Calculus of Variation and Path-Integrals with Non-Linear Generalized Functions

The calculus of variation and the construction of path integrals is revisited within the framework of non-linear generalized functions. This allows us to make a rigorous analysis of the variation of an action that takes into account the boundary effects, even when the approach with distributions has pathological defects. A specific analysis is provided for optimal control actions, and we show how such kinds of actions can be used to model physical systems. Several examples are studied: the harmonic oscillator, the scalar field, and the gravitational field. For the first two cases, we demonstrate how the boundary cost function can be used to assimilate the optimal control adjoint state to the state of the system, hence recovering standard actions of the literature. For the gravitational field, we argue that a similar mechanism is not possible. Finally, we construct the path integral for the optimal control action within the framework of generalized functions. The effect of the discretization grid on the continuum limit is also discussed.Comment: 39 pages, 2 figure

Exploring the limits of the generation of non-classical states of spins coupled to a cavity by optimal control

We investigate the generation of non-classical states of spins coupled to a common cavity by means of a collective driving of the spins. We propose a control strategy using specifically designed series of short coherent and squeezing pulses, which have the key advantage of being experimentally implementable with the state-of-the art techniques. The parameters of the control sequence are found by means of optimization algorithms. We consider the cases of two and four spins, the goal being either to reach a well-defined target state or a state maximizing a measure of non-classicality. We discuss the influence of cavity damping and spin offset on the generation of non-classical states. We also explore to which extent squeezing fields help enhancing the efficiency of the control process.Comment: 13 pages, 7 figure

Enhancing quantum exchanges between two oscillators

We explore the extent to which two quantum oscillators can exchange their quantum states efficiently through a three-level system which can be spin levels of colored centers in solids. High transition probabilities are obtained using Hamiltonian engineering and quantum control techniques. Starting from a weak coupling approximation, we derive conditions on the spin-oscillator interaction Hamiltonian that enable a high fidelity exhange of quanta. We find that these conditions cannot be fulfilled for arbitrary spin-oscillator coupling. To overcome this limitation, we illustrate how a time-dependent control field applied to the three-level system can lead to an effective dynamic that performs the desired exchange of excitation. In the strong coupling regime, an important loss of fidelity is induced by the dispersion of the excitation onto many Fock states of the oscillators. We show that this detrimental effect can be substantially reduced by suitable control fields, which are computed with optimal control numerical algorithms.Comment: 11 figure

Calculus of Variation and Path-Integrals with Non-Linear Generalized Functions

International audienceThe calculus of variation and the construction of path integrals is revisited within the framework of non-linear generalized functions. This allows us to make a rigorous analysis of the variation of an action that takes into account the boundary effects, even when the approach with distributions has pathological defects. A specific analysis is provided for optimal control actions, and we show how such kinds of actions can be used to model physical systems. Several examples are studied: the harmonic oscillator, the scalar field, and the gravitational field. For the first two cases, we demonstrate how the boundary cost function can be used to assimilate the optimal control adjoint state to the state of the system, hence recovering standard actions of the literature. For the gravitational field, we argue that a similar mechanism is not possible. Finally, we construct the path integral for the optimal control action within the framework of generalized functions. The effect of the discretization grid on the continuum limit is also discussed

A Probability-Based Algorithm for Evaluating Climbing Difficulty Grades

International audienceThis paper describes a new mathematical model for the estimation of the grade of a climbing route. The calculation is based on the association of several route and boulder sections separated by rests. Contrary to other similar methods, this model introduces a probabilistic approach describing the uncertainty that one can have about the grade and the different feelings that climbers can have on a route grade. Several aspects of the model are commented and studied. A short comparative study of some of the hardest routes in the world is also presented

Calculus of Variation and Path-Integrals with Non-Linear Generalized Functions

International audienceThe calculus of variation and the construction of path integrals is revisited within the framework of non-linear generalized functions. This allows us to make a rigorous analysis of the variation of an action that takes into account the boundary effects, even when the approach with distributions has pathological defects. A specific analysis is provided for optimal control actions, and we show how such kinds of actions can be used to model physical systems. Several examples are studied: the harmonic oscillator, the scalar field, and the gravitational field. For the first two cases, we demonstrate how the boundary cost function can be used to assimilate the optimal control adjoint state to the state of the system, hence recovering standard actions of the literature. For the gravitational field, we argue that a similar mechanism is not possible. Finally, we construct the path integral for the optimal control action within the framework of generalized functions. The effect of the discretization grid on the continuum limit is also discussed

