Control of ${}^{164}\mathrm{Dy}$ Bose-Einstein condensate phases and dynamics with dipolar anisotropy

We investigate the quench dynamics of quasi-one- and two-dimensional dipolar Bose-Einstein condensates of ${}^{164}\mathrm{Dy}$ atoms under the influence of a fast rotating magnetic field. The magnetic field thus controls both the magnitude and sign of the dipolar potential. We account for quantum fluctuations, critical to formation of exotic quantum droplet and supersolid phases in the extended Gross-Pitaevskii formalism, which includes the so-called Lee-Huang-Yang correction. An analytical variational ansatz allows us to obtain the phase diagrams of the superfluid and droplet phases. The crossover from the superfluid to the supersolid phase and to single and droplet arrays is probed with particle number and dipolar interaction. The dipolar strength is tuned by rotating the magnetic field with subsequent effects on phase boundaries. Following interaction quenches across the aforementioned phases, we monitor the dynamical formation of supersolid clusters or droplet lattices. We include losses due to three-body recombination over the crossover regime, where the three-body recombination rate coefficient scales with the fourth power of the scattering length $(a_s)$ or the dipole length $\left(a_{dd}\right)$. For fixed values of the dimensionless parameter, ${\epsilon }_{dd}=a_{dd}/a_s$, tuning the dipolar anisotropy leads to an enhancement of the droplet lifetimes.

Figure 1

Schematic illustration of the dBEC of ${}^{164}\mathrm{Dy}$ atoms trapped in (a) a circularly symmetric quasi-2D and (b) an elongated quasi-1D. A magnetic field is applied rotating at an angle $\varphi$ around the $z$ axis and aligning the atomic dipoles along its direction. (c) The table indicates presence or absence of density modulation (DM) and global phase coherence (GPC) within the superfluid (SF), supersolid (SS), single droplet (${\mathrm{DL}}_{\mathrm{S}}$), and multiple droplet (${\mathrm{DL}}_{\mathrm{M}}$) phases. Here we do not refer to typical density modulations originating from the external trap but to the ones stemming from the competition between short- and long-range interactions leading to spatially periodic density undulations.

Figure 2

Phases identified through the chemical potential of a circular quasi-2D dBEC in terms of the relative interaction strength parameter ${\epsilon }_{dd}=a_{dd}/a_s$ and the atom number for different magnetic field orientations, namely, (a) $\varphi =0^{\circ }$, (b) $\varphi ={30}^{\circ }$, (c) $\varphi ={60}^{\circ }$, and (d) $\varphi ={90}^{\circ }$. Apparently, as the magnetic field tends toward the $x\text{−}y$ plane ($\varphi ={90}^{\circ }$), the ${\mathrm{DL}}_{\mathrm{M}}$ and SS phases do not occur and solely the SF and ${\mathrm{DL}}_{\mathrm{S}}$ survive. The white dashed line indicates $\mu =0$, while the blue solid (dashed) line delimits the SS (${\mathrm{DL}}_{\mathrm{S}}$) phase. The dBEC resides in a quasi-2D trap characterized by $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$.

Figure 3

Ground-state density profiles (${\text{a}}_1$)–(${\text{a}}_3$) $n(x,y)$ and (${\text{a}}_4$)–(${\text{a}}_6$) $n(y,z)$ representing (${\text{a}}_1$), (${\text{a}}_4$) a SF, (${\text{a}}_2$), (${\text{a}}_5$) a SS, and (${\text{a}}_3$), (${\text{a}}_6$) a droplet (hexagonal) lattice. A SS state is characterized by overlapping density humps, while a droplet cluster has a crystal arrangement. The harmonically trapped quasi-2D dBEC with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$ has $N={10}^5$ particles and it is subjected to a magnetic field along the $z$ direction, i.e., $\varphi =0^{\circ }$. The color bar alternates for each panel and corresponds to the density changing from zero (black) to a maximum value (yellow) in units of $1/l_{\mathrm{osc}}^2=0.73\mu {\mathrm{m}}^{-2}$

Figure 4

Variation of the different energy components $E_{\sigma }$ (see the legend) as a function of the tilt angle $\varphi$ at a fixed ${\epsilon }_{dd}=1.87$ for the quasi-2D dBEC with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$. At a fixed ${\epsilon }_{dd}=1.87$, droplets are generated within the $\varphi <{22}^{\circ }$ and $\varphi >{\varphi }_m$ regions where $E_{\mathrm{DDI}}<0$. The green line refers to $E_{\mathrm{DDI}}/E_{\mathrm{CI}}$ (right axis) with CI denoting the contact interaction.

