Atomic loss and gain as a resource for nonequilibrium phase transitions in optical lattices

Recent breakthroughs in the experimental manipulation of strongly interacting atomic Rydberg gases in lattice potentials have opened an avenue for the study of many-body phenomena. Considerable efforts are currently being undertaken to achieve clean experimental settings that show a minimal amount of noise and disorder and are close to zero temperature. A complementary direction investigates the interplay between coherent and dissipative processes. Recent experiments have revealed a glimpse into the emergence of a rich nonequilibrium behavior stemming from the competition of laser excitation, strong interactions, and radiative decay of Rydberg atoms. The aim of the present theoretical work is to show that local incoherent loss and gain of atoms can in fact be the source of interesting out-of-equilibrium dynamics. This perspective opens up paths for the exploration of nonequilibrium critical phenomena and, more generally, phase transitions, some of which so far have been rather difficult to study. To demonstrate the richness of the encountered dynamical behavior we consider here three examples. The first two feature local atom loss and gain together with an incoherent excitation of Rydberg states. In this setting either a continuous or a discontinuous phase transition emerges with the former being reminiscent of genuine nonequilibrium transitions of stochastic processes with multiple absorbing states. The third example considers the regime of coherent laser excitation. Here the many-body dynamics is dominated by an equilibrium transition of the “model A” universality class.

Figure 1

Schematic representation of the system: (a),(b) an optical lattice is realized within a background cloud of atoms. Atoms within the sites undergo laser-induced coherent transitions between their ground state $\left|\downarrow \right.〉$ and a high-lying (Rydberg) state $\left|\uparrow \right.〉$. The corresponding Rabi frequency and laser detuning are $\mathrm{\Omega }$ and $\mathrm{\Delta }$, respectively. A third state, $\left|0\right.〉$, describes an empty or “inactive” site. Atoms are captured in and released from the sites with rates ${\gamma }_{\mathrm{B}\downarrow }$ (capturing a ground-state atom), ${\gamma }_{\mathrm{D}\downarrow }$ (losing a ground-state atom), and ${\gamma }_{\mathrm{D}\uparrow }$ (losing an excited atom). The atomic states are furthermore subject to dephasing at a rate $\mathrm{\Gamma }$. (c) Rydberg atoms interact with a van der Waals potential $V_{kq}$, whose value for nearest neighbors is denoted by $V_{\text{NN}}$. The corresponding energy shift of the Rydberg state in the vicinity of an excited atom is sketched. (d) Emergent branching processes in the limit of strong dephasing: in the case analyzed in Sec. 3 (square lattice) a single excitation has the potential for branching (with rate $\lambda$) which enables the production of larger clusters. In the case discussed in Sec. 4 (triangular lattice) the system parameters are chosen such that two nearby excitations are required for a cluster to grow.

Figure 2

Stationary density of excitations $n$ extracted from the mean-field equations (9) and (7b) for $-\mathrm{\Delta }=V_{\mathrm{NN}}=64\mathrm{\Gamma }, {\gamma }_{\mathrm{B}\downarrow }=0.01\mathrm{\Gamma }$, and ${\gamma }_{\mathrm{D}\uparrow }={\gamma }_{\mathrm{D}\downarrow }$. The data are shown as a function of the branching rate $\lambda$ for resonant processes and the density of active sites $\overline{\eta }={\gamma }_{\mathrm{B}\downarrow }/({\gamma }_{\mathrm{B}\downarrow }+{\gamma }_{\mathrm{D}\downarrow })$. The color scale is bounded by $n_{\text{max}}=0.5$. The red dashed line corresponds to the values taken by the critical rate ${\lambda }_c$ for different values of $\overline{\eta }$. A cross section is displayed in the inset for $\overline{\eta }=0.8$ (along the cyan horizontal line in the main figure), which highlights the mean-field scaling behavior $n\sim \lambda -{\lambda }_c$. The green, dashed line corresponds to the same curve calculated including the leading off-resonant processes relevant in a Rydberg gas. As expected, the introduction of the latter makes the transition smoother, but deviations are only visible in close vicinity to the critical point ${\lambda }_c$.

Figure 3

Phase diagrams of the pure and the Rydberg process (see text) in the $\overline{\eta }-\lambda$ plane for a 1D chain of 100 sites and a 2D square lattice of $20\times 20$ sites. The parameters are chosen as $-\mathrm{\Delta }=V_{\mathrm{NN}}=64\mathrm{\Gamma }, {\gamma }_{\mathrm{B}\downarrow }=0.01\mathrm{\Gamma }$, and ${\gamma }_{\mathrm{D}\downarrow }={\gamma }_{\mathrm{D}\uparrow }\equiv {\gamma }_{\text{D}}$. The color scale is set with respect to the maximal value the density can take, i.e., $n_{\text{max}}=1$ for the pure process and $1/2$ for the Rydberg one. Numerically computed exponents $\beta$ (static) and $\delta$ (dynamic) are displayed in the panels. The selected parameter ranges are shown as a cyan and a green line on the main plot. For the 1D pure process we also show (in log-log scale) the critical profiles of the stationary density $n$ as a function of ${\mathrm{\Delta }}_{\lambda }=\lambda -{\lambda }_c$ (lower-left inset) and of its evolution in time (upper-right inset) to highlight the scaling behavior. For comparison we provide the known DP exponents [34]: ${\beta }_{\text{1D}}=0.276, {\beta }_{\text{2D}}=0.584, {\delta }_{\text{1D}}=0.159$, and ${\delta }_{\text{2D}}=0.451$.

