arXiv subjects

David Viennot

Publications and source records attributed to David Viennot.

At least 19 recordsLinked to original sources

arXiv2025

Quasicoherent states of noncommutative D2-branes, Aharonov-Bohm effect and quantum Mobius strip

We find an analytical formula for the quasicoherent states of 3D fuzzy spaces defined by algebras generated by bosonic creation and annihilation operators. This one is expressed in a representation onto the coherent states of the CCR algebra. Such a fuzzy space can be assimilated to a noncommutative D2-brane of the M-theory (but also as a model of a qubit in contact with a bosonic environment). We apply this formula onto a D2-brane wrapped around an axis to obtain the geometry of a noncommutative cylinder. We show that the adiabatic transport of its quasicoherent states exhibits a topological effect similar to the Aharonov-Bohm effect. We study also a D2-brane wrapped and twisted to have the geometry of a noncommutative Mobius strip. Finally we briefly present the other two examples of a noncommutative torus and of a noncommutative Klein bottle.

math-ph↗

arXiv2025

Ergodic and synthetic Koopman analyses of cat maps onto classical 2-tori

We study classical continuous automorphisms of the torus (cat maps) from the viewpoint of the Koopman theory. We find analytical formulae for Koopman modes defined coherently on the whole of the torus, and their decompositions associated with the partition of the torus into ergodic components. The spectrum of the Koopman operator is studied in four cases of cat maps: cyclic, quasi-cyclic, critical (transition from quasi-cyclic to chaotic behaviour) and chaotic. The synthetic spectrum associated with the ergodic decomposition is also studied.

nlin.CD↗

arXiv2023

Metrics and geodesics on fuzzy spaces

We study the fuzzy spaces (as special examples of noncommutative manifolds) with their quasicoherent states in order to find their pertinent metrics. We show that they are naturally endowed with two natural "quantum metrics" which are associated with quantum fluctuations of "paths". The first one provides the length the mean path whereas the second one provides the average length of the fluctuated paths. Onto the classical manifold associated with the quasicoherent state (manifold of the mean values of the coordinate observables in the state minimising their quantum uncertainties) these two metrics provides two minimising geodesic equations. Moreover, fuzzy spaces being not torsion free, we have also two different autoparallel geodesic equations associated with two different adiabatic regimes in the move of a probe onto the fuzzy space. We apply these mathematical results to quantum gravity in BFSS matrix models, and to the quantum information theory of a controlled qubit submitted to noises of a large quantum environment.

math-ph↗

arXiv2023

Geometric phases, Everett's many-worlds interpretation of quantum mechanics, and wormholes

We present how the formalism of geometric phases in adiabatic quantum dynamics provides geometric realisations permitting to ``embody'' the Everett's many-worlds interpretation of quantum mechanics, including interferences between the worlds needed for the probability changes and the decoherence processes needed to solve the preferred basis problem. We show also that this geometric realisation is intimately related to quantum gravity (especially to matrix models), showing that the many-world interpretation can be consistent with quantum gravity. The concept of wormhole borrowed to general relativity is central in this geometric realisation. It appears not only as an image by analogy to help the interpretations, but also as a true physical model of quantum wormhole in quantum gravity, the two ones being consistent which each other.

quant-ph↗

arXiv2022

Consistency between adiabatic and nonadiabatic geometric phases for nonselfadjoint hamiltonians

We show that the adiabatic approximation for nonselfadjoint hamiltonians seems to induce two non-equal expressions for the geometric phase. The first one is related to the spectral projector involved in the adiabatic theorem, the other one is the adiabatic limit of the nonadiabatic geometric phase. This apparent inconsistency is resolved by observing that the difference between the two expressions is compensated by a small deviation in the dynamical phases.

quant-ph↗

arXiv2022

Competition between decoherence and purification: quaternionic representation and quaternionic fractals

We consider the competition between decoherence processes and an iterated quantum purification protocol. We show that this competition can be modelled by a nonlinear map onto the quaternion space. This nonlinear map has complicated behaviours, inducing a fractal border between the area of the quantum states dominated by the effects of the purification and the area of the quantum states dominated by the effects of the decoherence. The states on the border are unstable. The embedding in a 3D space of this border is like a quaternionic Julia set or a Mandelbulb with a fractal inner structure.

