David Viennot

Principal bundle structure of quantum adiabatic dynamics with a Berry phase which does not commute with the dynamical phase

A geometric model is proposed to describe the Berry phase phenomenon when the geometric phase does not commute with the dynamical phase. The structure used is a principal composite bundle in which the adiabatic transport appears as a horizontal lift. The formulation is applied to a simple quantum dynamical system controlled by two lasers.

The role of the geometric phases in adiabatic population tracking for non-Hermitian Hamiltonians

The definition of instantaneous eigenstate populations for a dynamical non- self-adjoint system is not obvious. The nadirect extension of the definition used for the self-adjoint case leads to inconsistencies; the resulting artifacts can induce a false inversion of population or a false adiabaticity. We show that the inconsistency can be avoided by introducing geometric phases in another possible definition of populations. An example is given which demonstrates both the anomalous effects and their removal by our approach.

Non-abelian higher gauge theory and categorical bundle

A gauge theory is associated with a principal bundle endowed with a connection permitting to define horizontal lifts of paths. The horizontal lifts of surfaces cannot be defined into a principal bundle structure. An higher gauge theory is an attempt to generalize the bundle structure in order to describe horizontal lifts of surfaces. A such attempt is particularly difficult for the non-abelian case. Some structures have been proposed to realize this goal (twisted bundle, gerbes with connection, bundle gerbe, 2-bundle). Each of them uses a category in place of the total space manifold of the usual principal bundle structure. Some of them replace also the structure group by a category (more precisely a Lie crossed module viewed as a category). But the base space remains still a simple manifold (possibly viewed as a trivial category with only identity arrows). We propose a new principal categorical bundle structure, with a Lie crossed module as structure groupoid, but with a base space belonging to a bigger class of categories (which includes non-trivial categories), that we called affine 2-spaces. We study the geometric structure of the categorical bundles built on these categories (which are a more complicated structure than the 2-bundles) and the connective structures on these bundles. Finally we treat an example interesting for quantum dynamics which is associated with the Bloch wave operator theory.

Holonomy of a principal composite bundle connection, non-Abelian geometric phases, and gauge theory of gravity

We show that the holonomy of a connection defined on a principal composite bundle is related by a non-Abelian Stokes theorem to the composition of the holonomies associated with the connections of the component bundles of the composite. We apply this formalism to describe the non-Abelian geometric phase (when the geometric phase generator does not commute with the dynamical phase generator). We find then an assumption to obtain a new kind of separation between the dynamical and the geometric phases. We also apply this formalism to the gauge theory of gravity in the presence of a Dirac spinor field in order to decompose the holonomy of the Lorentz connection into holonomies of the linear connection and of the Cartan connection.

The need for a flat higher gauge structure to describe a Berry phase associated with some resonance phenomena

In the presence of a resonance crossing producing splitting of the base manifold (for example, a circle crossing in a plane), we show that the rigorous geometrical structure within which the Berry phase arises may be a 2-bundle (a structure related to gerbes and to category theory) rather than a fiber bundle. The Bloch wave operator plays an important role in the associated theory.

Geometry of quantum active subspaces and of effective Hamiltonians

We propose a geometric formulation of the theory of effective Hamiltonians associated with active spaces. We analyze particularly the case of the time-dependent wave operator theory. This formulation is related to the geometry of the manifold of the active spaces, particularly to its Kahlerian structure. We introduce the concept of quantum distance between active spaces. We show that the time-dependent wave operator theory is, in fact, a gauge theory, and we analyze its relationship with the geometric phase concept.

A new kind of geometric phases in open quantum systems and higher gauge theory

A new approach is proposed, extending the concept of geometric phases to adiabatic open quantum systems described by density matrices (mixed states). This new approach is based on an analogy between open quantum systems and dissipative quantum systems which uses a C*-module structure. The gauge theory associated with these new geometric phases does not employ the usual principal bundle structure but a higher structure, a categorical principal bundle (the so-called principal 2-bundle or non-Abelian bundle gerbe) which is sometimes a non-Abelian twisted bundle. The need to site the gauge theory in this higher structure is a geometrical manifestation of the decoherence induced by the environment on the quantum system.

C*-geometric phase for mixed states: entanglement, decoherence and the spin system

We study a kind of geometric phase for entangled quantum systems, and particularly a spin driven by a magnetic field and entangled with another spin. The new kind of geometric phase is based on an analogy between open quantum systems and dissipative quantum systems which uses a C*-module structure. We show that the system presents, from the viewpoint of their geometric phases, two behaviours. The first one is identical to the behaviour of an isolated spin driven by a magnetic field, as the problem originally treated by Berry. The second one is specific to the decoherence process. The gauge structures induced by these geometric phases are then similar to a magnetic monopole gauge structure for the first case, and can be viewed as a kind of instanton gauge structure for the second case. We study the role of these geometric phases in the evolution of a mixed state, particularly by focusing on the evolution of the density matrix coherence. We investigate also the relation between the geometric phase of the mixed state of one of the entangled systems and the geometric phase of the bipartite system.

The topology of the adiabatic passage process for molecular photodissociative dynamics

We study population transfer processes for photodissociation, as described within the optical potential model, in the context of adiabatic or quasi-adiabatic dynamics. By a reformulation of the adiabatic passage theory in terms of the fibre bundle topology, we extend the analysis made by Yatsenko et al (2002 Phys. Rev. A 65 043407) from conservative to dissipative systems. We show that this topology is associated with the problem of eigenvalue labelling and hence with the continuous eigenvalue following process in the control parameter space. Problems involving adiabatic passage, the direct chirping process and non-adiabatic transitions are studied, with particular regard to the presence of resonance states. We also discuss the role played by the molecular continua.

Quantum chimera states

Abstract We study a theoretical model of closed quasi-hermitian chain of spins which exhibits quantum analogues of chimera states, i.e. long life classical states for which a part of an oscillator chain presents an ordered dynamics whereas another part presents a disordered dynamics. For the quantum analogue, the chimera behaviour deals with the entanglement between the spins of the chain. We discuss the entanglement properties, quantum chaos, quantum disorder and semi-classical similarity of our quantum chimera system. The quantum chimera concept is novel and induces new perspectives concerning the entanglement of multipartite systems.