Добірка наукової літератури з теми "Supersolidity, quantum gas, quantum fluctuations"

Here we develop a mathematical apparatus to describe quasi-classical fluctuations in a non-equilibrium electron gas with electron-electron collisions. We substantiate the method by deriving, from general principles of quantum kinetics, an equation recently proposed by us for an equal-time electron-electron correlation function. The derivation is performed using the kinetic diagram technique. In degenerate non-equilibrium gas, the theory predicts that there exists a specific equal-time correlation between electrons. Due to the prevalence of small-angle electron-electron scattering, the equation in question takes a rather simple and treatable form (the Coulomb-type electron-electron interaction stands out against the background of all other types of interaction as one that does not generate, in the framework of quasi-classical approach, any direct exchange effects).

The thermodynamics of excited nuclear systems allows the exploration of a phase transition in a two-component quantum mixture. Temperatures and densities are derived from quantum fluctuations of fermions. The pressures are determined from the grand partition function of Fisher's model. Critical scaling of observables is found for the first time for fragmenting systems which differ in neutron to proton concentrations thus constraining the equation of state (EOS) of asymmetric nuclear material. The derived critical exponent, β = 0.35 ±0.01, belongs to the liquid–gas universality class. The critical compressibility factor Pc/ρcTc increases with increasing neutron concentration, which could be due to finite-size and/or Coulomb effects.

In confined space, the thermodynamic potential is shape-dependent. Therefore, the pressure of ideal gases in confined space is anisotropic. We study this anisotropy in a thermodynamic manner and find that the thermodynamic pressures usually depend on the form of deformations, and hence are not equal to each other which is a natural representation of the anisotropic mechanical properties of a confined ideal gas. We also find that the boundary effects are much more significant than the statistical fluctuations under low-temperature and high-density conditions. Finally, we show that there is little difference between the boundary effects in 2D space and those in 3D space.

We consider the nonadiabatic energy fluctuations of a many-body system in a time-dependent harmonic trap. In the presence of scale-invariance, the dynamics becomes self-similar and the nondiabatic energy fluctuations can be found in terms of the initial expectation values of the second moments of the Hamiltonian, square position, and squeezing operators. Nonadiabatic features are expressed in terms of the scaling factor governing the size of the atomic cloud, which can be extracted from time-of-flight images. We apply this exact relation to a number of examples: the single-particle harmonic oscillator, the one-dimensional Calogero-Sutherland model, describing bosons with inverse-square interactions that includes the non-interacting Bose gas and the Tonks-Girdardeau gas as limiting cases, and the unitary Fermi gas. We illustrate these results for various expansion protocols involving sudden quenches of the trap frequency, linear ramps and shortcuts to adiabaticity. Our results pave the way to the experimental study of nonadiabatic energy fluctuations in driven quantum fluids.

The recent realization of Bose-Einstein condensation in atomic gases opens new possibilities for observation of macroscopic quantum phenomena. There are two important features of the system - weak interaction and significant spatial inhomogeneity. Because of this inhomogeneity a non-trivial "zeroth-order" theory exists, compared to the "first-order" Bogolubov theory. This theory is based on the mean-field Gross-Pitaevskii equation for the condensate ψ-function. The equation is classical in its essence but contains the ℏ constant explicitly. Phenomena such as collective modes, interference, tunneling, Josephson-like current and quantized vortex lines can be described using this equation. The study of deviations from the zeroth-order theory arising from zero-point and thermal fluctuations is also of great interest. Thermal fluctuations are described by elementary excitations which define the thermodynamic behaviour of the system and result in Landau-type damping of collective modes. Fluctuations of the phase of the condensate wave function restrict the monochromaticity of the Josephson current. Fluctuations of the numbers of quanta result in the quantum collapse-revival of the collective oscillations. This phenomenon is considered in some details. Collapse time for the JILA experimental conditions turns out to be of the order of seconds.

