Search

We construct and study the Kirkwood-Dirac (KD) representations naturally associated to the Fourier transform of finite abelian groups $G$. We identify all pure KD-positive states and all KD-real observables for these KD representations. We provide a necessary and sufficient condition ensuring that all KD-positive states are convex combinations of pure KD-positive states. We prove that for $G=\Z_{d}$, with $d$ a prime power, this condition is satisfied. We provide examples of abelian groups where it is not. In those cases, the convex set of KD-positive states contains states outside the convex hull of the pure KD-positive states.

We uncover new features of generalized contextuality by connecting it to the Kirkwood-Dirac (KD) quasiprobability distribution. Quantum states can be represented by KD distributions, which take values in the complex unit disc. Only for ``KD-positive'' states are the KD distributions joint probability distributions. A KD distribution can be measured by a series of weak and projective measurements. We design such an experiment and show that it is contextual iff the underlying state is not KD-positive. We analyze this connection with respect to mixed KD-positive states that cannot be decomposed as convex combinations of pure KD-positive states. Our result is the construction of a noncontextual experiment that enables an experimenter to verify contextuality., Comment: 17 pages, 4 figures

Given two observables $A$ and $B$, one can associate to every quantum state a Kirkwood-Dirac (KD) quasiprobability distribution. KD distributions are like joint classical probabilities except that they can have negative or nonreal values, which are associated to nonclassical features of the state. In the last decade, KD distributions have come to the forefront as a versatile tool to investigate and construct quantum advantages and nonclassical phenomena. KD distributions are also used to determine quantum-classical boundaries. To do so, one must have witnesses for when a state is KD nonpositive. Previous works have established a relation between the uncertainty of a pure state with respect to the eigenbases of $A$ and $B$ and KD positivity. If this $\textit{support uncertainty}$ is large, the state cannot be KD positive. Here, we construct two witnesses for KD nonpositivity for general mixed states. Our first witness is the convex roof of the support uncertainty; it is not faithful, but it extends to the convex hull of pure KD-positive states the relation between KD positivity and small support uncertainty. Our other witness is the convex roof of the total KD nonpositivity, which provides a faithful witness for the convex hull of the pure KD-positive states. This implies that the convex roof of the total nonpositivity captures the nonpositive nature of the KD distribution at the underlying pure state level., Comment: 16 pages, 2 figures

A central problem in quantum information is determining quantum-classical boundaries. A useful notion of classicality is provided by the quasiprobability formulation of quantum theory. In this framework, a state is called classical if it is represented by a quasiprobability distribution that is positive, and thus a probability distribution. In recent years, the Kirkwood-Dirac (KD) distributions have gained much interest due to their numerous applications in modern quantum-information research. A particular advantage of the KD distributions is that they can be defined with respect to arbitrary observables. Here, we show that if two observables are picked at random, the set of classical states of the resulting KD distribution is a simple polytope of minimal size. When the Hilbert space is of dimension $d$, this polytope is of dimension $2d-1$ and has $2d$ known vertices. Our result implies, $\textit{e.g.}$, that almost all KD distributions have resource theories in which the free states form a small and simple set., Comment: 12 pages

Recent years have seen the Kirkwood-Dirac (KD) distribution come to the forefront as a powerful quasi-probability distribution for analysing quantum mechanics. The KD distribution allows tools from statistics and probability theory to be applied to problems in quantum-information processing. A notable difference to the Wigner function is that the KD distribution can represent a quantum state in terms of arbitrary observables. This paper reviews the KD distribution, in three parts. First, we present definitions and basic properties of the KD distribution and its generalisations. Second, we summarise the KD distribution's extensive usage in the study or development of measurement disturbance; quantum metrology; weak values; direct measurements of quantum states; quantum thermodynamics; quantum scrambling and out-of-time-ordered correlators; and the foundations of quantum mechanics, including Leggett-Garg inequalities, the consistent-histories interpretation and contextuality. We emphasise connections between operational quantum advantages and negative or non-real KD quasi-probabilities. Third, we delve into the KD distribution's mathematical structure. We summarise the current knowledge regarding the geometry of KD-positive states (the states for which the KD distribution is a classical probability distribution), describe how to witness and quantify KD non-positivity, and outline relationships between KD non-positivity, coherence and observables' incompatibility., Comment: 42 pages, 14 figures; as published in NJP

