andrés; Berkelbach, Timothy C.; Reichman, David R.
  • The Journal of Chemical Physics, Vol. 143, Issue 19
  • DOI: 10.1063/1.4935443

Tunneling through molecules and quantum dots: Master-equation approaches
journal, May 2008


Bloch-Redfield equations for modeling light-harvesting complexes
journal, February 2015

  • Jeske, Jan; Ing, David J.; Plenio, Martin B.
  • The Journal of Chemical Physics, Vol. 142, Issue 6
  • DOI: 10.1063/1.4907370

Molecular model of J-aggregated pseudoisocyanine fibers
journal, July 2018

  • Bricker, William P.; Banal, James L.; Stone, Matthew B.
  • The Journal of Chemical Physics, Vol. 149, Issue 2
  • DOI: 10.1063/1.5036656

Staying positive: going beyond Lindblad with perturbative master equations
journal, April 2008


Accuracy assessment of perturbative master equations: Embracing nonpositivity
journal, January 2020


Non-Markovian quantum-state diffusion: Perturbation approach
journal, July 1999


Non-Markovian stochastic Schrödinger equations in different temperature regimes: A study of the spin-boson model
journal, March 2005

  • de Vega, Inés; Alonso, Daniel; Gaspard, Pierre
  • The Journal of Chemical Physics, Vol. 122, Issue 12
  • DOI: 10.1063/1.1867377

Tensor propagator for iterative quantum time evolution of reduced density matrices. I. Theory
journal, March 1995

  • Makri, Nancy; Makarov, Dmitrii E.
  • The Journal of Chemical Physics, Vol. 102, Issue 11
  • DOI: 10.1063/1.469508

Dynamics of the spin-boson model with a structured environment
journal, January 2004


Ultraslow quantum dynamics in a sub-Ohmic heat bath
journal, February 2010


Two-Time Correlation Functions of a System Coupled to a Heat Bath with a Gaussian-Markoffian Interaction
journal, April 1989

  • Tanimura, Yoshitaka; Kubo, Ryogo
  • Journal of the Physical Society of Japan, Vol. 58, Issue 4
  • DOI: 10.1143/JPSJ.58.1199

Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems
journal, August 2006


Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities
journal, July 2014

  • Tanimura, Yoshitaka
  • The Journal of Chemical Physics, Vol. 141, Issue 4
  • DOI: 10.1063/1.4890441

Time-dependent transport through quantum-impurity systems with Kondo resonance
journal, March 2015


The multi-configurational time-dependent Hartree approach
journal, January 1990


Multilayer formulation of the multiconfiguration time-dependent Hartree theory
journal, July 2003

  • Wang, Haobin; Thoss, Michael
  • The Journal of Chemical Physics, Vol. 119, Issue 3
  • DOI: 10.1063/1.1580111

Ultrafast Nonadiabatic Dynamics of Singlet Fission: Quantum Dynamics with the Multilayer Multiconfigurational Time-Dependent Hartree (ML-MCTDH) Method
journal, January 2016

  • Zheng, Jie; Xie, Yu; Jiang, Shengshi
  • The Journal of Physical Chemistry C, Vol. 120, Issue 3
  • DOI: 10.1021/acs.jpcc.5b09921

Hierarchy of Stochastic Pure States for Open Quantum System Dynamics
journal, October 2014


Non-Perturbative Calculation of Two-Dimensional Spectra Using the Stochastic Hierarchy of Pure States
journal, October 2016


Exact Open Quantum System Dynamics Using the Hierarchy of Pure States (HOPS)
journal, November 2017

  • Hartmann, Richard; Strunz, Walter T.
  • Journal of Chemical Theory and Computation, Vol. 13, Issue 12
  • DOI: 10.1021/acs.jctc.7b00751

Efficient non-Markovian quantum dynamics using time-evolving matrix product operators
journal, August 2018


Tensor network simulation of multi-environmental open quantum dynamics via machine learning and entanglement renormalisation
journal, March 2019

  • Schröder, Florian A. Y. N.; Turban, David H. P.; Musser, Andrew J.
  • Nature Communications, Vol. 10, Issue 1
  • DOI: 10.1038/s41467-019-09039-7

Theory of coherent resonance energy transfer for coherent initial condition
journal, October 2009

  • Jang, Seogjoo
  • The Journal of Chemical Physics, Vol. 131, Issue 16
  • DOI: 10.1063/1.3247899

