H ˜ ϵ = N ∑ i =1 [ − 1 2 τ 2 ∂ 2 ∂ ρ 2 i + 1 2 ρ 2 i ] + ∑ i>j 1 | ρ i − ρ $_{j}$| $_{3}$, (2) H$_{ij}$ =    − 2Δ + v$_{ij}$ Ω Ω 0 Ω − Δ 0 Ω Ω 0 − Δ Ω 0 Ω Ω 0   $^{,}$ (1) 2 ics simulations, and show the emergence of a dynamical thermalization timescale in these systems. FIG. 2: (color online) (a-d) Monte Carlo snapshots of the den- sity of particles in all mesoscopic phases for N = 13 dipoles, as a function of the effective mass τ . (a) superfluid; (b) su- persolid; (c) ring-like crystals; (d) classical crystal. Effective interaction potentials with suppressed deco- herence from atomic collisions and spontaneous emission are obtained as follows. In the presence of a homoge- neous electric field (field strength F , oriented along e $_{z}$) the energy levels of an alkali atom show the well-known Stark structure [6]. We are interested here in high an- gular momentum states which show a linear Stark ef- fect. The energy of these states is well approximated by ϵ$_{nn}$$_{1}$$_{n}$$_{2}$ = (3 / 2) ea$_{0}$n ( n$_{1}$ − n$_{2}$ ) F , with n$_{1}$ and n$_{2}$ parabolic quantum numbers, a$_{0}$ the Bohr radius and e the elec- tron charge. The highest-energy state | r 〉 of a given manifold n ( n$_{1}$ = n − 1, n$_{2}$ = 0) is energetically sep- arated from the next (lower lying) state in the mani- fold by Δ E$_{n}$ = e (3 / 2) a$_{0}$n F , while adjacent manifolds remain well separated by an energy Δ E$_{IT}$ ≫ Δ E$_{n}$ for field strengths smaller than F$_{IT}$ ≈ 1 3 n 5 m $^{2}$e5 128 π $^{3}$ϵ 3 $_{0}$ℏ 4 (Inglis- Teller limit). In our scheme, Fig. 1, each atom is ef- fectively reduced to a two-level system. Within the rotating-wave approximation and with a product ba- sis { | r 〉$_{i}$| r 〉$_{j}$ , | r 〉$_{i}$| g 〉$_{j}$ , | g 〉$_{i}$| r 〉$_{j}$ , | g 〉$_{i}$| g 〉$_{j}$ } the Hamiltonian governing the internal dynamics of two atoms is given by optical fields (e.g., optical lattice), and confinement to low-dimensional geometries. Collisional losses for R$_{ij}$ > R$_{0}$ are linked to popula- tion of the attractive part of the dipole-dipole interac- tion. Sampling of this attractive part can be suppressed by confining atoms to 2D using a tight (optical) trap- ping along z , with harmonic frequency ℏ ω$_{⊥}$ = ℏ $^{2}$/ma2 ⊥ > V 3D int ( R ), with m the atomic mass, and a$_{⊥}$ the transverse harmonic oscillator length. The BO potential is then purely repulsive in 3D: there is an energy barrier between the long-distance repulsion and the attractive short- distance regime. Residual losses are linked to tunneling below this energy barrier, and can be computed semiclas- sicaly as Γ$_{coll}$ = ω$_{att}$ exp[ − c (Ω $^{4}$Dm/ ℏ $^{2}$Δ$^{4}$a$_{⊥}$ ) 2 / $^{5}$], with c a constant of order unity, and ω$_{att}$ the attempt frequency, of order of the average kinetic energy in the gas. For R$_{ij}$ > R$_{0}$ the effective dynamics is purely 2D, with inter- actions V 2D int = (Ω / Δ) $^{4}$D/ρ3 ≡ ˜ D/ρ $^{3}$, with ρ = | ρ | , and ρ a vector in the ( x − y )-plane. with v$_{ij}$ = D (1 − 3 cos θ$_{ij}$ ) | R i − R $_{j}$| − 3 ≡ D ν$_{ij}$ R − 3 ij , the dipole-dipole interaction between two atoms at position R $_{i}$( R $_{j}$), and D = d 2 $_{0}$/ (4 πϵ$_{0}$ ). θ$_{ij}$ is the angle between the vector R i − R j and the dipole moment aligned par- allel to the z -axis. The interaction potential which is experienced by the dressed ground state atoms is given by the Born-Oppenheimer (BO) energy surface that adi- abatically connects to the energy of the product state | g 〉$_{i}$| g 〉$_{j}$ as Δ → ∞ . Here, we focus on the weak driv- ing limit Ω < Δ. For two atoms approaching each other and blue detuning Δ > 0, the groundstate BO potential will be approximately given by V 3D int ( R$_{ij}$, θ$_{ij}$ ) ≃ Dν$_{ij}$/R 3 ij for R$_{ij}$ < R$_{0}$ , with R$_{0}$ ≃ ( D/ ℏ Δ) 1 / 3 a resonant Condon point with typical values in the hundreds of nm, Fig. 1(b). Diabatic crossings with different potential surfaces of the same and different n -manifolds leading to collisional two- body losses will occur in this region R$_{ij}$ < R$_{0}$ , at distances of order R$_{n}$ ≃ ( D/ Δ E$_{n}$ ) 1 / 3 and R$_{IT}$ ≃ ( D/ Δ E$_{IT}$ ) 1 / $^{3}$, respectively, with R$_{0}$ > R$_{n}$ ≫ R$_{IT}$ , reminiscent of blue- shielding techniques [2, 9]. Mesoscopic crystals with Rydberg-dressed atoms: We consider a setup where N bosonic dressed atoms are con- fined to a 2D plane by applying a strong transverse trap- ping field [2], e.g a 1D optical lattice, and are aligned perpendicular to the plane [Fig. 1(c)]. We assume an additional in-plane parabolic trap with frequency ω , as realized in experiments by a magnetic dipole trap, or a single site of a large spacing optical lattice. Defin- ing length and energy scales r$_{0}$ = ( ˜ D/mω $^{2}$)1 / 5 and ˜ ϵ = mω $^{2}$r2 0 = ˜ D/r 3 0 = ( m $^{3}$ω6 ˜ D $^{2}$)1 / $^{5}$, respectively, the 2D Hamiltonian in dimensionless form reads For R$_{ij}$ ≳ R$_{0}$ , two-body diabatic losses are absent and V 3D int ( R$_{ij}$, θ$_{ij}$ ) ≃ (Ω / Δ) $^{4}$Dν$_{ij}$ /R 3 $_{ij}$. In this work we will focus on this parameter regime. This has two additional advantages: (i) spontaneous emission rates are strongly reduced to values γ$_{eff}$ and (ii) effective interactions are reduced to values compatible with trapping of atoms with