Quantum theory of a vortex line in an optical lattice

We investigate the quantum theory of a vortex line in a stack of weakly coupled two-dimensional Bose-Einstein condensates, that is created by a one-dimensional optical lattice. We derive the dispersion relation of the Kelvin modes of the vortex line and also study the coupling between the Kelvin and quadrupole modes. We solve the coupled dynamics of the vortex line and the quadrupole modes, both classically as well as quantum mechanically. The quantum mechanical solution reveals the possibility of generating nonequilibrium squeezed vortex states by strongly driving the quadrupole modes.

Figure 1

Frequencies of the $m=\pm 2$ quadrupole modes in units of ${\omega }_r$ for $k=0$ and $\Omega =0$ and as a function of interaction strength. The solid line is calculated using Eq. (31) and the open circles are calculated by solving the appropriate Bogoliubov-de Gennes equations numerically.

Figure 2

Demonstration of the resonance condition for the coupling between the kelvons and the quadrupole mode with $m=-2$ when $J=0.1$, $U=100$, and $\Omega =0$. The coupling becomes important when the kelvon frequency ${\omega }_K\left(k\right)={\omega }_{-2}\left(0\right)∕2$, i.e., when the lines cross. Since the lowest line does not cross the kelvon dispersion, the coupling between kelvons and the quadrupole mode with $m=2$ is very weak. The unit of frequency is ${\omega }_r$.

Figure 3

(a)–(c) The time evolution in units of $1∕{\omega }_r$ of the total kelvon number per site, the quadrupole density per site, and the number distribution of kelvons in the end of the simulation, when kelvons are treated classically and $N_s=101$. (d)–(f) These plots used the same parameters as the plots (a)–(c), except that the number of sites was larger, namely $N_s=751$. In both sets of simulations we had initially ${\mid q_{-2}\mid }^2∕N_s=\sqrt{1+2U}∕10$ and a very small kelvon population to act as a seed. Also, we used $J=0.1$ and $U=100$. In plots (c) and (f) the $+-$ signs indicate the actual modes used in the numerical solution, whereas the solid line is an interpolation between them.

Figure 4

Classical vortex line dynamics when $J=0.1$, $U=100$, and $N_s=101$. The initial state has ${\mid q_{-2}\mid }^2∕N_s=\sqrt{1+2U}∕10$ and a very small kelvon population to act as a seed. The unit of time is $1∕{\omega }_r$ and the unit of length is $\sqrt{\hslash ∕m{\omega }_r}$.

Figure 5

(a) Time evolution in units of $1∕{\omega }_r$ of the total average kelvon number per site for quantum mechanical kelvons. (b) Quadrupole density per site as a function of time. In both plots we used $J=0.1$, $U=100$, and $N_s=751$. Initially the system was prepared in the kelvon vacuum and with a quadrupole density ${\mid q_{-2}\mid }^2∕N_s=\sqrt{1+2U}∕10$.

Figure 6

(Color online) Kelvon-number distribution as a function of time in units of $1∕{\omega }_r$ when the kelvons are treated fully quantum mechanically. We used parameters $J=0.1$, $U=100$, and $N_s=751$. Initially, the system was prepared in the kelvon vacuum and with a quadrupole density ${\mid q_{-2}\mid }^2∕N_s=\sqrt{1+2U}∕10$. The unit of time is $1∕{\omega }_r$.

Figure 7

The time it takes for the quadrupole field density to halve from its initial value ${\mid q_{-2}\left(0\right)\mid }^2$. The solid line is based on the numerical solution of the quantum multimode problem, while the dashed line is based on Fermi’s golden rule. We used parameters $J=0.1$, $U=100$, $N_s=751$, and the unit of time is $1∕{\omega }_r$.

Figure 8

(a) The time evolution in units of $1∕{\omega }_r$ of the quadrature operator uncertainty $\sqrt{⟨{\widehat{b}}_{\mathrm{i}}^2⟩}$. We used parameters $N_s=100$, $N=100$, $J=0.1$, $U=100$, and ${\mid q_{-2}\left(0\right)\mid }^2∕N_s=\sqrt{1+2U}∕10$. The solid line is for the operator ${\widehat{b}}_1$ and the dashed line for the operator ${\widehat{b}}_2$. (b) The figure shows the time evolution for the product $\sqrt{⟨{\widehat{b}}_1^2⟩}\sqrt{⟨{\widehat{b}}_2^2⟩}$.

Figure 9

Deformation parameter of the uncertainty ellipse with two different numbers of lattice sites. The solid line was calculated with $N_s=101$ and the dashed line with $N_s=751$. In addition, we used parameters $J=0.1$, $U=100$, and ${\mid q_{-2}\left(0\right)\mid }^2∕N_s=\sqrt{1+2U}∕10$. The unit of time is $1∕{\omega }_r$.

Figure 10

Snapshots of the time evolution in units of $1∕{\omega }_r$ of the uncertainty ellipse when $N_s=101$. In addition, we used parameters $J=0.1$, $U=100$, and ${\mid q_{-2}\left(0\right)\mid }^2∕N_s=\sqrt{1+2U}∕10$. Dashed lines indicate the directions of the main axes of the uncertainty ellipse. The unit of time is $1∕{\omega }_r$ and the unit of length is $\sqrt{\hslash ∕m{\omega }_r}$.

Figure 11

Snapshots of the time evolution of the uncertainty ellipse with $N_s=751$. In addition, we used parameters $J=0.1$, $U=100$, and ${\mid q_{-2}\left(0\right)\mid }^2∕N_s=\sqrt{1+2U}∕10$. Dashed lines indicate the directions of the main axes of the uncertainty ellipse. The unit of time is $1∕{\omega }_r$ and the unit of length is $\sqrt{\hslash ∕m{\omega }_r}$.

Figure 12

Vortex-position distribution in units of $\sqrt{\hslash ∕m{\omega }_r}$ (a) before and (b) after the expansion, where condensate size increased by a factor of $3.5$. We used the parameters $J=0.1$, $U=100$, $\Omega =0$, and $N_s=101$. Initially, vortex positions were sampled from a two-dimensional Gaussian distribution with an asymmetry of $ϵ=0.9$ and the product of the widths was ${\sigma }_x{\sigma }_y=B_0$.