Periodically driven facilitated high-efficiency dissipative entanglement with Rydberg atoms

A time-dependent periodical field can be utilized to efficiently modify the Rabi coupling of system, exhibiting nontrivial dynamics. We propose a scheme to show that this feature can be utilized for speeding up the formation of dissipative steady entanglement based on the Rydberg antiblockade mechanism. When the modulation frequency is exactly equal to the central frequency of the driving field, a sufficient residence time of the two-excitation Rydberg state is allowed for an irreversible spontaneous decay onto the target state, leading to a steady entanglement with a high fidelity $\sim 0.98$ and a shorter generation time $<400\mu \mathrm{s}$. We show that a global maximal fidelity benefits from a consistence of microwave-field coupling and spontaneous decay strengths. The scheme manifests robust insensitivities towards the imperfect initialization, the fluctuations of van der Waals interaction, and the modulation frequency. This simple approach to facilitate the generation of dissipative entangled two-qubit states may guide an experimental direction in Rydberg quantum technology and quantum information.

Figure 1

(a), (b) Schematic of an effective three-level atom deriving from its original form of a four-level configuration. See texts for detailed parameter descriptions. (c) For example, the two hyperfine ground states $|1\rangle =|5s,f=1,m_f=0\rangle$, $|0\rangle =|5s,f=2,m_f=0\rangle$ are coupled by a weakly resonant Raman coupling with respect to the middle state $|p\rangle =|5p\rangle$, or by a direct microwave coupling $g$ between $|0\rangle$ and $|1\rangle$.

Figure 2

Schematic diagram of a high-fidelity facilitated entanglement preparation. (a) A detailed two-atom energy-level structure. For each atom, two ground states $|1\rangle$ and $|0\rangle$ are coupled by a continuous microwave field $g$, and $|1\rangle$ is far-off resonantly coupled to the Rydberg state $|r\rangle$ via a periodically modulated optical field with the Rabi frequency $\mathrm{\Omega }\left(t\right)$. Assuming that the detuning $\mathrm{\Delta }$ can compensate the interaction induced energy shift $U_{\mathrm{rr}}$ by satisfying the antiblockade condition $U_{\mathrm{rr}}=2\mathrm{\Delta }$ persistently, $|11\rangle$ is directly coupled to $|rr\rangle$ via a two-photon excitation with an effective value ${\mathrm{\Omega }}^2\left(t\right)/2\mathrm{\Delta }$. (b) The effective five-level diagram from a pair of three-level atoms shows the real atom-field interactions and spontaneous decays, where the four singly excited states (gray shadow area) are safely discarded due to $\mathrm{\Delta }\gg \mathrm{\Omega }\left(t\right)/\sqrt{2}$. $|S\rangle$ is a target dark entangled state, unidirectionally receiving the population from $|rr\rangle$ through spontaneous emission. The amplitude of ${\mathrm{\Omega }}_{\mathrm{eff}}\left(t\right)$ is illustrated in the inset where blue-solid and black-dashed curves stand for weak and strong frequency modulations, respectively. (c) The experimental setup (proposal; see Sec. 5 for detailed experimental parameters). Two atoms are trapped in dipolar traps formed by tightly focused laser beams (optical tweezers) with interatomic distance $R$ close to the critical value $R_c$, leading to the strength of van der Waals–type interaction $U_{\mathrm{rr}}=C_6/R^6$ between two individual atoms. They are also persistently driven by a microwave field $g$ as well as a periodically modulated optical field $\mathrm{\Omega }\left(t\right)$, realized by an acousto-optical modulator in the experimental implementation. The variation of interaction $\delta U\left(t\right)$ due to the imperfect position confinement in the foci of optical tweezers typically $<1\mu \mathrm{m}$ will be discussed in Sec. 6b.

Figure 3

Global output for (a) $P_{\mathrm{rr}}\left(t\right)$ and (b) ${\mathcal{F}}_{\mathrm{rr}}\left(\omega \right)$ versus the time $t\in (0,60)$ and the relative modulation frequency $\omega /{\omega }_0\in (0,2.5)$. Parameters are ${\mathrm{\Omega }}_0=57.6$, $U_{\mathrm{rr}}=1658.88$, and $\mathrm{\Delta }=829.44$ and ${\omega }_0$ (${\omega }_0^{-1}$) is the frequency (time) unit. Dissipation is ignored in calculating the unitary dynamics.

Figure 4

(a), (b) Unitary dynamics $P_{\mathrm{rr}}\left(t\right)$ and frequency spectrum ${\mathcal{F}}_{\mathrm{rr}}\left(\omega \right)$ are comparably presented for $\omega =0$ (blue-dotted), $\omega ={\omega }_0$ (black-solid), and $\omega =2{\omega }_0$ (red-dashed) within a short time period of ${\omega }_{0}t\in (0,15)$. Other parameters are the same as adopted in Fig. 3.