Ultracold quantum gases have nowadays become an invaluable tool in the study of quantum many-body problems. The high level of experimental control available on these systems and well established theoretical tools make ultracold quantum gases ideal platforms for quantum simulations of other systems currently inaccessible in experiments as well as for studies of fundamental properties of matter in the quantum degenerate regime. A key manifestation of quantum degeneracy in samples of ultracold bosonic neutral atoms is the formation of a Bose-Einstein condensate (BEC), a peculiar state of matter in which a macroscopic number of atoms occupy the same single-particle state. Bose-Einstein condensation occurs in extremely rarefied gases of bosonic atoms at temperatures around the nanoKelvin. At such temperatures, the equilibrium state of all known elements (except for helium) in ordinary conditions of density and pressure would be the solid phase. To obtain a BEC it is thus necessary to consider very dilute samples with a density of the order of 1014-1015 atoms/cm3, around eight orders of magnitude smaller then the density of ordinary matter. At such densities, the three-body recombination mechanisms responsible for the formation of molecules, that cluster to form solids, are suppressed. However, despite the extreme diluteness, two-body inter-atomic interactions play a prominent role in determining the physical properties of these systems. In the temperature and density regimes typical of BECs, the theoretical description of the system can be greatly simplified by noticing that the low-energy scattering properties of the real, generally involved, inter-atomic potential, can be perfectly reproduced by a simpler pseudo-potential, usually of the form of an isotropic contact repulsion, and described by a single parameter, the s-wave scattering length. Such parameter can even be tuned, in experiments, via the so-called Feschbach resonances. Despite its simplicity, this zero-range, isotropic interaction is responsible for an enormous variety of physical effects characterizing atomic BECs. This fact stimulated, over the last twenty years, the research of different possible types of interactions, that can eventually lead to the formation of new and exotic phases of matter. In this quest, the dipole-dipole interaction attracted great attention for different reasons. First, there are several experimental techniques to efficiently trap and cool atoms (or molecules) possessing a strong dipole moment. This led, for example, to the experimental realization of BECs of Chromium, Dysprosium and Erbium, which have, in the hyperfine state trapped for condensation, a magnetic dipole moment around ten times larger then the one typical of the particles in a BEC of alkali atoms. Moreover, since the dipole-dipole interaction is anisotropic and long-ranged, its low-energy scattering properties cannot be described by a simple short-range isotropic pseudo-potential. As a consequence, dipolar BECs show unique observable properties. The partially attractive nature of the dipole-dipole interaction can make a dipolar BEC unstable against collapse, similarly to the case of an ordinary (non-dipolar) BEC with negative scattering length. This happens, in particular, if a sample of magnetic atoms, polarized along a certain direction by some magnetic field, is not confined enough along such direction (for example via a harmonic potential). However, differently from ordinary BECs, where the collapse of the system is followed by a rapid loss of atoms and the destruction of the condensed phase, in the dipolar case such instability is followed by the formation of self-bound, (relatively) high density liquid-like droplets. If the geometry of the confinement potential allows it, the droplets spontaneously arrange into a regular, periodic configuration, in a sort of "droplet crystal". Moreover, by fine-tuning the interaction parameters, it is possible to achieve global phase coherence between these droplets. The spatially modulated, phase coherent system that forms is known as supersolid, and is a very peculiar system showing simultaneously the properties of a crystal and a superfluid. Ordinary mean-field theory, so successful in describing the vast phenomenology of ordinary BECs, fails in predicting the existence of the exotic phases of supersolids, quantum droplets and droplet crystals in a dipolar quantum gas. The state of the art description of dipolar BECs in such conditions is instead based on quantum fluctuations, taking into account the local density approximation of the first-order beyond-mean-field correction of the ground state energy of the system. This correction, known as the Lee-Huang-Yang correction, results in a repulsive energy term that balances the mean-field attraction at the relatively high densities that characterize the collapsing state. Using state-of-the-art simulation techniques, in this thesis I study the behavior of