quant-ph↗

arXiv2021

Fuzzy Schwarzschild (2+1)-spacetime

We present a toy model of fuzzy Schwarzschild space slice (as a noncommutative manifold) which quantum mean values and quantum quasi-coherent states (states minimizing the quantum uncertainties) have properties close to the classical slice of the $(r,θ)$ Schwarzschild coordinates (the so-called Flamm's paraboloid). This fuzzy Schwarzschild slice is built as a deformation of the noncommutative plane. Quantum time observables are introduced to add a time quantization in the model. We study the structure of the quasi-coherent state of the fuzzy Schwarzschild slice with respect to the quasi-coherent state and the deformation states of the noncommutative plane . The quantum dynamics of a fermion interacting with a fuzzy black hole described by the present model is studied. In particular we study the decoherence effects appearing in the neighborhood of the fuzzy event horizon. An extension of the model to describe a quantum wormhole is also proposed, where we show that fermions cross the wormhole not by traveling by its internal space but by quantum tunneling, in accordance with the non-traversable character of classical Einstein-Rosen bridges.

gr-qc↗

arXiv2021

Emergent gravity and D-brane adiabatic dynamics: emergent Lorentz connection

We explore emergent geometry of the spacetime at the microscopic scale by adiabatic transport of a quasi-coherent state of a fermionic string, with quantum spacetime described by the matrix theory (BFSS matrix model). We show that the generator of the Berry phase is the shift vector of the spacetime foliation by spacelike surfaces associated with the quasi-coherent state. The operator-valued generator of the geometric phase of weak adiabatic transport is the Lorentz connection of the emergent geometry which is not torsion free at the microscopic scale. The effects of the torsion seem consistent with the usual interpretation of the Berry curvature as a pseudo magnetic field.

gr-qc↗

arXiv2021

Effective Hamiltonians for almost-periodically driven quantum systems

We present an effective Hamiltonian theory available for some quasi-periodically driven quantum systems which does not need the knowledge of the Fourier frequencies of the control signal. It could also be available for some chaotically driven quantum systems. It is based on the Koopman approach which generalizes the Floquet approach used with periodically driven systems. We show the properties of the quasi-energy states (eigenvectors of the effective Hamiltonian) as quasi-recurrent states of the quantum system.

quant-ph↗

arXiv2019

Adiabatic Elimination and Sub-space Evolution of Open Quantum Systems

Efficient descriptions of open quantum systems can be obtained by performing an adiabatic elimination of the fast degrees of freedom and formulating effective operators for the slow degrees of freedom in reduced dimensions. Here, we perform the construction of effective operators in frequency space, and using the final value theorem or alternatively the Keldysh theorem, we provide a correction for the trace of the density matrix which takes into account the non trace-preserving character of the evolution. We illustrate our results with two different systems, ones where the eliminated fast subspace is constituted by a continuous set of states and ones with discrete states. Furthermore, we show that the two models converge for very large dissipation and at coherent population trapping points. Our results also provide an intuitive picture of the correction to the trace of the density matrix as a detailed balance equation.

quant-ph↗

arXiv2018

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

We propose a nonlinear Schrödinger equation in a Hilbert space enlarged with an ancilla such that the partial trace of its solution obeys to the Lindblad equation of an open quantum system. The dynamics involved by this nonlinear Schrödinger equation constitutes then a purification of the Lindbladian dynamics. This nonlinear equation is compared with other Schrödinger like equations appearing in the theory of open systems. We study the (non adiabatic) geometric phases involved by this purification and show that our theory unifies several definitions of geometric phases for open systems which have been previously proposed. We study the geometry involved by this purification and show that it is a complicated geometric structure related to an higher gauge theory, i.e. a categorical bibundle with a connective structure.

math-ph↗

arXiv2018

Chaos, decoherence and emergent extradimensions in D-brane dynamics with fluctuations

We study, by using tools of the dynamical system theory, a fermionic string streched from a non-commutative D2-brane (stack of D0-branes in the BFSS model) to a probe D0-brane as a quantum system driven by a chaotic system, the (classical and quantum) chaos being induced in the D2-brane dynamics by quantum fluctuations. We show that this dynamics with fluctuations induces a decoherence phenomenon on the reduced density matrix of the fermionic string which is characteristic of the chaotic behaviour since it presents an horizon of coherence. Moreover we show that this dynamics is associated with an invariant torus which involves extradimensions emerging from the fluctuations for the viewpoint of the fermionic string, extending a three dimensional space by six compact dimensions. The situation studied can be considered as a model of qubit (supported by the fermionic string) in interaction with a quantum black hole (modelized by the non-commutative D2-brane).

hep-th↗

arXiv2018

Schrödinger-Koopman quasienergy states of quantum systems driven by a classical flow

