Rotational tuning of the dipole-dipole interaction in a Bose gas of magnetic atoms

We investigate the dynamics of a Bose-Einstein condensate of magnetic atoms in which the dipoles are rotated by an external magnetic field. The time-averaged dipole-dipole interaction between the atoms is effectively tuned by this rotation, however, recent experimental and theoretical developments show that dynamic instabilities emerge that may cause heating. We present simulations of a realistic tuning sequence in this system, and characterize the system behavior and the emergence of instabilities. Our results indicate that the instabilities develop more slowly as the rotation frequency increases, and indicate that experiments with tuned dipole-dipole interactions should be feasible.

Figure 1

Schematic of the dipole tuning ramps we use in our simulations. The dipole direction $\mathbf{e}$ starts along $z$ and then spirals down, using a linear ramp of the polar angle $\phi$, over a time $t_{\mathrm{on}}$ into the $xy$ plane according to Eq. (1). For $t>t_{\mathrm{on}}$ the dipole executes uniform circular motion in the $xy$ plane with an angular velocity $\mathrm{\Omega }$.

Figure 2

System dynamics arising from a dipole tuning ramp with $\mathrm{\Omega }/2\pi =2\mathrm{kHz}$. Results are given for (a,b,c) $t_{\mathrm{on}}=20\mathrm{ms}$ and (d,e,f) $t_{\mathrm{on}}=10\mathrm{ms}$ ramps, with the polar angle shown in (a) and (d). The evolution of the system widths (b,e) and dipole energy (c,f). In (b) and (e) the instantaneous widths (solid lines: red ${\sigma }_x$, blue ${\sigma }_y$ and black ${\sigma }_z)$ are calculated from the time-dependent GP simulation $\psi$. We also show the ground-state $\overline{\psi }$ widths [dashed lines: red ${\overline{\sigma }}_x$, blue ${\overline{\sigma }}_y$, and black ${\overline{\sigma }}_z$, with ${\overline{\sigma }}_{\nu }={\left(\int d\mathbf{x}{\nu }^2{\left|\overline{\psi }\right|}^2/N\right)}^{1/2}]$ for the time-averaged interaction at the current value of $\phi$. In (c) and (f) we show the instantaneous dipole energy $E_D$ (solid red line) and the value of the time-averaged ground state ${\overline{E}}_D$ (dashed black line). The results are for a condensate of $N_c=2.5\times {10}^3{}^{164}\mathrm{Dy}$ atoms, with $a_{\mathrm{dd}}=130.8a_0$ and $a_s=174.4a_0$ (i.e., ${\epsilon }_{\mathrm{dd}}=0.75)$, confined in a harmonic trap with $({\omega }_x,{\omega }_y,{\omega }_z)/2\pi =(50,50,55){\mathrm{s}}^{-1}$. The initial state chemical potential is $\mu =h\times 264$ Hz. Noise is added with ${\epsilon }_{\mathrm{cut}}=2\mu$.

Figure 3

Dynamics of kinetic energy for $t_{\mathrm{on}}=20\text{ms}$ for a condensate of (a) $N_c=2.5\times {10}^3$ atoms giving $\mu =h\times 264\text{Hz}$, and (b) $N_c=5\times {10}^3$ atoms giving $\mu =h\times 342\text{Hz}$. Labels differentiate trajectories with different rotation frequencies $\mathrm{\Omega }$. All trajectories for each subplot use the same initial state. Other parameters are as in Fig. 2.

Figure 4

(a) Variation of instability time with trajectories. For each rotation frequency, we show $t_{\mathrm{inst}}$ for five trajectories seeded with independent initial state noise. Results here for $N_{\mathrm{cut}}=175$ with ${\epsilon }_{\mathrm{cut}}=2\mu$. (b) Dependence of the instability time on the amount of noise added. We show $t_{\mathrm{inst}}$ as a function of the number of modes $N_{\mathrm{cut}}$ to which noise is added. The two vertical dashed lines show where the associated single particle energy cutoff is equal to one and two times the chemical potential. The inset shows a similar calculation of the instability time for a condensate with $N_c=5\times {10}^3$ atoms. Parameters are as in Fig. 2.

Figure 5

(a) Bare and (b) scaled results for the instability time over a wide range of parameters. The parameters for the simulations are indicated in legend of subplot (b), with the parameter $f$ specifying the trap frequencies as $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi f\times (1,1,1.1)$. The DDI tuning ramp is as indicated in Fig. 2 and other parameters are the same as those used in Fig. 2. For each simulation case we take five trajectories to explore stochastic effects of the noise. The means are shown by the symbols and the vertical lines show the range of results. The slanted black line is a guide to the eye given by ${\omega }_x{t}_{\mathrm{inst}}=8\times {10}^3\hslash \mathrm{\Omega }/\mu N_c$. The chemical potential values, and other relevant quantities for the cases considered, are given in Appendix pp2.