Testing the Validity of the ‘Local’ and ‘Global’ GKLS Master Equations on an Exactly Solvable Model

Abstract

Abstract. When deriving a master equation for a multipartite weakly-interacting open quantum systems, dissipation is often addressed locally on each component, i.e. ignoring the coherent couplings, which are later added 'by hand'. Although simple, the resulting local master equation (LME) is known to be thermodynamically inconsistent. Otherwise, one may always obtain a consistent global master equation (GME) by working on the energy basis of the full interacting Hamiltonian. Here, we consider a two-node 'quantum wire' connected to two heat baths. The stationary solution of the LME and GME are obtained and benchmarked against the exact result. Importantly, in our model, the validity of the GME is constrained by the underlying secular approximation. Whenever this breaks down (for resonant weakly-coupled nodes), we observe that the LME, in spite of being thermodynamically flawed: (a) predicts the correct steady state, (b) yields with the exact asymptotic heat currents, and (c) reliably reflects the correlations between the nodes. In contrast, the GME fails at all three tasks. Nonetheless, as the inter-node coupling grows, the LME breaks down whilst the GME becomes correct. Hence, the global and local approach may be viewed as complementary tools, best suited to different parameter regimes.

Quantum current in dissipative systems

“…We conclude this subsection by emphasizing that caution must be maintained when using the global ME for computing the heat flow for small K. As discussed above, the secular approximation, indispensable for the applicability of the global ME, is compromised when K approaches 0. Given the results in [48,49] for the Caldeira-Leggett model, it is reasonable to expect that the local ME would provide a more reliable description in that regime. This issue can be decisively settled only upon solving the global, system-plus-baths dynamics in the limit of infinitely large baths, doing which, however, appears to be unfeasible [28].…”

“…This is typically expressed by the thermal state not being a steady-state solution of the local ME when the temperatures of the baths are equal. This brings about thermodynamically inconsistent behavior [45][46][47][48][49] such as non-zero particle or heat flow between two thermal baths at equal temperatures [46], or, when the temperatures are different, spontaneous heat flow against the temperature gradient [47]. Figure 1.…”

“…Given that there is no interaction between the particles when K=1, it is obvious that it is the description provided by the global ME that fails. Indeed, as K decreases, so do also some of the gaps in the spectrum of H ab , leading to the breakdown of the secular approximation [28,48]. In fact, in the weak coupling regime of some models [48,49], the local ME leads to a steady state closer to that obtained via an exact solution of the system as a whole, including the baths.…”

Non-additive dissipation in open quantum networks out of equilibrium

“…showing that local additive Lindblad models may perform well in non-equilibrium scenarios when the network nodes are nearly degenerate [52,53], but fail for large detunings [20].…”

“…In this appendix, we justify the EPA (53) and prove that the error is of order O G W ( ). The integral (24), with j j w ( )given by equation (35), can be estimated with high accuracy for G W  by analytically continuing the integrand into the complex ω plane.…”