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Denis Bernard

Publications and source records attributed to Denis Bernard.

At least 109 records · Page 6Linked to original sources
arXiv2011

Convergence of repeated quantum non-demolition measurements and wave function collapse

Motivated by recent experiments on quantum trapped fields, we give a rigorous proof that repeated indirect quantum non-demolition (QND) measurements converge to the collapse of the wave function as predicted by the postulates of quantum mechanics for direct measurements. We also relate the rate of convergence toward the collapsed wave function to the relative entropy of each indirect measurement, a result which makes contact with information theory.

math-ph↗
arXiv2011

Full Counting Statistics in the Resonant-Level Model

We derive the large deviation function, which provides the large-time full counting statistics for the charge transfer, in the non-equilibrium steady state of the resonant-level model. The general form of this function in free fermion models, in terms of transmission coefficients, was proposed by Levitov and Lesovik in 1993 using a particular measurement set-up involving an interacting spin. It was later suggested to hold as well for a proper quantum mechanical measurement of the transferred charge. We give a precise proof of both statements in the resonant-level model. We first give a full description of the model and its steady state. That is, we explain how the decoupled system prepared with a charge differential evolves, with the impurity coupling, towards the Hershfield non-equilibrium density matrix, in the sense of averages of finitely-supported operators. We describe how this holds both for the usual resonant-level model with a point-like impurity, and for a regularised model with an impurity spread on a finite region, shedding light on subtleties associated to the point-like impurity. We then prove Levitov-Lesovik formula by recasting the problem into calculating averages of finitely-supported operators.

math-ph↗
arXiv2011

Quantum Stochastic Processes: A Case Study

We present a detailed study of a simple quantum stochastic process, the quantum phase space Brownian motion, which we obtain as the Markovian limit of a simple model of open quantum system. We show that this physical description of the process allows us to specify and to construct the dilation of the quantum dynamical maps, including conditional quantum expectations. The quantum phase space Brownian motion possesses many properties similar to that of the classical Brownian motion, notably its increments are independent and identically distributed. Possible applications to dissipative phenomena in the quantum Hall effect are suggested.

math-ph↗
arXiv2010

Four-wave interaction in gas and vacuum: definition of a third-order nonlinear effective susceptibility in vacuum: $χ$ vacuum(3)

Semiclassical methods are used to study the nonlinear interaction of light in vacuum in the context of four wave mixing. This study is motivated by a desire to investigate the possibility of using recently developed powerful ultrashort (femtosecond) laser pulses to demonstrate the existence of nonlinear effects in vacuum, predicted by quantum electrodynamics (QED). An approach, similar to classical nonlinear optics in a medium, is developed in this article. A third-order nonlinear effective susceptibility of vacuum is then introduced.

physics.optics↗
arXiv2010

The Gaussian free field and SLE(4) on doubly connected domains

The level lines of the Gaussian free field are known to be related to SLE(4). It is shown how this relation allows to define chordal SLE(4) processes on doubly connected domains, describing traces that are anchored on one of the two boundary components. The precise nature of the processes depends on the conformally invariant boundary conditions imposed on the second boundary component. Extensions of Schramm's formula to doubly connected domains are given for the standard Dirichlet and Neumann conditions and a relation to first-exit problems for Brownian bridges is established. For the free field compactified at the self-dual radius, the extended symmetry leads to a class of conformally invariant boundary conditions parametrised by elements of SU(2). It is shown how to extend SLE(4) to this setting. This allows for a derivation of new passage probabilities a la Schramm that interpolate continuously from Dirichlet to Neumann conditions.

math-ph↗
arXiv2010

QCD and Hadronic Interactions with Initial-State-Radiation at B-Factories

The efforts to improve on the precision of the measurement and theoretical prediction of the anomalous magnetic moment of the muon a_mu have turned into a test of our understanding of the hadronic contribution to vacuum polarisation. I describe how recent measurements of hadron production in e+e- interactions with initial-state radiation provide precision measurements of the hadron cross section, and have improved on the contribution to the prediction of the value of a_mu that dominates the global uncertainty.

hep-ex↗
arXiv2009

Off-Critical SLE(2) and SLE(4): a Field Theory Approach

Using their relationship with the free boson and the free symplectic fermion, we study the off-critical perturbation of SLE(4) and SLE(2) obtained by adding a mass term to the action. We compute the off-critical statistics of the source in the Loewner equation describing the two dimensional interfaces. In these two cases we show that ratios of massive by massless partition functions, expressible as ratios of regularised determinants of massive and massless Laplacians, are (local) martingales for the massless interfaces. The off-critical drifts in the stochastic source of the Loewner equation are proportional to the logarithmic derivative of these ratios. We also show that massive correlation functions are (local) martingales for the massive interfaces. In the case of massive SLE(4), we use this property to prove a factorisation of the free boson measure.