Rigorous derivations of the approach of individual elements of large isolated systems to a state of thermal equilibrium, starting from arbitrary initial states, are exceedingly rare. This is particularly true for quantum mechanical systems. We demonstrate here how, through a mechanism of repeated scattering, an approach to equilibrium of this type actually occurs in a specific quantum system, one that can be viewed as a natural quantum analog of several previously studied classical models. In particular, we consider an optical mode passing through a reservoir composed of a large number of sequentially-encountered modes of the same frequency, each of which it interacts with through a beam splitter. We then analyze the dependence of the asymptotic state of this mode on the assumed stationary common initial state $\sigma$ of the reservoir modes and on the transmittance $\tau=\cos\lambda$ of the beam splitters. These results allow us to establish that at small $\lambda$ such a mode will, starting from an arbitrary initial system state $\rho$, approach a state of thermal equilibrium even when the reservoir modes are not themselves initially thermalized. We show in addition that, when the initial states are pure, the asymptotic state of the optical mode is maximally entangled with the reservoir and exhibits less nonclassicality than the state of the reservoir modes.

The Kirkwood-Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables $A$ and $B$. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In recent years, KD distributions have proven instrumental in mapping out nonclassical phenomena and quantum advantages. These quantum features have been connected to nonpositive entries of KD distributions. Consequently, it is important to understand the geometry of the KD-positive and -nonpositive states. Until now, there has been no thorough analysis of the KD positivity of mixed states. Here, we characterize how the full convex set of states with positive KD distributions depends on the eigenbases of $A$ and $B$. In particular, we identify three regimes where convex combinations of the eigenprojectors of $A$ and $B$ constitute the only KD-positive states: $(i)$ any system in dimension $2$; $(ii)$ an open and dense set of bases in dimension $3$; and $(iii)$ the discrete-Fourier-transform bases in prime dimension. Finally, we investigate if there can exist mixed KD-positive states that cannot be written as convex combinations of pure KD-positive states. We show that for some choices of observables $A$ and $B$ this phenomenon does indeed occur. We explicitly construct such states for a spin-$1$ system., Comment: 35 pages, 2 figures

We study modulational instability in a dispersion-managed system where the sign of the group-velocity dispersion is changed at uniformly distributed random distances around a reference length. An analytical technique is presented to estimate the instability gain from the linearized nonlinear Schr{\"o}dinger equation, which is also solved numerically. The comparison of numerical and analytical results confirms the validity of our approach. Modulational instability of purely stochastic origin appears. A competition between instability bands of periodic and stochastic origin is also discussed. We find an instability gain comparable to the conventional values found in a homogeneous anomalous dispersion fiber., Comment: 4 pages, 4 figure

Assessing whether a quantum state $\hat \rho$ is nonclassical ($\textit{i.e.}$, incompatible with a mixture of coherent states) is a ubiquitous question in quantum optics, yet a nontrivial experimental task because many nonclassicality witnesses are nonlinear in $\hat \rho$. In particular, if we want to witness or measure the nonclassicality of a state by evaluating its quadrature coherence scale, this $\textit{a priori}$ requires full state tomography. Here, we provide an experimental procedure for directly accessing this quantity with a simple linear interferometer involving two replicas (independent and identical copies) of the state $\hat \rho$ supplemented with photon-number-resolving measurements. This finding, which we interpret as an extension of the Hong-Ou-Mandel effect, illustrates the wide applicability of the multicopy interferometric technique in order to circumvent state tomography in quantum optics., Comment: Minor corrections in v2 to match the published version of the paper, 12 pages, 3 figures

We provide an in-depth study of the recently introduced notion of completely incompatible observables and its links to the support uncertainty and to the Kirkwood-Dirac nonclassicality of pure quantum states. The latter notion has recently been proven central to a number of issues in quantum information theory and quantum metrology. In this last context, it was shown that a quantum advantage requires the use of Kirkwood-Dirac nonclassical states. We establish sharp bounds of very general validity that imply that the support uncertainty is an efficient Kirkwood-Dirac nonclassicality witness. When adapted to completely incompatible observables that are close to mutually unbiased ones, this bound allows us to fully characterize the Kirkwood-Dirac classical states as the eigenvectors of the two observables. We show furthermore that complete incompatibility implies several weaker notions of incompatibility, among which features a strong form of noncommutativity.

