Fast and dephasing-tolerant preparation of steady Knill-Laflamme-Milburn states via dissipative Rydberg pumping

We propose a fast and dephasing-tolerant scheme for the preparation of steady Knill-Laflamme-Milburn (KLM) state between a pair of Rydberg atoms by dissipation. In this scheme, arbitrary initial state can be engineered to the KLM state with high fidelity (99.24%). Compared to most schemes based on Rydberg dissipation dynamics, the time to prepare steady state can be significantly reduced in our scheme $(80 \mu \mathrm{s})$ due to the use of strong resonant dipole-dipole interactions. In addition, the scheme can effectively suppress the dephasing error of the Rydberg atoms caused by interatomic distance fluctuations, which may severely disrupt the desired systematic dynamics. Thus the present scheme is of interest and deserves further experimental investigation to enrich the dissipation dynamics based on the Rydberg atoms.

Figure 1

(a) Imagined experimental model diagram. (b) Level diagrams of atoms. The gray and black curved arrows represent dissipation ways from Rydberg states ($|p\rangle$, $|r\rangle$, and $|f\rangle$) and intermediate exited state $|e\rangle$ to ground states, respectively. (c) Detailed schematic about how the two-photon transition and recycling lasers work. Only two-photon transition $|1\rangle \leftrightarrow |r\rangle$ is plotted because $|1\rangle \leftrightarrow |p\rangle$ is easy to be analogized. In addition, the dark blue and purple arrows, respectively representing two $\pi$-polarized lasers at about 786 nm and 773 nm, can recycle all the other Zeeman ground states to $|0\rangle$ and $|1\rangle$.

Figure 2

Schematic of the effective model described Eq. (9), in the absence of resonant part. Observe that the purple, green, yellow, and black allows represent lasers pulses with Rabi frequencies $\frac{{\mathrm{\Omega }}_2}{\sqrt{2}}$, $\frac{{\mathrm{\Omega }}_e}{\sqrt{2}}$, -$\frac{{\mathrm{\Omega }}_{\omega }}{\sqrt{2}}$, and $\frac{3{\mathrm{\Omega }}_1^2}{8\mathrm{\Delta }}$, respectively.

Figure 3

Illustrations of (a) $H_p$ driven system, (b) Rydberg anti-blockade system, and (c) Rydberg blockade system. The curved arrows represent dissipation ways caused by spontaneous emission of the excited states.

Figure 4

Populations of (a) $|00\rangle$, (b) $|01\rangle$, (c) $|10\rangle$, and (d) $|11\rangle$ vs time, under the effective pumping Hamiltonian $H_p$ driven and original pumping Hamiltonian $H-H_{\omega }^{\prime }$ driven. The relevant parameters read, ${\mathrm{\Omega }}_2=2\pi \times 290.35\mathrm{kHz}$, ${\mathrm{\Omega }}_e=1.5{\mathrm{\Omega }}_2=2\pi \times 435.53\mathrm{kHz}$, ${\mathrm{\Omega }}_1=24{\mathrm{\Omega }}_2=2\pi \times 6.97\mathrm{MHz}$, and $V_{\mathrm{dip}}=\mathrm{\Delta }=27{\mathrm{\Omega }}_1=2\pi \times 188.15\mathrm{MHz}$.

Figure 5

(a) Fidelities vs time with blue solid line (marking with triangles) and orange dash line (marking with circles) corresponding to full Hamiltonian driven and effective Hamiltonian driven systems, respectively. (b) Fidelities driven by the original Hamiltonian vs. time when the initial states are pure states $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$. Relevant parameters are shown in Eq. (16).

Figure 6

Fidelity (at 80 $\mu \mathrm{s}$) vs $\delta V_{\mathrm{dip}}/V_{\mathrm{dip}}$. The initial states for all the points are $\rho \left(0\right)=\left(\right|00\rangle \langle 00|+|01\rangle \langle 01|+|10\rangle \langle 10|+|11\rangle \langle 11\left|\right)/4$ and the relevant parameters follows Eq. (16) with differences ${\mathrm{\Omega }}_1=40{\mathrm{\Omega }}_2$ and $\mathrm{\Delta }=15{\mathrm{\Omega }}_1$.

Figure 7

Fidelity (at 80 $\mu \mathrm{s}$) vs ${\mathrm{\Delta }}_c$. The initial states for all the points are $\rho \left(0\right)=\left(\right|00\rangle \langle 00|+|01\rangle \langle 01|+|10\rangle \langle 10|+|11\rangle \langle 11\left|\right)/4$ and the relevant parameters exactly follow Eq. (16). Observe that the influence of ${\mathrm{\Delta }}_c$ is so tiny that the fidelity exceed 99.24% (when ${\mathrm{\Delta }}_c=0$) at some points ${\mathrm{\Delta }}_c\ne 0$.

Figure 8

Fidelities (at 80 $\mu \mathrm{s}$) vs (a) ${\eta }_1$, (b) ${\eta }_2$, (c) ${\eta }_e$, and (d) ${\eta }_{\omega }$. Similarly, the influence of ${\eta }_q$ is so slight that at some points ${\eta }_q\ne 0$, the fidelity exceed which at point ${\eta }_q=0$. The initial states of all the points are still $\rho \left(0\right)=\left(\right|00\rangle \langle 00|+|01\rangle \langle 01|+|10\rangle \langle 10|+|11\rangle \langle 11\left|\right)/4$ and the relevant parameters follow Eq. (16).

Figure 9

Fidelity (at 80 $\mu \mathrm{s}$) vs ${\delta }^{\prime }/\delta$. The initial states of all the points are $\rho \left(0\right)=\left(\right|00\rangle \langle 00|+|01\rangle \langle 01|+|10\rangle \langle 10|+|11\rangle \langle 11\left|\right)/4$ and the relevant parameters follow Eq. (16).

Figure 10

(a) Fidelity and (b) corresponding evolution time vs $G_p(p=5^{m-1}+n$). The relevant parameters are the same as that shown in Eq. (16).

Figure 11

(a) Fidelity vs $V_{\mathrm{dip}}/{\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2$ at 80 $\mu \mathrm{s}$. (b) The 3000 power of fidelity vs $V_{\mathrm{dip}}/{\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2$ at 80 $\mu \mathrm{s}$, which is exhibited here for visually finding the points with higher fidelities.

Figure 12

Comparative diagram of operation time between the present scheme and the scheme of Ref. [35]. Note that we choose the same dipole-dipole interaction energy, $V_{\mathrm{dip}}=2\pi \times 188.15\mathrm{MHz}$, to plot this comparative diagram and the parameters relations we used here to plot the fidelity curve of the scheme of Ref. [35] is the same as that in Fig. 5 in Ref. [35], i.e., $V_{\mathrm{dip}}=90\mathrm{\Omega }=18000\gamma$. The initial states are all $\rho \left(0\right)=\left(\right|00\rangle \langle 00|+|01\rangle \langle 01|+|10\rangle \langle 10|+|11\rangle \langle 11\left|\right)/4$.