Dissipation-sensitive multiphoton excitations of strongly interacting Rydberg atoms

We theoretically investigate the effect of dissipation on multiphoton excitation of Rydberg atoms. The steady states and the dynamics are compared via two types of four-level excitation schemes with different dissipative paths of spontaneous emission. We find that in the case of strong Rydberg-Rydberg interaction, the schemes will settle in several different nonequilibrium steady states. The interesting aspect is that there exist the multistable steady states, which reveals the competition between interaction-induced nonlinearity and dissipation caused by spontaneous emission. A numerical simulation on the Rydberg population dynamics in the bistable region exhibits different features existing in the two schemes even with the same initial conditions, which accounts for the influence of the dissipation on the dynamics.

Figure 1

$N$-type and cascade atomic multilevel systems. States $\left|g\right.〉$, $\left|e\right.〉$, $\left|s\right.〉$, and $\left|r\right.〉$ respectively show the ground state, two intermediate states, and Rydberg state, which are successively coupled by laser Rabi frequencies ${\Omega }_g$, ${\Omega }_s$, and ${\Omega }_r$. The three-photon laser detuning $\Delta$ can be controlled in experiment. Other parameters are described in the text.

Figure 2

Phase diagram in the $\left({\Omega }_r,\Delta \right)$ space, indicating the influence of decay and nonlinearity on the phase of the Rydberg atoms, with the numbers of the stable solutions marked in different gray shades. The left column is for the $N$-type system and the right column is for the cascade one. From top to bottom, we fix the Rabi frequencies ${\Omega }_g=5.0$, ${\Omega }_s=2.0$, and change the strength of decay and Rydberg interaction: (a) and (d) $V_{dd}=-50$ and $\gamma =0.01$; (b) and (e) $V_{dd}=-50$ and $\gamma =1.0$; (c) and (f) $V_{dd}=-10$ and $\gamma =1.0$. (All parameters are scaled by $\Gamma$.)

Figure 3

The steady-state populations of state $|g\rangle$ (first row), state $|e\rangle$ (second row), state $|s\rangle$ (third row), and $|r\rangle$ (last row) are plotted as a function of three-photon detuning $\Delta$, with the weak Rydberg-state coupling case (${\Omega }_r/\Gamma =1.0$) in thick black curves and the strong coupling case (${\Omega }_r/\Gamma =8.0$) in thin blue (gray) curves. The dashed curves correspond to the values in the unstable phases. The left and right columns are for the $N$-type system and the cascade system, respectively. Other parameters are same as Figs. 2 and 2.

Figure 4

The final Rydberg state population (${\rho }_{rr}^{t\to \infty }$) for (a) the $N$-type system and (b) the cascade system, as a function of initial population on the $s$ state (${\rho }_{ss}^{t=0}$) and $g$ state (${\rho }_{gg}^{t=0}$). Dark and light colors represent the low and high Rydberg-state occupancy, respectively. The colorless regions are unstable and without stationary solutions. Panels (c) and (d) show population evolution in the bistable region for the $N$-type system and the cascade system, respectively, where ${\rho }_{gg}^{t=0}=0.1$ and ${\rho }_{ss}^{t=0}=0.4$ (dashed blue [gray] line) and $0.7$ (solid black line). See the text for other parameters.