a dipolar Bose gas confined in a variety of trapping configurations, focusing on ground-state properties, elementary excitations, and the dynamical behavior under several kinds of external perturbations, focusing in particular on the supersolid phase. After reviewing the basic theory of dipolar Bose gases, setting the theoretical background, and describing the numerical techniques used, I first study the behavior of the dipolar Bose gas in an ideal situation, namely when the gas is confined in a harmonic trap along the polarization direction of the dipoles as well as one of the orthogonal directions. Along the unconfined direction, instead, I set periodic boundary conditions, in order to simulate the geometry of a ring. I study in particular the phase diagram of the system, focusing on how the ground state evolves from a superfluid, homogeneous along the ring, to the supersolid regime, and eventually to an array of independent droplets, by tuning a single interaction parameter, namely the s-wave scattering length. The superfluid phase is here characterized by the occurrence of a roton minimum in the energy-momentum dispersion relation. The energy of the roton, called roton gap, decreases when the s-wave scattering length of the system is decreased and the dipole-dipole interaction becomes the dominant interaction mechanism. When the roton minimum touches the zero-energy axis, the superfluid system is not stable anymore against mechanical collapse. The system thus tend to form denser clusters of atoms, regularly arranged in an equally-spaced array of droplets, whose relative distance is fixed by the inverse of the roton momentum. Such droplets are stabilized by quantum fluctuations, which enters in the energy functional of the system via the Lee-Huang-Yang correction. The density profiles of these droplets maintain a finite overlap if the scattering length is not too small. The phase characterized by overlapping, dense droplets of dipolar atoms is called supersolid. The main signatures of supersolid behavior, which in the thesis are shown to occur in this system, are 1. The occurrence of two Goldstone modes, associated with the two symmetries spontaneously broken in the supersolid, namely the symmetry for continuous translations, which is broken in favor of a discrete one, and the U(1) symmetry associated with Bose-Einstein condensation. 2. The manifestation of Non-Classical Rotational Inertia, due to the partially superfluid character of the system. Simply speaking, since the system behaves only partially as a superfluid, any rotational perturbation drags only the non superfluid part of the system. Hence, any measurement of the moment of inertia would give a value which is smaller then the one of a classical system with the same density distribution. Having studied the behavior of the dipolar Bose gas in a ring trap, I move on to explore possible manifestations of supersolid behavior in a fully trapped configuration, namely when the system is confined in an elongated (cigar-shaped) harmonic trap, with the long axis orthogonal to the polarization direction. Part of the results obtained in the three-dimensional harmonic trap have been compared with the first available experiments. The two key signatures of supersolid behavior, namely the occurrence of two Goldstone modes and Non-Classical Rotational Inertia, can be detected, in this case, by studying the low-energy collective oscillations of the system. First, a behavior equivalent to the one of the two Goldstone modes predicted in the ring trap, can be found in the axial compressional oscillations of the harmonically trapped system, which bifurcate at the superfluid-supersolid phase transition. When the system is driven through the supersolid-independent droplet transition, the lower-energy mode, associated with phase coherence, tends to disappear, while the higher energy mode, associated with lattice excitations, tends to assume a constant frequency. This behavior is specular to the one of the two Goldstone modes in the ideal system, and thus signal the presence of supersolidity in the trapped system. Important experimental confirmation of the predictions reported in the thesis have already been found. Instead, as shown in the thesis, a key manifestation non-classical inertia in a trapped dipolar supersolid can be found by studying the rotational oscillation mode known as scissors mode, whose frequency is directly related to the value of the moment of inertia (similar to the frequency of oscillation of a torsional pendulum for a classical system). Studying the behavior of the frequency of the scissors mode across the superfluid-supersolid-independent droplets phase transitions, I demonstrate the actual occurrence of non-classical inertia in a harmonically trapped dipolar supersolid. Another key manifestation of superfluidity in general many-body systems is given by the occurrence of quantized vortices, which I study in the case of the trapped