Ground-state phase diagram of a dipolar condensate with quantum fluctuations

We consider the ground state properties of a trapped dipolar condensate under the influence of quantum fluctuations. We show that this system can undergo a phase transition from a low density condensate state to a high density droplet state, which is stabilized by quantum fluctuations. The energetically favored state depends on the geometry of the confining potential, the number of atoms, and the two-body interactions. We develop a simple variational ansatz and validate it against full numerical solutions. We produce a phase diagram for the system and present results relevant to current experiments with dysprosium and erbium condensates.

Figure 1

The quantum fluctuation parameter ${\gamma }_{\mathrm{QF}}$ for each species is indicated for (solid line) ${}^{164}\mathrm{Dy}$ and (solid line with symbols) ${}^{168}\mathrm{Er}$ cases as a function of $a_s$ (where $a_0$ is the Bohr radius). The values where ${\epsilon }_{\mathrm{dd}}=1$ for each species are indicated by small red boxes.

Figure 2

Stationary solution properties in an oblate trap as a function of the $s$-wave scattering length. (a) Energy, (b) width parameters, and (c) peak density. The variational and GPE solutions are given as solid and dotted lines respectively. Insets to (a) show contours of $E({\sigma }_{\rho },{\sigma }_z)$ highlighting the minima, for the $a_s$ values indicated [also shown as vertical dashed lines in (a)]. Vertical dashed lines in (b) and (c) represent the cases studied in Fig. 3. Results are for the case of $N=15\times {10}^3 {}^{164}\mathrm{Dy}$ atoms with $a_{\mathrm{dd}}=130a_0$, where $a_0$ is the Bohr radius, and $\{{\omega }_{\rho },{\omega }_z\}=2\pi \times \{45,133\}{\text{s}}^{-1}$.

Figure 3

Density slices of stationary state solutions to the generalized GPE in the $y=0$ plane for the LDP [(a)–(c)] and the HDP [(d)–(f)] for three regimes $a_s/a_0=78$, 81, and 86 indicated as vertical dashed lines in Figs. 2 and 2. For $a_s/a_0=78$ the LDP exhibits a “blood-cell” profile, while for $a_s/a_0=81$ and 86 the HDP has a Saturn-ring-like feature. (g) The effective potential as defined in Eq. (12) for the case of $a_s/a_0=86$.

Figure 4

Surface marking the phase transition between the low density (LDP) and high density (HDP) phases of a dysprosium dipolar condensate. The critical line is indicated as a thick black line. Parameters are for ${}^{164}\mathrm{Dy}$, and the harmonic trap is taken to have a fixed geometric mean trap frequency of $\overline{\omega }/2\pi =64.6$ Hz. The phase diagram was calculated using the variational ansatz. Examples of two paths are indicated as vertical lines that correspond to an $s$-wave quench that (1) passes across the phase transition and (2) avoids the phase transition by evolving continuously from the LDP to HDP outside the critical line. Path (1) is the interaction quench used in Ref. [19]. The two curves which lie on the surface are the cuts that are explored further in Figs. 5 and 6.

Figure 5

Phase diagram as a function of $s$-wave scattering length and trap aspect ratio for (a) dysprosium and (b) erbium. The shading represents the peak density of the ground state. The black line indicates the phase transition, the large dot marks the critical point (CP), and the gray lines bound the region of metastable coexistence. Note that for erbium the density eventually exceeds the maximum given in the color bar but we allow this saturation to facilitate the comparison with dysprosium. The HDP generally exists at smaller $a_s$ than the LDP. Parameters: $N=15000, \overline{\omega }/2\pi =64.6$ Hz. The red line with dots indicates the quench path used in Ref. [19].