Figure 5

Population of target entangled state $P_S\left(t\right)$ versus time $t$ under the modulation frequency $\omega ={\omega }_0$ (red-solid) and $\omega =0$ (green-dashed). A detailed plot of the relationship of the accumulated population in every duration $\tau$ on $|rr\rangle$ and unitary dynamics $P_{\mathrm{rr}}\left(t\right)$ are comparably represented for time $t\in [0,20]$ in the inset below. Inset inside shows an extensive plot by enlarging the time range to ${10}^4$ where the final population for the case of no modulation $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0$ is kept to be a saturation $\sim 0.96$. Except the optimal parameters as discussed in Sec. 3, here we choose $\gamma =0.4$, $g=0.85$ and set $P_{11}(t=0)=1.0$ at the initial time.

Figure 6

Fidelity of final target entangled state $F_S[=P_S(t=100)$] versus the simultaneous adjustments for microwave coupling $g$ and spontaneous decay $\gamma$. A global maximum of $F_S$ is denoted in the plot. Here ${\omega }_0$ (${\omega }_0^{-1}$) is the frequency (time) unit.

Figure 7

Time-dependent population of the target state $P_S\left(t\right)$ versus $t(\mu \mathrm{s}$) for different ${\mathrm{\Omega }}_0$ and $\mathrm{\Delta }$ values. Note that the conditions of $U_{\mathrm{rr}}=2\mathrm{\Delta }$ and $\omega ={\omega }_0$ are kept.

Figure 8

Final fidelity ${\overline{F}}_S$ for a duration of $t\in [T_S-100,T_S]\mu \mathrm{s}$, versus the variation of initial population probability on ground states $|00\rangle$, $|T\rangle$, and $|11\rangle$, defined by $P_{00}(t=0)$, $P_T(t=0)$, and $P_{11}(t=0)$, satisfying the conserved normalization $P_{11}\left(0\right)+P_T\left(0\right)+P_{00}\left(0\right)=1.0$. The target state $|S\rangle$ is initially unoccupied.

Figure 9

Real time dependence of population $P_S\left(t\right)$ (red solid) on the target state $|S\rangle$ with variable vdW interactions $U_{\mathrm{rr}}\left(t\right)$ realized by a slight change $\delta U\left(t\right)$ around its critical value $U_{rr,c}$. In case (i) $\delta U\left(t\right)$ shows a sinusoidal function with amplitudes (a) $U_0=0.1U_{rr,c}$ and (b) $U_0=0.2U_{rr,c}$. Turning to case (ii), $\delta U\left(t\right)$ is obtained randomly from $[-U_0,U_0]$ with (c) $U_0=0.1U_{rr,c}$ and (d) $U_0=0.2U_{rr,c}$.

Figure 10

Final fidelity $F_S$ at $T_S=366.3\mu \mathrm{s}$ versus a small variation of perturbation $\delta {\omega }_0$ to the periodic driving frequency ${\omega }_0$, denoted by $\delta {\omega }_0/{\omega }_0\in [-0.1,0.1]$, i.e., the driving frequency changes from $0.9{\omega }_0$ to $1.1{\omega }_0$. Inset stands for an extensive range by modifying the range to $\delta {\omega }_0/{\omega }_0\in [-1.0,1.0]$.

Figure 11

Entire energy-level diagram of a pair of original four-level atoms. The unitary dynamics between $|11\rangle$ and $|rr\rangle$ as shown in the yellow box is mediated by two laser fields ${\mathrm{\Omega }}_{p,s}$ and detunings ${\mathrm{\Delta }}_{1,2}$. An effective five-level model that consists of states $\left\{\right|11\rangle ,|T\rangle ,|S\rangle ,|00\rangle ,|rr\rangle \}$ is finally obtained in the main text, by discarding all middle states for the off-resonant detunings $|{\mathrm{\Delta }}_{1,2}|\gg {\mathrm{\Omega }}_p,{\mathrm{\Omega }}_s$. (b), (c) Two different designs of laser fields ${\mathrm{\Omega }}_{p,s}$ and detuning $\delta =|{\mathrm{\Delta }}_{1,2}|$, coinciding with the parameters $\mathrm{\Omega }\left(t\right)$, $\mathrm{\Delta }\left(t\right)$ in the effective two three-level atom system.

Figure 12

Unitary dynamics of $P_{\mathrm{rr}}\left(t\right)$ (red dashed) in an original pair of four-level atoms versus time (units of $\mu \mathrm{s}$) under (a) a periodically modulated field ${\mathrm{\Omega }}_s\left(t\right)$ and (b) a periodically modulated field ${\mathrm{\Omega }}_p\left(t\right)$. Parameters are described in the text. The black solid curve stands for the unitary dynamics $P_{\mathrm{rr}}\left(t\right)$ in the case of an effective system the same as presented in Fig. 4.