A general approach to quantum dynamics using a variational master equation: Application to phonon-damped Rabi rotations in quantum dots
journal, August 2011


Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations
journal, July 2013


Open-quantum-system dynamics: Recovering positivity of the Redfield equation via the partial secular approximation
journal, July 2019


Completely positive master equation for arbitrary driving and small level spacing
journal, February 2020


Partial secular Bloch-Redfield master equation for incoherent excitation of multilevel quantum systems
journal, March 2015

  • Tscherbul, Timur V.; Brumer, Paul
  • The Journal of Chemical Physics, Vol. 142, Issue 10
  • DOI: 10.1063/1.4908130

Preservation of positivity by dynamical coarse graining
journal, August 2008


Entangling two unequal atoms through a common bath
journal, January 2010


Non-rotating-wave master equation
journal, April 1996


Positivity preserving non-Markovian master equations
journal, December 2000


Lindblad equations for strongly coupled populations and coherences in photosynthetic complexes
journal, May 2009

  • Palmieri, Benoit; Abramavicius, Darius; Mukamel, Shaul
  • The Journal of Chemical Physics, Vol. 130, Issue 20
  • DOI: 10.1063/1.3142485

Phenomenological position and energy resolving Lindblad approach to quantum kinetics
journal, January 2018


Universal Lindblad equation for open quantum systems
journal, September 2020


Statistical mechanics of Floquet systems: The pervasive problem of near degeneracies
journal, May 2009


Quantum adiabatic Markovian master equations
journal, December 2012


Introduction to quantum noise, measurement, and amplification
journal, April 2010


Quantum trajectories and open many-body quantum systems
journal, March 2014


Environment-induced entanglement in a refined weak-coupling limit
journal, October 2009


Refined weak-coupling limit: Coherence, entanglement, and non-Markovianity
journal, April 2017


Environment Induced Entanglement in Markovian Dissipative Dynamics
journal, August 2003


Entangling two atoms via spontaneous emission
journal, March 2004


A Mechanism of Magnetic Hysteresis in Heterogeneous Alloys
journal, May 1948

  • Stoner, E. C.; Wohlfarth, E. P.
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 240, Issue 826
  • DOI: 10.1098/rsta.1948.0007

Thermal Fluctuations of a Single-Domain Particle
journal, June 1963


Identification of the Dominant Precession-Damping Mechanism in Fe, Co, and Ni by First-Principles Calculations
journal, July 2007


Adiabatic drag and initial slip in random processes
journal, December 1983


Slippage of initial conditions for the Redfield master equation
journal, October 1999

  • Gaspard, P.; Nagaoka, M.
  • The Journal of Chemical Physics, Vol. 111, Issue 13
  • DOI: 10.1063/1.479867

Markovian Approximation in the Relaxation of Open Quantum Systems
journal, November 2005

  • Cheng, Y. C.; Silbey, R. J.
  • The Journal of Physical Chemistry B, Vol. 109, Issue 45
  • DOI: 10.1021/jp051303o

Memory effects in the relaxation of quantum open systems
journal, October 1992

  • Suárez, Alberto; Silbey, Robert; Oppenheim, Irwin
  • The Journal of Chemical Physics, Vol. 97, Issue 7
  • DOI: 10.1063/1.463831

Post-Markov master equation for the dynamics of open quantum systems
journal, February 2000


Variational calculation of the dynamics of a two level system interacting with a bath
journal, March 1984

  • Silbey, Robert; Harris, Robert A.
  • The Journal of Chemical Physics, Vol. 80, Issue 6
  • DOI: 10.1063/1.447055

Kaleidoscope of quantum phases in a long-range interacting spin-1 chain
journal, May 2016


Tensor Network Renormalization
journal, October 2015


Driven quantum tunneling
journal, October 1998


Effective Floquet–Gibbs states for dissipative quantum systems
journal, May 2016


Single-shot read-out of an individual electron spin in a quantum dot
journal, July 2004

  • Elzerman, J. M.; Hanson, R.; Willems van Beveren, L. H.
  • Nature, Vol. 430, Issue 6998
  • DOI: 10.1038/nature02693

Single-shot readout of an electron spin in silicon
journal, September 2010

  • Morello, Andrea; Pla, Jarryd J.; Zwanenburg, Floris A.
  • Nature, Vol. 467, Issue 7316
  • DOI: 10.1038/nature09392