Quantum Nonequilibrium Steady States Induced by Repeated Interactions

Abstract

We study the steady state of a finite XX chain coupled at its boundaries to quantum reservoirs made of free spins that interact one after the other with the chain. The two-point correlations are calculated exactly and it is shown that the steady state is completely characterized by the magnetization profile and the associated current. Except at the boundary sites, the magnetization is given by the average of the reservoirs' magnetizations. The steady state current, proportional to the difference in the reservoirs' magnetizations, shows a non-monotonous behavior with respect to the system-reservoir coupling strength, with an optimal current state for a finite value of the coupling. Moreover, we show that the steady state can be described by a generalized Gibbs state.Understanding non-equilibrium behavior of quantum systems on the basis of general principles is one of the more challenging prospects of statistical physics. In particular, the complete characterization of the so-called Quantum Non-Equilibrium Steady-States (QNESS), i. e. stationary current full quantum states, is of primer and central focus since they are possible candidates for playing a role similar to Gibbs states in constructing a nonequilibrium statistical mechanics [1,2,3,4,5,6,7,8].To elucidate the general guiding principles for a nonequilibrium statistical mechanics, exactly solvable models play a central role. Among many models, the XX quantum chain is one of the simplest non-trivial manybody system. Its N -sites Hamiltonian is given bywhere the σs are the usual Pauli matrices, J is the exchange coupling and h a transverse (possibly external) magnetic field. Due to the fact that its dynamics can be described in an explicit way, the one-dimensional XX model has been extensively studied in various nonequilibrium contexts [9]. On the experimental side, the most promising perspectives for these studies come from the ultra-cold atoms community since the XX model (1) Antal et al.[13] studied the ground state of the Ising and XX chain Hamiltonian with the addition of a magnetization or energy current J via a Lagrange multiplier. The ground state of the effective Hamiltonian H S − λJ was interpreted as a non-equilibrium stationary current full state. Such an effective Hamiltonian was supposed to capture locally the essential features of a finite chain coupled at its boundary sites to quantum reservoirs. Soon after, Ogata [14], Aschbacher and Pillet [15] considered the anisotropic XY steady state induced by the unitary dynamics, U (t) = e −itHS , starting with an initial state in which the left and right halves are set at inverse temperature β L and β R . They showed that the QNESS can be effectively described by a generalized Gibbs state ∼ e −βHS +δY whereβ = (β L +β R )/2 is the average inverse temperature, δ = (β L − β R )/2 is a driving force coupled to a long-range operator Y commuting with H S . The operator Y is given by Y = ∞ l=1 µ l Y l where Y l are currents operators associated to lth sites conserved quantities and where the coefficients ...

Quantum transport efficiency and Fourier's law

“…where σ z k , σ + k , and σ − k are the Pauli-z, raising, and lowering operators in the basis of ground and excited states of the kth two-level system, respectively, with on-site energy ω and coupling strength g. Similar simple models of coupled effective two-level systems are used in recent analyses of energy transfer in photosynthetic complexes [10][11][12] and spin transport in models for magnetism [16][17][18]. The influence of the two heat baths is modeled by incoherently coupling each of the terminal sites to a bosonic heat bath described by a master equation of Lindblad form.…”

A matrix product solution for a nonequilibrium steady state of an XX chain

“…An explicit solution for the first two orders in the driving can be also obtained as a particular case of an analytic solution for the XX chain with dephasing [14]. Slightly different model resulting in the same solution, in which XX chain repeatedly interacts with independent spins of the bath, has been explicitly solved in [15]. For studies of nonequilibrium states in a doubly infinite chain see [17,16,18].…”

Quantum quench from a thermal tensor state: Boundary effects and generalized Gibbs ensemble

“…In other words, such infinitely extended system will always have, at each instant of time, two infinite portions (the far left and the far right playing the role of reservoirs) that remain equilibrated at the two initial temperatures. Between these two reservoirs, the (finite portion of the) system will show nonlocal properties reflecting the non-local structure of the non-equilibrium steady state (which is widely believed conjecture) [49][50][51][52][53][54][55][56] . Recently, by means of thermodynamic Bethe anstaz 57 , the energy current in non-equilibrium steady states of integrable models of relativistic quantum field theory has been evaluated 58 and interestingly, seems to be in disagreement with what has been numerically found in Ref.…”