Figure 4

Stationary density of excitations $n$ (we select the stable solution with highest value) extracted from the mean-field equations (13) and (7b). The data are shown as a function of $\lambda$ (branching rate) and $\overline{\eta }$ (stationary density of active sites) for $-\mathrm{\Delta }=2V_{\text{NN}}=64\mathrm{\Gamma }, {\gamma }_{\mathrm{B}\downarrow }=0.01\mathrm{\Gamma }$, and ${\gamma }_{\mathrm{D}\uparrow }={\gamma }_{\mathrm{D}\downarrow }$. Here $n_{\text{max}}=0.5$. The dashed line corresponds to the critical rate ${\lambda }_c$ for different values of $\overline{\eta }$. In the inset we plot the profiles of the three stationary solutions of Eq. (13) for $\overline{\eta }\approx 0.65$ in a parameter interval indicated by the cyan horizontal line in the main figure. The absorbing solution is displayed as a solid, cyan line which never leaves 0. The upper solid orange line represents the stable solution with finite density which only occurs for $\lambda >{\lambda }_c$. The dashed green line corresponds to the third, unstable solution.

Figure 5

(a) Stationary density profile in the $\lambda -\overline{\eta }$ plane for the branching process defined on a $25\times 25$ triangular lattice. The color scale extends up to $n_{\text{max}}=0.5$. As in the mean-field treatment in Fig. 4, we fix $-\mathrm{\Delta }=2V_{\text{NN}}=64\mathrm{\Gamma }, {\gamma }_{\mathrm{B}\downarrow }=0.01\mathrm{\Gamma }$, and ${\gamma }_{\mathrm{D}\uparrow }={\gamma }_{\mathrm{D}\downarrow }\equiv {\gamma }_{\text{D}}$. For a sufficiently large population of trapped atoms (sufficiently small loss rate ${\gamma }_{\text{D}}$) and large enough $\lambda$ ($\propto {\mathrm{\Omega }}^2$), the process becomes clearly capable of maintaining a finite density of excitations for long times. (b) Counting statistics $p\left(n\right)$ of the final density $n\left(t_{\text{max}}\right)$ at fixed $\mathrm{\Omega }=0.125\mathrm{\Gamma }$ and $\mathrm{\Gamma }{t}_{\text{max}}=25000$ as a function of $\overline{\eta }$. Bimodality is a signature of the discontinuity of the phase transition and becomes clearly visible in the inset section taken along a section at $\overline{\eta }\approx 0.7194$.

Figure 6

Stationary phase diagram of the mean-field equations of motion (15a)–(15d) for three different detunings $\mathrm{\Delta }$ in the plane spanned by $\mathrm{\Omega }/V$ and ${\gamma }_{\mathrm{D}\downarrow }/V={\gamma }_{\mathrm{D}\uparrow }/V\equiv {\gamma }_{\text{D}}/V$. The shaded areas correspond to the bistable regions for $\mathrm{\Gamma }=0.01V$ and ${\gamma }_{\mathrm{B}\downarrow }={\gamma }_{\text{D}}$. Points lying outside represent choices for which there is only one physical solution. From top to bottom, the detuning is fixed at $-0.255V, -0.22V$, and $-0.15V$. The boundaries of each shaded region meet with vanishing net angle at the corresponding critical point (red disk). The solid black line displays the path taken by the critical point as $\mathrm{\Delta }$ is varied. The dashed lines show how the profile of this path shrinks when the dephasing rate $\mathrm{\Gamma }$ is increased. From the outermost to the innermost, the values correspond to $\mathrm{\Gamma }=0.001V, 0.01V$ (solid line), $0.04V$, and $0.08V$. The transition disappears when $\mathrm{\Gamma }/V\ge {\gamma }_{\mathrm{B}\downarrow }/4({\gamma }_{\mathrm{B}\downarrow }+{\gamma }_{\text{D}})=0.125$. The bistable region is also absent when $\mathrm{\Delta }/V$ lies outside the interval $[-9{\gamma }_{\mathrm{B}\downarrow }/\left[16({\gamma }_{\mathrm{B}\downarrow }+{\gamma }_{\text{D}})\right],0]=[-9/32,0]$.