Two-dimensional miscible-immiscible supersolid and droplet crystal state in a homonuclear dipolar bosonic mixture

The recent realization of a binary dipolar Bose-Einstein condensate [A. Trautmann et al., Phys. Rev. Lett. 121, 213601 (2018)] presents new exciting aspects for studying quantum droplets and supersolids in a binary mixture. Motivated by this experiment, we study the ground-state phases and dynamics of a Dy-Dy mixture. The dipolar bosonic mixture exhibits qualitatively novel and rich physics. Relying on the three-dimensional numerical simulations in the extended Gross-Pitaevskii framework, we unravel the ground-state phase diagrams and characterize their different possible phases. The emergent phases include single-droplet (SD), multiple-droplets (MD), doubly supersolid (SS), and superfluid (SF) states in both miscible and immiscible phases. Intriguing mixed ground states are observed for an imbalanced binary mixture, including a combination of SS-SF, SS-MD, and SS-SS phases. We also explore the dynamics across the phase boundaries by linear quenches of interspecies and intraspecies scattering lengths. During these dynamical processes, we observe an abrupt change in phase which initially results in some instability in the system and forms some metastable states in the intermediate timescale. However, the state produced in long-time evolution is similar to our predicted ground state. Although we demonstrate the possible results for a Dy-Dy mixture and for a specific parameter range of intraspecies and interspecies scattering lengths, our results are in general valid for other dipolar mixtures and may become an important benchmark for future experimental scenarios.

Figure 1

Ground-state phase diagrams in an oblate harmonic trap of a balanced Dy-Dy mixture as a function of the number of particles and (a) intracomponent and (b) intercomponent $s$-wave scattering length. The color bar corresponds to the value of the overlap integral $\mathrm{\Lambda }$. The white solid contour has been drawn at $\mathrm{\Lambda }=0.5$ and indicates the phase transition from a miscible to an immiscible ground state. The two other white dashed and dash-dotted lines are drawn at $\mathrm{\Lambda }=0.9$ and 0.1, respectively. The black dashed lines separate the phase diagram in SF, SS, and single and multiple insulating droplet states. Examples of immiscible 3D isosurfaces of (c) SD $(•)$, (d) MD $(•)$, (e) SS $(•)$, and (f) SF $(•)$ states are shown. Ground-state density profiles of a miscible (g) SD $(\blacksquare )$, (h) MD $(\blacksquare )$, and (i) SS $(\blacksquare )$, and (j) SF $(\blacksquare )$ states are shown. All these states are highlighted by the corresponding marker in the ground-state phase diagrams [see (a) and (b)]. Here $n_i$ is the density of species $i$, where $i=1,2$ (both the components have identical shapes). Results are for $N_1=N_2=N$ and (a) $a_{12}=90a_B$ and (b) $a_{ii}=90a_B$, with $a_{ii}^{\text{dd}}=131a_B$ (${\mu }_i^m=9.93{\mu }_B$), where $a_B$ is the Bohr radius and $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)$ Hz.

Figure 2

Ground-state phase diagram of a Dy-Dy mixture as a function of intracomponent $s$-wave scattering length $a_{ii}$ and trap aspect ratio $\lambda$. The shading represents the peak density of the combined binary system. The black dashed lines indicate phase boundaries between the superfluid state (SF) with a low-peak density and a supersolid (SS) and a droplet state (SD and MD) with a high peak density. The black circle with a red border marks the critical point (CP) and the white solid line is drawn at $\mathrm{\Lambda }=0.5$, representing the immiscibility phase boundary. Results are for the case of $N_1=N_2=6\times {10}^4, {}^{164}\mathrm{Dy}$ atoms with $a_{ii}^{\text{dd}}=131a_B$ (${\mu }_i^m=9.93{\mu }_B$), where $a_B$ is the Bohr radius and ${\mu }_B$ the Bohr magneton.

