Annulled van der Waals interaction and fast Rydberg quantum gates

A pair of neutral atoms separated by several microns and prepared in identical $s$ states of large principal quantum number experience a van der Waals interaction. If microwave fields are used to generate a superposition of $s$ states with different principal quantum numbers, a null point may be found at which a specific superposition state experiences no van der Waals interaction. An application of this Rydberg state in a quantum controlled-z gate is proposed, which takes advantage of GHz rate transitions to nearby Rydberg states. A gate operation time in the tens of nanoseconds is predicted.

Figure 1

Superposition states are illustrated by half disks and eigenstates by disks. (a) Dressed-state picture for three Rydberg states coupled by two microwave fields. The state $|1\rangle$ is a dark state, and $|p_{\pm }\rangle$ are defined below Eq. (7). (b) A schematic of the atomic levels of ${}^{87}\mathrm{Rb}$ used in preparing the state $|1\rangle$ along with the laser and microwave fields. Inset: illustrates that the $\mathsf{Y}$ configuration of lasers produces an effective two-state system. (c) State evolution of one atom during the $C_{\text{z}}$ gate protocol involving two pairs of microwave pulses, denoted by i and ii, separated by a wait period.

Figure 2

The vdWI coefficient ${\overline{C}}_6$ (see text) as a function of $\beta$ for different pairs of principal quantum numbers $(n_1,n_2)$. The Rydberg states have a common magnetic quantum number $m_J=-\frac{1}{2}$.

Figure 3

Atomic levels involved in the $C_{\text{z}}$ gate protocol and loss channels. The two thick dashed curves denote the two chosen transitions, the two thin dashed curves denote a resonant two-photon process, while all other solid curves denote one-photon loss channels, with corresponding transition frequencies given (in units of GHz).

Figure 4

(a) Performance of a $C_{\text{z}}$ gate realized as a function of $\mathbb{B}$ when an optimal $\mathrm{\Omega }$ is used for each $\mathbb{B}$. $T_{\text{g}}$: gate time, $E_0$: fidelity error (excluding ${\overline{E}}_{\mathcal{L}}$). Here, $\delta /\left(2\pi \right)=0.26$ GHz. (b) Contribution (scaled by ${10}^3$) to the gate fidelity error due to fluctuation of atomic positions as a function of the trap depth when $L=4.4\mu \text{m}$.

Figure 5

Energy levels of the atom-photon system for a single Rydberg atom with the microwave field dressing.

Figure 6

Numerical evidence of the dark state $|11\rangle$ with $(n_1,n_2)=(57,59)$, and the two-atom distance $L=4.4\mu \mathrm{m}$. Here, $p={10}^4{[1-|\langle \psi \left(t\right)|11\rangle |}^2]$ is the population error of the dark state $|11\rangle$ scaled by ${10}^4$, while ${\phi }_i/100$, where $i=1,\cdots ,4$, represents the phase of the state $|11\rangle$. (a) and (b) present the short and long time scale results, respectively.

Figure 7

Population of $|\underline{1}\rangle$ as a function of time $t$ when a single atom is excited to the Rydberg state with Rabi frequency ${\mathrm{\Omega }}_{\text{ls}}$ and detuning ${\mathrm{\Delta }}_{\text{ls}}$. The initial state is $|\underline{1}\rangle$.

Figure 8

Sum of populations on the states $|\underline{1}\underline{1}\rangle$ and $|\underline{1}1\rangle$ as a function of time $t$ when each of the two interacting atoms are excited to the Rydberg state $|1\rangle$ with a Rabi frequency ${\mathrm{\Omega }}_{\text{ls}}$ and two-atom Rydberg blockade $B$. For $B=0$, the ground-state population of one atom has the same time evolution as that of an isolated atom, while for $B\ne 0$, the time evolution deviates from the single-atom case. The initial state is $|\underline{1}\rangle \left(\right|\underline{1}\underline{1}\rangle )$ for a one (two) atom system.

Figure 9

Simulated population, scaled by 100, on the level $62p$, the levels $\{62p,68d,77f,\cdots ,156i\}$, and the levels $\{68d,77f,\cdots ,156i\}$ as a function of time starting from the initial state $sin\beta |n_{1}s\rangle$.

Figure 10

Probability distribution of ${\overline{C}}_6$ when the dressing microwave Rabi frequency ${\mathrm{\Omega }}_j^{\prime }$ obeys a Gaussian distribution with a standard deviation ${\varsigma }_j{\mathrm{\Omega }}_j$, where $j=1$ or 2. ${\varsigma }_j$ is $1\%,1.5\%$, and $2\%$ in (a), (b), and (c), respectively.

Figure 11

Schematic of position fluctuation of atoms. An atom is mainly trapped inside the ellipse. ${\sigma }_j^2$ is the variance of the atomic position along the $j$ axis, where $j=x,y$, or $z$. Here, ${\sigma }_x={\sigma }_y$.

Figure 12

(a) Scaled gate fidelity error due to the fluctuation of atomic positions as a function of the trap depth when $L=5.2\mu \text{m}$. (b) Total gate fidelity error $E_0+{\overline{E}}_{\mathcal{L}}$ scaled by ${10}^3$ when $L=5.2\mu \text{m}$, where $E_0=3.17\times {10}^{-3}$.