Vortex formation and quantum turbulence with rotating paddle potentials in a two-dimensional binary Bose-Einstein condensate

We conduct a theoretical study of the creation and dynamics of vortices in a two-dimensional binary Bose-Einstein condensate with a mass imbalance between the species. To initiate the dynamics, we use one or two rotating paddle potentials in one species, while the other species is influenced only via the interspecies interaction. In both species, the number and the dominant sign of the vortices are determined by the rotation frequency of the paddle potential. Clusters of positive and negative vortices form at a low rotation frequency comparable to that of the trap when using the single paddle potential. In contrast, vortices of the same sign tend to dominate as the rotation frequency of the paddle increases, and the angular momentum reaches a maximum value at a paddle frequency where the paddle velocity becomes equal to the sound velocity of the condensate. When the rotation frequency is sufficiently high, the rapid annihilation of vortex-antivortex pairs significantly reduces the number of vortices and antivortices in the system. For two paddle potentials rotating in the same direction, the vortex dynamics phenomenon is similar to that of a single paddle. However, when the paddle potentials are rotated in the opposite direction, both positive- and negative-signed vortices occur at all rotational frequencies. At the low rotation frequencies, the cluster of like-signed vortices produces the $k^{-5/3}$ and $k^{-3}$ power laws in the incompressible kinetic energy spectrum at low and high wave numbers, respectively, a hallmark of the quantum turbulent flows.

Figure 1

Snapshot of (a i)–(e i) density $n_{\mathrm{A}}$ and (a ii)–(e ii) vorticity ${\mathrm{\Omega }}_{\mathrm{A}}$ profiles of species A at different instants of time: (a) $t=0$ s, (b) $t=0.08$ s, (c) $t=0.43$ s, (d) $t=2.01$ s, and (e) $t=3.48$ s. The binary BECs are made of ${}^{133}\mathrm{Cs}{\text{−}}^{87}\mathrm{Rb}$ atoms. An elliptical paddle potential characterized by the parameters $\eta =0.05$ and $d=0.1l$ is rotated with the angular frequency ${\omega }_p=\omega$ within species A (${}^{133}\mathrm{Cs}$) in order to trigger the dynamics. The color bars of the top and bottom rows represent the number density $n$ in $\mu {\mathrm{m}}^{-2}$ and the vorticity $\mathrm{\Omega }$, respectively. The binary BECs are initialized in a two-dimensional harmonic potential with frequency $\omega /2\pi =30.832$ Hz, $\lambda =100$ and having intra- and interspecies scattering lengths $a_{\mathrm{AA}}=280a_0$, $a_{\mathrm{BB}}=100.4a_0$, and $a_{\mathrm{AB}}=0$. The number of atoms for both species are $N_{\mathrm{A}}=N_{\mathrm{B}}=60000$.

Figure 2

Variation of the number of vortices $N_{+}^{\mathrm{A}}$ and antivortices $N_{-}^{\mathrm{A}}$ of species A at steady state as a function of the paddle frequency ${\omega }_p$. The other parameters are the same as in Fig. 1.

Figure 3

Snapshots of the vorticity profiles of species A $\left({\mathrm{\Omega }}_{\mathrm{A}}\right)$ taken at $t=3.5\mathrm{s}$ for two different paddle frequencies (a) ${\omega }_p=3\omega$ and (b) ${\omega }_p=10\omega$ of rotation of the paddle potential. Shown also is (c) the time evolution (logarithmic scale) of angular momentum $\left(L_z^{\mathrm{A}}\right)$ for different values of ${\omega }_p$ (see the legends). The vertical lines in (c) represent the times when the amplitude of the paddle started to ramp off and become zero, respectively. The color bar of (a) and (b) represents the vorticity $\mathrm{\Omega }$. The other parameters are the same as in Fig. 1.

Figure 4

Snapshots of density profiles of (a) species A $\left(n_{\mathrm{A}}\right)$ and (b) species B $\left(n_{\mathrm{B}}\right)$, with interspecies scattering length $a_{\mathrm{AB}}=80a_0$ and ${\omega }_p=3\omega$ at $t=3.5\mathrm{s}$. Also shown are the variations of angular momentum (c) $L_z^{\mathrm{A}}$ and (d) $L_z^{\mathrm{B}}$ varying with paddle frequency ${\omega }_p$ and time $t$. The color bars represent the number densities (a) $n_{\mathrm{A}}$ and (b) $n_{\mathrm{B}}$ in $\mu {\mathrm{m}}^{-2}$ and (c) and (d) the angular momentum in units of $\hslash$.

