Number Of Particles In Condensate Research Articles

Number-conserving master equation theory for a dilute Bose-Einstein condensate

We describe the transition of $N$ weakly interacting atoms into a Bose-Einstein condensate within a number-conserving quantum master equation theory. Based on the separation of time scales for condensate formation and noncondensate thermalization, we derive a master equation for the condensate subsystem in the presence of the noncondensate environment under the inclusion of all two-body interaction processes. We numerically monitor the condensate particle number distribution during condensate formation, and derive a condition under which the unique equilibrium steady state of a dilute, weakly interacting Bose-Einstein condensate is given by a Gibbs-Boltzmann thermal state of $N$ noninteracting atoms.

Manipulating nonclassical quantum statistical properties of light field by an f-deformed Bose–Einstein condensate

We consider the interaction between an f-deformed Bose–Einstein condensate and a single-mode quantized light field. By using the Gardiner’s phonon operators, we find that there exists a natural deformation in the model which modifies the Bogoliubov approximation under the condition of large but finite number of particles in condensate. This approach introduces an intrinsically deformed Bose–Einstein condensate, where the deformation parameter, well-defined by the particle number N in condensate, controls the strength of the associated nonlinearity. By introducing the deformed Gardiner’s phonon operators we modify the very dilute-gas approximation through including atomic collisions in condensate. The rate of atomic collisions κ , as a new deformation parameter in the deformed Bose–Einstein condensate, controls the nonlinearity related to the atomic collisions. We show that by controlling the nonlinearities in the f-deformed atomic condensate through the two atomic parameters N and κ , it is possible to generate and manipulate the nonclassical quantum statistical properties of radiation field, such as, the sub-Poissonian photon statistics and quadrature squeezing. Also, it is possible to control the collapses and revivals phenomena in the average number of photons by atomic parameters N and κ .

Bogoliubov Transformations and Quantum Decoherence in an Atomic Bose-Einstein Condensate

In this brief review, we discuss quantum decoherence of an atomic Bose-Einstein condensate using Bogoliubov transformation. In particular, condensate phase is introduced and its diusion is studied. Validity of the Bogoliubov prescription is investigated and restored via the condensate phase. Role of quantum state of the condensate on the phase collapse is examined. Relation between the uctuations in the condensate particle number and the dynamics of the condensate phase is claried.

Number-conserving approach to a minimal self-consistent treatment of condensate and noncondensate dynamics in a degenerate Bose gas

We describe a number-conserving approach to the dynamics of Bose-Einstein condensed dilute atomic gases. This builds upon the works of Gardiner [Phys. Rev. A 56, 1414 (1997)] and Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. We consider what is effectively an expansion in powers of the ratio of noncondensate to condensate particle numbers, rather than inverse powers of the total number of particles. This requires the number of condensate particles to be a majority, but not necessarily almost equal to the total number of particles in the system. We argue that a second-order treatment of the relevant dynamical equations of motion is the minimum order necessary to provide consistent coupled condensate and noncondensate number dynamics for a finite total number of particles, and show that such a second-order treatment is provided by a suitably generalized Gross-Pitaevskii equation, coupled to the Castin-Dum number-conserving formulation of the Bogoliubov--de Gennes equations. The necessary equations of motion can be generated from an approximate third-order Hamiltonian, which effectively reduces to second order in the steady state. Such a treatment as described here is suitable for dynamics occurring at finite temperature, where there is a significant noncondensate fraction from the outset, or dynamics leading to dynamical instabilities, where depletion of the condensate can also lead to a significant noncondensate fraction, even if the noncondensate fraction is initially negligible.

Condensation ofNInteracting Bosons: A Hybrid Approach to Condensate Fluctuations

We present a new method of calculating the distribution function and fluctuations for a Bose-Einstein condensate (BEC) of N interacting atoms. The present formulation combines our previous master equation and canonical ensemble quasiparticle techniques. It is applicable both for ideal and interacting Bogoliubov BEC and yields remarkable accuracy at all temperatures. For the interacting gas of 200 bosons in a box we plot the temperature dependence of the first four central moments of the condensate particle number and compare the results with the ideal gas. For the interacting mesoscopic BEC, as with the ideal gas, we find a smooth transition for the condensate particle number as we pass through the critical temperature.

Fluctuation statistics of mesoscopic Bose-Einstein condensates: Reconciling the master equation with the partition function to reexamine the Uhlenbeck-Einstein dilemma

The atom fluctuation statistics of an ideal, mesoscopic, Bose-Einstein condensate are investigated from several different perspectives. By generalizing the grand canonical analysis (applied to the canonical ensemble problem), we obtain a self-consistent equation for the mean condensate particle number that coincides with the microscopic result calculated from the laser master equation approach. For the case of a harmonic trap, we obtain an analytic expression for the condensate particle number that is very accurate at all temperatures, when compared with numerical canonical ensemble results. Applying a similar generalized grand canonical treatment to the variance, we obtain an accurate result only below the critical temperature. Analytic results are found for all higher moments of the fluctuation distribution by employing the stochastic path integral formalism, with excellent accuracy. We further discuss a hybrid treatment, which combines the master equation and stochastic path integral analysis with results obtained based on the canonical ensemble quasiparticle formalism [Kocharovsky et al., Phys. Rev. A 61, 053606 (2000)], producing essentially perfect agreement with numerical simulation at all temperatures.