Figure 6

Upper: Peak density versus scattering length. (a) Dysprosium atom numbers from black to red (gray): 500, 1000, 1250, $5\times {10}^4$. (b) Erbium atom numbers from black to red (gray): 1500, 3000, 4000, $5\times {10}^4$. Solid curves represent variational solutions while points are from the full GPE, the latter of which is only shown for the smallest and largest atom numbers. The vertical solid lines illustrate ${\epsilon }_{\mathrm{dd}}=1$ and ${\epsilon }_{\mathrm{dd}}=0.6$. Lower: (c) Dysprosium phase diagram in $a_s-N$ space demonstrating a critical point at $N$ = 1110, $a_s/a_0=71.1$. (d) Erbium critical point at $N$ = 3380, $a_s/a_0=44.1$. The shading represents the peak density of the ground state. The black lines indicate the phase transition, the large dots mark the critical points, and the gray lines bound the region of metastable coexistence. Note that for erbium the density eventually exceeds the maximum given in the color bar, but we allow this saturation to facilitate the comparison with dysprosium. Trap parameters are the same for all panels: ${\omega }_{\rho }=2\pi \times 81.4{\mathrm{s}}^{-1}, {\omega }_z=2\pi \times 40.7 {\mathrm{s}}^{-1}$.

Figure 7

Properties of a single droplet of Dy atoms in a prolate trap as a function of $a_s$: (a) radial width, (b) axial width, (c) peak density. Variational solutions are given as lines while GPE results are shown as points. Black is for $N={10}^4$ atoms and red (gray) for $N=5\times {10}^4$. Vertical dashed lines represent the scattering lengths of the curves shown in Fig. 8. Vertical solid line marks ${\epsilon }_{\mathrm{dd}}=1$. Trap: ${\omega }_{\rho }=2\pi \times 81.4 {\mathrm{s}}^{-1}, {\omega }_z=2\pi \times 40.7 {\mathrm{s}}^{-1}$.

Figure 8

Properties of a single droplet of Dy atoms in a prolate trap as a function of $N$: (a) radial width, (b) axial width, (c) peak density. Variational solutions are given as lines while GPE results are shown as points. Black is for $a_s/a_0=100$ and red (gray) for $a_s/a_0=70$. Thin horizontal lines show the analytic prediction of Eq. (13). Vertical dashed lines represent the $N$ values of the curves in Fig. 7. Trap: ${\omega }_{\rho }=2\pi \times 81.4 {\mathrm{s}}^{-1}, {\omega }_z=2\pi \times 40.7 {\mathrm{s}}^{-1}$.

Figure 9

Cutoff regions with $\{{\sigma }_{\rho },{\sigma }_z\}=\{1/6,1\}\mu \mathrm{m}$ with $k_0$ from $n=1.5\times {10}^{15} {\mathrm{cm}}^{-3}$ and $a_s=70a_0$ [26], giving $\{k_{c,\rho },k_{z,\rho }\}=\{2.1,0.35\}{k}_0$. Lines indicate the $k_c$ boundaries (see text) used for Cutoff I (blue/gray), Cutoff II (red/light gray), and Cutoff III (with ${\epsilon }_{\mathrm{dd}}=1.87$, black).

Figure 10

Results for the real part of ${\mathcal{Q}}_5^{\prime }\left({\epsilon }_{\mathrm{dd}}\right)$ for the cutoff options from Fig. 9 (same colors). Also shown is the result with no cutoff (green dash-dotted) and the approximate result ${\mathcal{Q}}_5\left({\epsilon }_{\mathrm{dd}}\right)=1+\frac{3}{2}{\epsilon }_{\mathrm{dd}}^2$ (black dashed). At ${\epsilon }_{\mathrm{dd}}=1.87,\mathrm{Im}\left\{{\mathcal{Q}}_5\left({\epsilon }_{\mathrm{dd}}\right)\right\}\approx 0.137$ and $\mathrm{Im}\left\{{\mathcal{Q}}_5^{\prime }\left({\epsilon }_{\mathrm{dd}}\right)\right\}\approx 0.013$ for Cutoff I.