Zero-point energy of ultracold atoms

Abstract

We analyze the divergent zero-point energy of a dilute and ultracold gas of atoms in D spatial dimensions. For bosonic atoms we explicitly show how to regularize this divergent contribution, which appears in the Gaussian fluctuations of the functional integration, by using three different regularization approaches: dimensional regularization, momentum-cutoff regularization and convergence-factor regularization. In the case of the ideal Bose gas the divergent zero-point fluctuations are completely removed, while in the case of the interacting Bose gas these zero-point fluctuations give rise to a finite correction to the equation of state. The final convergent equation of state is independent of the regularization procedure but depends on the dimensionality of the system and the two-dimensional case is highly nontrivial. We also discuss very recent theoretical results on the divergent zero-point energy of the D-dimensional superfluid Fermi gas in the BCS-BEC crossover. In this case the zero-point energy is due to both fermionic single-particle excitations and bosonic collective excitations, and its regularization gives remarkable analytical results in the BEC regime of composite bosons. We compare the beyond-mean-field equations of state of both bosons and fermions with relevant experimental data on dilute and ultracold atoms quantitatively confirming the contribution of zero-point-energy quantum fluctuations to the thermodynamics of ultracold atoms at very low temperatures.

Two-dimensional quantum droplets in dipolar Bose gases

“…Indeed, obtaining reliable estimate for the beyond mean field correction to the equation of state (EoS) in low dimensions is challenging even for Bose systems with contact interactions [27][28][29]. At zero temperature, the LHY corrections to the EoS can be written [10,22] ò dm…”

Nonuniversal Equation of State of the Two-Dimensional Bose Gas

“…(17). This divergence can be regularized with dimensional regularization, where the space dimension D is analytically continued [22,32,33]. To this end we extend the two-dimensional integral to a generic complex D = 2 − ε dimension, and then take the limit ε → 0.…”

“…which runs by changing κ and depends on the dimension D through ε = 2 − D [30,32,33]. To remove the divergence 1/ε in Eq.…”

Condensation and superfluidity of dilute Bose gases with finite-range interaction

“…In particular, we obtain an adimensional integral using the integration variable t k g n mg n , 4 2 2 2 0 0 0  l = ( ) ( ), then we extend the spatial dimension D to the complex value D  e = -. We remark that this additional step is needed because the dimensional regularization procedure is not always able to heal the ultraviolet divergence of the integrals [35]. After the integration, we obtain f n g 0 0 ( ) ( ) in the form…”

“…We remark that we take the derivative of the grand potential with respect to the chemical potential first and then we substitute the mean field value of the chemical potential μ e =g 0 ψ 0 2 , with the identification for the condensate density n 0 0 2 y = . This procedure can be justified considering that the same procedure is implemented to calculate the condensate fraction of a noninteracting Bose gas [35]. With this identification, we express the momentum density   as…”