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

Dragomir Davidović

Georgia Institute of Technology, Atlanta, Georgia, United States

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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.

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.

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