Roton immiscibility in a two-component dipolar Bose gas

We characterize the immiscibility-miscibility transition (IMT) of a two-component Bose-Einstein condensate (BEC) with dipole-dipole interactions. In particular, we consider the quasi-two-dimensional geometry, where a strong trapping potential admits only zero-point motion in the trap direction, while the atoms are more free to move in the transverse directions. We employ the Bogoliubov treatment of the two-component system to identify both the well-known long-wavelength IMT in addition to a rotonlike IMT, where the transition occurs at finite-wave number and is reminiscent of the roton softening in the single-component dipolar BEC. Additionally, we verify the existence of the roton IMT in the fully trapped, finite systems by direct numerical simulation of the two-component coupled nonlocal Gross-Pitaevskii equations.

Figure 1

The phase diagram of the balanced two-component system in the q2D geometry with dipole lengths $b_{11}/l_z=2/3$ and scattering lengths $a_{11}=a_{22}=1$ and $a_{12}=1.05$ as a function of the dipole length of component 2, $b_{22}/l_z$, and quasiparticle wave number $q{l}_z$. The familiar long-wavelength immiscible region and the roton-immiscible region are labeled and colored, and the white region signifies the existence of a stable miscible state. The roton-immiscible region extends down to $b_{22}/l_z=0$. The contours in the shaded region mark increments of $0.1{\omega }_z$.

Figure 2

Dispersions of the balanced, miscible, two-component q2D DBEC with $a_{11}/l_z=a_{22}/l_z=1$, $a_{12}/l_z=1.05$, $b_{11}/l_z=2/3$, and $b_{22}/l_z=0$. The lower branch of the two-component dispersion ${\omega }_{-}\left(\mathbf{q}\right)$, corresponding to out-of-phase quasiparticle excitations, exhibits a rotonlike instability corresponding to a transition to immiscibility. The imaginary part of the dispersion is shaded in teal.

Figure 3

Critical interspecies $s$-wave scattering length $a_{12}/l_z$ (top panel) and wave number (bottom panel) for the balanced two-component q2D DBEC as a function of the dipole length of component 1, $b_{11}/l_z$, with $a_{11}/l_z=a_{22}/l_z=1$ and $b_{22}/l_z=0$. The parameters for which the system is immiscible are shaded teal in the top panel. The nondipolar IMT is seen for $b_{11}/l_z=0$, where the IMT threshold occurs at $a_{12}/l_z=1$ ($\Delta =0$) and $q_{\mathrm{crit}}{l}_z=0$. The roton immiscibility emerges for nonzero $b_{11}/l_z$ and decreases in characteristic wavelength as $b_{11}/l_z$ is increased. The black “x” in the top panel marks $b_{11}/l_z=2/3$ and $a_{12}/l_z=1.05$, the parameters for the example roton IMT used in Figs. 2 and 5.

Figure 4

Densities $n_1\left(\mathit{\rho }\right)={\left|{\varphi }_1\left(\mathit{\rho }\right)\right|}^2$ (left panel) and $n_2\left(\mathit{\rho }\right)={\left|{\varphi }_2\left(\mathit{\rho }\right)\right|}^2$ (right panel) corresponding to stationary solutions of the coupled GPEs (16) (converged in energy to a part in ${10}^8$ in imaginary time evolution) for the nondipolar balanced system, with $a_{11}/l_z=a_{22}/l_z=1$, $a_{12}/l_z=1.05$, and $b_{11}/l_z=b_{22}/l_z=0$. Here, the immiscibility is phononlike and occurs on the longest length scale available in the system.

Figure 5

Densities $n_1\left(\mathit{\rho }\right)={\left|{\varphi }_1\left(\mathit{\rho }\right)\right|}^2$ (left panel) and $n_2\left(\mathit{\rho }\right)={\left|{\varphi }_2\left(\mathit{\rho }\right)\right|}^2$ (right panel) corresponding to stationary solutions of the coupled GPEs (16) (converged in energy to a part in ${10}^8$ in imaginary time evolution) for the balanced system, with $a_{11}/l_z=a_{22}/l_z=1$, $a_{12}/l_z=1.05$, $b_{22}/l_z=0$, and $b_{11}/l_z=2/3$. Here the immiscibility is rotonlike and occurs on the length scale $\sim$2 $2\pi l_z$.