Photon addition and subtraction render Gaussian states non-Gaussian. We provide a quantitative analysis of the change in nonclassicality produced by these processes by analyzing the Wigner negativity and quadrature coherence scale (QCS) of the resulting states. The QCS is a recently introduced measure of nonclassicality [PRL 122, 080402 (2019), PRL 124, 090402 (2020)], that we show to undergo a relative increase under photon addition/subtraction that can be as large as 200\%. This implies that the degaussification and the concomitant increase of nonclassicality come at a cost. Indeed, the QCS is proportional to the decoherence rate of the state so that the resulting states are considerably more prone to environmental decoherence. Our results are quantitative and rely on explicit and general expressions for the characteristic and Wigner functions of photon added/subtracted single- and multi-mode Gaussian states for which we provide a simple and straightforward derivation. These expressions further allow us to certify the quantum non-Gaussianity of the photon-subtracted states with positive Wigner function., Comment: Considerably expanded version with study of Wigner negative volume and of genuine non-Gaussianity of photon-added/subtracted Gaussian states

We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD). We consider Gaussian and dichotomous colored stochastic processes. We resort to different analytical methods (namely, the cumulant expansion and the functional approach) and assess their reliability in estimating the MI gain of stochastic origin. If the power spectral density (PSD) of the GVD fluctuations is centered at null wavenumber, we obtain low-frequency MI sidelobes which converge to those given by a white noise perturbation when the correlation length tends to 0. If instead the stochastic processes are modulated in space, one or more MI sidelobe pairs corresponding to the well-known parametric resonance (PR) condition can be found. A transition from small and broad sidelobes to peaks nearly indistinguishable from PR-MI is predicted, in the limit of large perturbation amplitudes and correlation lengths of the random process. We find that the cumulant expansion provides good analytical estimates for small PSD values and small correlation lengths, when the MI gain is very small. The functional approach is rigorous only for the dichotomous processes, but allows us to model a wider range of parameters and to predict the existence of MI sidelobes comparable to those observed in homogeneous fibers of anomalous GVD, Comment: 12 pages, 6 figures submitted

Given two orthonormal bases in a d-dimensional Hilbert space, one associates to each state its Kirkwood-Dirac (KD) quasi-probability distribution. KD-nonclassical states - for which the KD-distribution takes on negative and/or nonreal values - have been shown to provide a quantum advantage in quantum metrology and information, raising the question of their identification. Under suitable conditions of incompatibility between the two bases, we provide sharp lower bounds on the support uncertainty of states that guarantee their KD-nonclassicality. In particular, when the bases are completely incompatible, a notion we introduce, states whose support uncertainty is not equal to its minimal value d+1 are necessarily KD-nonclassical. The implications of these general results for various commonly used bases, including the mutually unbiased ones, and their perturbations, are detailed.

We study modulational instability (MI) in optical fibers with random group velocity dispersion (GVD) generated by sharply localized perturbations of a normal GVD fiber that are either randomly or periodically placed along the fiber and that have random strength. This perturbation leads to the appearance of low frequency MI side lobes that grow with the strength of the perturbations, whereas they are faded by randomness in their position. If the random perturbations exhibit a finite average value, they can be compared with periodically perturbed fibers, where Arnold tongues appear. In that case, increased randomness in the strengths of the variations tends to affect the Arnold tongues less than increased randomness in their positions.