dipolar Bose gas in a harmonic trap which is isotropic in the plan orthogonal to the polarization direction. I study in particular the size of the core of the vortex as function of the interaction parameters, showing that, in the superfluid phase, it increases as the superfluid-supersolid phase transition is approached. Then, in the supersolid phase, I show that quantized vortices settle in the interstices between the density peaks, and their size and even their shape are fixed respectively by the droplet distance and the shape of the lattice cell. I also study the critical frequency for the vortex nucleation under a rotating quadrupolar deformation of the trap, showing that it is related to the frequency of the lower-energy quadrupole mode, associated with the partial superfluid character of the system. In fact, in this configuration, the quadrupole mode splits into three modes, two of which can be associated to lattice excitations, and one to superfluid excitations. I find that the critical rotational frequency for vortex nucleation is related to the lower frequency quadrupole mode only, i.e. the one related to the superfluid character of the system. In ordinary BECs, when many vortices nucleates, they typically tend to arrange in a trinagular lattice. In a supersolid, however, vortices do not form on top of a uniform superfluid background, but rather on the background of the supersolid lattice, which is itself typically triangular. I thus show that the lattice formed by the vortices in the supersolid lattice is not triangular, but rather hexagonal, since the vortices settle in the interstices between the density peaks. Finally, I show that all these features can be observed in an expansion experiment. In the last part of the thesis, I study the behavior of the dipolar Bose gas confined by hard walls. In particular, I investigate the novel density distributions, with special focus on the effects of supersolidity. Differently from the case of harmonic trapping, in this case, the ground state density shows a strong depletion in the bulk region and an accumulation of atoms near the walls, well separated from the bulk, as a consequence of the competition between the attractive and the repulsive nature of the dipolar force. In a quasi two-dimensional geometry characterized by cylindrical box trapping, the consequence is that the superfluid accumulating along the walls forms spontaneously a ring shape, showing eventually also supersolidity. For sufficiently large values of the atom density, also the bulk region can exhibit supersolidity, the resulting geometry reflecting the symmetry of the confining potential even for large systems.

Nous présentons dans ce manuscrit des mesures de corrélations spatiales à un et deux corps effectuées sur un gaz de bosons unidimensionnel et ultra-froid piégé à la surface d'une microstructure. Les corrélations à deux corps sont mises en évidence par des mesures de fluctuations de densité in situ ; les corrélations à un corps sont sondées grâce à des mesures de distributions en impulsion. Nous avons observé des fluctuations de densité sub-poissoniennes dans le régime d'interactions faibles, mettant ainsi en évidence pour la première fois le sous-régime du régime de quasi-condensat dans lequel la fonction de corrélation à deux corps est dominée par les fluctuations quantiques. Nous avons également observé des fluctuations sub-poissoniennes quelle que soit la densité dans le régime d'interactions fortes ; notre mesure constitue la première observation d'un unique gaz de bosons unidimensionnel dans ce régime. Le piège magnétique que nous avons utilisé est un piège modulé qui possède la propriété remarquable de découplage entre confinements transverse et longitudinal. Cette spécificité nous a permis de façonner à volonté la forme du confinement longitudinal. En particulier, nous avons pu obtenir des pièges harmoniques et quartiques. Nous avons également utilisé les propriétés de ce piège modulé afin de réaliser une lentille magnétique longitudinale. Cette technique nous a permis de mesurer la distribution en impulsion du gaz, dans le régime d'interactions faibles. Nous présentons deux résultats, obtenus de part et d'autre de la transition molle entre les régimes de gaz de Bose idéal et de quasi-condensat. Sur le plan théorique, nous montrons qu'une théorie de champ classique ne suffit pas à décrire quantitativement cette transition molle pour les paramètres typiques de l'expérience. Nous avons donc recours à des calculs Monte-Carlo quantiques. La température extraite de l'ajustement de nos donnée par ces calculs est en bon accord avec celle obtenue en ajustant les fluctuations de densité in situ avec la thermodynamique de C. N. Yang et C. P. Yang. Enfin, nous démontrons une méthode de compensation de la gravité (piégeage harmonique résiduel) lors de la phase de lentille magnétique, qui nous permet d'améliorer considérablement la résolution en impulsion de cette technique.