Superfluidity without symmetry breaking: The time-dependent Hartree-Fock approximation for Bose-condensed condensates

We develop a time-dependent Hartree-Fock approximation that is appropriate for Bose-condensed systems. Defining a {\it depletion Green's function} allows the construction of condensate and depletion particle densities from eigenstates of a single time-dependent Hamiltonian, guaranteeing that our approach is a conserving approximation. The poles of this Green's function yield the energies of number-changing excitations for which the condensate particle number is held fixed, which we show has a gapped spectrum in the superfluid state. The linearized time-dependent version of this has poles at the collective frequencies of the system, yielding the expected zero sound mode for a uniform infinite system. We show how the approximations may be expressed in a general linear response formalism.

Number-of-particle fluctuations and stability of Bose-Einstein-condensed systems

In this paper we show that a normal total number-of-particle fluctuation can be obtained consistently from the static thermodynamic relation and dynamic compressibility sum rule. In models using the broken U(1) gauge symmetry, in order to keep the consistency between statics and dynamics, it is important to identify the equilibrium state of the system with which the density response function is calculated, so that the condensate particle number ${N}_{0}$, the number of thermal depletion particles $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}$, and the number of noncondensate particles ${N}_{\mathrm{NC}}$ can be unambiguously defined. We also show that the chemical potential determined from the Hugenholtz-Pines theorem should be consistent with that determined from the equilibrium equation of state. The ${N}^{4∕3}$ anomalous fluctuation of the number of noncondensate particles is an intrinsic feature of the broken U(1) gauge symmetry. However, this anomalous fluctuation does not imply the instability of the system. Using the random phase approximation, which preserves the U(1) gauge symmetry, such an anomalous fluctuation of the number of noncondensate particles is completely absent.

Sub-shot-noise phase sensitivity with a Bose-Einstein condensate Mach-Zehnder interferometer

Bose Einstein Condensates, with their coherence properties, have attracted wide interest for their possible application to ultra precise interferometry and ultra weak force sensors. Since condensates, unlike photons, are interacting, they may permit the realization of specific quantum states needed as input of an interferometer to approach the Heisenberg limit, the supposed lower bound to precision phase measurements. To this end, we study the sensitivity to external weak perturbations of a representative matter-wave Mach-Zehnder interferometer whose input are two Bose-Einstein condensates created by splitting a single condensate in two parts. The interferometric phase sensitivity depends on the specific quantum state created with the two condensates, and, therefore, on the time scale of the splitting process. We identify three different regimes, characterized by a phase sensitivity $\Delta \theta$ scaling with the total number of condensate particles $N$ as i) the standard quantum limit $\Delta \theta \sim 1/N^{1/2}$, ii) the sub shot-noise $\Delta \theta \sim 1/N^{3/4}$ and the iii) the Heisenberg limit $\Delta \theta \sim 1/N$. However, in a realistic dynamical BEC splitting, the 1/N limit requires a long adiabaticity time scale, which is hardly reachable experimentally. On the other hand, the sub shot-noise sensitivity $\Delta \theta \sim 1/N^{3/4}$ can be reached in a realistic experimental setting. We also show that the $1/N^{3/4}$ scaling is a rigorous upper bound in the limit $N \to \infty$, while keeping constant all different parameters of the bosonic Mach-Zehnder interferometer.

Phase dynamics after connection of two separate Bose-Einstein condensates

We study the dynamics of the relative phase following the connection of the two independently formed Bose-Einstein condensates. Dissipation is assumed to be due to the creation of quasiparticles induced by a fluctuating condensate particle number. The coherence between different values of the phase, which is characteristic of the initial Fock state, is quickly lost after the net exchange of a few atoms has taken place. This process effectively measures the phase and marks the onset of a semiclassical regime in which the system undergoes Bloch oscillations around the initial particle number. These fast oscillations excite quasiparticles within each condensate and the system relaxes at a longer time scale until it displays low-energy, damped, Josephson plasma oscillations, eventually coming to a halt when the equilibrium configuration is finally reached.

Many-particle entanglement in two-component Bose-Einstein condensates

We investigate schemes to dynamically create many particle entangled states of a two component Bose-Einstein condensate in a very short time proportional to 1/N where $N$ is the number of condensate particles. For small $N$ we compare exact numerical calculations with analytical semiclassical estimates and find very good agreement for $N \geq 50$. We also estimate the effect of decoherence on our scheme, study possible scenarios for measuring the entangled states, and investigate experimental imperfections.

Influence of boundary conditions on statistical properties of ideal Bose-Einstein condensates.