The quantum nature of the state of a bosonic quantum field manifests itself in its entanglement, coherence, or optical nonclassicality which are each known to be resources for quantum computing or metrology. We provide quantitative and computable bounds relating entanglement measures with optical nonclassicality measures. These bounds imply that strongly entangled states must necessarily be strongly optically nonclassical. As an application, we infer strong bounds on the entanglement that can be produced with an optically nonclassical state impinging on a beam splitter. For Gaussian states, we analyze the link between the logarithmic negativity and a specific nonclassicality witness called "quadrature coherence scale"., Comment: 13 pages, 2 figures, change of notation in v2

We study the dynamics of the many-body atomic kicked rotor with interactions at the mean-field level, governed by the Gross-Pitaevskii equation. We show that dynamical localization is destroyed by the interaction, and replaced by a subdiffusive behavior. In contrast to results previously obtained from a simplified version of the Gross-Pitaevskii equation, the subdiffusive exponent does not appear to be universal. By studying the phase of the mean-field wave function, we propose a new approximation that describes correctly the dynamics at experimentally relevant times close to the start of subdiffusion, while preserving the reduced computational cost of the former approximation., Comment: v1) 5 pages, 4 figures; v2) 7 pages, 4 figures

We introduce, for each state of a bosonic quantum field, its quadrature coherence scale (QCS), a measure of the range of its quadrature coherences. Under coupling to a thermal bath, the purity and QCS are shown to decrease on a time scale inversely proportional to the QCS squared. The states most fragile to decoherence are therefore those with quadrature coherences far from the diagonal. We further show a large QCS is difficult to measure since it induces small scale variations in the state's Wigner function. These two observations imply a large QCS constitutes a mark of "macroscopic coherence". Finally, we link the QCS to optical classicality: optical classical states have a small QCS and a large QCS implies strong optical nonclassicality., Comment: 12 pages, 5 figures. New version to match the published version. Minor errors were corrected

We introduce the thermal-difference states (TDS), a three-parameter family of single-mode non-Gaussian bosonic states whose density operator is a weighted difference of two thermal states. We show that the states of "heralded photons" generated via parametric down-conversion (PDC) are precisely those among the TDS that are nonclassical, meaning they have a negative $P$-function. The three parameters correspond in that context to the initial brightness of PDC and the transmittances, characterizing the linear loss in the signal and the idler channels. At low initial brightness and unit transmittances, the heralded photon state is known to be a single-photon state. We explore the influence of brightness and linear loss on the heralded state of the signal mode. In particular, we analyze the influence of the initial brightness and the loss on the state nonclassicality by computing several measures of nonclassicality, such as the negative volume of the Wigner function, the sum of quantum Fisher information for two quadratures, and the ordering sensitivity, introduced recently by us [Phys. Rev. Lett. 122, 080402 (2019)]. We argue finally that the TDS provide new benchmark states for the analysis of a variety of properties of single-mode bosonic states., Comment: 13 pages, 7 figures, final accepted version

We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasi-probabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-based measure of nonclassicality. We analyse the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct.

Boltzmann provided a scenario to explain why individual macroscopic systems composed of a large number $N$ of microscopic constituents are inevitably (i.e., with overwhelming probability) observed to approach a unique macroscopic state of thermodynamic equilibrium, and why after having done so, they are then observed to remain in that state, apparently forever. We provide here rigourous new results that mathematically prove the basic features of Boltzmann's scenario for two classical models: a simple boundary-free model for the spatial homogenization of a non-interacting gas of point particles, and the well-known Kac ring model. Our results, based on concentration inequalities that go back to Hoeffding, and which focus on the typical behavior of individual macroscopic systems, improve upon previous results by providing estimates, exponential in $N$, of probabilities and time scales involved.

We show that the nonlinear stage of modulational instability induced by parametric driving in the {\em defocusing} nonlinear Schr\"odinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis.

We consider the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We prove a persistence result for such relative equilibria, present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and show how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of relative equilibria of nonlinear Schr{\"o}dinger and Manakov equations. We provide a comparison of our approach to the one by Grillakis-Shatah-Strauss.

We are interested in a kinetic equation intended to describe the interaction of particles with their environment. The environment is modeled by a collection of local vibrational degrees of freedom. We establish the existence of weak solutions for a wide class of initial data and external forces. We also identify a relevant regime which allows us to derive, quite surprisingly, the attractive Vlasov--Poisson system from the coupled Vlasov-Wave equations.