The physics of ultracold quantum gases has been the subject of a long-lasting and intense research activity, which started almost a century ago with purely theoretical studies and had a fluorishing experimental development after the implementation of laser and evaporative cooling techniques that led to the first realization of a Bose Einstein condensate (BEC) over 25 years ago. In recent years, a great interest in ultracold atoms has developed for their use as platforms for quantum technologies, given the high degree of control and tunability offered by ultracold atom systems. These features make ultracold atoms an ideal test bench for simulating and studying experimentally, in a controlled environment, physical phenomena analogous to those occurring in other, more complicated, or even inaccessible systems, which is the idea at the heart of quantum simulation. In the rapidly developing field of quantum technologies, it is highly important to acquire an in-depth understanding of the state of the quantum many-body system that is used, and of the processes needed to reach the desired state. The preparation of the system in a given target state often involves the crossing of second order phase transitions, bringing the system strongly out-of-equilibrium. A better understanding of the out-of-equilibrium processes occurring in the vicinity of the transition, and of the relaxation dynamics towards the final equilibrium condition, is crucial in order to produce well-controlled quantum states in an efficient way. In this thesis I present the results of the research activity that I performed during my PhD at the BEC1 laboratory of the BEC center, working on ultracold gases of 23Na atoms in an elongated harmonic trap. This work had two main goals: the accurate determination of the equilibrium properties of a Bose gas at finite temperature, by the measurement of its equation of state, and the investigation of the out-of-equilibrium dynamics occurring when a Bose Einstein condensate is prepared by cooling a thermal cloud at a finite rate across the BEC phase transition.To study the equilibrium physics of a trapped atomic cloud, it is crucial to be able to observe its density distribution in situ. This requires a high optical resolution to accurately obtain the density profile of the atomic distribution, from which thermodynamic quantities can then be extracted. In particular, in a partially condensed atomic cloud at finite temperature, it is challenging to resolve well also the boundaries of the BEC, where the condensate fraction rapidly drops in a narrow spatial region. This required an upgrade of the experimental apparatus in order to obtain a high enough resolution. I designed, tested and implemented in the experimental setup new imaging systems for all main directions of view. Particular attention was paid for the vertical imaging system, which was designed to image the condensates in trap with a resolution below 2 μm, with about a factor 4 improvement compared to the previous setup. The implementation of the new imaging systems involved a partial rebuilding of the experimental apparatus used for cooling the atoms. This created the occasion for an optimization of the whole system to obtain more stable working conditions. Concurrently I also realized and included in the experiment an optical setup for the use of a Digital Micromirror Device (DMD) to project time-dependent arbitrary light patterns on the atoms, creating optical potentials that can be controlled at will. The use of this device opens up exciting future scenarios where it will be possible to locally modify the trapping potential and to create well-controlled barriers moving through the atomic cloud. Another challenge in imaging the density distribution in situ is determined by the fact that the maximum optical density (OD) of the BEC, in the trap center, exceeds the low OD of the thermal tails by several orders of magnitude. In order to obtain an accurate image of the whole density profile, we developed a minimally destructive, multi-shot imaging technique, based on the partial transfer of a fraction of atoms to an auxiliary state, which is then probed. Taking multiple images at different extraction fractions, we are able to reconstruct the whole density profile of the atomic cloud avoiding saturation and maintaining a good signal to noise ratio. This technique, together with the improvements in the imaging resolution, has allowed us to accurately obtain the optical density profile of the Bose gas in trap, from which the 3D density profile was then calculated applying an inverse Abel transform, taking advantage of the symmetry of the trap. From images of the same cloud after a time-of-flight expansion, we measured the temperature of the gas. From these quantities we could find the pressure as a function of the density and temperature, determining the canonical equation of state of the weakly interacting Bose gas in equilibrium at finite temperature. These measurements also allowed us to clearly observe the non-monotonic temperature behavior of the chemical potential near the critical point for the phase transition, a feature that characterizes also other superfluid systems, but that had never been observed before in weakly interacting Bose gases. The second part of this thesis work is devoted to the study of the dynamical processes that occur during the formation of the