Adiabatic theorem for bipartite quantum systems in weak coupling limit

Abstract

Abstract. We study the adiabatic approximation of the dynamics of a bipartite quantum system with respect to one of the components, when the coupling between its two components is perturbative. We show that the density matrix of the considered component is described by adiabatic transport formulae exhibiting operator-valued geometric and dynamical phases. The present results can be used to study the quantum control of the dynamics of qubits and of open quantum systems where the two components are the system and its environment. We treat two examples, the control of an atomic qubit interacting with another one and the control of a spin in the middle of a Heisenberg spin chain.

Purification of Lindblad dynamics, geometry of mixed states and geometric phases

“…It is moreover the non-adiabatic generalisation of the C * -geometric phase introduced in ref. [19,20,26,37]. We can also show that…”

“…The geometric and dynamical phase generators appearing in the adiabatic theorem ref. [37] are clearly the adiabatic limit of the generators introduced in theorem 3. Let G {Φ bβ } bβ (t) ⊂ U (H S ) be the group of operators of H S acting on the eigenvectors of H U as phase changes.…”

“…The adiabatic geometric phase generator A (1) α found in ref. [37] is the sum of the two geometric phase generators found theorem 3 with the "density eigenmatrix" ρ aα playing the role ofρ(t). Let P •α = b |Φ bα Φ * bα | be the eigenprojection in H S ⊗ H A associated with the eigenvectors of H U related to ξ α by perturbation.…”

“…This excludes the more interesting cases where the relaxation rates are sufficiently large to deviate the dynamics or where the interaction duration is sufficiently large to the dynamics goes to the steady state (with the open quantum systems there is a competition between the adiabatic and the relaxation processes). In second order of perturbation the adiabatic approximation deals with mixed state [37] but the nonlinearity could induces strong difficulties to apply this approach. A more general approach of the adiabatic geometric phases could consists to use the notion of noncommutative eigenvalues as introduced in ref.…”

Almost quantum adiabatic dynamics and generalized time-dependent wave operators

“…The approximate dynamics is then governed by the effective Hamiltonian H ef f = P 0 HP 0 − ı Ṗ 0 P 0 (where P 0 is the orthogonal projector onto the adiabatic active space spaned by the few instantaneous eigenvectors, and the dot denotes the time derivative). ı Ṗ 0 P 0 is associated with the geometric (Berry) phase [7,8]. The conditions of the adiabatic approximation can be very restrictive (the Hamiltonian variations must be slow and a gap condition between the eigenvalues is required).…”