Repeated interactions in open quantum systems

Abstract

Analyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the dynamics of quantum coherences (decoherence) and quantum correlations (entanglement), or the emergence of heat and particle fluxes in non-equilibrium situations. From the mathematical physics perspective, one of the main challenges is to derive the irreversible dynamics of the open system, starting from a unitary dynamics of the system and its environment. The repeated interactions systems considered in these notes are models of non-equilibrium quantum statistical mechanics. They are relevant in quantum optics, and more generally, serve as a relatively well treatable approximation of a more difficult quantum dynamics. In particular, the repeated interaction models allow to determine the large time (stationary) asymptotics of quantum systems out of equilibrium.Comment: Lecture notes of the summer school "Non-equilibrium statistical mechanics" (Montreal, July 2011

Quantum master equations for entangled qubit environments

“…We derive dynamical maps and master equations within the structure of repeated quantum interactions [12,14,24,26]. In this formalism, a quantum system in Hilbert space H S couples to an environment which is comprised of a stream of identical and independent quantum systems such that H E ≡ l H (l) E .…”

Collision models in quantum optics

“…Along this line, an emerging approach is to use quantum collision models (CMs) or, better to say, NM generalisations of CMs . The basic version of a CM [29][30][31][32][33][34][35] considers a system S in contact with a bath B, the latter being made up of a large number of smaller non-interacting particles or "an-cillas". The dynamics proceeds through successive pairwise "collisions" between S and the bath ancillas, each collision being typically modeled as a unitary operation on S and the involved ancilla.…”

Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

“…Consider a continuously parameterized family of channels A (s) with s ∈ [0, 1] and with no rotating points. Instead of thinking of these as repeated applications of random channels [83,84], we assume that A (s) is smoothly varying. Starting with an initially fixed-point state of ρ ∞ = A (0) (ρ ∞ ), we determine how this state evolves under the map…”