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Markovian master equations for quantum thermal machines: local versus global approach

Hofer, P., Martí Perarnau-Llobet, L David M Miranda, Haack, G., Ralph Silva, Brask, J., Brunner, N.

Published Jan 1, 2017• 206 citations
OpenAlex

Abstract

The study of quantum thermal machines, and more generally of open quantum systems, often relies on master equations.Two approaches are mainly followed.On the one hand, there is the widely used, but often criticized, local approach, where machine sub-systems locally couple to thermal baths.On the other hand, in the more established global approach, thermal baths couple to global degrees of freedom of the machine.There has been debate as to which of these two conceptually different approaches should be used in situations out of thermal equilibrium.Here we compare the local and global approaches against an exact solution for a particular class of thermal machines.We consider thermodynamically relevant observables, such as heat currents, as well as the quantum state of the machine.Our results show that the use of a local master equation is generally well justified.In particular, for weak inter-system coupling, the local approach agrees with the exact solution, whereas the global approach fails for non-equilibrium situations.For intermediate coupling, the local and the global approach both agree with the exact solution and for strong coupling, the global approach is preferable.These results are backed by detailed derivations of the regimes of validity for the respective approaches.

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The study of quantum thermal machines, and more generally of open quantum systems, often relies on master equations.Two approaches are mainly followed.On the one hand, there is the widely used, but often criticized, local approach, where machine sub-systems locally couple to thermal baths.On the other hand, in the more established global approach, thermal baths couple to global degrees of freedom of the machine.There has been debate as to which of these two conceptually different approaches should be used in situations out of thermal equilibrium.Here we compare the local and global approaches against an exact solution for a particular class of thermal machines.We consider thermodynamically relevant observables, such as heat currents, as well as the quantum state of the machine.Our results show that the use of a local master equation is generally well justified.In particular, for weak inter-system coupling, the local approach agrees with the exact solution, whereas the global approach fails for non-equilibrium situations.For intermediate coupling, the local and the global approach both agree with the exact solution and for strong coupling, the global approach is preferable.These results are backed by detailed derivations of the regimes of validity for the respective approaches.
The study of quantum thermal machines, and more generally of open quantum systems, often relies on master equations. Two approaches are mainly followed. On the one hand, there is the widely used, but often criticized, local approach, where machine sub-systems locally couple to thermal baths. On the other hand, in the more established global approach, thermal baths couple to global degrees of freedom of the machine. There has been debate as to which of these two conceptually different approaches should be used in situations out of thermal equilibrium. Here we compare the local and global approaches against an exact solution for a particular class of thermal machines. We consider thermodynamically relevant observables, such as heat currents, as well as the quantum state of the machine. Our results show that the use of a local master equation is generally well justified. In particular, for weak inter-system coupling, the local approach agrees with the exact solution, whereas the global approach fails for non-equilibrium situations. For intermediate coupling, the local and the global approach both agree with the exact solution and for strong coupling, the global approach is preferable. These results are backed by detailed derivations of the regimes of validity for the respective approaches.
We investigate the dynamics of a qubit chain locally coupled to a thermal reservoir, modeled through repeated collisions with particles drawn from a heat bath. Under suitable conditions, the resulting Lindblad equation is thermodynamically consistent -- it drives the system toward thermal equilibrium -- while remaining local at short times. This framework reveals a crossover between the global Lindblad equation derived from the secular approximation in weak-coupling theory and the local dissipative models often employed in the literature, which generally fail to ensure thermodynamic consistency.
Abstract Open systems of coupled qubits are ubiquitous in quantum physics. Finding a suitable master equation to describe their dynamics is therefore a crucial task that must be addressed with utmost attention. In the recent past, many efforts have been made toward the possibility of employing local master equations, which compute the interaction with the environment neglecting the direct coupling between the qubits, and for this reason may be easier to solve. Here, we provide a detailed derivation of the Markovian master equation for two coupled qubits interacting with common and separate baths, considering pure dephasing as well as dissipation. Then, we explore the differences between the local and global master equation, showing that they intrinsically depend on the way we apply the secular approximation. Our results prove that the global approach with partial secular approximation always provides the most accurate choice for the master equation when Born–Markov approximations hold, even for small inter-system coupling constants. Using different master equations we compute the stationary heat current between two separate baths, the entanglement dynamics generated by a common bath, and the emergence of spontaneous synchronization, showing the importance of the accurate choice of approach.
Advances in experimental techniques have opened new routes for harnessing non-equilibrium dynamics in mesoscopic quantum systems. In this context, we study the impact of on-site interactions on the transport properties of a continuous quantum thermal machine composed of two coupled oscillators connected to two thermal reservoirs. In the weak system-reservoir coupling regime, where a long-standing debate concerns which reduced description should be preferred, we first show that the Redfield master equation (RME) provides an accurate and unifying framework that interpolates between two well-known limits: the \textit{local} and \textit{global} master equations. By relying on the Hierarchy of Pure States (HOPS), a numerically exact stochastic method, we then explore the full parameter space and show that interactions can be leveraged to tune the efficiency of the thermal machine at high temperatures (while leaving it essentially unchanged at low temperatures), induce non-reciprocal transport under asymmetric reservoir couplings, and generate steady-state entanglement within the junction. We derive expressions for system-bath correlators, such as heat and particle currents, consistently across different frameworks. Our work features on-site interactions to enhance the versatility of quantum thermodynamic junctions and clarifies the role of non-Markovianity and non-linearities in quantum transport.
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.
Identifying which master equation is preferable for the description of a multipartite open quantum system is not trivial and has led in recent years to the local vs global debate in the context of Markovian dissipation. We treat here a paradigmatic scenario in which the system is composed of two interacting harmonic oscillators A and B, with only A interacting with a thermal bath---collection of other harmonic oscillators---and we study the equilibration process of the system initially in the ground state with the bath finite temperature. We show that the completely positive version of the Redfield equation obtained using coarse-grain and an appropriate time-dependent convex mixture of the local and global solutions give rise to the most accurate approximations of the whole exact system dynamics, i.e., both at short and at long timescales, outperforming the local and global approaches.