Supersolid droplet crystal in a dipole-blockaded gas

Supersolid droplet crystal in a dipole-blockaded gas


A novel supersolid phase is predicted for an ensemble of Rydberg atoms in the dipole-blockade regime, interacting via a repulsive dipolar potential “softened” at short distances. Using exact numerical techniques, we study the low temperature phase diagram of this system, and observe an intriguing phase consisting of a crystal of mesoscopic superfluid droplets. At low temperature, phase coherence throughout the whole system, and the ensuing bulk superfluidity, are established through tunnelling of identical particles between neighbouring droplets.

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The search for novel phases of matter drives much of the current research in condensed matter physics. Of particular interest are phases simultaneously displaying different types of order. A chief example, of great current interest, is the so-called supersolid, namely a phase featuring crystalline order, and also capable of sustaining dissipation-less flow. Attempts to observe experimentally a supersolid phase of matter, primarily in a crystal of solid helium, have spanned four decades since early theoretical predictions Andreev and Lifshitz (1969). The most credible claim of such an observation to date Kim and Chan (2004a, b), has been subjected to in-depth scrutiny over the past few years, and it seems fair to state that agreement is lacking at the present time, as to whether experimental findings indeed signal a supersolid phenomenon Prokof’ev (2007).
A new, fascinating avenue to the observation of supersolid and other phases of matter not yet observed (or even thought of), is now opened by advances in cold atom physics, providing not only remarkably clean and controlled experimental systems, but also allowing one to “fashion” artificial inter-particle potentials, not arising in any known condensed matter system. This allows one to address a key theoretical question, namely which two-body interaction potential(s), if any, can lead to the occurrence of a supersolid phase in free space (i.e., not on a lattice).

In a recent article Henkel et al. (2010), Henkel et al. have proposed, based on a mean-field treatment, that a Bose condensate of particles interacting through an effective potential which flattens off at short distance, might support a density modulation. In this Letter, we show by first principle numerical simulations that interaction potentials which combine a long-distance repulsion with a short-distance cutoff, lead in fact to the appearance of a novel self-assembled crystalline phase of mesoscopic superfluid droplets in a system of bosons. Furthermore, such a crystal can turn supersolid in the limit, as tunneling of particles across neighbouring droplets takes place, and superfluid phase coherence is established across the whole system, as individual separate Bose condensates (droplets) organize into a single, global condensate. Specifically, we consider the following two-body potential:


being the characteristic strength of the interaction. This kind of interaction potential can be realized with cold dipole-blockaded Rydberg atoms Lukin et al. (2001); Tong et al. (2004); Vogt et al. (2006); Heidemann et al. (2008); Pritchard et al. (2010). The parameters and above can be controlled with external fields Pupillo et al. (2010); Henkel et al. (2010) (we come back to this point below).

Our system of interest comprises identical bosons of mass , confined to two dimensions 1. The many-body Hamiltonian is the following (in dimensionless form):


where is the distance between particles and , and is given by Eq. (1). All lengths are expressed in terms of the characteristic length , and we introduce a dimensionless cutoff for the potential (1). The system is enclosed in a square cell of area , with periodic boundary conditions. The particle density is , but we shall express our results in terms of the (dimensionless) inter-particle distance =. The energy scale is .
The low-temperature phase diagram of such a system has been explored by means of first principles numerical simulations, based on the Continuous-space Worm Algorithm Boninsegni et al. (2006a, b). Numerical results shown here pertain to simulations with a number of particles varying between 50 and 400, in order to carry out extrapolation of the results to the thermodynamic limit. Our ground state estimates are obtained as extrapolations of results at finite temperature. Details of the simulations are standard, as the use of the potential (1) entails no particular technical difficulty.
In the limit , the truncation of the dipolar potential at short distances does not play an important role, and the low temperature phase diagram of (2) is that of purely dipolar bosons in two dimensions, investigated previously by several authors Büchler et al. (2007); Astrakharchik et al. (2007). It is known that for   = 0.06 the ground state of the system is a triangular crystal, whereas for   = 0.08 it is a uniform superfluid (in the intermediate density range a more complex scenario is predicted Spivak and Kivelson (2004)). As we show below, a very different physics sets in when , in the density ranges which correspond to either the crystalline or superfluid phase in the purely dipolar system.

