Variational approach for the two-dimensional trapped Bose-Einstein condensate

We construct a many-body Gaussian variational approach for the two-dimensional trapped Bose gas in the condensate phase. Interaction between particles is modelized by a generalized pseudopotential of zero range that allows recovering perturbative results in the ultradilute limit, while back action of noncondensate particles on the condensate part is taken into account for higher density. As an application, we derive the equation of state and solve stability problems encountered in similar mean-field formalisms for a single vortex configuration.

Figure 1

Equation of state of the 2D homogeneous Bose gas. Dotted line: EOS deduced from Schick’s formula $\mu =4\pi {\hslash }^{2}n∕\left[m\ln \right(n{a}_{2D}^2\left)\right]$ [25]. Full line: full variational approach (HFB). Dashed-dotted line: perturbative EOS [10, 14].

Figure 2

Typical evolution of the anomalous mode energy of a single vortex in a nonrotating trap as a function of the temperature. The presence of a negative energy mode in the spectrum leads to the so-called thermodynamical instability of the single vortex [27]. Dashed line: the coupling constant used in the gapless equations is arbitrarily evaluated using the two-body $T$ matrix at energy $-2{\mu }_{\text{loc}}$. Full line: full variational scheme; at finite temperature, the anomalous mode energy is not stabilized.

Figure 3

Typical density profile of a single vortex at finite temperature $(k_{B}T=20\hslash {\omega }_L)$, obtained using the full variational approach when the anomalous mode is not thermalized. Inset: mean-field coupling constant in harmonic oscillator units, obtained by solving Eq. (12).