We present a general approach to calculating the entanglement of formation for superpositions of two-mode coherent states, placed equidistantly on a circle in the phase space. We show that in the particular case of rotationally-invariant circular states the Schmidt decomposition of two modes, and therefore the value of their entanglement, are given by analytical expressions. We analyse the dependence of the entanglement on the radius of the circle and number of components in the superposition. We also show that the set of rotationally-invariant circular states creates an orthonormal basis in the state space of the harmonic oscillator, and this basis is advantageous for representation of other circular states of light., Comment: 10 pages, 6 figures

Demonstrating how microscopic dynamics cause large systems to approach thermal equilibrium remains an elusive, longstanding, and actively-pursued goal of statistical mechanics. We identify here a dynamical mechanism for thermalization in a general class of two-component dynamical Lorentz gases, and prove that each component, even when maintained in a non-equilibrium state itself, can drive the other to a thermal state with a well-defined effective temperature., Comment: 5 pages, 5 figures

An otherwise free classical particle moving through an extended spatially homogeneous medium with which it may exchange energy and momentum will undergo a frictional drag force in the direction opposite to its velocity with a magnitude which is typically proportional to a power of its speed. We study here the quantum equivalent of a classical Hamiltonian model for this friction phenomenon that was proposed in [11]. More precisely, we study the spectral properties of the quantum Hamiltonian and compare the quantum and classical situations. Under suitable conditions on the infrared behaviour of the model, we prove that the Hamiltonian at fixed total momentum has no ground state except when the total momentum vanishes, and that its spectrum is otherwise absolutely continuous., Comment: 40 pages

It is shown that sufficiently large periodic modulations in the coefficients of a nonlinear Schr{\"o}dinger equation can drastically impact the spatial shape of the Peregrine soliton solutions: they can develop multiple compression points of the same amplitude, rather than only a single one, as in the spatially homogeneous focusing nonlinear Schr{\"o}dinger equation. The additional compression points are generated in pairs forming a comb-like structure. The number of additional pairs depends on the amplitude of the modulation but not on its wavelength, which controls their separation distance. The dynamics and characteristics of these generalized Peregrine soliton are analytically described in the case of a completely integrable modulation. A numerical investigation shows that their main properties persist in nonintegrable situations, where no exact analytical expression of the generalized Peregrine soliton is available. Our predictions are in good agreement with numerical findings for an interesting specific case of an experimentally realizable periodically dispersion modulated photonic crystal fiber. Our results therefore pave the way for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in the wide class of physical systems modeled by the nonlinear Schr{\"o}dinger equation.

We study, both theoretically and experimentally, modulational instability in optical fibers that have a longitudinal evolution of their dispersion in the form of a Dirac delta comb. By means of Floquet theory, we obtain an exact expression for the position of the gain bands, and we provide simple analytical estimates of the gain and of the bandwidths of those sidebands. An experimental validation of those results has been realized in several microstructured fibers specifically manufactured for that purpose. The dispersion landscape of those fibers is a comb of Gaussian pulses having widths much shorter than the period, which therefore approximate the ideal Dirac comb. Experimental spontaneous MI spectra recorded under quasi continuous wave excitation are in good agreement with the theory and with numerical simulations based on the generalized nonlinear Schr\"odinger equation.

We study the long time behaviour of the speed of a particle moving in $\mathbb{R}^d$ under the influence of a random time-dependent potential representing the particle's environment. The particle undergoes successive scattering events that we model with a Markov chain for which each step represents a collision. Assuming the initial velocity is large enough, we show that, with high probability, the particle's kinetic energy $E(t)$ grows as $t^{\frac25}$ when $d>5$.

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system.

In the single band tight-binding approximation, we consider the transport properties of an electron in a homogeneous static electric field. We show that repeated interactions of the electron with two-level systems in thermal equilibrium suppress the Bloch oscillations and induce a steady current, the statistical properties of which we study.

