We show that the phase of a condensate in a finite temperature gas spreads linearly in time at long times rather than in a diffusive way. This result is supported by classical field simulations, and analytical calculations which are generalized to the quantum case under the assumption of quantum ergodicity in the system. This superdiffusive behavior is intimately related to conservation of energy during the free evolution of the system and to fluctuations of energy in the prepared initial state.
We show that the phase of a condensate in a finite temperature gas spreads linearly in time at long times rather than in a diffusive way. This result is supported by classical field simulations, and analytical calculations which are generalized to the quantum case under the assumption of quantum ergodicity in the system. This superdiffusive behavior is intimately related to conservation of energy during the free evolution of the system and to fluctuations of energy in the prepared initial state.
A condensate of pairs in an isolated, homogeneous, unpolarised, finite-size spin 1/2 Fermi gas at low nonzero temperature T, undergoes with time a phase change with a random component, due to coupling to the gas thermal phonons. With the quantum second Josephson relation relating the derivative of the condensate phase operator to the phonon occupation numbers, and linearised kinetic equations giving the evolution of occupation number fluctuations, we access the behaviour of the phase change variance at times much longer than the phonon collision time. The case of a convex phonon branch is similar to the Bose gas case: the leading collisional processes are the Beliaev-Landau 3-phonons processes, and the variance is the sum of a ballistic term and of a delayed diffusive term, whose analytical expressions are given in the thermodynamic limit. The concave case is more exotic. It is analysed at time scales T−9, allowing one to restrict to 2 phonons → 2 phonons small-angle Landau-Khalatnikov processes. The total number of phonons is conserved and the phonon mean occupation numbers at equilibrium can exhibit a negative chemical potential, assumed isotropic. The phase change variance is then the sum of a ballistic term, of a diffusive term, of exotic subsubleading terms and of a constant term. The analytic expression of some of the corresponding coefficients is obtained, as well as the diverging leading behavior of the other ones when the phonon chemical potential tends to 0. When this chemical potential is 0, the variance sub-ballistic part becomes superdiffusive, with an exponent 5/3 and an exactly-known coefficient. For a nonzero infinitesimal phonon chemical potential, a law is found, interpolating between superdiffusive and diffusive phase spreading. Also new results are obtained on the phonon Landau-Khalatnikov damping rate, in particular at negative phonon chemical potential.
Temporal coherence is a fundamental property of macroscopic quantum systems, such as lasers in optics and Bose-Einstein condensates in atomic gases and it is a crucial issue for interferometry applications with light or matter waves. Whereas the laser is an ``open'' quantum system, ultracold atomic gases are weakly coupled to the environment and may be considered as isolated. The coherence time of a condensate is then intrinsic to the system and its derivation is out of the frame of laser theory. Using quantum kinetic theory, we predict that the interaction with noncondensed modes gradually smears out the condensate phase, with a variance growing as At2+Bt+C at long times t, and we give a quantitative prediction for A, B, and C. Whereas the coefficient A vanishes for vanishing energy fluctuations in the initial state, the coefficients B and C are remarkably insensitive to these fluctuations. The coefficient B describes a diffusive motion of the condensate phase that sets the ultimate limit to the condensate coherence time. We briefly discuss the possibility to observe the predicted phase spreading, also including the effect of particle losses.
Within the classical field model, we find that the phase of a uniform Bose-Einstein condensate at nonzero temperature undergoes a true diffusive motion in the microcanonical ensemble, the variance of the condensate phase change between time zero and time t growing linearly in t. The phase diffusion coefficient obeys a simple scaling law in the double thermodynamic and Bogoliubov limit. We construct an approximate calculation of the diffusion coefficient, in fair agreement with the numerical results over the considered temperature range, and we extend this approximate calculation to the quantum field.
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].
The central problem of this chapter is temporal coherence of a three-dimensional spatially homogeneous Bose-condensed gas, initially prepared at finite temperature and then evolving as an isolated interacting system. A first theoretical tool is a number-conserving Bogoliubov approach that allows to describe the system as a weakly interacting gas of quasi-particles. This approach naturally introduces the phase operator of the condensate: a central actor since loss of temporal coherence is governed by the spreading of the condensate phase-change. A second tool is the set of kinetic equations describing the Beliaev-Landau processes for the quasi-particles. We find that in general the variance of the condensate phase-change at long times t is the sum of a ballistic term ∝t2 and a diffusive term ∝t with temperature and interaction dependent coefficients. In the thermodynamic limit, the diffusion coefficient scales as the inverse of the system volume. The coefficient of t2 scales as the inverse volume squared times the variance of the energy of the system in the initial state and can also be obtained by a quantum ergodic theory (the so-called eigenstate thermalisation hypothesis).
In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.