Fragmented-condensate solid of dipolar excitons

We discuss a possible link between the recently observed macroscopic ordering of ultracold dipolar excitons (MOES) and the phenomenon of supersolidity. In the dilute limit we predict a stable supersolid state for a quasi-one-dimensional system of bosonic dipoles characterized by two- and three-body contact repulsion. We phenomenologically extend our theory to the strongly-correlated regime and find a critical value of the contact interaction parameter at which the supersolid exhibits a quantum phase transition to a fragmented state. The wavelength of the fragmented-condensate solid is defined by the balance between the quantum pressure and the entropy due to fluctuations of the relative phases between the fragments. Our model appears to be in good agreement with the relevant experimental data, including the very recent results on commensurability effect and wavelength of the MOES.

Figure 1

Change of the shape of the elementary excitation spectrum of a dilute 1D dipolar condensate produced by the variation of the density $n_1$. We use Eq. (22) with $\zeta =50$. For $n_1=n_r\equiv {[2x_{*}ln(\zeta /e^{3/2})]}^{-1}$ the spectrum starts to develop a roton-maxon structure (solid thick line). An increase of $n_1$ pushes the roton minimum towards zero, as is illustrated by the thin line taken at $n_1=(n_r+30n_c)/31$. At $n_1=n_c\equiv {[2x_{*}ln(\zeta /2e)]}^{-1}$ the roton touches zero and for larger densities the uniform condensate becomes dynamically unstable.

Figure 2

The dimensionless energy $\tilde{\mathcal{E}}=\mathcal{E}{x}_{*}^2/V_0^{1\mathrm{D}}$ of a quasi-one-dimensional condensate of bosonic dipoles as a function of the density ${\tilde{n}}_1=n_1{x}_{*}$. The uniform state (U) ceases to be the ground state at $n_1=n_c^{\left(1\right)}$. At this point the system undergoes a first order quantum phase transition to a supersolid state (S) via a collapse (shaded area) accompanied by an increase of the instability magnitude $\theta$ and wave vector $k_0$. For small $\theta$ the corresponding dependence $\tilde{\mathcal{E}}\left({\tilde{n}}_1\right)$ is given by (28) with the substitution (31) and is shown by the red line. As $k_0$ approaches $a_y^{-1}$, three-body repulsive forces become more efficient and, eventually, stabilize the system in a supersolid phase (S). The dependence of the energy of this phase on the density is given by (35) and is shown by the blue line. The new stable point $n_c^{\left(2\right)}$ is defined by the requirement $\mu \left(n_c^{\left(2\right)}\right)=\mu \left(n_c^{\left(1\right)}\right)$, where $\mu =\partial \mathcal{E}/\partial n_1$ is the chemical potential of the system. We take $\zeta ={10}^4,x_{*}/a_y=0.1$, and $\gamma =1/4$ (for $\alpha =8)$. We find ${\tilde{n}}_c^{\left(1\right)}=0.12,{\tilde{n}}_c^{\left(2\right)}=0.49$, and $k_0\left(n_c^{\left(2\right)}\right)=0.48a_y^{-1}$.

Figure 3

Inhomogeneous state of the 2D cigar as a chain of trapped 2D Bose-Einstein condensates. The maxima of the density wave (44) [short dashed gray line] are approximated by the Thomas-Fermi paraboloids (51) (red and blue solid lines). The oscillator frequencies ${\omega }_j$ are chosen such that the corresponding harmonic traps intersect the cosine profile (46) at the bending points defined as ${\partial }^2{V}_{\mathrm{auto}}\left(x\right)/\partial x^2=0$. In the $y$ direction the width of the condensates is equal to the width of the cigar. In the boundary region in between the adjacent beads the Thomas-Fermi approximation fails. In this region the density profile takes universal form (blue and red dash-dot lines) which does not depend neither on the actual shape of the autolocalizing potential $V_{\mathrm{auto}}(x,y)$, nor on the parameter $V_0$ characterizing the contact part of the exciton-exciton interaction [57]. These enter the expression for the characteristic length $d_x=\left(2m\right|F_x{|/{\hslash }^2)}^{-1/3}$ (which we conveniently use as a unit length), and the transformation ${\mathrm{\Pi }}_j(x,y=0)=\sqrt{\pi /12}{\left(\eta d_x\right)}^{-1}{\pi }_{k_0}(x/d_x)$ for the trial function. We have taken $d_x=R_x/4$ in order to better visualize the boundary region in such specific scale. In practice, one expects $d_x\ll R_x$.

Figure 4

Topological transformation of the fragmented-condensate solid conserving simultaneously the total number of particles $N$, the chemical potential $\mu$, and the total energy of the system $E$. In the Thomas-Fermi limit the density profile of each bead $|{\mathrm{\Pi }}_j{(x,y)|}^2$ takes the form on an inverted paraboloid. The height of the paraboloid is defined by the chemical potential according to $|{\mathrm{\Pi }}_j{(0,0)|}^2=\mu /{\tilde{V}}_0$. Providing that the condition (56) is satisfied, the total energy and the number of particles in the green paraboloid is the same as in a system of the two blue ones. We have used the same units of length, density, and energy as in Fig. 3.

Figure 5

Dependence of the width (a), the wavelength (b), and the critical temperature (c) of the MOES on the gate voltage $V_g$. Squares are the experimental data [41]. In (a) the solid line is the cubic function (83) with $c_3,c_2,c_1$, and $c_0$ being the fitting parameters. We substitute thus obtained analytic equation for the MOES width into Eq. (82). The formula (82) is then used to fit the corresponding dependence of the wavelength in (b). We fix $x=4$ and $V_0=6.8\mu \mathrm{eV}\times \mu m^2$, consistently with the earlier estimates [38, 68]. The bath temperature is $T=1.6$ K. The only adjustable parameters are $n\left(V_g^{*}\right)$ and $dn\left(V_g^{*}\right)/d{V}_g$, which enter the expansion (84) for the exciton density $n$ around $V_g^{*}=1.17$ V. The obtained dependence of $n$ on $V_g$ is used to plot the values of the critical temperature [Eq. (85)] at which the MOES is expected to disappear [solid line in (c)].