We prove approach to thermal equilibrium for the fully Hamiltonian dynamics of a dynamical Lorentz gas, by which we mean an ensemble of particles moving through a $d$-dimensional array of fixed soft scatterers that each possess an internal harmonic or anharmonic degree of freedom to which moving particles locally couple. We establish that the momentum distribution of the moving particles approaches a Maxwell-Boltzmann distribution at a certain temperature $T$, provided that they are initially fast and the scatterers are in a sufficiently energetic but otherwise arbitrary stationary state of their free dynamics--they need not be in a state of thermal equilibrium. The temperature $T$ to which the particles equilibrate obeys a generalized equipartition relation, in which the associated thermal energy $k_{\mathrm B}T$ is equal to an appropriately defined average of the scatterers' kinetic energy. In the equilibrated state, particle motion is diffusive.

We study numerically and theoretically the $d$-dimensional Hamiltonian motion of fast particles through a field of scatterers, modeled by bounded, localized, (time-dependent) potentials, that we refer to as (in)elastic non-dissipative Lorentz gases. We illustrate the wide applicability of a random walk picture previously developed for a field of scatterers with random spatial and/or time-dependence by applying it to four other models. First, for a periodic array of spherical scatterers in $d\geq2$, with a smooth (quasi)periodic time-dependence, we show Fermi acceleration: the ensemble averaged kinetic energy $\left<\|p(t)\|^2\right>$ grows as $t^{2/5}$. Nevertheless, the mean squared displacement $\left<\|q(t)\|^2\right>\sim t^2$ behaves ballistically. These are the same growth exponents as for random time-dependent scatterers. Second, we show that in the soft elastic and periodic Lorentz gas, where the particles' energy is conserved, the motion is diffusive, as in the standard hard Lorentz gas, but with a diffusion constant that grows as $\|p_0\|^{5}$, rather than only as $\|p_0\|$. Third, we note the above models can also be viewed as pulsed rotors: the latter are therefore unstable in dimension $d\geq 2$. Fourth, we consider kicked rotors, and prove them, for sufficiently strong kicks, to be unstable in all dimensions with $\left<\|p(t)\|^2\right>\sim t$ and $\left<\|q(t)\|^2\right>\sim t^3$. Finally, we analyze the singular case $d=1$, where $\left< \|p(t)\|^2\right>$ remains bounded in time for time-dependent non-random potentials whereas it grows at the same rate as above in the random case.

We study the long time motion of fast particles moving through time-dependent random force fields with correlations that decay rapidly in space, but not necessarily in time. The time dependence of the averaged kinetic energy and mean-squared displacement is shown to exhibit a large degree of universality; it depends only on whether the force is, or is not, a gradient vector field. When it is, p^{2}(t) ~ t^{2/5} independently of the details of the potential and of the space dimension. Motion is then superballistic in one dimension, with q^{2}(t) ~ t^{12/5}, and ballistic in higher dimensions, with q^{2}(t) ~ t^{2}. These predictions are supported by numerical results in one and two dimensions. For force fields not obtained from a potential field, the power laws are different: p^{2}(t) ~ t^{2/3} and q^{2}(t) ~ t^{8/3} in all dimensions d\geq 1.

We study the Hamiltonian motion of an ensemble of unconfined classical particles driven by an external field F through a translationally-invariant, thermal array of monochromatic Einstein oscillators. The system does not sustain a stationary state, because the oscillators cannot effectively absorb the energy of high speed particles. We nonetheless show that the system has at all positive temperatures a well-defined low-field mobility over macroscopic time scales of order exp(-c/F). The mobility is independent of F at low fields, and related to the zero-field diffusion constant D through the Einstein relation. The system therefore exhibits normal transport even though the bath obviously has a discrete frequency spectrum (it is simply monochromatic) and is therefore highly non-Ohmic. Such features are usually associated with anomalous transport properties.