Emergent gravity from the correlation of spin-12\tfrac{1}{2} systems coupled with a scalar field

International audienceThis paper introduces several ideas of emergent gravity, which come from a system similar to an ensemble of quantum spin-$\tfrac{1}{2}$ particles. To derive a physically relevant theory, the model is constructed by quantizing a scalar field in curved space-time. The quantization is based on a classical discretization of the system, but contrary to famous approaches, like loop quantum gravity or causal triangulation, a Monte-Carlo based approach is used instead of a simplicial approximation of the space-time manifold. This avoids conceptual issues related to the choice of the lattice. Moreover, this allows us to easily encode the geometric structures of space, given by the geodesic length between points, into the mean value of a correlation operator between two spin-like systems. Numerical investigations show the relevance of the approach, and the presence of two regimes: a classical and a quantum regime. The latter is obtained when the density of points reaches a given threshold. Finally, a multi-scale analysis is given, where the classical model is recovered from the full quantum one. Each step of the classical limit is illustrated with numerical computations, showing the very good convergence towards the classical limit and the computational efficiency of the theory

Emergent gravity from the correlation of spin-12\tfrac{1}{2} systems coupled with a scalar field

International audienceThis paper introduces several ideas of emergent gravity, which come from a system similar to an ensemble of quantum spin-$\tfrac{1}{2}$ particles. To derive a physically relevant theory, the model is constructed by quantizing a scalar field in curved space-time. The quantization is based on a classical discretization of the system, but contrary to famous approaches, like loop quantum gravity or causal triangulation, a Monte-Carlo based approach is used instead of a simplicial approximation of the space-time manifold. This avoids conceptual issues related to the choice of the lattice. Moreover, this allows us to easily encode the geometric structures of space, given by the geodesic length between points, into the mean value of a correlation operator between two spin-like systems. Numerical investigations show the relevance of the approach, and the presence of two regimes: a classical and a quantum regime. The latter is obtained when the density of points reaches a given threshold. Finally, a multi-scale analysis is given, where the classical model is recovered from the full quantum one. Each step of the classical limit is illustrated with numerical computations, showing the very good convergence towards the classical limit and the computational efficiency of the theory

Loop quantum gravity with optimal control path integral, and application to black hole tunneling

International audienceThis paper presents a novel path integral formalism for Einstein’s theory of gravitation from the viewpoint of optimal control theory. Despite its close connection to the well-known variational principle of physicists, optimal control turns out to be more general. Within this context, a Lagrangian which is different from the Einstein-Hilbert Lagrangian is defined. Einstein’s field equations are recovered exactly with variations of the new action functional. The quantum theory is obtained using Ashtekar variables and the loop scalar product. As an illustrative example, the tunneling process of a black hole into another black hole or into a white hole is investigated with a toy model

Quantum States Seen by a Probe: Partial Trace Over a Region of Space

The partial trace operation is usually considered in composite quantum systems, to reduce the state on a single subsystem. This operation has a key role in the decoherence effect and quantum measurements. However, partial trace operations can be defined in more generic situations. In particular, it can be used to restrict a quantum state (for a single or several quantum entities) on a specific region of space, the rest of the universe being treated as an environment. The reduced state is then interpreted as the state that can be detected by an ideal probe with a limited spatial extent. In this paper, such an operation is investigated for systems defined on a Fock Hilbert space. A generic expression of the reduced density matrix is computed, and it is applied to several case studies: eigenstates of the number operator, coherent states, and thermal states. These states admit very different behaviors. In particular, (i) a decoherence effect happens on eigenstates of the number operator (ii) coherent or thermal states remain coherent or thermal, but with an amplitude/temperature reduced non-trivially by the overlap between the field and the region of interest