Vladimir V. Kocharovsky

Fluctuations in Ideal and Interacting Bose–Einstein Condensates: From the Laser Phase Transition Analogy to Squeezed States and Bogoliubov Quasiparticles ⁎

We review the phenomenon of equilibrium fluctuations in the number of condensed atoms n 0 in a trap containing N atoms total. We start with a history of the Bose–Einstein distribution, a similar grand canonical problem with an indefinite total number of particles, the Einstein–Uhlenbeck debate concerning the rounding of the mean number of condensed atoms n ¯ 0 near a critical temperature T c , and a discussion of the relations between statistics of BEC fluctuations in the grand canonical, canonical, and microcanonical ensembles. First, we study BEC fluctuations in the ideal Bose gas in a trap and explain why the grand canonical description goes very wrong for all moments 〈 ( n 0 − n ¯ 0 ) m 〉 , except of the mean value. We discuss different approaches capable of providing approximate analytical results and physical insight into this very complicated problem. In particular, we describe at length the master equation and canonical-ensemble quasiparticle approaches which give the most accurate and physically transparent picture of the BEC fluctuations. The master equation approach, that perfectly describes even the mesoscopic effects due to the finite number N of the atoms in the trap, is quite similar to the quantum theory of the laser. That is, we calculate a steady-state probability distribution of the number of condensed atoms p n 0 ( t = ∞ ) from a dynamical master equation and thus get the moments of fluctuations. We present analytical formulas for the moments of the ground-state occupation fluctuations in the ideal Bose gas in the harmonic trap and arbitrary power-law traps. In the last part of the review, we include particle interaction via a generalized Bogoliubov formalism and describe condensate fluctuations in the interacting Bose gas. In particular, we show that the canonical-ensemble quasiparticle approach works very well for the interacting gases and find analytical formulas for the characteristic function and all cumulants, i.e., all moments, of the condensate fluctuations. The surprising conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit. We also resolve the Giorgini, Pitaevskii and Stringari (GPS) vs. Idziaszek et al. debate on the variance of the condensate fluctuations in the interacting gas in the thermodynamic limit in favor of GPS. Furthermore, we clarify a crossover between the ideal-gas and weakly-interacting-gas statistics which is governed by a pair-correlation, squeezing mechanism and show how, with an increase of the interaction strength, the fluctuations can now be understood as being essentially 1/2 that of an ideal Bose gas. We also explain the crucial fact that the condensate fluctuations are governed by a singular contribution of the lowest energy quasiparticles. This is a sort of infrared anomaly which is universal for constrained systems below the critical temperature of a second-order phase transition.

Self-similar analytical solution of the critical fluctuations problem for the Bose–Einstein condensation in an ideal gas

Paper is published in J. Phys. A: Math. Theor. 43 (2010) 225001, doi:10.1088/1751-8113/43/22/225001. Exact analytical solution for the universal probability distribution of the order parameter fluctuations as well as for the universal statistical and thermodynamic functions of an ideal gas in the whole critical region of Bose-Einstein condensation is obtained. A universal constraint nonlinearity is found that is responsible for all nontrivial critical phenomena of the BEC phase transition. Simple analytical approximations, which describe the universal structure of the critical region in terms of confluent hypergeometric or parabolic cylinder functions, as well as asymptotics of the exact solution are derived. The results for the order parameter, all higher-order moments of BEC fluctuations, and thermodynamic quantities, including specific heat, perfectly match the known asymptotics outside critical region as well as the phenomenological renormalization-group ansatz with known critical exponents in the close vicinity of the critical point. Thus, a full analytical solution to a long-standing problem of finding a universal structure of the lambda-point for BEC in an ideal gas is found.

Microscopic theory of a phase transition in a critical region: Bose-Einstein condensation in an interacting gas

Abstract We present a microscopic theory of the second-order phase transition in an interacting Bose gas that allows one to describe formation of an ordered condensate phase from a disordered phase across an entire critical region continuously. We derive the exact fundamental equations for a condensate wave function and the Greens functions, which are valid both inside and outside the critical region. They are reduced to the usual Gross–Pitaevskii and Beliaev–Popov equations in a low-temperature limit outside the critical region. The theory is readily extendable to other phase transitions, in particular, in the physics of condensed matter and quantum fields.

