Population trapping in a pair of periodically driven Rydberg atoms

We study the population trapping extensively in a periodically driven Rydberg pair. The periodic modulation of the atom-light detuning effectively suppresses the Rabi couplings and, together with Rydberg-Rydberg interactions, leads to the state-dependent population trapping. We identify a simple yet general scheme to determine population trapping regions using driving-induced resonances, the Floquet spectrum, and the inverse participation ratio. Contrary to the single-atom case, we show that the population trapping in the two-atom setup may not necessarily be associated with level crossings in the Floquet spectrum. Further, we discuss under what criteria population trapping can be related to dynamical stabilization, taking specific and experimentally relevant initial states, which include both product and the maximally entangled Bell states. The behavior of the entangled states is further characterized by the bipartite entanglement entropy.

Figure 1

Floquet mode properties of a driven single two-level atom with $\omega =8\mathrm{\Omega }$. (a) The quasienergies ${\epsilon }_k$ and (b) IPR (${\mathrm{\Pi }}_1^{|g\rangle }$) as a function of ${\mathrm{\Delta }}_0$ for $\delta =15\mathrm{\Omega }$. (c) The quasienergies ${\epsilon }_k$ and (d) IPR (${\mathrm{\Pi }}_1^{|g\rangle }$) as a function of $\alpha =\delta /\omega$ for the resonance ${\mathrm{\Delta }}_0=\omega$ ($n_1=1$). In (d), we also show the Bessel function, $J_1\left(\alpha \right)$. Its zeros coincide with ${\mathrm{\Pi }}_1^{|g\rangle }=0$ indicating the population trapping. The parameter $\alpha$ is varied by changing $\delta$. Plots (e) and (f) show the results for the case of primary resonance, ${\mathrm{\Delta }}_0=0$ ($n_1=0$). The crossings of ${\epsilon }_k$ in (e) and the zeros of ${\mathrm{\Pi }}_1^{|g\rangle }$ in (f) coincide with the zeros of $J_0\left(\alpha \right)$. The plots (e) and (f) are the special case of population trapping corresponding to the dynamical stabilization. The parameter $\alpha$ is varied by changing $\delta$ and keeping $\omega$ constant.

Figure 2

The IPR (${\mathrm{\Pi }}_1^{|g\rangle }$) as a function of $\alpha$ and ${\mathrm{\Delta }}_0$ for $\omega =8\mathrm{\Omega }$. The pearl-stripes are along the $\alpha$ axis at the resonances $n\omega ={\mathrm{\Delta }}_0$. The local minima (${\mathrm{\Pi }}_1^{|g\rangle }=0$) along the first stripe are the points of DS for which $J_0\left(\alpha \right)=0$. The parameter $\alpha$ is varied by changing $\delta$ and keeping $\omega$ constant.

Figure 3

Population dynamics for the resonance type R1 ($n_1\omega ={\mathrm{\Delta }}_0$) for the initial states (a) $|I\rangle =|gg\rangle$ and (b) $|I\rangle =|ee\rangle$. The same, but with the resonance type R2 ($n_2\omega ={\mathrm{\Delta }}_0-V_0$) for the initial state (c) $|I\rangle =|gg\rangle$ and (d) $|I\rangle =|ee\rangle$ with ${\mathrm{\Delta }}_0=2\mathrm{\Omega }$. In (a) we see the Rabi oscillations between $|gg\rangle$ and $|+\rangle$ states, whereas in (b) we observe no dynamics. Similarly, (c) shows the absence of dynamics, and the Rabi oscillations between $|+\rangle$ and $|ee\rangle$ states are shown in (d). We took $V_0=10\mathrm{\Omega }, \delta =15\mathrm{\Omega }$, and $\omega =8\mathrm{\Omega }$ for all plots. The value of ${\mathrm{\Delta }}_0$ is taken such that $n_1=1$ for (a) and (b), and for (c) and (d) we have $n_2=-1$.

Figure 4

(a) The quasienergy spectrum for $N=2$ as a function of ${\mathrm{\Delta }}_0$ for $V_0=10\mathrm{\Omega }, \delta =15\mathrm{\Omega }$, and $\omega =8\mathrm{\Omega }$. Plots (b) and (c) show ${\mathrm{\Pi }}_2^{|gg\rangle }$ and ${\mathrm{\Pi }}_2^{|ee\rangle }$, respectively. The peaks in ${\mathrm{\Pi }}_2$ and the avoided crossings in ${\epsilon }_k$ indicate the three different resonant transitions: (R1) $n_1\omega ={\mathrm{\Delta }}_0$, (R2) $n_2\omega ={\mathrm{\Delta }}_0-V_0$, and (R3) $n_3\omega =2{\mathrm{\Delta }}_0-V_0$, labeled by $n_1, n_2$, and $n_3$, respectively.