We investigate the probability distribution that governs the number of ground-state particles in a partially condensed ideal Bose gas confined to a cubic volume within the canonical ensemble. Imposing either periodic or Dirichlet boundary conditions, we derive asymptotic expressions for all its cumulants. Whereas the condensation temperature becomes independent of the boundary conditions in the large-system limit, as implied by Weyl's theorem, the fluctuation of the number of condensate particles and all higher cumulants remain sensitive to the boundary conditions even in that limit. The implications of these findings for weakly interacting Bose gases are briefly discussed.

Master equation vs. partition function: canonical statistics of ideal Bose–Einstein condensates

Within the canonical ensemble, a partially condensed ideal Bose gas with arbitrary single-particle energies is equivalent to a system of uncoupled harmonic oscillators. We exploit this equivalence for deriving a formula which expresses all cumulants of the canonical distribution governing the number of condensate particles in terms of the poles of a generalized Zeta function provided by the single-particle spectrum. This formula lends itself to systematic asymptotic expansions which capture the non-Gaussian character of the condensate fluctuations with utmost precision even for relatively small, finite systems, as confirmed by comparison with exact numerical calculations. We use these results for assessing the accuracy of a recently developed master equation approach to the canonical condensate statistics; this approach turns out to be quite accurate even when the master equation is solved within a simple quasithermal approximation. As a further application of the cumulant formula we show that, and explain why, all cumulants of a homogeneous Bose–Einstein condensate “in a box” higher than the first retain a dependence on the boundary conditions in the thermodynamic limit.

Stability of Attractive Bose–Einstein Condensates

We propose the critical nonlinear Schrodinger equation with a harmonic potential as a model of attractive Bose–Einstein condensates. By an elaborate mathematical analysis we show that a sharp stability threshold exists with respect to the number of condensate particles. The value of the threshold agrees with the existing experimental data. Moreover with this threshold we prove that a ground state of the condensate exists and is orbital stable. We also evaluate the minimum of the condensate energy.

Condensate Fluctuations in Trapped Bose Gases: Canonical vs. Microcanonical Ensemble

We study the fluctuation of the number of particles in ideal Bose–Einstein condensates, both within the canonical and the microcanonical ensemble. Employing the Mellin–Barnes transformation, we derive simple expressions that link the canonical number of condensate particles, its fluctuation, and the difference between canonical and microcanonical fluctuations to the poles of a Zeta function that is determined by the excited single-particle levels of the trapping potential. For the particular examples of one- and three-dimensional harmonic traps we explore the microcanonical statistics in detail, with the help of the saddle-point method. Emphasizing the close connection between the partition theory of integer numbers and the statistical mechanics of ideal Bosons in one-dimensional harmonic traps, and utilizing thermodynamical arguments, we also derive an accurate formula for the fluctuation of the number of summands that occur when a large integer is partitioned.

Maxwell's Demon at work: Two types of Bose condensate fluctuations in power-law traps

After discussing the idea underlying the Maxwell's Demon ensemble, we employ this ensemble for calculating fluctuations of ideal Bose gas condensates in traps with power-law single-particle energy spectra. Two essentially different cases have to be distinguished. If the heat capacity is continuous at the condensation point, the fluctuations of the number of condensate particles vanish linearly with temperature, independent of the trap characteristics. In this case, microcanonical and canonical fluctuations are practically indistinguishable. If the heat capacity is discontinuous, the fluctuations vanish algebraically with temperature, with an exponent determined by the trap, and the micro-canonical fluctuations are lower than their canonical counterparts.

Stability of Bose condensed atomic 7Li.

We study the stability of a Bose condensate of atomic $^7$Li in a (harmonic oscillator) magnetic trap at non-zero temperatures. In analogy to the stability criterion for a neutron star, we conjecture that the gas becomes unstable if the free energy as a function of the central density of the cloud has a local extremum which conserves the number of particles. Moreover, we show that the number of condensate particles at the point of instability decreases with increasing temperature, and that for the temperature interval considered, the normal part of the gas is stable against density fluctuations at this point.

Dielectric approach to a dense charged boson gas atT=0

The generalized dielectric constant is calculated to the order next to the Bogoliubov approximation. This is done by using the analogy between the condensed boson system and a fictitious fermion system with spin degeneracy equal to the total number of particles (instead of two). From the zero of the dielectric constant, we have calculated the first-order correction to the Bogoliubov plasmon energy and the half linewidth of the plasmon states. The real part of the solution is examined from a graphical view point to show that up to the specified order of approximation there exists only one mode of elementary excitation. The screening of the system to a static impurity charge is shown to be exponential at a long distance. The response of the system to a static transverse vector potential shows a perfect Meissner effect at long- and short-wavelength limits. We have also examined the diagrammatic structure for the number of particles in condensate. We show that the series contains only terms of integer powers of ${r}_{s}^{\frac{3}{4}}$. The entire treatment is fully number conserving.