Figure 6

The shaded region shows the nonzero imaginary part of the two-component dispersion ${\omega }_{-}\left(\mathbf{q}\right)$ in the direction perpendicular to the polarization tilt for $a_{11}/l_z=a_{22}/l_z=1$, $a_{12}/l_z=1.05$, $b_{11}/l_z=1/4$, and $b_{22}/l_z=0$ as a function of polarization angle $\alpha$ and quasiparticle wave number $q{l}_z$. The miscibility of the system for $\alpha =0$ is reflected in the purely real character of the dispersion at this polarization angle, and the roton immiscibility is seen as the emergence of a nonzero imaginary part of ${\omega }_{-}\left(\mathbf{q}\right)$ at finite wave number $q{l}_z\sim 1$ at $\alpha /\pi \sim 0.05$. The contours in the shaded region mark increments of $0.1{\omega }_z$.

Figure 7

The critical polarization angle $\alpha$ (top panel) and the critical wave number $q_{\mathrm{crit}}{l}_z$ (bottom panel) for the balanced two-component q2D DBEC with $a_{11}/l_z=a_{22}/l_z=1$ and $b_{22}/l_z=0$ for various dipole lengths of component 1, $b_{11}/l_z$, as a function of the interspecies $s$-wave scattering length $a_{12}/l_z$. The immiscible parameters are shaded in the top panel. Here, the roton immiscibility emerges as a function of polarization angle for ${\mathbf{q}}_{\mathrm{crit}}$ perpendicular to the direction of tilt.

Figure 8

Condensate densities $n_1(\mathit{\rho },t)={\left|{\varphi }_1(\mathit{\rho },t)\right|}^2$ (top row) and $n_2(\mathit{\rho },t)={\left|{\varphi }_2(\mathit{\rho },t)\right|}^2$ (bottom row) as a function of time, where the polarization angle is tilted from a value where the system is miscible [$\alpha =0$ at $t=0$, column (a)] to a value where the system is roton immiscible (${\alpha }_{\mathrm{hold}}=\pi /4$) at an angle $\eta =\pi /5$ off of the $x$ axis. In (a), the system is clearly miscible. Over a time $t_{\mathrm{ramp}}{\omega }_z=200$, $\alpha$ is linearly ramped from $\alpha =0$ to ${\alpha }_{\mathrm{hold}}=\pi /4$. It is held at ${\alpha }_{\mathrm{hold}}$ for $t_{\mathrm{hold}}{\omega }_z=400$, then is linearly ramped back to $\alpha =0$ over $t_{\mathrm{ramp}}{\omega }_z=200$. In column (b), the densities are shown at time $t{\omega }_z=120$, where the system is just starting to exhibit immiscible character. The densities are shown at time $t{\omega }_z=200$ in column (c), where the system is fully immiscible on the roton length scale. The densities are shown for $t{\omega }_z=760$ in column (d), as the angle has returned to a value corresponding to miscibility, and column (e) shows the densities at time $t{\omega }_z=1200$, where the polarization angle has returned to $\alpha =0$ but the system is not rethermalized due to the introduction of strong phase fluctuations in the process of spatial separation en route to miscibility.

Figure 9

Stationary densities $n_1\left(\mathit{\rho }\right)={\left|{\varphi }_1\left(\mathit{\rho }\right)\right|}^2$ (left column) and $n_2\left(\mathit{\rho }\right)={\left|{\varphi }_2\left(\mathit{\rho }\right)\right|}^2$ (right column) of the balanced two-component DBEC in a trap with aspect ratio $\lambda =10$ for $a_{11}/l_z=a_{22}/l_z=1$, $a_{12}/l_z=1.1$, and $b_{22}/l_z=0$. The rows show solutions for various dipole lengths of component 1, being (a) $b_{11}/l_z=0$, (b) $b_{11}/l_z=1/4$, (c) $b_{11}/l_z=1/2$, and (d) $b_{11}/l_z=2/3$. In row (a), the system is nondipolar and the interactions are thus balanced, resulting in an immiscibility on the length scale of the trap. In row (b), the presence of the relatively small dipole moment in component 1 breaks this symmetry, but the immiscibility is still long-wavelength and component 1 thus engulfs component 2 due to its greater self-repulsion. The roton immiscibility emerges for $b_{11}/l_z\gtrsim 1/2$, as is seen in rows (c) and (d). While both immiscible states exhibit ordering near the roton length scale, the length scale is smaller in (d) where the dipole length is larger, as suggested by the results shown in Fig. 3.