Fast Rydberg antiblockade regime and its applications in quantum logic gates

Unlike the Rydberg blockade regime, the Rydberg antiblockade regime (RABR) allows more than one Rydberg atom to be excited, which can bring other interesting phenomena and applications. We propose an alternative scheme to quickly achieve the RABR. The proposed RABR can be implemented by adjusting the detuning of the classical driving field, which is, in turn, based on the former numbers of the excited Rydberg atoms. In contrast to the former schemes, the current one enables more than two atoms to be excited to Rydberg states in a short period of time and thus is useful for large-scale quantum information processing. The proposed RABR can be used to construct two- and multiqubit quantum logic gates. In addition, a Rydberg excitation superatom, which can decrease the blockade error and enlarge the blockade radius for Rydberg blockade-based schemes, is constructed based on the suggested RABR and used to realize a more robust quantum logic gate. The mechanical effect and the ionization are discussed, and the performance is investigated using the master-equation method. Finally, other possible applications of the present RABR are also given.

Figure 1

Illustration of Rydberg blockade [(a), (b)] and antiblockade [(c), (d)] regimes with simultaneous driving. Each of the two atoms has one ground state $|1\rangle$ and one Rydberg state $|r\rangle$. The strength of the Rydberg interaction is $V$. The coupling strength between the Rydberg atom and the classical field is $\mathrm{\Omega }$. (a) [(c)] shows two interacting Rydberg atoms with resonant (large-detuned) driving. The transition $|1\rangle \leftrightarrow |r\rangle$ couples the laser resonantly [(a)] or dispersively [(c)] with blue detuning $\mathrm{\Delta }$. (b) [(d)] shows the effective coupling processes of (a) [(c)] in the two-atom basis {$|11\rangle ,|T\rangle ,|rr\rangle$}, in which $|T\rangle =\left(\right|1r\rangle +|r1\rangle )/\sqrt{2}$.

Figure 2

Illustration of the RABR. Each of the atoms has one Rydberg state $|r\rangle$ and one ground state $|1\rangle$. (a) Two-qubit case. The transition ${|1\rangle }_{1\left(2\right)}\leftrightarrow {|r\rangle }_{1\left(2\right)}$ is driven by a classical laser resonantly (dispersively with blue detuning $\mathrm{\Delta }$), in which the numbers ($k$ = 1, 2) mean the transition of atom $k$. (b) Multiqubit case. We suppose RRI strengths (denoted by $V$) between any two atoms equal each other. The antiblockade condition is, for the $i\mathrm{th}$ atom, ${\mathrm{\Delta }}_i=(i-1)V$. The circles with numbers denote the steps.

Figure 3

Variations of populations of relevant states over time for the present (a) two-, (b) three-, and (c) four-qubit RABR schemes, respectively, with $\gamma$ being set as zero. (d) Populations of collective Rydberg excitation of $n$-qubit RABR at the proper time versus $\gamma$. For the case of $n=1$, strictly speaking, there is no such thing as the Rydberg blockade and RABR, which require at least two Rydberg atoms. The rest of the parameters are chosen as ${\mathrm{\Delta }}_i=(i-1)V$ and $V=10\mathrm{\Omega }$.

Figure 4

Illustration of the two-qubit quantum logic gate. Each of the atoms has one Rydberg state $|r\rangle$ and two ground states, $|0\rangle$ and $|1\rangle$. The transition ${|1\rangle }_{1\left(2\right)}\leftrightarrow {|r\rangle }_{1\left(2\right)}$ is driven by a classical laser resonantly (dispersively with blue detuning $\mathrm{\Delta }$), in which the subscript $i$ ($i$ = 1, 2) means the transition of atom $i$. The circles with numbers denote the steps.

Figure 5

(a) Three interacting Rydberg atoms in a 2D optical lattice. (b) Three Rydberg atoms with nearest-neighboring interaction in a 1D optical lattice. (c) Illustration of a three-qubit controlled phase gate. In order to facilitate the analysis and design, we let the RRI strength $V_{12}=V_{23}=V_{13}=V$. ${\mathrm{\Delta }}_s$ denotes the detuning between atom $s$ and the corresponding transition ${|1\rangle }_s\to {|r\rangle }_s$. The antiblockade conditions for (a) and (b) are ${\mathrm{\Delta }}_s=(s-1)V(s=1,2,3)$ and ${\mathrm{\Delta }}_s=V(s\ne 1)$ and ${\mathrm{\Delta }}_s=0(s=1)$, respectively.

Figure 6

(a) Average fidelity of the final step of two- and three-qubit controlled-phase gates, with the RIS method and the TPQO method. The parameters are set as $V=12\mathrm{\Omega },\mathrm{\Delta }=V$ and $\gamma =0$. We have reset the initial time of the final step to be zero for clarity. (b) Average fidelity with respect to $\gamma$ at the optimal time for two- and three-qubit cases under the RIS and TPQO methods. The parameters are set as $V=12\mathrm{\Omega },{\mathrm{\Delta }}_i=(i-1)V$. For the RIS method, we use 1000 (500) groups of random initial states to test the performance of a two- (three-) qubit logic gate.

Figure 7

Controlled-phase gate based on the RESA. (a) RESA interacting with a Rydberg atom. The logic qubits of the RESA are $\left\{\right|\overline{0}\rangle \equiv {|00\rangle }_{\mathrm{AB}},|\overline{1}{\rangle \equiv |11\rangle }_{\mathrm{AB}}\}$. (b) The process to realize the RESA-based Rydberg blockade controlled-phase gate. Quantum information of the RESA is stored in the basis states $|\overline{0}\rangle , |\overline{1}\rangle$. $\mathrm{\Omega }$ is the Rabi frequency between the corresponding ground state and Rydberg state.

Figure 8

(a) Average fidelity $\overline{F}$ (${\overline{F}}_1$) of the RESA-based (traditional) blockade two-qubit controlled-phase gate. (b) $\overline{F}-{\overline{F}}_1$ versus $V$ and $\gamma$.

Figure 9

Performance of the two-qubit controlled-phase gate with the parameters extracted from experiment [52]. (a) Average fidelity versus $\gamma$ at time $4\pi /\mathrm{\Omega }$. The relevant parameters are taken from Table 1. The mean maximal fidelities versus scaled standard deviations of inhomogeneous parameters (b) $\mathrm{\Omega }$ and (c) $V$. The error bars show the standard deviation of $\left\{F_{\max }^j\right\}$. The rest of the parameters in (b) and (c) are the same as in (a), and the TPQO method defined in Eq. (35) is used.

Figure 10

Performance of the RABR-based two- and three-qubit controlled-phase gates with dephasing error.