Figure 5

Impact of the field orientation $\varphi$ on the spatial distribution of the ${\mathrm{DL}}_{\mathrm{S}}$ state. Densities (${\text{a}}_1$)–(${\text{a}}_3$) $n(x,y)$ and (${\text{a}}_4$)–(${\text{a}}_6$) $n(y,z)$ of a dBEC with $N=6\times {10}^4$ in a quasi-2D trap of $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$. The single droplet becomes narrower (wider) for larger $\varphi$ in the interval $\varphi <{\varphi }_m$ ($\varphi >{\varphi }_m$) and becomes elongated along the $z$ ($y$) direction due to the combined effect of the polarization of the magnetic field and ${\epsilon }_{dd}$. The characteristic length scale set by the trap is $l_{\mathrm{osc}}=1.17\mu \mathrm{m}$. The color bar expressed in units of $1/l_{\mathrm{osc}}^2=0.73\mu {\mathrm{m}}^{-2}$ is different for each panel and denotes the density with intensity from zero (black) to a maximum value (yellow).

Figure 6

Integrated density profiles $n(x,y)$ of the dBEC containing $N=2.5\times {10}^5$ atoms with ${\epsilon }_{dd}=1.75$ for (${\mathrm{a}}_1$) $\varphi =0^{\circ }$, (${\mathrm{a}}_2$) $\varphi ={15}^{\circ }$, and (${\mathrm{a}}_3$) $\varphi ={20}^{\circ }$. $n(x,y)$ for (${\mathrm{b}}_1$) ${\epsilon }_{dd}=2.11$, (${\mathrm{b}}_2$) ${\epsilon }_{dd}=1.82$, and (${\mathrm{b}}_3$) ${\epsilon }_{dd}=1.55$ with a fixed $\varphi ={10}^{\circ }$. Various bound state configurations can be created by either tuning the tilt angle, e.g., hexagonal lattices, or adjusting ${\epsilon }_{dd}$ such as different polygons. The external quasi-2D geometry is characterized by $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$ defining a length scale $l_{\mathrm{osc}}=1.17\mu \mathrm{m}$. The color bar (units of $1/l_{\mathrm{osc}}^2=0.73\mu {\mathrm{m}}^{-2}$) refers to the density which is distinct for each panel and has a gradient from zero (black) to a maximum value (yellow).

Figure 7

Snapshots of the dBEC density $n(x,y)$ following an interaction quench from a SF state with ${\epsilon }_{dd}=1.1$ to (${\text{a}}_1$)–(${\text{a}}_6$) ${\epsilon }_{dd}=1.45$ and (${\text{b}}_1$)–(${\text{b}}_6$) ${\epsilon }_{dd}=1.87$. The initially 2D smooth density profile deforms in the course of the evolution toward a SS and a droplet lattice, respectively. The dBEC consists of $N=6\times {10}^4$ atoms and it is confined in a quasi-2D harmonic trap characterized by $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$. The color bar denotes the density in units of $1/l_{\mathrm{osc}}^2=0.73\mu {\mathrm{m}}^{-2}$ and differs for each panel while featuring a gradient from zero (black) to a maximum value (yellow). The system's characteristic timescale set by the trap is ${\omega }_x^{-1}=3.5\mathrm{ms}$ and the length scale defined through the harmonic oscillator length $l_{\mathrm{osc}}=1.17\mu \mathrm{m}$.

Figure 8

Characteristic phase profiles (${\text{a}}_1$)–(${\text{a}}_3$) $\mathrm{\Phi }(x,y,z=0)$ at a specific time instant (see the legends) in the long-time dynamics after the quench. The pink circles designate the edges of the dBEC cloud. In all cases, a quench of a quasi-2D dBEC from its SF state at ${\epsilon }_{dd}=1.1$ is considered toward the (${\text{a}}_1$) SF with ${\epsilon }_{dd}=1.2$, (${\text{a}}_2$) the SS having ${\epsilon }_{dd}=1.45$ and (${\text{a}}_3$) the ${\mathrm{DL}}_{\mathrm{S}}$ with ${\epsilon }_{dd}=1.87$ phase. The phase undulations designate the SS and the droplet states in contrast to the smooth phase of a SF. (b) The time evolution of the global phase coherence ${\beta }_c\left(t\right)$ is presented for different postquench ${\epsilon }_{dd}$ values in the quasi-2D geometry with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$. The dashed lines in (b) represent the time evolution of ${\beta }_c\left(t\right)$ when ${\epsilon }_{dd}\left(t\right)$ increases in a linear manner with ramp time $\tau =120\mathrm{ms}\gg {\omega }_x^{-1}$. Coherence is completely lost in the droplet regime, while it is almost perfectly maintained following an adiabatic ramp toward the SF and the SS phases. The colors of the dashed lines refer to the same postquench ${\epsilon }_{dd}$ values with the ones indicated by the solid lines. Other system parameters are the same as in Fig. 7.

