Applications of the modified Rydberg antiblockade regime with simultaneous driving

Through analyzing the effective dynamics of two Rydberg atoms under the dispersive coupling process, we give a method to modify the traditional antiblockade regime with simultaneous driving without adding any extra controls and resources. The modified regime can be used to construct controlled-PHASE and controlled-NOT gates in one step without manipulating the shape and amplitude, and tailoring sequences of the driving pulses. For the controlled-PHASE gate, only one pumping process from the ground state to the Rydberg state of each atom is required thus the resource is minimal. And the atomic addressability is not necessary. The scheme can be improved a lot after adding single-qubit operations since arbitrary phases can be achieved accurately with a short period of evolution time. For one-step construction of controlled-NOT gate, two schemes with high fidelities are given and discussed. Besides, the modified regime can also be used to improve the performance of the dissipation-based quantum entanglement preparation scheme. The optimal value of the modified condition for entanglement preparation is discussed for a wide range of parameters. Our study enriches the physics and applications of the simultaneous-driving-based Rydberg antiblockade regime without adding any operation complexity.

Figure 1

(a) Schematic representation of two trapped Rydberg atoms. (b) Diagram of two four-energy-level Rydberg atoms. Each atom has two ground states $|0\rangle$ and $|1\rangle$, one short-lived state $|p\rangle$ and one Rydberg state $|r\rangle$. The ground state $|1\rangle$ is coupled to the Rydberg level $|r\rangle$ by a two-photon process via the intermediate energy level $|p\rangle$. The corresponding Rabi frequencies of the transition $|1\rangle \to |p\rangle$ and $|p\rangle \to |r\rangle$ are ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$, respectively. (c) Diagram of two three-energy-level Rydberg atoms. Each atom has two ground states $|0\rangle$ and $|1\rangle$, and one Rydberg state $|r\rangle$. The transition from ground state $|1\rangle$ to $|r\rangle$ is coupled directly by a laser with Rabi frequency $\mathrm{\Omega }$. Experimentally, in Ref. [41], researchers used a 319-nm laser to couple the ${}^{133}\mathrm{Cs}$ atom directly from the ground state to the Rydberg state, in a single-photon transition [58].

Figure 2

Phase of $|11\rangle$ and the absolute value of the amplitude of $|rr\rangle$ versus $\mu$, respectively, at the time $t=8\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$. The conditional phase is accurate only when the amplitude of $|rr\rangle$ equals zero.

Figure 3

(a) Imaginary part of the coefficient of $|11\rangle$ versus $\mu$ with the precondition $t=8\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$ that keeps $|01\rangle$ and $|10\rangle$ invariant being satisfied. (b) and (c) Real part of the coefficient of $|11\rangle$ versus $\mu$ with (b) the precondition $t=8\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$ or $t=16\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$ and (c) the precondition $t=12\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$ or $t=20\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$ being satisfied, respectively.

Figure 4

Gate fidelity versus $\mu$ at different evolution times with $\mathrm{\Delta }=20\mathrm{\Omega }$. The initial state is supposed as $|{\psi }_0\rangle =\left(\right|00\rangle +|01\rangle +|10\rangle +|11\rangle )/2$. The fidelity is plotted via Eqs. (9) and (10). (a) and (b) Fidelity of the standard controlled-$\pi$ gate. (c) and (d) Fidelity of the gate $\widehat{U}=|00\rangle \langle 00|-|01\rangle \langle 01|-|10\rangle \langle 10|-|11\rangle \langle 11|$. (e) Fidelity of the gate $\widehat{U}=|00\rangle \langle 00|+|01\rangle \langle 01|+|10\rangle \langle 10|+e^{-i\pi /2}|11\rangle \langle 11|$.

Figure 5

(a), (c), and (f) Gate fidelity plotted versus $\mu$ under effective (solid line) and full (dashed line) Hamiltonians with ${\mu }_2\nu =2\pi (\pi ,3\pi )$ and $\mathrm{\Delta }=12\mathrm{\Omega }$. The initial state is set as $\left(\right|00\rangle +|01\rangle +|10\rangle +|11\rangle )/2$. (b), (d), and (f) Conditional phase plotted by effective (solid line) and full (dashed line) Hamiltonians versus $\mu$ with ${\mu }_2\nu =2\pi (\pi ,3\pi )$ and $\mathrm{\Delta }=12\mathrm{\Omega }$.

Figure 6

Energy diagram for one-step implementation of controlled-NOT gate between two Rydberg atoms. (a) Constructing the controlled-NOT gate approximately with economic resources. (b) Analytical model to construct the controlled-NOT gate through adding one laser field on control atom and one Raman process on target atom, respectively.

Figure 7

(a) Values of the real part of $[m-2e^{-i{\mathrm{\Omega }}^{2}t/\left(4\mathrm{\Delta }\right)}]/4$ versus $\mu$ at two different evolution times. (b) and (c) Gate fidelity versus $\mu$ with the condition $\mathrm{\Delta }=20\mathrm{\Omega }$ for two different evolution times with the specific initial state $|{\psi }_0\rangle =\left(0.1\right|00\rangle +0.1|01\rangle +|10\rangle +0.05|11\rangle )/N$ ($N$ is the normalization coefficient). The fidelity is plotted via Eqs. (9) and (10). (b) Fidelity of the standard controlled-NOT gate. (c) Fidelity of the controlled-NOT${}^{\prime }$ gate $\widehat{U}=|00\rangle \langle 00|+|01\rangle \langle 01|-|11\rangle \langle 10|-|10\rangle \langle 11|$.

