Quantum fluctuations of vortex lattices in ultracold gases

We discuss the effects of quantum fluctuations on the properties of vortex lattices in rapidly rotating ultracold atomic gases. We develop a variational method that goes beyond the Bogoliubov theory by including the effects of interactions between the quasiparticle excitations. These interactions are found to have significant quantitative effects on physical properties even at relatively large filling factors. We use our theory to predict the expected experimental signatures of quantum fluctuations of vortices and to assess the competition of the triangular vortex lattice phase with other phases in finite-sized systems.

Figure 1

Vortex core particle density as a function of the inverse filling factor, for the full Bogoliubov theory (solid line), the long-wavelength theory of Ref. [13] (dashed line), and our variational wave function in Eq. (29) for $N_{\mathrm{v}}=90$ (diamonds). The relation given in Appendix was used to map from the mean-squared vortex core displacement $\langle \delta {\mathbf{r}}^2\rangle$ calculated in Ref. [13] to vortex core density. $n_{\mathrm{c}}$ (dotted line) is the critical core density at which the lattice should “melt” according to a Lindemann criterion with parameter $\langle \delta {\mathbf{r}}^2\rangle =0.145l^2$.

Figure 2

Comparison of our results with ED data for $N_{\mathrm{v}}=6$ and $L_y/L_x=1/\sqrt{3}$. As expected and explained in Sec. 4, the indefinite-particle-number trial wave function overestimates the energy, and this overestimate is larger at low filling factors. In all the other figures our variational results were obtained for the definite-number trial wave function in Eq. (29).

Figure 3

The results for the largest studied system, i.e., for $N_{\mathrm{v}}=90$ ($L_y/L_x=\frac{5}{9}\sqrt{3}$ for the triangular lattice and $L_y/L_x=9/10$ for the square lattice). The variational energies were calculated using the definite-number trial wave function.

Figure 4

Condensate depletion in a range of system sizes. The symbols correspond to our variational results for the definite-number state and the lines without symbols show the depletion calculated in the Bogoliubov approximation.

Figure 5

Shear modulus ($C_2$) of the triangular vortex lattice as a function of the filling factor for $N_{\mathrm{v}}=90$.

Figure 6

Energy of the smectic state (circles) compared against the energy of the triangular vortex lattice (triangles) for $N_{\mathrm{v}}=8,L_y/L_x=\sqrt{3}/4$ and for $N_{\mathrm{v}}=24,L_y/L_x=4\sqrt{3}/9$.