Dipole-dipole-interaction–driven antiblockade of two Rydberg atoms

Resonant laser excitation of multiple Rydberg atoms are prohibited, leading to Rydberg blockade, when the long-range van der Waals interactions are stronger than the laser-atom coupling. Rydberg blockade can be violated, i.e., simultaneous excitation of more than one Rydberg atom, by off-resonant laser excitation, causing an excitation antiblockade. Rydberg antiblockade gives rise to strongly correlated many-body dynamics and spin-orbit coupling and also finds quantum computation applications. Instead of commonly used van der Waals interactions, we investigate antiblockade dynamics of two Rydberg atoms interacting via dipole-dipole exchange interactions. We study typical situations in current Rydberg atom experiments, where different types of dipole-dipole interactions can be achieved by varying Rydberg state couplings. An effective Hamiltonian governing underlying antiblockade dynamics is derived. We illustrate that geometric gates can be realized with the Rydberg antiblockade which is robust against the decay of Rydberg states. Our study may stimulate new experimental and theoretical exploration of quantum optics and strongly interacting many-body dynamics with Rydberg antiblockade driven by dipole-dipole interactions.

Figure 1

(a) Two-body interaction strength for Rb atoms excited to Rydberg state $|100s\rangle$ versus interatomic distance $d$. $R_c$ denotes the crossover distance between DD and vdW interactions [4]. (b) The dynamics of Rydberg blockade and antiblockade with vdW-type RRI. $E_{gg}$, $E_{rg}$, and $E_{rr}$ denote the energies of the two-atom state $|gg\rangle$, $|gr\rangle$ $\left(\right|rg\rangle )$, and $|rr\rangle$, respectively. $E_{rr}^{\prime }$ denotes the energy when both atoms are excited in Rydberg states but excluding two-body interactions. The resonant laser excitation (dotted-dashed line) leads to the Rydberg blockade [2, 3, 4]. The middle excitation process (dotted line) is the conventional Rydberg antiblockade with simultaneous driving [20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32]. The right one (solid line) is the RAB with sequential driving [29].

Figure 2

(a) Left panel shows two Rydberg atoms with resonant RRI. $|0\rangle$ and $|1\rangle$ are two ground states. $|p\rangle$, $|d\rangle$, and $|f\rangle$ are three Rydberg states with the Förster resonance interaction ${\widehat{H}}_d=V_d\left(\right|dd\rangle \langle pf|+|dd\rangle \langle fp|+\mathrm{H}.\mathrm{c}.)$. Right panel gives the effective RAB process in the dressed-state basis. (b) Populations of difference states for RAB scheme in Sec. 2a during one evolution period $T=2\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$ with the consideration of practical atomic spontaneous emission ${\gamma }_p=1.89\mathrm{kHz}$, ${\gamma }_d=4.55\mathrm{kHz}$, and ${\gamma }_f=7.69\mathrm{kHz}$. The inset shows that the dressed state decays to zero at the end of the laser pulse. Parameters are $\mathrm{\Omega }=2\pi \times 5$ MHz and $\mathrm{\Delta }$ is set to satisfy the antiblockade condition. The initial state is $|11\rangle$ and the interatomic distance is $3\mu \mathrm{m}$.

Figure 3

(a) Left panel shows two Rydberg atoms with DD interactions. $|0\rangle$ and $|1\rangle$ are two ground states. $|p\rangle$ and $|d\rangle$ are two Rydberg states with the spin-exchange interaction ${\widehat{H}}_d=V_d\left(\right|pd\rangle \langle dp|+\mathrm{H}.\mathrm{c}.)$. Right panel shows the effective RAB process in the dressed state basis. (b) Populations of the states for RAB scheme in Sec. 2b under one evolution period $T=2\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$ with the consideration of practical atomic spontaneous emission ${\gamma }_p=1.69\mathrm{kHz}$ and ${\gamma }_d=4\mathrm{kHz}$. Parameters are chosen as $\mathrm{\Omega }=2\pi \times 5$ MHz and $\mathrm{\Delta }$ is set to satisfy the antiblockade condition. The initial state is set as $|11\rangle$ and the interatomic distance is set as $3\mu \mathrm{m}$.

