Abstract

We review the theory of the polariton condensate taking into account exciton–exciton interactions.

It is known [1] that coherent electromagnetic radiation resonant with an isolated exciton energy level excites in the crystal a coherent polariton wave with the wave vector ko ≠ 0 — the non-equilibrium polariton condensate. Different scattering processes accompanying its propagation lead to the loss of the initial coherence of the polariton wave, complete or partial depletion of the condensate, excitation of polaritons with wave vector k ≠ 0, and other phenomena.

In the present paper the effect of exciton–exciton scattering processes on the properties of a coherently excited polariton system are discussed. This scattering mechanism is of considerable interest due to recent experimental investigations [2] and many interesting results (see e.g. Refs. [3, 4]) obtained in theoretical study of dynamic and kinetic processes in a system of interacting polaritons.

According to Refs. [4, 5], exciton–exciton scattering is very important when coherent polaritons are excited in a certain spectral region in which energy and momentum conservation laws allow real processes of two-quantum excitation of polaritons from the condensate. These processes lead to the instability of the condensed state of the polariton system. The existence of this spectral region situated around the isolated exciton resonance is due to the peculiarities of the polariton dispersion relation.

In [5] the energy spectrum of non-condensate polaritons, arising as the result of decay of the coherent polariton wave, is studied.

Abstract

The influence of the radiative decay of excitonic molecules on a possible quasiequilibrium Bose–Einstein condensation (BEC) of excitonic molecule's is examined with respect to the radiative renormalization of the excitonic molecule energy (excitonic molecule Lamb shift). For the excitonic molecule wave function, a Schrödinger equation which contains polariton effects is derived and analyzed. Both the inverse excitonic molecule radiative lifetime γm and the biexciton Lamb shift Δm depend strongly on the total excitonic molecule momentum K. The energy renormalization Δm(K) leads to the excitonic molecule effective mass modification and can result in a camel-back structure at K = 0, which opposes a BEC of excitonic molecules at K = 0.

Introduction

Observations of a quasiequilibrium Bose–Einstein condensation (BEC) of excitonic molecules (EM) have been attempted [1, 2, 3] following its theoretical prediction [4, 5] (for reviews see, e.g. [6, 7]). Recent approaches [8, 9] with high-precision techniques renewed the interest in this phenomenon. Traditionally, one tries to detect BEC of the EMs in luminescence. In the new approach the appearance of coherence in the thermal system of the EMs has been tested by means of four-wave mixing and treated as a fundamental manifestation of BEC. (See the paper by Hasuo et al. in this book.) Both optical methods for the BEC detection imply that the optical transition to the corresponding intermediate exciton (IE) state is dipole-active. In this case the EM state is unstable against optical decay with a “giant” oscillator strength [6].

Abstract

The two-particle spectrum in a dense Fermion system is treated using a thermodynamic Green function approach. A self-consistent description of possible bound states and a superfluid condensate in a correlated medium is given.

A detailed understanding of superfluidity and superconductivity in correlated Fermion systems, especially the transition from the Cooper-paired state (weak-coupling limit) to the Bose condensed state of tightly bound pairs of Fermions (strong coupling limit) [1], is of great interest for very different physical systems. The problem of a unified treatment of Bose–Einstein condensation (BEC) and the Bardeen–Cooper–Schrieffer (BCS) phase arises not only in describing the electron structure of strongly correlated electron superfluids such as superconductors [2], the electronhole system in semiconductors [3], spin-polarized hydrogen and liquid He [4], but also in the theory of nuclear matter [5] and quark–gluon systems [6].

Recently, there have been several new approaches to this stimulating problem. A Monte-Carlo simulation of a finite He system at zero temperature has been performed in Ref. [7]. The microscopic theory of strongly coupled quantum fluids has been treated within the Jastrow approximation (cf. Ref. [8]) to obtain the ground state and low-lying excited states of strongly correlated boson quantum fluids. The crossover from weak to strong coupling superconductivity has been considered using a functional integral representation [2] (see also Ref. [7]).

Abstract

Using the exciton-mediated photovoltaic effect, we examine exciton transport over large distances in Cu2O as a function of temperature and particle density. Evidence for a phase transition at low temperatures and high densities is attributed to the onset of excitonic superfluidity.

We have performed exciton transport measurements over a range of temperatures and exciton densities in ultrapure, oriented large Cu2O single crystals. A sketch of the experimental method is shown in Fig. 1. The crystal is illuminated on the back surface by 10 ns pulses from a frequency-doubled YAG laser (λ = 532 nm). The initial exciton density created over an absorption depth (about one micron at λ = 532 nm) can be varied by inserting calibrated neutral density filters in the laser beam, reaching values of up to 1019 cm−3. The excitons which have migrated to the opposite face of the crystal are dissociated into free carriers by the high electric field near the Cu Schottky contact [1] deposited in a comb configuration together with an ohmic Au electrode, resulting in an external current. A time-resolved measurement of that current will give the velocity distribution of the excitons migrating through the crystal. This method of detection – as opposed to photoluminescence – is particularly well suited to the study of optically inactive paraexcitons in Cu2O; moreover, since the migration time is of the order of one microsecond as compared to the lifetime of 13 μs [2] for paraexcitons, recombination processes have little influence on the measurements.