BEC order parameter within a thermal cloud. The cooling at finite rate across the Bose-Einstein condensation transition brings the system in a strongly out-of-equilibrium state, which is worth investigating, together with the subsequent relaxation towards an equilibrium state. This is of interest also in view of achieving a better understanding of second order phase transitions in general, since such phenomena are ubiquitous in nature and relevant also in other platforms for quantum technologies. A milestone result in the study of second order phase transitions is given by the Kibble-Zurek mechanism, which provides a simple model capturing important aspects of the evolution of a system that crosses a second-order phase transition at finite rate. It is based on the principle that in an extended system the symmetry breaking associated with a continuous phase transition can take place only locally. This causes the formation of causally disconnected domains of the order parameter, at the boundaries of which topological defects can form, whose number and size scale with the rate at which the transition is crossed, following a universal power law. It was originally developed in the context of cosmology, but was later successfully tested in a variety of systems, including superfluid helium, superconductors, trapped ions and ultracold atoms. The BEC phase transition represents in this context a paradigmatic test-bench, given the high degree of control at which this second-order phase transition can be crossed by means of cooling ramps at different rates. Already early experiments investigated the formation of the BEC order parameter within a thermal cloud, after quasi-instantaneous temperature quenches or very slow evaporative cooling. In the framework of directly testing the Kibble-Zurek mechanism, further experiments were performed, both in 2D and 3D systems, focusing on the emergence of coherence and on the statistics of the spontaneously generated topological defects as a function of the cooling rate. The Kibble-Zurek mechanism, however, does not fully describe the out-of-equilibrium dynamics of the system at the transition, nor the post-quench interaction mechanisms between domains that lead to coarse-graining. Most theoretical models are based on a direct linear variation of a single control parameter, e.g. the temperature, across the transition. In real experiments, the cooling process is controlled by the tuning of other experimental parameters and a global temperature might not even be well defined, in a thermodynamic sense, during the whole process. Moreover, the temperature variation is usually accompanied by the variation of other quantities, such as the number of atoms and the collisional rate, making it difficult to accurately describe the system and predict the post-quench properties. Recent works included effects going beyond the Kibble-Zurek mechanism, such as the inhomogeneity introduced by the trapping potential, the role of atom number losses, and the saturation of the number of defects for high cooling rates. These works motivate further studies, in particular of the dynamics taking place at early times, close to the crossing of the critical point. The aim of the work presented in this thesis is to further investigate the timescales associated to the formation and evolution of the BEC order parameter and its spatial fluctuations, as a function of the rate at which the transition point is crossed. We performed experiments producing BECs by means of cooling protocols that are commonly used in cold-atom laboratories, involving evaporative cooling in a magnetic trap. We explored a wide range of cooling rates across the transition and found a universal scaling for the growth of the BEC order parameter with the cooling rate and a finite delay in its formation. The latter was already observed in earlier works, but for a much more limited range of cooling rates. The evolution of the fluctuations of the order parameter was also investigated, with an analysis of the timescale of their decay during the relaxation of the system, from an initial strongly out-of-equilibrium condition to a final equilibrium state. This thesis is structured as follows: The first chapter presents the theoretical background, starting with a brief introduction to the concept of Bose Einstein condensation and a presentation of different models describing the thermodynamics of an equilibrium Bose gas. The second part of this chapter then deals with the out-of-equilibrium dynamics that is inevitably involved in the crossing of a second-order phase transition such as the one for Bose-Einstein condensation. The Kibble-Zurek mechanism is briefly reviewed and beyond KZ effects are pointed out, motivating a more detailed investigation of the timescales involved in the BEC formation. In the second chapter, I describe the experimental apparatus that we use to cool and confine the atoms. Particular detail is dedicated to the parts that have been upgraded during my PhD, such as the imaging system. In the third chapter I show our experimental results on the measurement of the equation of state of the weakly interacting uniform Bose gas at finite temperature. In the fourth chapter I present our results on the out-of-equilibrium dynamics in the formation of the condensate order parameter and its spatial fluctuations, as a function of different cooling rates.

