Decay-dephasing-induced steady states in bosonic Rydberg-excited quantum gases in an optical lattice

We investigate the possibility of realizing supersolid quantum phases in bosonic Rydberg-excited quantum lattice gases in the presence of nonunitary processes, by simulating the dynamical evolution starting from initial preparation in nondissipative equilibrium states. Within Gutzwiller theory, we first analyze the many-body ground state of a bosonic Rydberg-excited quantum gas in a two-dimensional optical lattice for variable atomic hopping rates and Rabi detunings. Furthermore, we perform time evolution of different supersolid phases using the Lindblad-master equation. With the inclusion of two different nonunitary processes, namely, spontaneous decay from a Rydberg state to the ground state and dephasing of the addressed Rydberg state, we study the effect of nonunitary processes on those quantum phases and observe long-lived states in the presence of decay and dephasing. We find that long-lived supersolid quantum phases are observable within a range of realistic decay and dephasing rates, while high rates cause any initial configuration to homogenize quickly, preventing possible supersolid formation.

Figure 1

Various primitive cells, corresponding to different ordered structures: The spanning vectors $\mathbf{A}$ and $\mathbf{B}$ starting at an arbitrary site $i$ point to the next equivalent site, thus creating the boundaries of the primitive cell (colored in orange). The sites within are labeled by increasing numbers, going from left to right, up to down.

Figure 2

Gutzwiller ground-state phase diagram: (a) the inverse density $1/\overline{n}$ of the many-body ground state; (b)–(e) mean expectation values of the observables ${\overline{\varphi }}_{\nu }$ and ${\overline{n}}_{\nu }$ for varying detuning $\mathrm{\Delta }/\mathrm{\Omega }$ and ground-state hopping $t_g/\mathrm{\Omega }$, and the drawn white phase boundaries act as rough guides to the eye. The fixed parameters for the calculations are $U_g=0.1,U_{ge}=5, U_e=100, V=10000, \mu =0.25, t_e=0$ with $\mathrm{\Omega }$ as scale.

Figure 3

Devil's staircase of the primitive cell size $V_{\mathrm{prim}}$ (blue crosses) and the Rydberg fraction on the occupied site $n_1^e$ (orange diamonds) in the “frozen” limit ($t_g/\mathrm{\Omega }=0$). Additionally, the real-space distributions of Rydberg excitations of various representative quantum phases are displayed in the $x$ and $y$ directions in units of the lattice spacing $a$.

Figure 4

Initial CB-SS quantum phase with average filling $\overline{n}=1$. The primitive cell of the system consists of two sites. The parameters for the Gutzwiller calculations are $U_g=0.1,U_{ge}=5, U_e=100, V=10000, \mu =0.1, t_g=0.0414, t_e=0, \mathrm{\Delta }=0.069$ with $\mathrm{\Omega }$ as scale.

Figure 5

(a) imbalance $\mathrm{\Delta }n\left(t_f\right)$ as a function of the decay and dephasing rates; (b) the exponential decay constant of the mean condensate order parameter of the ground state $\gamma$ as a function of the decay and dephasing rates ($t_f=100\mu s$).

Figure 6

Time evolution of relevant observables for an initial CB-SS state shown in Fig. 4 for different decay and dephasing rates $\mathrm{\Gamma }/\mathrm{\Omega }$ and $\kappa /\mathrm{\Omega }$. Low rates cause a slow decay of the condensate and increase the imbalance $\mathrm{\Delta }n$ of the system, moderate rates enhance the speed with which the condensate drops, and high rates lead to homogeneity.

Figure 7

Initial SS quantum phase with average filling $\overline{n}\approx 1.2$. The primitive cell of the system consists of eight sites. The parameters for the Gutzwiller calculations are $U_g=0.1,U_{ge}=5, U_e=100, V=10000, \mu =0.25, t_g=0.099, t_e=0, \mathrm{\Delta }=2.18$ with $\mathrm{\Omega }$ as scale.

Figure 8

Time evolution of $n_g$ up to $1000\mu s$ starting with the SS configuration shown in Fig. 7 for $\mathrm{\Gamma }/\mathrm{\Omega }=1$ and $\kappa /\mathrm{\Omega }=0$. Starting with an initial SS, the spontaneous emission process drives the system into a new SS quantum phase with a checkerboard-like long-range order.

Figure 9

Time evolution of relevant observables for an initial SS state shown in Fig. 7 for different decay and dephasing rates $\mathrm{\Gamma }/\mathrm{\Omega }$ and $\kappa /\mathrm{\Omega }$. A low spontaneous emission rate induces a transition into a SS with new crystalline structure, while strong decay leads to the homogenization of the system. Dephasing does not destroy long-range order.

Figure 10

Time evolution of relevant observables for an initial SS state shown in Fig. 7 for fixed rates $\mathrm{\Gamma }/\mathrm{\Omega }=0.1$ and $\kappa /\mathrm{\Omega }=0.1$. Fluctuations are strong, but the depletion of the condensate is slow and the system remains a SS for long times.

Figure 11

Frequency spectrum of the site-averaged time evolution of the ground state occupation number $n_g$ depicted in Fig. 10 for various time windows of length $200\mu s$ with varying starting time $t$. Dominant frequencies are found around $0.5\mathrm{\Omega }$ and $\mathrm{\Omega }$, and a small peak is hinted at around $2\mathrm{\Omega }$.

Figure 12

Comparison between the decay-dephasing-quench time evolution of the CB-SS [dashed blue and red lines from Figs. 6–6] and the detuning-quench time evolution of an initially homogeneous system (solid green and orange lines) for different decay and dephasing rates $\mathrm{\Gamma }/\mathrm{\Omega }$ and $\kappa /\mathrm{\Omega }$.

Figure 13

Time evolution of relevant observables for an initial homogeneous state with the parameters used in Fig. 7 for fixed rates $\mathrm{\Gamma }/\mathrm{\Omega }=0.1$ and $\kappa /\mathrm{\Omega }=0.1$. The system goes through two symmetry breakings and converges to the evolution of the initial quantum phase with initial supersolid configuration (compare with Fig. 10).