Figure 8

Fidelity of the standard controlled-NOT gate versus evolution time. The initial state is chosen as $|{\psi }_0\rangle =\left(0.2\right|00\rangle +0.2|01\rangle +0.5|10\rangle +0.05|11\rangle )/N$ ($N$ is the normalization coefficient). For the curve plotted with the full Hamiltonian, $\mathrm{\Delta }=20\mathrm{\Omega }$ is used.

Figure 9

(a) Energy diagram of the dissipative scheme to prepare steady entanglement. The coupling between $|1\rangle$ and $|r\rangle$ is dispersive with detuning $-\mathrm{\Delta }$ and Rabi frequency $\mathrm{\Omega }$. And the transition between two ground states is driven by the resonant microwave field with Rabi frequency $\omega$. The Rydberg state $|r\rangle$ decays to two ground states with a total spontaneous rate $\gamma$. (b) Diagram of the effective process of the dynamics under the two-atom basis.

Figure 10

(a) Fidelity of the steady entanglement versus $\mu$ under the modified RAB condition. (b) Fidelities of the entanglement versus evolution time with optimal $\mu =0.465$ and $\mu =0$ (traditional RAB regime). The parameters are chosen as $\mathrm{\Omega }/2\pi =0.036$ MHz, $\omega /2\pi =1.45\times {10}^{-4}$ MHz, $\mathrm{\Delta }=3$ MHz, and $\gamma =1.257$ KHz. The initial state is chosen as $|11\rangle$, and the target state is $|S\rangle$. For simplicity, we have assumed the Rydberg state has equal spontaneous rates on two ground states.

Figure 11

(a) and (d) Probabilities of $|11\rangle$ ($|rr\rangle$) versus time under the traditional RAB regime. (b) and (c) Probabilities of $|rr\rangle$ ($|11\rangle$) versus time under the modified RAB regime with $\mu =0.465$. We have set $|rr\rangle$ as the initial state. On the other hand, if $|11\rangle$ is considered as the initial state, the curve of the probability of $|rr\rangle$ is the same as (a) (traditional Rydberg antiblockade regime) or (c) (modified RAB regime).

Figure 12

(a1)–(c1) Optimal values of $\mu$ versus $\mathrm{\Omega }(\mathrm{\Delta },V)$. (a2)–(c2)] Fidelity of the scheme versus $\mathrm{\Omega }(\mathrm{\Delta },V)$ with $\mu$ being optimal. (d) Fidelity to reveal the optimal $\mu$ versus $\gamma$ with parameters the same as that in Figs. 10. (e)–(g) Fidelity of the optimal $\mu$ versus $\gamma$, and the rest of the parameters are chosen the same as that in Fig. 10. For each $\mathrm{\Omega }(\mathrm{\Delta },V, \gamma )$, we scan 200 groups of $\mu$ from 0 to 1 and find the top two fidelities. ${\mathrm{\Omega }}_0$ and ${\mathrm{\Delta }}_0$ are the same as $\mathrm{\Omega }$ and $\mathrm{\Delta }$ in Fig. 10, and $V_0=2{\mathrm{\Delta }}_0$ are supposed.

Figure 13

(a) Average fidelity of the approximate controlled-PHASE gates versus $\gamma$ at the time $8\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$: (i) $\mu =7.75$, controlled-$-\frac{\pi }{2}$ gate; (ii) $\mu =2.25$, controlled-$\frac{\pi }{2}$ gate. (b) Average fidelity of the approximate controlled-${\mathrm{NOT}}^{\prime }$ gate versus $\gamma$ with $\mu =7.5$ at the time $8\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$. (c) Average fidelity of the accurate controlled-NOT gate versus $\gamma$ with $\mu =2$ at the optimal time $t=\sqrt{2}\pi \mathrm{\Delta }/{\mathrm{\Omega }}^2$. (d) Average fidelity of the accurate controlled-$\pi$ gate. $\mathrm{\Delta }=10\mathrm{\Omega }$ is assumed for the above simulations.

Figure 14

Average fidelity of the modified RAB-regime-based scheme with 1000 groups of the initially mixed state and pure state, respectively. The parameters are the same as those used in Fig. 10.

Figure 15

Average fidelity of the modified RAB-regime-based quantum logic gates versus scaled standard deviations of inhomogeneous parameters $V$ with $m=100$ at the optimal evolution time. (a) The approximate controlled- $-\frac{\pi }{2}$ gate. (b) and (c) The controlled-NOT gate with the schematic in Figs. 6 and 6. (d) The accurate controlled-$\pi$ gate. Parameters are chosen as $\mathrm{\Delta }=10\mathrm{\Omega },\gamma ={10}^{-4}\mathrm{\Omega }$, (a) $\mu =7.75$, (b) $\mu =7.5$, (c) $\mu =2$, and (d) $\mu =\sqrt{4/3}$.

Figure 16

(a) and (b) Fidelities of the accurate controlled-PHASE gate (without loss of generality, we here consider $\theta =\pi$) versus the deviation of $\mathrm{\Delta }$ ($\mathrm{\Omega }$) under Eq. (30). (c) and (d) Fidelities of the accurate controlled-NOT gate versus deviation of $\mathrm{\Delta }$ ($\mathrm{\Omega }$) with the initial state $|{\psi }_0\rangle =\left(0.1\right|00\rangle +0.1|01\rangle +0.4|10\rangle +0.1|11\rangle )/N$ ($N$ is the normalization coefficient). $\mathrm{\Delta }=10\mathrm{\Omega }$ and $\gamma ={10}^{-4}\mathrm{\Omega }$ are used.