Typical equilibrium state of an embedded quantum system

Abstract

We consider an arbitrary quantum system coupled non perturbatively to a large arbitrary and fully quantum environment. In [G. Ithier and F. Benaych-Georges, Phys. Rev. A 96, 012108 (2017)] the typicality of the dynamics of such an embedded quantum system was established for several classes of random interactions. In other words, the time evolution of its quantum state does not depend on the microscopic details of the interaction. Focusing at the long time regime, we use this property to calculate analytically a new partition function characterizing the stationary state and involving the overlaps between eigenvectors of a bare and a dressed Hamiltonian. This partition function provides a new thermodynamical ensemble which includes the microcanonical and canonical ensembles as particular cases. We check our predictions with numerical simulations.In what state of equilibrium can a quantum system be? Does this state have universal properties and what are the conditions for its emergence? These questions are not new, dating even from the very birth of quantum theory [1] and are surprisingly open [2,3]. Indeed, the foundations of statistical physics still rely today on a static Bayesian point of view assuming the equiprobability of the accessible states defining the microcanonical ensemble. Assuming temperature and chemical potential can be defined then the canonical and grand canonical ensembles can be derived, allowing to calculate all relevant macroscopic quantities in the thermodynamical limit [4][5][6]. In order to link theoretical predictions calculated with averages over these ensembles to experimental quantities measured on a single system, an assumption of ergodicity is made. Despite being broadly accepted, this assumption is not justified in a satisfactory manner (see, e.g., the discussion in Ref. [7]). Triggered by recent progress in the quantum engineering of mesoscopic systems [8,9], some theoretical progress has been achieved for attempting to explain thermodynamical equilibrium with a purely quantum point of view.From the early work of von Neumann on quantum ergodicity [1,10], most theoretical studies aiming at understanding thermalisation as a quantum and universal [11] process have focused on looking for signatures of thermalisation on physical observables of large quantum systems [12][13][14][15], for instance with the Eigenstate Thermalisation Hypothesis (ETH) surmise [16][17][18]. Instead of observables, one can also focus on the state of a system embedded in a larger one for which a "canonical typicality" property has been established: the overwhelming majority of pure quantum states of the composite system are locally [19] canonical [20][21][22]. This static "typicality" has been extended to the dynamics of embedded quantum systems (two-level [23], four-level [24] and arbitrary [25] quantum systems). We apply here this "dynamical typicality" property in order to calculate analytically and with full generality the stationary state of an embedded quantum system at long time. We find a new thermody...

Statistical diagonalization of a random biased Hamiltonian: the case of the eigenvectors

“…Mat. of [8]). E[G n,m (z 1 )G p,q (z 2 )] is zero ∀z 1 , z 2 , except if • (n = m and p = q): these are correlations between the diagonal terms of the resolvent, E[G n,n (z 1 )G p,p (z 2 )].…”

“…Mat. of [8]). This implies that all extra diagonal mean Green functions are zero: E[G n,m (z)] = 0 ∀z for n = m when Ŵ is Gaussian.…”

Dynamical typicality of embedded quantum systems

“…Second they provide the rigorous ground for an averaging procedure over random interactions which can be used for analytical calculations performed with full generality i.e. for arbitrary system, environment, and initial state (see [17] for an application to the equilibrium state of an embedded quantum system). More generally, this work provide a rigorous justification for a new kind of ergodicity partly envisioned in the early work of Wigner and Dyson [18] when modeling entire nuclear Hamiltonians using random matrices.…”

“…As a consequence, it should be noted that this phenomenon provides an approximate way of calculation of s (t) simply by averaging: s (t) = Tr e ( (t)) ≈ E[Tr e ( (t))] = Tr e (E[ (t)]), where E is the average over the set of interaction Hamiltonians considered. This property will be used in [17] in order to calculate analytically the equilibrium state of an embedded quantum system when thermalisation takes place. Finally, it is important to stress that the upper bound in Eq.…”

“…This pattern is a signature of the microscopic structure of the interaction Hamiltonian. The theoretical prediction (black dashed line) is provided by [17] where an averaging procedure over W (justified by measure concentration) allows to calculate analytically the typical stationary state of S: for the set of parameters considered here, one has p0 ∝ ρe( e + ∆) and p1 ∝ ρe( e). It should be noted that remarkably in this case, this prediction matches the one of the microcanonical ensemble.…”