3 g ( r ) 0 1 2 3 4 0 0.5 1 1.5 2 2.5 τ=1 τ =2.5 τ =5 τ =20 0 0.1 0.2 0.3 f n’ 1 7 13 τ =1 τ =2.5 r / r$_{0}$ crystalline RC phase, particles are arranged in concentric rings with a fixed number of particles per ring. For small enough temperatures, these rings are free to rotate in- dependently of each other. In the crystalline CC phase, atoms are arranged in the classical crystal configuration, at fixed relative positions. Mesoscopic crystals have been also found in excitonic materials in Refs.[12]. In the following we provide example results for N = 13 atoms, which we found to display all general features of mesoscopic clusters with N ≲ 40. We find that for N ≫ 40 the system resembles the homogeneous situation, with a sharp crossover between SF and CC phases around τ$_{c}$ . FIG. 3: (color online) Radial density profiles g ( r ) for the cases of Fig. 3. Inset: statistics of computed particle exchanges f as a function of the number of particles n ′ participating to the exchange for the cases of finite superfluid fraction τ = 1 and 2.5. In a superfluid, f is finite for all n ′ ≤ N . The supersolid phase with τ = 2 . 5 has finite density modulation (figure) and the same f -distribution of τ = 1 (inset). Panels (a-d) in Fig. 3 are snapshots of the particle density for N = 13 and 1 ≤ τ ≤ 20 in the four meso- scopic phases described above. These results correspond to a low enough temperature T ≪ ˜ ϵ , that they can be regarded as groundstate estimates. Panel (a) shows a SF for τ = 1 with a flat density profile (see also g ( r ) in Fig. 3), where the various particle probability clouds overlap. This overlap is directly connected to superfluid- ity [13], and consistently we here measure ρ$_{s}$ = 1. The emergence of a MS phase for τ = 2 . 5 ≲ τ$_{c}$ is signaled by a distinguishable density modulation in Fig. 1(b) (see also Fig. 3), combined to a measured ρ$_{s}$ = 1. We find that the superfluid properties are completely unaltered by the increased strength of interactions with respect to the case τ = 1. This is quantified in the inset of Fig. 3 by measuring the statistics of exchange cycles f in a many- particle path [13]: in a superfluid, the probability f that n ′ particles exchange is finite for all n ′ ≤ N , while in a crystal is approximately zero. In the inset of Fig. 3 it is shown that f is finite and equal for τ = 1 and 2.5, computed at corresponding low temperatures (i.e., same fractions of ˜ ϵ ). where τ ≡ ˜ ϵ/ ℏ ω = ( r$_{0}$/ℓ ) 2 = ( m ˜ D/ ℏ $^{2}$ℓ ) 2 / 5 characterizes the strength of the dipole-dipole interactions in the trap with ℓ = √ ℏ /mω the harmonic oscillator length. Equa- tion (2) shows that τ plays the role of an effective mass representing a control parameter, which can be increased by increasing the strength of dipole-dipole interactions, or by compressing the trap. For small τ ≲ 1, we expect the kinetic energy to dom- inate, and the cluster to be in a weakly interacting su- perfluid (SF) phase, while for τ ≫ 1 the kinetic energy becomes negligible, and the system ground state should resemble the classical crystalline (CC) configuration ob- tained by minimizing the last two terms of Eq. (2). We obtain an estimate of the critical τ$_{c}$ for the crossover from the superfluid to the crystal by noting that for a homo- geneous system the superfluid-crystal transition occurs at r$_{QM}$ = Dm/ ℏ $^{2}$a = 18 ± 4 [2, 10], where r$_{QM}$ repre- sents the ratio of the dipolar interactions D/a 3 to the kinetic energy ℏ $^{2}$/ma2 with a the mean interparticle dis- tance. By rewriting τ$_{c}$ = ( r$_{QM}$a/ℓ ) 2 / $^{5}$, and approximat- ing a ∼ ℓ , we obtain the prediction τ$_{c}$ ≃ 3, which is essentially N - independent . For τ ≳ τ$_{c}$ [panels (c-d)] we observe crystallization of the atomic cloud, with ρ$_{s}$ = 0 and finite density modu- lations, consistently with the estimates above. In partic- ular, for τ = 5 we obtain a RC phase, with concentric, independent rings [14], while for τ = 20 particles are ar- ranged in the CC configuration for N = 13, with 4 par- ticles at the center, and 9 outside. This classical crystal, characterized by large peaks in g ( r ) in Fig. 3, is found to be the groundstate configuration for all τ ≳ 20. We determine the generic zero-temperature phase di- agram for trapped dipolar atoms by means of Quantum Monte Carlo simulations based on the continuous-space Worm Algorithm [11]. We find that for mesoscopic clus- ters with N ≲ 40 four different quantum phases exist: (i) a SF for τ ≪ τ$_{c}$ , (ii) a mesoscopic supersolid (MS) for τ ≲ τ$_{c}$ , (iii) a ring-shaped crystal (RC) for τ ≳ τ$_{c}$ , and (iv) a CC for τ ≫ τ$_{c}$ . These phases are distinguished by measuring the superfluid fraction ρ$_{s}$ and the radial den- sity profile g ( r ): the SF and MS phases are superfluid with ρ$_{s}$ = 1, while the RC and CC phases have ρ$_{s}$ = 0; the SF has a flat, featureless g ( r ), while the MS, RC and CC phases have significant density modulations. In the Effective heating : The residual effective single-particle spontaneous emission rate is γ$_{eff}$ ∼ (Ω / Δ) $^{2}$γ$_{r}$ . In this work, we focus on the case where after the spontaneous emission the atom is found in | g 〉 , and then instanta- neously re-dressed, which is the relevant situation for n ≲ 30 [15]. Spontaneous emission results in the release of a photon, with momentum ℏ k and recoil energy E$_{R}$ , in the tens of kHz. For dipoles parallel to F , emission will be preferentially along z , and E$_{R}$ will be absorbed by the confining lattice potential with ℏ ω$_{⊥}$ > E$_{R}$ . However, any finite component of the momentum ℏ k$_{xy}$ of the emit- ted photon in the ( x − y )-plane will result in a random ”kick” of the atoms, which will ultimately translate into