Figure 9

The same as in Fig. 7 but for the quasi-1D setting. The generation of (${\text{a}}_1$)–(${\text{a}}_6$) SS for ${\epsilon }_{dd}=1.45$ and (${\text{b}}_1$)–(${\text{b}}_6$) droplet arrays when ${\epsilon }_{dd}=1.87$ is illustrated. The dBEC consists of $N=6\times {10}^4$ atoms and it resides in a quasi-1D trap with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (227,37,135)\mathrm{Hz}$. The color bar in units of $1/l_{\mathrm{osc}}^2=1.33\mu {\mathrm{m}}^{-2}$ refers to the density while it is different for each panel and changes from zero (black) to a maximum value (yellow). The characteristic length scale is the harmonic oscillator length $l_{\mathrm{osc}}=0.85\mu \mathrm{m}$ and the corresponding timescale is ${\omega }_x^{-1}=0.7\mathrm{ms}$.

Figure 10

(a) Dynamics of the particle loss for different postquench values of ${\epsilon }_{dd}$ and tilt angles $\varphi$ (see legend). Droplets feature the largest losses at $\varphi =0$, while a finite field orientation leads to a SF state exhibiting more dramatic decay processes. Insets: Corresponding density snapshots $n(x,y)$ after the quench from ${\epsilon }_{dd}=1.1$ visualizing the formation of SS (upper panel) and droplets (lower panel) at short timescales. (b) Same as in (a) but for several $\varphi$ (see legend) and a fixed post-quench ${\epsilon }_{dd}=2.18$ residing in the droplet phase. The emergent patterns persist for longer timescales by means of a tilted magnetic field. The droplet lifetime is enhanced for fixed ${\epsilon }_{dd}<a_{dd}/D$ and increasing $\varphi$ as long as $\varphi <{\varphi }_m$. The dBEC with $N=6\times {10}^4$ experiences a quasi-2D trap of $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$ setting a characteristic timescale ${\omega }_x^{-1}=3.5\mathrm{ms}$.

Figure 11

Energy of the quasi-2D dBEC as predicted within the variational approach [Eq. (A7)] in terms of the widths ${\sigma }_z$ and ${\sigma }_x$. Different panels refer to field orientations (${\text{a}}_1$) $\varphi =0^{\circ }$, $\left({\text{a}}_2\right)\varphi ={30}^{\circ }$, $\left({\text{a}}_3\right)\varphi ={60}^{\circ }$, and $\left({\text{a}}_4\right)\varphi ={90}^{\circ }$. The white crosses denote the location of the respective energy minimum associated with the equilibrium set (${\sigma }_x$, ${\sigma }_z$). The sign of the energy subsequently characterizes the equilibrium state as SF or droplet. The equilibrium state energy between the variational method and the eGPE results are in good agreement, exhibiting an average error not more than $10\%$. (b) The response of (left axis) ${\sigma }_{\eta }$ with $\eta \equiv \{x,y,z\}$ and (right axis) the ratio ${\sigma }_y/{\sigma }_z$ for varying $\varphi$ in the quasi-2D regime with fixed ${\epsilon }_{dd}=1.75$ and $N=6\times {10}^4$. The results obtained through the variational principle (dashed lines) and the eGPE method (solid lines) are in agreement.

Figure 12

Time evolution of the dBEC widths [(${\text{a}}_1$), (${\text{a}}_3$), (${\text{a}}_5$)] ${\sigma }_x$ and [(${\text{a}}_2$), (${\text{a}}_4$), (${\text{a}}_6$)] ${\sigma }_z$ quantifying the breathing motion of the background caused by the quench. A quench from a SF state with ${\epsilon }_{dd}=1.1$ toward $\left({\text{a}}_1\right)\text{and}\left({\text{a}}_2\right)$ the SF, $\left({\text{a}}_3\right)\text{and}\left({\text{a}}_4\right)$ the SS, and $\left({\text{a}}_5\right)\text{and}\left({\text{a}}_6\right)$ the ${\mathrm{DL}}_{\mathrm{M}}$ phases characterized by specific postquench values of ${\epsilon }_{dd}$ (see legend). Apparently, a saturation trend of the dynamically formed droplet lattice occurs [(${\text{a}}_5$) and (${\text{a}}_6$)]. The dBEC consists of $N=6\times {10}^4$ atoms confined in a quasi-2D trap with $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (45,45,133)\mathrm{Hz}$.