Figure 1: Snapshots of a system of bosons interacting via potential (1), at the four different temperatures 200 (a), 20 (b), 1.0 (c) and 0.1 (d), expressed in units of . Points shown are taken along individual particle world lines. The nominal value of in this case is 0.14, whereas the cutoff of the potential (1) is =0.3.
Figure 2: (Color online) Results shown are for =0.14 and =0.3. Temperature is in units of . (a): Pair correlation function at a temperature = 200 (triangles), 20 (squares), 1.0 (diamonds) and 0.1 (circles). The simulated system comprises =200 particles. (b): Superfluid density vs. for systems with =100 (square), and 200 (diamond) particles. (c): Frequency of occurrence of permutation cycles of length at the same four temperatures reported in panel (a). Longer permutation cycles occur at lower temperature.

Fig. 1 shows typical configurations (i.e., particle world lines) produced by Monte Carlo simulations of a system of bosons interacting via the potential (1), at a nominal density corresponding to , at different temperatures spanning three orders of magnitude. The value of the cutoff in this case is 0.3. At the highest temperature, a simple classical gas phase is observed, as shown by the pair correlation function , shown in Fig. 2 (a), which is just a constant (note that does not vanish at the origin, owing to the flattening off of the potential at short distance). As is decreased, an intriguing effect takes place, namely particles bunch into mesoscopic droplets, in turn forming a regular (triangular) crystal. This is shown qualitatively in the snapshots in Fig. 1, but also confirmed quantitatively by the structure of the as well (Fig. 2(a)), which displays pronounced, broad maxima, as well as well-defined minima, where the function approaches zero. We henceforth refer to this phase as the droplet-crystal phase.
The formation of such droplets is a purely classical effect, that depends on the flattening off of the repulsive inter-particle potential below the cutoff distance. In fact, a simple estimate of the number of particles per droplet, can be obtained by considering a triangular lattice of point-like dipoles, each one of strength (as it comprises particles), and by minimizing with respect to the potential energy per particle, for a fixed density. The result is


where . Eq. (3) furnishes a fairly accurate estimate of for the (wide) range of values of the parameters and explored here. For instance, using the parameters of Fig. 1, we find from (3) , which agrees quite well with our simulation result. It is worth noting that a similar sort of pattern formation, due to competing interactions, has been previously established for classical colloidal systems Liu et al. (2008); Archer et al. (2008).
In the limit, long exchanges of identical particles can take place, as a result of particles tunneling from one droplet to an adjacent one. Long exchanges of particles can result in a finite superfluid response throughout the whole system, and indeed for we observe such a bulk superfluid signal, in a range of values of in the vicinity of . A typical result is shown in Fig. 2(b) 2. Because superfluidity arises in concomitance with the droplet-crystal structure, the denomination supersolid seems appropriate.

In order to establish that droplets are individually superfluid, one may consider the statistics of permutation cycles. Fig. 2 (c) shows the frequency of occurrence of exchange cycles involving a varying number of particles (), at three different temperatures, at the physical conditions of Fig. 1. As one can see, as the temperature is lowered exchange cycles involving growing numbers of particles occur. At low temperature they involve almost all the particles in the system; however, even at a higher temperature (e.g., =20 in Fig. 2(c)) one observes exchanges comprising a number of particles up to , i.e., particles inside an individual droplet. This is evidence that droplets are individually Bose condensed and superfluid, even though the system as a whole does not display superfluidity. That droplets should be superfluid at low is not surprising, given that particles in a droplet are essentially non-interacting, due to the flatness of the potential at short distance. However, that droplets are themselves superfluid does not imply that a bulk supersolid phase will always occur in the 0 limit, as discussed above.

Figure 3: Schematic ground state phase diagram of (2) as a function of . The superfluid droplet crystal (SDC) is sandwiched between an insulating droplet crystal (IDC) and a superfluid (SF). For   , an IDC, a single-particle crystal (C) and a superfluid phases are observed. The widths of the SDC and C regions depend on the value of .

At = 0, the supersolid phase is sandwiched between an insulating droplet crystal at high density (i.e., lower ) and a homogeneous superfluid phase at lower density. For   , only two insulating phases are observed, namely the insulating droplet crystal at high density and the crystal of single particles, already detected in Refs. Büchler et al. (2007); Astrakharchik et al. (2007), as well as a superfluid phase at lower density. All of this is summarized in the schematic phase diagram shown in Fig. 3. It is important to stress that supersolid behaviour in this system originates from tunnelling of particles between droplets which are themselves individually superfluid, so that the individual superfluid droplets connect to form a bulk superfluid. This is reminiscent of the phase-locking mechanism in a (self-assembled) array of Josephson junctions.