Figure 5

Variation of the absolute value of the time-averaged angular momentum $|{\overline{L}}_z^{\mathrm{A}}|$ of species A at $a_{\mathrm{AB}}=0\text{and}80a_0$, and $|{\overline{L}}_z^{\mathrm{B}}|$ of species B at $a_{\mathrm{AB}}=80a_0$ as a function of paddle frequency (see the legend). The markers and solid curves show the values of angular momentum from the simulations and the fittings with skewed normal distribution, respectively. Here the paddle configuration is the same as that described in Fig. 1.

Figure 6

(a) Time evolution (logarithmic scale) of angular momentum $L_z^i$ for species $i=\mathrm{A},\mathrm{B}$ at interspecies interaction $a_{\mathrm{AB}}=80a_0$ for different paddle frequencies ${\omega }_p=\omega$, $2\omega$, $3\omega$, $4\omega$, and $10\omega$. Here two identical paddle rotates in species A in the clockwise direction. The insets depict the snapshot of vorticity profiles of (b) species A (${\mathrm{\Omega }}_{\mathrm{A}}$) and (c) species B (${\mathrm{\Omega }}_{\mathrm{B}}$) at $t=3.5\mathrm{s}$ with ${\omega }_p=4\omega$.

Figure 7

Snapshot of the densities of (a) species A, $n_{\mathrm{A}}$, and (b) species B, $n_{\mathrm{B}}$, with interspecies interaction $a_{\mathrm{AB}}=80a_0$ at different instants of time: (a i) and (b i) $t=0$ s, (a ii) and (b ii) $t=0.04$ s, (a iii) and (b iii) $t=0.4$ s, (a iv) and (b iv) $t=1.0$ s, and (a v) and (b v) $t=2.0$ s. Two elliptic paddles characterized by the parameters $\eta =0.1$ and $d=0.1l$ and rotating oppositely to each other with frequency ${\omega }_p=5\omega$ within species A (${}^{133}\mathrm{Cs}$) are used to trigger the dynamics. The color bars represent the number density $n$ in $\mu {\mathrm{m}}^{-2}$. The vortices (antivortices) are marked with red (blue) in (a iii)–(a v) and (b iii)–(b v) [vortex identification is not done in (a ii) and (b ii) due to the large number].

Figure 8

Variation of (a) time-averaged angular momentum ${\overline{L}}_z^i$, (b) its standard deviation ${\sigma }_i$, (c) vortex imbalance $|N_{+}^i-N_{-}^i|$ at steady state, and (d) $\sqrt{|{\mathrm{\Gamma }}_i|}$ (${\mathrm{\Gamma }}_i$ is the circulation at steady state), as a function of ${\omega }_p$ for the $i\mathrm{th}$ species ($i=\mathrm{A},\mathrm{B}$). The scattering lengths are given by $a_{\mathrm{AA}}=280a_0$, $a_{\mathrm{BB}}=100a_0$, and $a_{\mathrm{AB}}=80a_0$. The dynamics has been triggered by employing two paddle potentials rotating counterclockwise. The paddle configuration is described in Fig. 7.

Figure 9

Incompressible kinetic energy spectra of species A, $E_{\mathrm{A}}^{\mathrm{ic}}\left(k\right)$, at different time instants (see the legend) for different paddle frequencies (a) ${\omega }_p=\omega$, (b) ${\omega }_p=3\omega$, and (c) ${\omega }_p=8\omega$. The solid and dashed lines represent the slopes of $k^{-5/3}$ and $k^{-3}$, respectively. The interspecies scattering length $a_{\mathrm{AB}}=0$. The dashed vertical lines define the positions at $k=2\pi /R_{\mathrm{A}}$ ($R_{\mathrm{A}}=10l$ is the Thomas-Fermi radius), $k_p$ ($2\pi /a$, with $a$ the semimajor axis of the paddle), ${\xi }_{\mathrm{A}}^{-1}$, and $2\pi /{\xi }_{\mathrm{A}}$, respectively. Here the healing length ${\xi }_{\text{A}}$ of species A is $0.062l$. Except for ${\omega }_p$, all other parameters are the same as in Fig. 1.