The nature and properties of the vacuum as well as the meaning and localization properties of one or many particle states have attracted a fair amount of attention and stirred up sometimes heated debate in relativistic quantum field theory over the years. I will review some of the literature on the subject and will then show that these issues arise just as well in non-relativistic theories of extended systems, such as free bose fields. I will argue they should as such not have given rise either to surprise or to controversy. They are in fact the result of the misinterpretation of the vacuum as ``empty space'' and of a too stringent interpretation of field quanta as point particles. I will in particular present a generalization of an apparently little known theorem of Knight on the non-localizability of field quanta, Licht's characterization of localized excitations of the vacuum, and explain how the physical consequences of the Reeh-Schlieder theorem on the cyclicity and separability of the vacuum for local observables are already perfectly familiar from non-relativistic systems of coupled oscillators.

We give a complete and rigorous proof of the Unruh effect, in the following form. We show that the state of a two-level system, uniformly accelerated with proper acceleration $a$, and coupled to a scalar bose field initially in the Minkowski vacuum state will converge, asymptotically in the detector's proper time, to the Gibbs state at inverse temperature $\beta=\frac{2\pi}{a}$. The result also holds if the field and detector are initially in an excited state. We treat the problem as one of return to equilibrium, exploiting in particular that the Minkowski vacuum is a KMS state with respect to Lorentz boosts. We then use the recently developed spectral techniques to prove the stated result.

These notes contain an extended version of lectures given at the ``Summer School on Large Coulomb Systems'' in Nordfjordeid, Norway, in august 2003. They furnish a short introduction to the theory of quantum harmonic systems, or free bose fields. The main issue addressed is the one of local states. I will adopt the definition of Knight of ``strictly local excitation of the vacuum'' and will then state and prove a generalization of Knight's Theorem which asserts that finite particle states cannot be perfectly localized. It will furthermore be explained how Knight's a priori counterintuitive result can be readily understood if one remembers the analogy between finite and infinite dimensional harmonic systems alluded to above. I will also discuss the link between the above result and the so-called Newton-Wigner position operator thereby illuminating, I believe, the difficulties associated with the latter. I will in particular argue that those difficulties do not find their origin in special relativity or in any form of causality violation, as is usually claimed.

We study the closed Hamiltonian dynamics of a free particle moving on a ring, over one section of which it interacts linearly with a single harmonic oscillator. On the basis of numerical and analytical evidence, we conjecture that at small positive energies the phase space of our model is completely chaotic except for a single region of complete integrability with a smooth sharp boundary showing no KAM-type structures of any kind. This results in the cleanest mixed phase space structure possible, in which motions in the integrable region and in the chaotic region are clearly separated and independent of one another. For certain system parameters, this mixed phase space structure can be tuned to make either of the two components disappear, leaving a completely integrable or completely chaotic phase space. For other values of the system parameters, additional structures appear, such as KAM-like elliptic islands, and one parameter families of parabolic periodic orbits embedded in the chaotic sea. The latter are analogous to bouncing ball orbits seen in the stadium billiard. The analytical part of our study proceeds from a geometric description of the dynamics, and shows it to be equivalent to a linked twist map on the union of two intersecting disks., Comment: 17 pages, 11 figures Typos corrected to display section labels

We study the Hamiltonian dynamics of a free particle injected onto a chain containing a periodic array of harmonic oscillators in thermal equilibrium. The particle interacts locally with each oscillator, with an interaction that is linear in the oscillator coordinate and independent of the particle's position when it is within a finite interaction range. At long times the particle exhibits diffusive motion, with an ensemble averaged mean-squared displacement that is linear in time. The diffusion constant at high temperatures follows a power law D ~ T^{5/2} for all parameter values studied. At low temperatures particle motion changes to a hopping process in which the particle is bound for considerable periods of time to a single oscillator before it is able to escape and explore the rest of the chain. A different power law, D ~ T^{3/4}, emerges in this limit. A thermal distribution of particles exhibits thermally activated diffusion at low temperatures as a result of classically self-trapped polaronic states., Comment: 15 pages, 4 figures Submitted to Physical Review E

We show that on a suitable time scale, logarithmic in $\hbar$, the coherent states on the two-torus, evolved under a quantized perturbed hyperbolic toral automorphism, equidistribute on the torus. We then use this result to obtain control on the possible strong scarring of eigenstates of the perturbed automorphisms by periodic orbits. Our main tool is an adapted Egorov theorem, valid for logarithmically long times.