Ultra-relativisitic heavy ion collisions produce quark gluon plasma-a hot and dense soup of deconfined quarks and gluons akin to the early universe. We study two models in the context of these collisions namely, Polyakov Quark Meson Model (PQM) and Hadron Resonance Gas Model (HRGM).The PQM Model provides us with a simple and intuitive understanding of the QCD equation of state and thermodynamics at non zero temperature and baryon density while the HRGM is the principle model to analyse the hadron yields measured in these experiments across the entire range of beam energies. We study the effect of including the commonly neglected fermionic vacuum fluctuations to the (2+1) flavor PQM model. The conventional PQM model suffers from a rapid phase transition contrary to what is found through lattice simulations. Addition of the vacuum term tames the rapid transition and significantly improves the model’s agreement to lattice data. We further investigate the role of the vacuum term on the phase diagram. The smoothening effect of the vacuum term persists even at non zero . Depending on the value of the mass of the sigma meson, including the vacuum term results in either pushing the critical end point into higher values of the chemical potential or excluding the possibility of a critical end point altogether. We compute the fluctuations(correlations) of conserved charges up to sixth(fourth) order. Comparison is made with lattice data wherever available and overall good qualitative agreement is found, more so for the case of the normalised susceptibilities. The model predictions for the ratio of susceptibilities approach to that of an ideal gas of hadrons as in HRGM at low temperatures while at high temperature the values are close to that of an ideal gas of massless quarks. We examine the stability of HRGMs by extending them to take care of undiscovered resonances through the Hagedorn formula. We find that the influence of unknown resonances on thermodynamics is large but bounded. We model the decays of resonances and investigate the ratios of particle yields in heavy-ion collisions. We find that extending these models do not have much effect on hydrodynamics but the hadron yield ratios show better agreement with experiment. In principle HRGMs are internally consistent up to a temperature higher than the cross over temperature in QCD; but by examining quark number susceptibilities we find that their region of applicability seems to end even below the QCD cross over.

This chapter contains the essential statistical mechanics required to understand the inner workings of, and interpretation of results from, computer simulations. The microcanonical, canonical, isothermal–isobaric, semigrand and grand canonical ensembles are defined. Thermodynamic, structural, and dynamical properties of simple and complex liquids are related to appropriate functions of molecular positions and velocities. A number of important thermodynamic properties are defined in terms of fluctuations in these ensembles. The effect of the inclusion of hard constraints in the underlying potential model on the calculated properties is considered, and the addition of long-range and quantum corrections to classical simulations is presented. The extension of statistical mechanics to describe inhomogeneous systems such as the planar gas–liquid interface, fluid membranes, and liquid crystals, and its application in the simulation of these systems, are discussed.