Microscopic theory of phase transitions in a critical region

The problem of finding a microscopic theory of phase transitions across a critical point is a central unsolved problem in theoretical physics. We find a general solution to that problem and present it here for the cases of Bose-Einstein condensation in an interacting gas and ferromagnetism in a lattice of spins, interacting via a Heisenberg or Ising Hamiltonian. For Bose-Einstein condensation, we present the exact, valid for the entire critical region, equations for the Green functions and order parameter, that is a critical-region extension of the Beliaev-Popov and Gross-Pitaevskii equations. For the magnetic phase transition, we find an exact theory in terms of constrained bosons in a lattice and obtain similar equations for the Green functions and order parameter. In particular, we outline an exact solution for the three-dimensional Ising model.

Coherent radiation from neutral molecules moving above a grating.

We predict and study the effect of parametric self-induced excitation of a molecule moving above the dielectric or conducting medium with periodic grating. In this case the radiation reaction force modulates the molecular transition frequency which results in a parametric instability of dipole oscillations even from the level of quantum or thermal fluctuations. The present mechanism of instability of electrically neutral molecules is different from that of the well-known Smith-Purcell and transition radiation in which a moving charge and its oscillating image create an oscillating dipole. We show that parametrically excited molecular bunches can produce an easily detectable coherent radiation flux of up to a microwatt.

The breaks and the hidden components in the power-law spectra of synchrotron radiation of the self-consistent current structures

Widespread use of a broken-power-law description of the spectra of synchrotron emission of various plasma objects requires an analysis of origin and a proper interpretation of spectral components. We show that, for a self-consistent magnetic configuration in a collisionless plasma, these components may be angle-dependent according to an anisotropic particle momentum distribution and may have no counterparts in a particle energy distribution. That has never been studied analytically and is in contrast to a usual model of synchrotron radiation, assuming an external magnetic field and a particle ensemble with isotropic momentum distribution. We demonstrate that for the wide intervals of observation angle the power-law spectra and, in particular, the positions and number of spectral breaks may be essentially different for the cases of the self-consistent and not-self-consistent magnetic fields in current structures responsible for the synchrotron radiation of the ensembles of relativistic particles with the multi-power-law energy distributions.

Intracavity nonlinear optics of semiconductor nanostructures and new mid/far-infrared laser schemes

Recent theoretical and experimental results in nonlinear optics of semiconductor nanostructures and intracavity nonlinear mixing of laser modes are reviewed. A comparative analysis of the implemented and newly suggested schemes for the difference- and sum-frequency, second-harmonic, and parametric generation is presented. In particular, the problems of nonlinear photonic crystals, true- and quasi-phase-matching designs, and the use of the longitudinal, transverse, and 2D (metal or dielectric) gratings in the surface-emitting semiconductor devices are discussed. The most promising schemes of nonlinear-mixing lasers, both quantum cascade (intersubband) and diode-type (interband), are described. Finally, numerous applications of nonlinear-mixing lasers and open issues in physics and technology posed by the problem are given.

Terahertz sources based on the intracavity wave mixing of self-generated fields in semiconductor lasers

A concept of intracavity nonlinear wave mixing using modes generated in semiconductor injection lasers as an intracavity optical pump for the mixing is developed. This approach utilizes very high nonlinear susceptibilities of III-v semiconductors that cannot be used as a passive externally pumped crystal because of the strong absorption of the optical pump and the absence of a convenient phase-matching scheme. In particular, it opens prospects for mastering the terahertz frequency range on the basis of a standard heterolaser technology. We review recent theoretical and experimental investigations in the field and discuss applications.

Anomalous Statistics of Bose-Einstein Condensate in an Interacting Gas: An Effect of the Trap’s Form and Boundary Conditions in the Thermodynamic Limit

We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap’s form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit—only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap’s form and boundary conditions.

An approach of the space-time empirical modes to the nonlinear phenomena in lasers with low-q cavities

Highly nonlinear spatial-temporal patterns are inevitable features of lasing in the low-Q (bad) cavities where a photon lifetime, TE, is less than a polarization relaxation time (a lifetime of the optical dipole oscillations), T2, of the individual active centers excited by pumping [1]. A theoretical analysis of the spatial-temporal dynamics of the field and its spectral and correlation features in such lasers, known as the superradiant lasers, cannot be based on a standard decomposition on neither ‘cold’ no ‘hot’ modes which have fixed spatial profiles and are defined, respectively, by a cavity without or with taking into account the active medium. A progress in modern technologies, especially in the field of semiconductor heterostructures, leaves no doubts about near fabrication of these lasers and their applications in the optical information processing, the wideband dynamical spectroscopy, and diagnostics of many-particle systems in condensed active media.