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Register a new userBose-Einstein condensation with a finite number of particles in a power-law trap
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Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number $N$ of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature $T_c$ in terms of a power series in $x_0=epsilon_0/k_BT_c$, where $epsilon_0$ denotes the zero-point energy of the trapping potential. This expression, which applies in cartesian, cylindrical and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of $x_0$ up to the second order. We show that, for a harmonic trap, the well known first order shift of the critical temperature $Delta T_c/T_c propto N^{-1/3}$ is inaccurate when $N leqslant 10^{5}$, the next order (proportional to $N^{-1/2}$) being significant. We also show that finite size effects on the condensation temperature cancel out in a cubic trapping potential, e.g. $V(mathbi{r}) propto r^3$. Finally, we show that in a generic power-law potential of higher order, e.g. $V(mathbi{r}) propto r^alpha$ with $alpha > 3$, the shift of the critical temperature becomes positive. This effect provides a large increase of $T_c$ for relatively small atom numbers. For instance, an increase of about +40% is expected with $10^4$ atoms in a $V(mathbi{r}) propto r^{12}$ trapping potential.
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In this reply we show that the criticisms raised by J. Noronha are based on a misapplication of the model we have proposed in [A. Jaouadi, M. Telmini, E. Charron, Phys. Rev. A 83, 023616 (2011)]. Here we explicitly discuss the range of validity of the approximations underlying our analytical model. We also show that the discrepancies pointed out for very small atom numbers and for very anisotropic traps are not surprising since these conditions exceed the range of validity of the model.
The recent experimental condensation of ultracold atoms in a triangular optical lattice with negative effective tunneling energies paves the way to study frustrated systems in a controlled environment. Here, we explore the critical behavior of the chiral phase transition in such a frustrated lattice in three dimensions. We represent the low-energy action of the lattice system as a two-component Bose gas corresponding to the two minima of the dispersion. The contact repulsion between the bosons separates into intra- and inter-component interactions, referred to as $V_{0}$ and $V_{12}$, respectively. We first employ a Huang-Yang-Luttinger approximation of the free energy. For $V_{12}/V_{0} = 2$, which corresponds to the bare interaction, this approach suggests a first order phase transition, at which both the U$(1)$ symmetry of condensation and the $mathbb{Z}_2$ symmetry of the emergent chiral order are broken simultaneously. Furthermore, we perform a renormalization group calculation at one-loop order. We demonstrate that the coupling regime $0<V_{12}/V_0leq1$ shares the critical behavior of the Heisenberg fixed point at $V_{12}/V_{0}=1$. For $V_{12}/V_0>1$ we show that $V_{0}$ flows to a negative value, while $V_{12}$ increases and remains positive. This results in a breakdown of the effective quartic field theory due to a cubic anisotropy, and again suggests a discontinuous phase transition.
The partition function and specific heat of a system consisting of a finite number of bosons confined in an external potential are calculated in canonical ensemble. Using the grand partition function as the generating function of the partition function, an iterative scheme is established for the calculation of the partition function of system with an arbitrary number of particles. The scheme is applied to finite number of bosons confined in isotropic and anisotropic parabolic traps and in rigid boxes. The specific heat as a function of temperature is studied in detail for different number of particles, different degrees of anisotropy, and different spatial dimensions. The cusp in the specific heat is taken as an indication of Bose-Einstein condensation (BEC).It is found that the results corresponding to a large number of particles are approached quite rapidly as the number of bosons in the system increases. For large number of particles, results obtained within our iterative scheme are consistent with those of the semiclassical theory of BEC in an external potential based on the grand canonical treatment.
Bose-Einstein condensates (BECs) are macroscopic coherent matter waves that have revolutionized quantum science and atomic physics. They are essential to quantum simulation and sensing, for example underlying atom interferometers in space and ambitious tests of Einsteins equivalence principle. The key to dramatically increasing the bandwidth and precision of such matter-wave sensors lies in sustaining a coherent matter wave indefinitely. Here we demonstrate continuous Bose-Einstein condensation by creating a continuous-wave (CW) condensate of strontium atoms that lasts indefinitely. The coherent matter wave is sustained by amplification through Bose-stimulated gain of atoms from a thermal bath. By steadily replenishing this bath while achieving 1000x higher phase-space densities than previous works, we maintain the conditions for condensation. This advance overcomes a fundamental limitation of all atomic quantum gas experiments to date: the need to execute several cooling stages time-sequentially. Continuous matter-wave amplification will make possible CW atom lasers, atomic counterparts of CW optical lasers that have become ubiquitous in technology and society. The coherence of such atom lasers will no longer be fundamentally limited by the atom number in a BEC and can ultimately reach the standard quantum limit. Our development provides a new, hitherto missing piece of atom optics, enabling the construction of continuous coherent matter-wave devices. From infrasound gravitational wave detectors to optical clocks, the dramatic improvement in coherence, bandwidth and precision now within reach will be decisive in the creation of a new class of quantum sensors.
We analyze free expansion of a trapped one-dimensional Bose gas after a sudden release from the confining trap potential. By using the stationary phase and local density approximations, we show that the long-time asymptotic density profile and the momentum distribution of the gas are determined by the initial distribution of Bethe rapidities (quasimomenta) and hence can be obtained from the solutions to the Lieb-Liniger equations in the thermodynamic limit. For expansion from a harmonic trap, and in the limits of very weak and very strong interactions, we recover the self-similar scaling solutions known from the hydrodynamic approach. For all other power-law traps and arbitrary interaction strengths, the expansion is not self-similar and shows strong dependence of the density profile evolution on the trap anharmonicity. We also characterize dynamical fermionization of the expanding cloud in terms of correlation functions describing phase and density fluctuations.
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