After three papers on statistical mechanics, mostly duplicating work by Boltzmann and Gibbs, Einstein relied heavily on arguments from statistical mechanics in the most revolutionary of his famous 1905 papers, the one introducing the light‐quantum hypothesis. He showed that the equipartition theorem inescapably leads to the classical Rayleigh‐Jeans law for black‐body radiation and the ultraviolet catastrophe (as Ehrenfest later called it). Einstein and Ehrenfest were the first to point this out but the physics community only accepted it after the venerable H.A. Lorentz, came to the same conclusion in 1908. The central argument for light quanta in Einstein’s 1905 paper involves a comparison between fluctuations in black‐body radiation in the Wien regime and fluctuations in an ideal gas. From this comparison Einstein inferred that black‐body radiation in the Wien regime behaves as a collection of discrete, independent, and localized particles. We show that the same argument works for non‐localized quantized wave modes. Although nobody noticed this flaw in Einstein’s reasoning at the time, his fluctuation argument, and several others like it, failed to convince anybody of the reality of light quanta. Even Millikan’s verification of Einstein formula for the photoelectric effect only led to the acceptance of the formula, not of the theory behind it. Einstein’s quantization of matter was better received, especially his simple model of a solid consisting of quantized oscillators. This model could explain why the specific heats of solids fall off sharply as the temperature is lowered instead of remaining constant as it should according to the well‐known Dulong‐Petit law, which is a direct consequence of the equipartition theorem. The confirmation of Einstein’s theory of specific heats by Nernst and his associates was an important milestone in the development of quantum theory and a central topic at the first Solvay conference of 1911, which brought the fledgling theory to the attention of a larger segment of the physics community. Returning to the quantum theory after spending a few years on the development of general relativity, Einstein combined his light‐quantum hypothesis with elements of Bohr’s model of the atom in a new quantum radiation theory.

Nowadays, due to the advances in nanolithography technology it is possible to fabricate structures whose electronic properties correspond to that of a quasi-one-dimensional electron gas. Such structures allow us to observe ballistic quantum transport at low temperatures, and remarkable experimental observations have resulted1. Many theoretical studies have investigated conductance fluctuations2 and voltage controlled defects. Cahay et al3 studied the problem of localization associated with the conductance fluctuations of an array of elastic scatterers. Joe et al4 discussed the effects of a voltage controlled impurity for the conductance of a single open quantum box. As the impurity size is changed, it causes conductance oscillations due to the interference of circulating and bound states of the quantum box. In this paper we analyze how changes in geometry of a structure with three open dots affect its electronic properties.

We have investigated both the linear and nonlinear absorption of CuBr microcrystallites (MC’s). The linear absorption spectrum is dominated by pronounced exciton peaks similar to bulk CuBr. The quantum confinement is evidenced by the increasing blue shift of the peaks with decreasing MC radius. We concentrate on the situation of RMC≫REXCITON (experiment : factor 10), where fluctuations of RMC are smaller than 10% as found from X-ray studies and where the experimental spectra can be well explained in terms of the exciton centre-of-mass quantum confinement. At resonant excitation with dye laser pulses we observe a pronounced bleaching of the exciton absorption with rising intensity accompanied by a blue shift of the absorption maximum. We have calculated the energy-shift of the exciton resonance for a high-density exciton gas confined to a MC in Hartree-Fock approximation and good agreement with the experimental data is obtained. At 77K we find a Lorentzian saturation intensity of about 100 kW/cm2 and a contrast as large as 30. Operation of the nonlinearity at room temperature with no significant difference to 77K is demonstrated. The results evidence that excitons in MC's give rise to larger optical nonlinearities than in the bulk material.

We develop a statistical theory for laser operation with external optical feedback. The theory is applicable to a broad class of singlemode lasers including gas lasers and semiconductor lasers. We introduce a characteristic potential function in which each minimum corresponds to a possible mode frequency, while the corresponding second derivative is an immediate measure for the linewidth. The minima in this potential are shifted in position with respect to the power maxima by an amount mainly determined by the amplitudephase coupling or linewidth enhancement parameter. This explains why in highly dispersive lasers the preferred and most stable mode is not the one with highest power. A diffusion model is developed in which the spontaneous emission related quantum fluctuations are the driving noise forces. This leads to a Fokker-Pianck equation for the phase probability distribution function, which is the key equation for investigating dynamical stability, lifetimes and switching times. We determine the stationary probability distribution and calculate external mode lifetimes and hopping rates.