Figure 10

Incompressible kinetic energy spectra of species B, $E_{\mathrm{B}}^{\mathrm{ic}}\left(k\right)$, at different time instants (see the legend) for different paddle frequencies (a) ${\omega }_p=\omega$, (b) ${\omega }_p=3\omega$, and (c) ${\omega }_p=8\omega$. The solid and dashed lines represent the slopes of $k^{-5/3}$ and $k^{-3}$, respectively. The interspecies scattering length $a_{\mathrm{AB}}=80a_0$. The dashed vertical lines define the positions at $k=2\pi /R_{\mathrm{B}}$ ($R_{\mathrm{B}}\approx 10l$ is the Thomas-Fermi radius), $k_p$ ($2\pi /a$, with $a$ the semimajor axis of the paddle), ${\xi }_{\mathrm{B}}^{-1}$, and $2\pi /{\xi }_{\mathrm{B}}$. Here the healing length ${\xi }_A$ of species B is $0.099l$. Except for ${\omega }_p$, all other parameters are the same as in Fig. 4.

Figure 11

Compressible kinetic energy spectra of species A, $E_{\mathrm{A}}^{\mathrm{c}}\left(k\right)$, at (a) ${\omega }_p=\omega$ and (b) ${\omega }_p=8\omega$, without interspecies interaction $a_{\mathrm{AB}}=0$ and of species B, $E_{\mathrm{B}}^{\mathrm{c}}\left(k\right)$, at (c) ${\omega }_p=6\omega$ with $a_{\mathrm{AB}}=80a_0$ at different time instants (see the legend). Black solid and dashed lines represent the slopes of $k^{-7/2}$ and $k$, respectively, and the blue solid line in (c) represents the slope of $k^{-3/2}$. The dashed vertical lines of (a) and (b) and of (c) are described in Figs. 9 and 10, respectively. All the parameters are the same as discussed in Sec. 3a.

Figure 12

Vorticity ${\mathrm{\Omega }}_{\mathrm{A}}$ of species A at scattering lengths (a) $a_{\mathrm{AB}}=0$ and (b) $a_{\mathrm{AB}}=80a_0$ at ${\omega }_p=\omega$. Also shown are the density profiles (c) $n_{\mathrm{A}}$ and (d) $n_{\mathrm{B}}$ of species A and B, respectively, for $a_{\mathrm{AB}}=80a_0$ and ${\omega }_p=3\omega$. All the snapshots shown are at $t=3.5$ s. The binary BEC is realized at two hyperfine levels of ${}^{87}\mathrm{Rb}$ atoms. To trigger the dynamics, an optical paddle potential is rotated in species A. The color bars represent (a) and (b) the vorticity ${\mathrm{\Omega }}_{\mathrm{A}}$ and the number densities (c) $n_{\mathrm{A}}$ and (d) $n_{\mathrm{B}}$ in $\mu {\mathrm{m}}^{-2}$.

Figure 13

Snapshot of (a i)–(e i) density $\left(n_{\mathrm{A}}\right)$ species and (a ii)–(e ii) vorticity $\left({\mathrm{\Omega }}_{\mathrm{A}}\right)$ profiles of ${}^{133}\mathrm{Cs}$ at different instants of time: (a) $t=0$ s, (b) $t=0.1$ s, (c) $t=0.5$ s, (d) $t=1.0$ s, and (e) $t=2.1$ s. An elliptical paddle potential with amplitude $V_0=-10{\mu }_{\mathrm{A}}$ and characterized by the parameters $\eta =0.05$ and $d=0.1l$ is rotated with the angular frequency ${\omega }_p=4\omega$ within species A. The color bars represent (a i)–(e i) the number density $n$ in $\mu {\mathrm{m}}^{-2}$ and (a ii)–(e ii) the vorticity $\mathrm{\Omega }$. The binary BECs are initialized in a two-dimensional harmonic potential with frequency $\omega /2\pi =30.832\mathrm{Hz}$, $\lambda =100$, and having intra- and interspecies scattering lengths $a_{\mathrm{AA}}=280a_0$, $a_{\mathrm{BB}}=100.4a_0$, and $a_{\mathrm{AB}}=0$. The numbers of atoms for both species are $N_{\mathrm{A}}=N_{\mathrm{B}}=60000$.