Abstract

We study necessary conditions for the observation of Bose–Einstein condensation in a magnetically trapped sample of atomic cesium gas. These constraints are due to interatomic collisions in the sample. We show that the prospects for observing Bose–Einstein condensation are favorable for a gas of ground-state Cs atoms in the highest state of the lowest hyperfine manifold. An interesting aspect of the calculations is that the scattering length for this f = 3, mf = −3 hyperfine state shows pronounced resonance structures as a function of applied magnetic field leading to variations of two orders of magnitude. Most importantly, the scattering length can change sign near the resonances. This suggests a controllable means to change the behavior of the Bose condensate because for negative values a condensate is unstable and other (quantum-)collective effects might be observed. The origin of the resonances is understood from the bound singlet and triplet rovibrational Cs2 states which are perturbed due to the hyperfine and Zeeman interactions.

It is a long standing goal to achieve quantum-collective effects in atomic Fermi- or Bose gases. The prominent reasons are that for a relatively simple system with low density, a microscopic theoretical treatment of the phase transition is still feasible, and to have an experimental testing ground for more complicated quantum-coherent effects such as superfluidity in 4He and superconductivity in metals. Here, we focus on atomic species which behave as (composite) bosons.

Abstract

Various sum rules, accounting for the coupling between density and particle excitations and emphasizing in an explicit way the role of Bose–Einstein condensation, are discussed. Important consequences on the fluctuations of the particle operator as well as on the structure of elementary excitations are reviewed. These include a recent generalization of the Hohenberg–Mermin–Wagner theorem holding at zero temperature.

Introduction

The sum rule approach has been employed extensively in the literature in order to explore various dynamic features of quantum many body systems from a microscopic point of view (see [1] and references therein). An important merit of the method is its explicit emphasis on the role of conservation laws and of the symmetries of the problem. Furthermore, the explicit determination of sum rules is relatively easy and often requires only a limited knowledge of the system. Usually the sum rule approach is, however, employed without giving special emphasis to the possible occurrence of (spontaneously) broken symmetries. For example, the most famous f-sum rule [2] holding for a large class of systems is not affected by the existence of an order parameter in the system.

The purpose of this paper is to discuss a different class of sum rules which are directly affected by the presence of a broken symmetry. These sum rules can be used to predict significant properties of the system which are the consequence of the existence of an order parameter.

Abstract

The kinetics of the formation of coherent correlation properties associated with Bose condensation is studied in detail. The evolution of a nonequilibrium state with no “condensate-seed” is related to a hierarchy of relaxation times. At the first stage, a particle flux in energy space toward low energies sets in. The evolution in this case is described by a nonlinear Boltzmann equation, with a characteristic time given by interparticle collisions. When the particles which will later form the condensate have a kinetic energy which is less than the potential energy, a quasicondensate starts to form. In this stage, fluctuations of the density (but not of the phase) are suppressed and short-range coherent correlation properties are governed by the equation of motion for a quasiclassical complex field. The next stage is connected with the formation of the long-range order. The time for forming topological order and therefore genuine superfluidity proves to be dependent on the system size. The off-diagonal long-range order, arising after the attenuation of long-wave phase fluctuations, has a size-dependent relaxation time as well.

Introduction

The problem of Bose–Einstein Condensation (BEC) kinetics, being interesting in itself, has acquired a special significance in connection with experimental efforts to observe this condensation in a number of systems with particles with a finite lifetime. Such systems include spin-polarized atomic hydrogen [1], excitons [2] and biexcitons [3] in semiconductors and, more recently, laser-cooled atomic systems [4].

Abstract

Using magnetically trapped atomic hydrogen as an example, we investigate the prospects of achieving Bose–Einstein condensation in a dilute Bose gas. We show that, if the gas is quenched sufficiently far into the critical region of the phase transition, the typical time scale for the nucleation of the condensate density is short and of O(ħ/kBTc). As a result we find that thermalizing elastic collisions act as a bottleneck for the formation of the condensate. In the case of doubly polarized atomic hydrogen these occur much more frequently than the inelastic collisions leading to decay and we are led to the conclusion that Bose–Einstein condensation can indeed be achieved within the lifetime of the gas.

Introduction

In the last few years it has been clearly demonstrated that not only charged ions but also neutral atoms can be conveniently trapped and cooled by means of electro-magnetic fields. Although the physics of the various ingenious scenarios developed to accomplish this is already interesting in itself [1], the opportunities offered by an atomic gas sample at very low temperatures are exciting in their own right. Examples in this respect are the performance of high-precision spectroscopy, the search for a violation of CP invariance by measuring the electric dipole moment of atomic cesium [2], the construction of an improved time standard based on an atomic fountain [3] and the achievement of Bose–Einstein condensation in a weakly interacting gas.