Figure 3

Miscible and immiscible SD state in an imbalanced mixture. (a) Integrated axial density of species 1 (violet) and species 2 (green) of (a i) a miscible and (a ii) an immiscible SD state. (b) Corresponding chemical potential densities in the axial direction ($z$ axis). Results are for the case of $a_{11}=70a_B, a_{22}=80a_B, N_1={10}^4, N_2=5\times {10}^3$, and (a i) and (b i) $a_{12}=65a_B$ ($<\sqrt{a_{11}{a}_{22}}$) and (a ii) and (b ii) $a_{12}=75a_B$ ($>\sqrt{a_{11}{a}_{22}}$). In both cases the binary mixture is confined in an oblate shape harmonic trap with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)$ Hz.

Figure 4

Three-dimensional isosurface density profiles of [species-1 (red) and species-2 (violet)] immiscible ground states in an imbalanced binary mixture, drawn at $10\%$ of the respective peak densities. (a i) Immiscible SD state in the $z$ direction, with the same parameters as in Fig. 3. The (a ii) SS-SF, (a iii) SS-MD, and (a iv) SS-SS mixed ground states are shown in the $x\text{−}y$ plane. (b) Different pattern formation at the interface of two species in an imbalanced mixture. Results are for the case of a binary DBEC confined in a harmonic trap with $\{{\omega }_x,{\omega }_y,{\omega }_z\}=2\pi \times \{45,45,133\}$ Hz and (a) $a_{11}=90a_B, a_{22}=95a_B, a_{12}=95a_B$, and $N_1=6\times {10}^4$ and (b) $a_{11}=90a_B, a_{22}=100a_B$, and $a_{12}=100a_B$. The other parameters are (a ii) $N_2=5\times {10}^3$, (a iii) $N_2=2\times {10}^4$, (a iv) $N_2=5\times {10}^4$, (b i) $N_1=6\times {10}^4$ and $N_2={10}^3$, (b ii) $N_1=6\times {10}^4$ and $N_2=5\times {10}^3$, (b iii) $N_1=8\times {10}^4$ and $N_2=1.5\times {10}^4$, and (b iv) $N_1={10}^5$ and $N_2=3\times {10}^4$.

Figure 5

(a) and (b) Effective potential experienced by the species 1 and specis 2 due to the presence of the other component of a SS-SF mixed state in the $x-y$ plane. The color bar represents the effective potential in units of $\hslash {\omega }_x$. (c) Corresponding chemical potential densities of species 1 (red) and species 2 (light green) along the $x$ axis. The results are for the case of $a_{11}=90a_B, a_{22}=100a_B, a_{12}=100a_B, N_1={10}^5$, and $N_2=3\times {10}^4$ and the imbalanced binary mixture is confined in a harmonic trap potential with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)$ Hz.

Figure 6

Tuning the interspecies scattering lengths $a_{12}$ for the fixed values of $a_{ii}=90a_B$. (a) Linear ramp with ramp time $t_{\text{on}}=100$ ms. From an initial value of $a_{12}=70a_B$, the interspecies scattering lengths are tuned to the final values $a_{12}=95a_B$ and $100a_B$, respectively, through this ramp. (b) Corresponding variation of the overlap integral with time. Snapshots of the density profile of (c i)–(f i) species 1 $n_1(x,z,y=0)$, (c ii)–(f ii) species 2 $n_2(x,z,y=0)$, and (c iii)–(f iii) the total system $n(x,z,y=0)$ are shown, following a quench from a miscible MD state with $a_{12}=70a_B$ to an immiscible MD state with $a_{12}=100a_B$. Both condensates consist of $N=6\times {10}^4$ atoms and the balanced binary mixture is confined in a harmonic trap with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)$ Hz. The color bar denotes the density in units of $\mu {\mathrm{m}}^{-2}$.