Figure 4: (Color online) Left panels: Monte Carlo snapshots of a system of particles, interacting via the potential (1), confined in a harmonic trap of strength , at the two temperatures =100 (a) and =0.5 (b). Right panels: corresponding momentum distributions, all normalized to unity for comparison purposes. The value of in this case is 0.3. The development of secondary peaks at low temperature signals the occurrence of a supersolid phase.

The results discussed so far pertain to numerical simulation of the system described by Eq. (2) in its bulk phase. However, in any experiment aimed at probing the physics of such a system, the assembly of particles must necessarily be finite (a few thousand particles is a typical number for current experiments with cold dipolar atoms), confined by an external potential. In order to enable a direct comparison with possible future experiments, we have performed simulations of the same system spatially confined in-plane by a harmonic trap, i.e., the term is added to Eq. (2), being the strength of the trap.
Fig. 4 shows typical many-particle configurations of a trapped system comprising =400 particles, at two different temperatures. Also shown are the associated momentum distributions , which are obtained by Fourier transforming of the spherically and translationally averaged one-body density matrix, computed by Monte Carlo.
Here too, droplets with a well-defined average number of particles form, and organize themselves on a triangular lattice. Correspondingly, the momentum distribution, which is directly observable experimentally by time-of-flight measurements Greiner et al. (2002), develops a sharp central peak, with additional structure on its sides. The secondary peaks correspond to oscillations in the one-body density matrix, in turn reflecting particle tunnelling to adjacent droplets. They are therefore connected to the appearance of the supersolid phase, as explained above.

Summarizing, accurate numerical simulations of a system of dipolar particles interacting via a potential softened at short distance, reveal the existence of a low temperature crystalline phase of superfluid droplets. This phase turns superfluid (supersolid) at through a mechanism of tunnelling of particles between adjacent droplets. At higher density this tunnelling is suppressed and an insulating droplet crystal occurs, a phase which has not previously been predicted. The interaction that underlies such intriguing, until now unobserved physical behaviour, can be realized with dipolar atoms in the dipole-blockade regime.

A comment is on order, concerning the dependence of the results on the particular form of potential utilized here, namely Eq. (1) with its abrupt, sharp cutoff at . First off, the superfluid droplet crystal phase does not crucially depend on the dipolar form of the interaction at long distances. Indeed, it is also observed in our simulations for a van der Waals-like potentials (i.e., ). Secondly, we have obtained qualitatively similar results with different model potentials, featuring a more realistic “flat” region at short distances, as well as a smoother merge of long- and short-range behaviours. For example, we considered the potential , which is naturally realized in a cold gas of alkali atoms by weakly dressing the groundstate of each atom with an excited Rydberg state with a large dipole moment , in the kDebye range Pupillo et al. (2010).
While several dressing schemes are possible Henkel et al. (2010); Honer et al. (2010), here we consider as the lowest-energy state of a Rydberg manifold with principal quantum number for an atom in the presence of a homogeneous electric field in the linear Stark regime, with the Inglis-Teller limit. For a laser with (effective) Rabi frequency and red detuning , the dressed groundstate reads , and thus , with , while the cutoff arises because of the Rydberg-blockade mechanism Lukin et al. (2001). Spontaneous emission rates from are strongly reduced to values of at most . Observability of the phases above benefits from large values of , and in particular of the ratio , favoring comparatively small values of  3. For example, for Rb atoms in an electric field kV/m, with , MHz and GHz, we obtain = 1450 Debye, 400 nm, 210 nm, , and 120 nK. Collective many-body effects in the Rydberg-blockade regime Honer et al. (2010) not described by Eq. (2) should be negligible provided , which is readily satisfied for parameters as in Fig. 1.

This work was supported in part by the Natural Science and Engineering Research Council of Canada under research grant 121210893, and by the Alberta Informatics Circle of Research Excellence (iCore), IQOQI, the FWF, MURI, U.Md. PFC/JQI, EOARD (grant FA8655-10-1-3081), NAME-QUAM. Useful discussions with I. Lesanovsky, N. Prokof’ev and T. Pohl are gratefully acknowledged.


  1. Confinement to two dimensions can be achieved using a tight (magnetic or optical) trap along .
  2. We compute directly the bulk superfluid fraction through the usual winding number estimator for the bulk system.
  3. 2D scattering requires , with the transverse harmonic oscillator length, in the tens of nm range.


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