Abstract

We show that the natural variable to follow the crossover from Cooper-pair-based superconductivity to Bose–Einstein condensation within the model of Noziéres and Schmitt-Rink is the product kFξ, where kF is the Fermi wave vector and ξ, is the coherence length for two-electron correlation. In terms of this product, the results of the model do not depend on the detailed form of the (separable) pairing potential, and the crossover turns out to be restricted to the universal region π−1≲kFξ ≲ 2π. Experimental estimates indicate that kFξ ≈10 (> 2π) for high-Tc superconductors.

Evolution from weak to strong coupling superconductivity has been considered by Nozières and Schmitt-Rink [1] (hereafter referred to as NSR) following the pioneering work by Leggett [2]. After the discovery of high- Tc superconductivity, the interest in this problem has grown, and many papers on this subject have appeared [3]. In the present work, we show that working within the simplified treatment by NSR, it is already possible to isolate the essential features of the crossover.

Central to the work of NSR and Leggett is the argument [4] that the BCS wave function has the Bose–Einstein condensation (BEC) built in as a limiting case. (See the review by Randeria in this volume.) NSR study the evolution from BCS to BEC through the increase of the coupling strength associated to an effective fermionic attractive potential, and conclude that the evolution is “smooth”.

In recent years, the phenomenon of Bose-Einstein condensation (BEC) has become an increasingly active area of research, both experimentally and theoretically. Long associated with the study of superfluid 4He and 3He-4He mixtures, current research increasingly deals with other condensed matter systems, including spin-polarized hydrogen, excitons, laser-cooled atoms, high-temperature superconductors, and subatomic matter.

The present volume contains a series of authoritative review articles on current BEC research and related phenomena. The editors invited the leading experts to review their research field in such a way that their articles would introduce non-experts to current research and at the same time highlight some of the most promising topics for study in the next decade. These articles contain new material which is not available elsewhere. This is the first book devoted to BEC as an inter-disciplinary subject in its own right. It covers research in atomic and molecular physics, laser physics, low temperature physics, subatomic physics and astrophysics.

In the opening chapter, Snoke and Baym introduce the various review articles by discussing them in the context of the dominant themes in current BEC studies. These themes include broken gauge invariance, phase coherence in equilibrium and non-equilibrium situations, time scales for the formation of a Bose condensate, Bose particles with a finite lifetime, and BEC vs. BCS superfluidity in fermionic systems with attractive interactions.

The interdisciplinary character of modern BEC research has led to its being largely ignored by the regular low temperature conferences.

Abstract

Liquid helium is the prototypical example of a superfluid – a liquid that flows without viscosity and transfers heat without a temperature gradient. These properties are intimately related to the Bose condensation that occurs in this strongly interacting liquid. Bose condensation is most directly observed in the single particle atomic momentum distribution, where the Bose condensate appears as a delta function singularity. In this article, we discuss the experimental techniques used to observe the condensate and the current status of measurments of the Bose condensate in liquid helium.

Introduction

Liquid helium (4He) has fascinated physicists ever since Kammerlingh-Onnes liquified the last of the so called permanent gases in 1908. However, evidence of what is without doubt the most fascinating property of this unique liquid, superfluidity, was not reported until almost 25 years later [1]. The superfluid phase, where heat is transferred without a thermal gradient and mass flows without a driving pressure, is a macroscopic manifestation of microscopic quantum effects governing the behavior of atoms [2]. Bose–Einstein condensation was first proposed by London [3] as the microscopic explanation for these fascinating phenomena.

Helium is unique among condensed atomic systems since the bulk properties of the liquid are dominated by quantum effects. The most important of these are the statistical effects that arise for identical particles. For bosons, such as 4He atoms, there is no limitation on the number of particles that can occupy a single quantum state.

Abstract

The case of excitons as candidates for Bose–Einstein condensation is discussed, and experimental results in CuCl and Cu2O are presented. In CuCl, spectral analysis of the luminescence from biexcitons as a function of their density reveals a gradual evolution from classical statistics towards a quantum degenerate regime. The appearance of a sharp emission line below a critical temperature and above a critical density is attributed to the presence of a laser-induced Bose–Einstein condensate of excitonic molecules. This interpretation is supported by pump-probe experiments which show that additional particles injected in the presence of a biexciton condensate are drawn into it.

In Cu2O, free exciton luminescence spectral analysis of ortho- and paraexcitons reveals a gradual evolution from a classical to a Bose quantum degenerate regime with increasing particle densities. Orthoexciton densities close to the critical density for condensation are obtained at high incoherent excitation. Under similar pumping, paraexciton densities exceeding the critical value are inferred from luminescence intensity ratios. Anomalous transport properties of paraexcitons, such as ballistic propagation over macroscopic distances and formation of soliton-like excitonic packets are discussed as evidence for excitonic superfluidity.

Introduction

Excitons

The lowest electronically excited state of a non-metallic crystal corresponds to the promotion of one electron from the top of the highest fully occupied valence band to the bottom of the next empty conduction band. A correct evaluation of the required energy must include the Coulomb correlation between the promoted electron and all other electrons left behind in the valence band.