Figure 5

The quasienergy spectrum ${\epsilon }_k$ and IPR (${\mathrm{\Pi }}_2^{|gg\rangle }, {\mathrm{\Pi }}_2^{|ee\rangle }$) for $N=2, {\mathrm{\Delta }}_0=0$, and $\omega =8\mathrm{\Omega }$, as a function of $\alpha$ for different $V_0$. Plot (a) shows ${\epsilon }_n$ for $V_0=0\mathrm{\Omega }$ (dashed lines) and $V_0=0.2\mathrm{\Omega }$ (solid lines), and (b) and (c) show the same for $V_0=2\mathrm{\Omega }$ and $V_0=8\mathrm{\Omega }$, respectively. Since ${\mathrm{\Delta }}_0=0$, in (a) and (b) the level crossings take place at the zeros of $J_0\left(\alpha \right)$. In (a)–(c) the color bar indicates the probability of finding the state $|gg\rangle$ in each of the Floquet modes. The dashed vertical lines in (c) mark $J_0\left(\alpha \right)=0$, and at those points the central Floquet mode consists purely of the $|gg\rangle$ state, which indicates dynamical stabilization. Plots (d) and (e) show the IPR ${\mathrm{\Pi }}_2^{|gg\rangle }$ and ${\mathrm{\Pi }}_2^{|ee\rangle }$, respectively. In (f), we show the Bessel functions $J_0\left(\alpha \right)$ (solid line) and $J_{-1}\left(\alpha \right)$ (dashed line). The parameter $\alpha$ is varied by changing $\delta$ and keeping $\omega$ constant.

Figure 6

The IPR (a) ${\mathrm{\Pi }}_2^{|gg\rangle }$ and (b) ${\mathrm{\Pi }}_2^{|ee\rangle }$ as a function of $V_0$ and $\alpha$ for $N=2, {\mathrm{\Delta }}_0=0$ (R1 resonance), and $\omega =8\mathrm{\Omega }$. The regions of ${\mathrm{\Pi }}_2^{|gg\rangle }=0$ correspond to the dynamical stabilization of $|gg\rangle$, those where both ${\mathrm{\Pi }}_2^{|gg\rangle }\sim 1$ and ${\mathrm{\Pi }}_2^{|ee\rangle }\sim 0$ indicate the population trapping of $|ee\rangle$, and ${\mathrm{\Pi }}_2^{|gg\rangle }=2$ represents the Rydberg antiblockade in which the system exhibits Rabi oscillations between $|gg\rangle$ and $|ee\rangle$ via the intermediate state $|+\rangle$. The intricate patterns arise due to the competition between the Rabi couplings for the transitions $|gg\rangle \leftrightarrow |+\rangle [\propto J_{n_1}\left(\alpha \right)]$ and $|+\rangle \leftrightarrow |ee\rangle [\propto J_{n_2}\left(\alpha \right)]$. If R2 is satisfied with $V_0={\mathrm{\Delta }}_0$ instead of R1, (a) is ${\mathrm{\Pi }}_2^{|ee\rangle }$ and (b) is ${\mathrm{\Pi }}_2^{|gg\rangle }$. The parameter $\alpha$ is varied by changing $\delta$ and keeping $\omega$ constant.

Figure 7

(a) The IPR (${\mathrm{\Pi }}_2^{|gg\rangle }$) as a function of $\alpha$ for $\omega =30\mathrm{\Omega }$ for different $V_0$ satisfying the R3 resonance with $n_3=0$, i.e., $2{\mathrm{\Delta }}_0=V_0$. (b) The same as in (a), but for different $\omega$ and $V_0=6\mathrm{\Omega }$. In (c), we show the dynamics for the initial state $|gg\rangle$ for the two different resonances: R1 (solid line with $n_1=0$) and R3 (dashed line with $n_3=0$) at the first root of $J_0\left(\alpha \right), \omega =15\mathrm{\Omega }$, and $V_0=6\mathrm{\Omega }$. In (d), we show the same as in (c), except that the initial state is $|ee\rangle$, and for the resonances: R2 (solid line with $n_2=0$) and R3 (dashed line with $n_3=0$). The parameter $\alpha$ is varied by changing $\delta$ and keeping $\omega$ constant.

Figure 8

(a) IPR ${\mathrm{\Pi }}_2^{|+\rangle }$ as a function of $\alpha$ and $V_0$ for $\omega =8\mathrm{\Omega }$ and ${\mathrm{\Delta }}_0=0$. The parameter $\alpha$ is varied by changing $\delta$ and keeping $\omega$ constant. (b) The general behavior of the dynamics of the entanglement entropy ${\mathcal{S}}_A$ for ${\mathrm{\Pi }}_2^{|+\rangle }=0$ (solid line), indicating dynamical stabilization and for ${\mathrm{\Pi }}_2^{|+\rangle }=1$ (dashed line).