In this paper we construct a sequence of eigenfunctions of the ``quantum Arnold's cat map'' that, in the semiclassical limit, show a strong scarring phenomenon on the periodic orbits of the dynamics. More precisely, those states have a semiclassical limit measure that is the sum of 1/2 the normalized Lebesgue measure on the torus plus 1/2 the normalized Dirac measure concentrated on any a priori given periodic orbit of the dynamics. It is known (the Schnirelman theorem) that ``most'' sequences of eigenfunctions equidistribute on the torus. The sequences we construct therefore provide an example of an exception to this general rule. Our method of construction and proof exploits the existence of special values of Planck's constant for which the quantum period of the map is relatively ``short'', and a sharp control on the evolution of coherent states up to this time scale. We also provide a pointwise description of these states in phase space, which uncovers their ``hyperbolic'' structure in the vicinity of the fixed points and yields more precise localization estimates., Comment: LaTeX, 49 pages, includes 10 figures. I added section 6.6. To be published in Commun. Math. Phys

We show that in the semi-classical limit the eigenfunctions of quantized ergodic symplectic toral automorphisms can not concentrate in measure on a finite number of closed orbits of the dynamics. More generally, we show that, if the pure point component of the limit measure has support on a finite number of such orbits, then the mass of this component must be smaller than two thirds of the total mass. The proofs use only the algebraic (i.e. not the number theoretic) properties of the toral automorphisms together with the exponential instability of the dynamics and therefore work in all dimensions., Comment: Latex file, 19 pages

We study the behaviour, in the simultaneous limits \hbar going to 0, t going to \infty, of the Husimi and Wigner distributions of initial coherent states and position eigenstates, evolved under the quantized hyperbolic toral automorphisms and the quantized baker map. We show how the exponential mixing of the underlying dynamics manifests itself in those quantities on time scales logarithmic in \hbar. The phase space distributions of the coherent states, evolved under either of those dynamics, are shown to equidistribute on the torus in the limit \hbar going to 0, for times t between |\log\hbar|/(2\gamma) and |\log|\hbar|/\gamma, where \gamma is the Lyapounov exponent of the classical system. For times shorter than |\log\hbar|/(2\gamma), they remain concentrated on the classical trajectory of the center of the coherent state. The behaviour of the phase space distributions of evolved position eigenstates, on the other hand, is not the same for the quantized automorphisms as for the baker map. In the first case, they equidistribute provided t goes to \infty as \hbar goes to 0, and as long as t is shorter than |\log\hbar|/\gamma. In the second case, they remain localized on the evolved initial position at all such times.

We study the one-dimensional random dimer model, with Hamiltonian $H_\omega=\Delta + V_\omega$, where for all $x\in\Z, V_\omega(2x)=V_\omega(2x+1)$ and where the $V_\omega(2x)$ are i.i.d. Bernoulli random variables taking the values $\pm V, V>0$. We show that, for all values of $V$ and with probability one in $\omega$, the spectrum of $H$ is pure point. If $V\leq1$ and $V\neq 1/\sqrt{2}$, the Lyapounov exponent vanishes only at the two critical energies given by $E=\pm V$. For the particular value $V=1/\sqrt{2}$, respectively $V=\sqrt{2}$, we show the existence of additional critical energies at $E=\pm 3/\sqrt{2}$, resp. E=0. On any compact interval $I$ not containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all $q>0$ and for all $\psi\in\ell^2(\Z)$ with sufficiently rapid decrease: $$ \sup_t r^{(q)}_{\psi,I}(t) \equiv \sup_t < P_I(H_\omega)\psi_t, |X|^q P_I(H_\omega)\psi_t > <\infty. $$ Here $\psi_t=e^{-iH_\omega t} \psi$, and $P_I(H_\omega)$ is the spectral projector of $H_\omega$ onto the interval $I$. In particular if $V>1$ and $V\neq \sqrt{2}$, these results hold on the entire spectrum (so that one can take $I=\sigma(H_\omega)$)., Comment: 14 pages