We study the properties of the quasienergy states of a quantum system driven by a classical dynamical system. The quasienergies are defined in a same manner as in light-matter interaction but where the Floquet approach is generalized by the use of the Koopman approach of dynamical systems. We show how the properties of the classical flow (fixed and cyclic points, ergodicity, chaos) influence the driven quantum system. This approach of the Schrödinger-Koopman quasienergies can be applied to quantum control, quantum information in presence of noises, and dynamics of mixed classical-quantum systems. We treat the example of a kicked spin ensemble where the kick modulation is governed by discrete classical flows as the Arnold's cat map and the Chirikov standard map.

math-ph↗

arXiv2017

Exotic states in the strong field control of H$_2^+$ dissociation dynamics: From exceptional points to zero-width resonances

H$_2^+$ is an ideal candidate for a detailed study of strong field coherent control strategies inspired by basic mechanisms referring to some specific photodissociation resonances. Two of them are considered in this work, namely: Zero-width resonances (ZWR) on one hand, and coalescing pairs of resonances at exceptional points (EP) on the other hand. An adiabatic transport theory based on Floquet Hamiltonian formalism is developed within the challenging context of multiphoton dynamics involving nuclear continua. It is shown that a rigorous treatment is only possible for ZWRs, whereas adiabatic transport mediated by EPs is subjected to restrictions. Numerical maps of resonance widths and non-adiabatic couplings in the laser parameter plane help in optimally shaping control pulses. Full time-dependent wavepacket dynamics shows the possibility of selective, robust filtration and vibrational population transfers, within experimental feasibility criteria.

quant-ph↗

arXiv2017

Adiabatic transport of qubits around a black hole

We consider localized qubits evolving around a black hole following a quantum adiabatic dynamics. We develop a geometric structure (based on fibre bundles) permitting to describe the quantum states of a qubit and the spacetime geometry in a single framework. The quantum decoherence induced by the black hole on the qubit is analysed in this framework (the role of the dynamical and geometric phases in this decoherence is treated), especially for the quantum teleportation protocol when one qubit falls to the event horizon. A simple formula to compute the fidelity of the teleportation is derived. The case of a Schwarzschild black hole is analysed.

gr-qc↗

arXiv2016

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 call affine 2-spaces. We study the geometric structure of the categorical bundles built on these categories (which is 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.

math-ph↗

arXiv2016

Controlling vibrational cooling with Zero-Width Resonances: An adiabatic Floquet approach

In molecular photodissociation, some specific combinations of laser parameters (wavelength and intensity) lead to unexpected Zero-Width Resonances (ZWR), with in principle infinite lifetimes. Their interest in inducing basic quenching mechanisms have recently been devised in the laser control of vibrational cooling through filtration strategies [O. Atabek et al., Phys. Rev. A87, 031403(R) (2013)]. A full quantum adiabatic control theory based on the adiabatic Floquet Hamiltonian is developed to show how a laser pulse could be envelop-shaped and frequency-chirped so as to protect a given initial vibrational state against dissociation, taking advantage from its continuous transport on the corresponding ZWR, all along the pulse duration. As compared with previous control scenarios actually suffering from non-adiabatic contamination, drastically different and much more efficient filtration goals are achieved. A semiclassical analysis helps in finding and interpreting a complete map of ZWRs in the laser parameter plane. In addition, the choice of a given ZWR path, among the complete series identified by the semiclassical approach, amounts to be crucial for the cooling scheme, targeting a single vibrational state population left at the end of the pulse, while all others have almost completely decayed. The illustrative example, offering the potentiality to be transposed to other diatomics, is Na2 prepared by photoassociation in vibrationally hot but translationally and rotationally cold states.

quant-ph↗

arXiv2016

Global integration of the Schrödinger equation within the wave operator formalism: The role of the effective Hamiltonian in multidimensional active spaces

A global solution of the Schrödinger equation, obtained recently within the wave operator formalism for explicitly time-dependent Hamiltonians [J. Phys. A: Math. Theor. 48, 225205 (2015)], is generalized to take into account the case of multidimensional active spaces. An iterative algorithm is derived to obtain the Fourier series of the evolution operator issuing from a given multidimensional active subspace and then the effective Hamiltonian corresponding to the model space is computed and analysed as a measure of the cyclic character of the dynamics. Studies of the laser controlled dynamics of diatomic models clearly show that a multidimensional active space is required if the wavefunction escapes too far from the initial subspace. A suitable choice of the multidimensional active space, including the initial and target states, increases the cyclic character and avoids divergences occuring when one-dimensional active spaces are used. The method is also proven to be efficient in describing dissipative processes such as photodissociation.

quant-ph↗