Figure 4

(a) Left panel shows level scheme of the two Rydberg atoms. $|0\rangle$ and $|1\rangle$ are two ground states. $|s\rangle$ and $|p\rangle$ are two Rydberg states for the left atom, and $|s^{\prime }\rangle$ and $|p^{\prime }\rangle$ are two Rydberg states for the right atom. These two Rydberg atoms are interacting each other through the spin-exchange interaction ${\widehat{H}}_d=V_d\left(\right|s{s}^{\prime }\rangle \langle p{p}^{\prime }|+\mathrm{H}.\mathrm{c}.)$. Right panel shows the effective RAB process in the dressed state basis. (b). Population dynamics for RAB scheme in Sec. 2c under one evolution period $T=2\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$ with atomic spontaneous emission rate ${\gamma }_s=8.33\mathrm{kHz}$, ${\gamma }_{s^{\prime }}=7.69\mathrm{kHz}$, ${\gamma }_p=4\mathrm{kHz}$, ${\gamma }_{p^{\prime }}=3.7\mathrm{kHz}$. Parameters are $\mathrm{\Omega }=2\pi \times 5$ MHz and $\mathrm{\Delta }$ is set to satisfy the antiblockade condition. The initial state is $|11\rangle$ and the interatomic distance is $2\mu \mathrm{m}$.

Figure 5

(a) Characteristic interatomic distance $R_c$ for the state $|n{S}_{1/2},m_J=1/2\rangle$ [95]. Fidelities of the controlled-$Z$ gate versus fluctuations of the laser parameters, i.e., deviations (b) $d\mathrm{\Omega }$ and (c) $d\mathrm{\Delta }$. For the DD interaction, the energy levels are same as those in Fig. 2 with $\mathrm{\Omega }=2\pi \times 9.9$ MHz, ${\gamma }_p=1.89\mathrm{kHz}$, ${\gamma }_d=4.55\mathrm{kHz}$, and ${\gamma }_f=7.69\mathrm{kHz}$. The interatomic distance is $3\mu \mathrm{m}$, and $\mathrm{\Delta }$ is determined through the RAB condition given in Sec. 2a. For the vdW interaction, the energy levels are the same as those in Fig. 2 without considering $|p\rangle$ and $|f\rangle$ states. The vdW interaction is given by the Hamiltonian $H_{\mathrm{vdW}}=C_6/r^6|d\rangle \langle d|\otimes |d\rangle \langle d|$ with $C_6=1700\text{GHz}\mu {\text{m}}^6$ [58] with interatomic distance $6.6\mu \text{m}$. Besides, $\mathrm{\Omega }=2\pi \times 2.2$ MHz, ${\gamma }_d=4.55\mathrm{kHz}$, and the RAB condition in Ref. [86] are considered. For panels (b) and (c), the gate time is determined by $T=2\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$.

Figure 6

Fidelities of the controlled-$Z$ gate versus Rabi frequency at the gate time $T=2\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$. For DD interactions, the energy level and parameters are chosen as that in Fig. 5, except that the Rabi frequency is varied from $2\pi \times 0.5$ to $2\pi \times 30$ MHz. The detuning is chosen to satisfy the RAB condition given in Sec. 2a. For vdW interactions, the energy level is the same as that in Fig. 2 without considering $|p\rangle$ and $|f\rangle$ and the RAB condition in given in Ref. [86]. In both cases, the arrow indicates the range of Rabi frequencies where the gate fidelity is larger than 0.9.

Figure 7

Fidelity of the RAB-based gates with respect to deviation of interatomic distances at the gate time $T=2\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$. Panel (a) [(b)] corresponds to case (i) [(ii)] in Sec. 3c. Here, $\mathrm{\Omega }=2\pi \times 6.663$ MHz, $\mathrm{\Delta }=10\mathrm{\Omega }$, and $r_d=3\mu \text{m}$. $V_d=2\pi \times 94$ MHz. For panel (a) $V_{\mathrm{vdw}}=V_d$, while for panel (b), $V_{\mathrm{vdw}}$ is shown in the legend of the figure.

Figure 8

(a) Evolution of the fidelity of the geometric controlled-arbitrary-phase gate. The parameters are the same as in Fig. 2. (b) Bloch-sphere representation of the geometric quantum operation. The coupling to the dressed state via RAB gives rise to the desired phase shift to the computational basis at the end of the gate operation.

Figure 9

(a) Dynamical processes to generate the steady entangled state by combining the unitary and dissipative dynamics. (b) Infidelity of the steady entangled state $\left(\right|01\rangle -|10\rangle )/\sqrt{2}$ versus $\omega /{\mathrm{\Omega }}_{\mathrm{eff}}^{\prime }$. The interatomic distance is $3\mu \mathrm{m}$, and the Rabi frequency is s $\mathrm{\Omega }=2\pi \times 1$ MHz. $\mathrm{\Delta }$ is determined through $V_d=\sqrt{2}\mathrm{\Delta }$.

Figure 10

(left) Dynamical process of Eq. (3) when the initial state is $|11\rangle$. (right) Dynamical process of Eq. (4). The right panel is the effective process of the left panel if the antiblockade condition $\sqrt{2}{V}_d-\mathrm{\Delta }=\mathrm{\Delta }$ is satisfied, which is similar to the “two-photon process.”