4 can be imaged using, e.g., tightly focused beams. The authors thank H.P. B¨ uchler and F. Ferlaino for discussions. This work was supported by IQOQI, the Austrian FWF, the US through MURI and EOARD, the EU through STREP FP7-ICT-2007-C project NAME- QUAM, the Canadian NSERC through G121210893. an effective intrinsic heating rate for the many-body sys- tem. We conservatively estimate the crystalline phases ( τ ≫ 1) to be observable up to an average “melting” time t$_{m}$ ∝ T$_{M}$/ ( γ$_{eff}$E$_{R}$ ), with T$_{M}$ the (classical) melting tem- perature in the trap. We computed T$_{M}$ for all N < 30 by means of molecular dynamics simulations for the 2D system in the classical limit ( τ → ∞ ) [16]. By monitor- ing the behavior of the specific heat and density-density correlations, we determine the value T$_{M}$ = 0 . 4(2)˜ ϵ . This can result in characteristic lifetimes of order of several milliseconds. For example, for $^{87}$Rb atoms coupled to the Rydberg-state ( n = 20, n$_{1}$ = 19, n$_{2}$ = 0) with d$_{0}$ ≈ 1 . 45kDebye, γ$_{r}$ ∼ 100kHz, E$_{R}$ ≃ 25kHz, via a laser with Ω / 2 π = 100MHz, Δ / 2 π = 1GHz and a DC field F = 25kV/m, for ω/ 2 π = 5kHz we obtain r$_{0}$ ≃ 680nm, τ ≃ 20, ˜ ϵ ≃ 4 E$_{R}$ , and t$_{m}$ ≳ 5ms. We further investigated the melting dynamics. Spon- taneous emission was simulated by applying random in- plane kicks to each particle with momentum ℏ k$_{xy}$ = (2 mE$_{Rxy}$ ) 1 / $^{2}$, with fixed E$_{Rxy}$ ≲ E$_{R}$ , at an average rate γ$_{eff}$ . Panels (a) and (b) in Fig. 4 show our results for E$_{Rxy}$ = 0 . 5˜ ϵ and 2.5 ˜ ϵ as a function of time t , respec- tively, for N = 13 particles [see Fig. 3]. Thin lines are single classical-dynamics trajectories, while thick dashed lines are averages over many trajectories. We find that the average energy E$_{tot}$ increases linearly with time t , as expected. For the smaller E$_{Rxy}$ of panel (a), we find that the crystal melts roughly at the expected time t$_{m}$ (cor- responding to several spontaneous emission processes), while in panel (b) the observed melting time t$_{eq}$ is signif- icantly longer than t$_{m}$ (roughly the time of one sponta- neous emission). This prolonged stability is a dynamical process: classical melting can occur only after the system has reached thermal equilibrium. The latter corresponds to the time t$_{eq}$ at which the average potential energy E$_{pot}$ and kinetic energy E$_{kin}$ become equal and equipartition of energy applies. t$_{eq}$ , and thus the crystal lifetime, can be (much) longer than t$_{m}$ . Since interparticle distances can be of the order of a µ m or more, the spatial structure of the crystalline phases 0 0.2 0.6 (a) N =13 γ$_{eff }$=0.32ω E $_{R}$=0.5ε N =13 γ$_{eff }$=0.64ω E $_{r}$=2.5ε E kin E pot E tot E kin E pot E tot ω t /2π E/N ε 0 0.1 0.15 0.25 0.2 0 0.2 0.1 0.4 0.5 0 1 3 (b) ω t /2π E/N ε t$_{m}$ t$_{eq}$ t$_{m}$ t$_{eq}$ FIG. 4: (color online) Molecular dynamics (MD) simulation of crystal heating. Black, red and blue lines are total en- ergy E tot , kinetic E kin and potential E pot energies vs. time t , respectively. Thin continuous lines: single MD trajectories. Thick dashed lines: averages over many MD trajectories. t eq and t m are the equilibration and melting time, respectively. [1] T. Lahaye et al. , Rep. Prog. Phys. 72 , 126401 (2009); R.V. Krems, W.C. Stwalley and B. Friedrich, Cold Molecules: Theory Experiment, Applications (CRC Press, 2009). [2] H. P. B¨ uchler et al. , Phys. Rev. Lett. 98 , 060404 (2007); A. Micheli et al. , Phys. Rev. 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