math-ph↗
arXiv2008

Conditioning SLEs and loop erased random walks

We discuss properties of dipolar SLE(k) under conditioning. We show that k=2, which describes continuum limits of loop erased random walks, is characterized as being the only value of k such that dipolar SLE conditioned to stop on an interval coincides with dipolar SLE on that interval. We illustrate this property by computing a new bulk passage probability for SLE(2).

math-ph↗
arXiv2007

LERW as an example of off-critical SLEs

Two dimensional loop erased random walk (LERW) is a random curve, whose continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter kappa=2. In this article we study ``off-critical loop erased random walks'', loop erasures of random walks penalized by their number of steps. On one hand we are able to identify counterparts for some LERW observables in terms of symplectic fermions (c=-2), thus making further steps towards a field theoretic description of LERWs. On the other hand, we show that it is possible to understand the Loewner driving function of the continuum limit of off-critical LERWs, thus providing an example of application of SLE-like techniques to models near their critical point. Such a description is bound to be quite complicated because outside the critical point one has a finite correlation length and therefore no conformal invariance. However, the example here shows the question need not be intractable. We will present the results with emphasis on general features that can be expected to be true in other off-critical models.

math-ph↗
arXiv2006

Are Domain Walls in Spin Glasses Described by Stochastic Loewner Evolutions?

Domain walls for spin glasses are believed to be scale invariant invariant; a stronger symmetry, conformal invariance, has the potential to hold. The statistics of zero-temperature Ising spin glass domain walls in two dimensions are used to test the hypothesis that these domain walls are described by a Schramm-Loewner evolution SLE$_κ$. Multiple tests are consistent with SLE$_κ$, where $κ=2.30(5)$. Both conformal invariance and the domain Markov property are tested. The latter does not hold in small systems, but detailed numerical evidence suggests that it holds in the continuum limit.

cond-mat.dis-nn↗
arXiv2006

2D growth processes: SLE and Loewner chains

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.

math-ph↗
arXiv2005

Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales

A statistical mechanics argument relating partition functions to martingales is used to get a condition under which random geometric processes can describe interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs to satisfy this condition leads to some natural processes, which we study in this note. We give examples of such multiple SLEs and discuss how a choice of conformal block is related to geometric configuration of the interfaces and what is the physical meaning of mixed conformal blocks. We illustrate the general ideas on concrete computations, with applications to percolation and the Ising model.

math-ph↗
arXiv2004

Loewner Chains

These lecture notes on 2D growth processes are divided in two parts. The first part is a non-technical introduction to stochastic Loewner evolutions (SLEs). Their relationship with 2D critical interfaces is illustrated using numerical simulations. Schramm's argument mapping conformally invariant interfaces to SLEs is explained. The second part is a more detailed introduction to the mathematically challenging problems of 2D growth processes such as Laplacian growth, diffusion limited aggregation (DLA), etc. Their description in terms of dynamical conformal maps, with discrete or continuous time evolution, is recalled. We end with a conjecture based on possible dendritic anomalies which, if true, would imply that the Hele-Shaw problem and DLA are in different universality classes.

cond-mat.stat-mech↗
arXiv2004

SLE, CFT and zig-zag probabilities

The aim of these notes is threefold. First, we discuss geometrical aspects of conformal covariance in stochastic Schramm-Loewner evolutions (SLEs). This leads us to introduce new ``dipolar'' SLEs, besides the known chordal, radial or annular SLEs. Second, we review the main features of our approach connecting SLEs to conformal field theories (CFTs). It is based on using boundary CFTs to probe the SLE hulls. Finally, we study zig-zag probabilities and their relation with CFT correlation functions. We suggest a putative link between the braiding of SLE samples and that of CFT correlation functions.

math-ph↗
arXiv2003

A Classification of random Dirac fermions

We present a detailed classification of random Dirac hamiltonians in two spatial dimensions based on the implementation of discrete symmetries. Our classification is slightly finer than that of random matrices, and contains thirteen classes. We also extend this classification to non-hermitian hamiltonians with and without Dirac structure.

cond-mat.dis-nn↗