Figure 7

Time evolution of the momentum space density of (a) species 1 ${\tilde{n}}_1\left(k_x\right)$, (b) species 2 ${\tilde{n}}_2\left(k_x\right)$, and (c) the total system $\tilde{n}\left(k_x\right)$, following a linear quench with ramp time $t_{\text{on}}=100$ ms from a miscible SF state with $a_{ii}=120a_B$ to a immiscible MD state with $a_{ii}=85a_B$. Both condensates consist of $N=6\times {10}^4$ atoms, with $a_{12}=90a_B$, and the balanced binary mixture is confined in a harmonic trap with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)$ Hz. The color bar denotes the density in the $k_x$ space, in units of ${10}^3\mu \mathrm{m}$.

Figure 8

Snapshots of the density profile of the composite binary system in momentum space following a linear quench with ramp time $t_{\text{on}}=100$ ms from a miscible SF state with $a_{ii}=120a_B$ to an immiscible MD state with $a_{ii}=85a_B$, keeping the interspecies scattering length $a_{12}$ fixed at $90a_B$. (a) Initially at $t=0$ ms, a single peak corresponds to the SF state at $k_x=0\mu {\mathrm{m}}^{-1}$. (b) At $t=55.7$ ms, two symmetric rotons peak at $k_x=\pm 3.05\mu {\mathrm{m}}^{-1}$. The insets show the corresponding density distribution $\tilde{n}(k_x,k_y,k_z=0)$ in the $k_x-k_y$ plane. The parameters are the same as in Fig. 7.

Figure 9

Variation of energy with the interspecies scattering length $a_{12}$ of a balanced binary mixture. The mixture is confined in a harmonic trap with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)$ Hz and the other parameters fixed at $a_{11}=a_{22}=90a_B$ and $N_1=N_2={10}^4$ atoms. The results are obtained through a variational approach within the SSA framework (dashed line) and the coupled EGPE solutions (solid line) are in agreement.

Figure 10

Ground-state phase diagrams in an oblate harmonic trap of a Dy-Dy mixture as a function of the number of particles in each condensate and (a) intraspecies and (b) interspecies scattering length. The color domains on the phase plots correspond to the different contrast regimes as labeled above the phase diagrams. The other parameters are the same as in Fig. 1.

Figure 11

Effective potential due to the presence of the other component of an immiscible SD state in the $x-z$ plane. The color bar represents the effective potential in units of $\hslash {\omega }_x$. The other parameters are the same as in Fig. 3.

Figure 12

Quenching intraspecies scattering lengths for a fixed values of $a_{12}=90a_B$. (a) Linear ramp with ramp time $t_{\text{on}}=100$ ms. From an initial value of $a_{ii}=120a_B$, the intraspecies scattering lengths are quenched to final values $a_{ii}=85a_B$ and $95a_B$, respectively, through this ramp. (b) Corresponding variation of the overlap integral with time. Snapshots of the density profile of (c i)–(e i) species 1 $n_1(x,y,z=0)$, (c ii)–(e ii) species 2 $n_2(x,y,z=0)$, and (c iii)–(e iii) the total system $n(x,y,z=0)$ are shown, following a quench from a miscible SF state ($a_{ii}=120a_B$) to a miscible SS state ($a_{ii}=95a_B$). Both condensates consist of $N=6\times {10}^4$ atoms and the balanced binary mixture is confined in a harmonic trap with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)$ Hz. The color bar denotes the density in units of $\mu {\mathrm{m}}^{-2}$.

Figure 13

Characteristic phase profiles of the composite binary mixture in the $x-y$ plane at a specific time instant following a linear quench with ramp time 100 ms from a miscible SF state with $a_{ii}=120a_B$ to an immiscible MD state with $a_{ii}=85a_B$, keeping the interspecies scattering length $a_{12}$ fixed at $90a_B$. The black dashed circles delineate the edges of the dipolar binary mixture. The phase profiles correspond to (a) the miscible SF state, (b) the miscible SS state, and (c) the immiscible MD state. The parameters are the same as in Fig. 7.