Trapping potentials and quantum gates for microwave-dressed Rydberg atoms on an atom chip

TL;DR

The paper addresses coherent control of Rydberg atoms near an atom-chip surface where adsorbate-induced inhomogeneous electric fields would otherwise degrade coherence. It introduces a microwave-dressing strategy that creates trap potentials at prescribed distances by coupling opposite-dipole Rydberg states, enabling surface-adjacent qubits and multi-state trapping for pairwise addressing. It then demonstrates a cavity-mediated SWAP gate between distant Rydberg qubits coupled to a planar resonator, optimizing detuning to tolerate finite-temperature photons and showing high fidelity (>$0.95$) up to $\bar{n}_{th}=10$. The approach promises a scalable platform for quantum computation or simulation by combining strong on-chip coupling, flexible qubit encoding, and cavity-mediated interactions in a cryogenic, chip-based environment.

Abstract

Rydberg atoms in dc electric fields acquire static dipole moments. When the atoms are close to a surface producing an inhomogeneous electric field, such as by the adsorbates on an atom chip, depending on the sign of the dipole moment of the Rydberg-Stark eigenstate, the atoms may experience a force toward or away from the surface. We show that by applying a bias electric field and coupling a desired Rydberg state by a microwave field of proper frequency to another Rydberg state with opposite sign of the dipole moment, we can create a trapping potential for the atom at a prescribed distance from the surface. Perfectly overlapping trapping potentials for several Rydberg states can also be created by multicomponent microwave fields. A pair of such trapped Rydberg states of an atom can represent a qubit. Finally, we discuss an optimal realization of the SWAP gate between pairs of such atomic Rydberg qubits separated by a large distance but interacting with a common mode of a planar microwave resonator at finite temperature.

Figures (5)

• Figure 1: (a) Illustration of a portion of the integrated atom chip with a patch of adsorbates (red disk) on the surface of dielectric gap (blue) between the superconductors (silver) confining the resonator field. The adsorbates, modelled as a uniformly charged disk (centered at $x,y,z=0$) of radius $83\:\mu$m and charge density $6.45 \times 10^{-20}\:\mathrm{C} / \mu \mathrm{m}^2$, produce a spatially inhomogeneous electric field $F_{\mathrm{ad}}$ which is partially compensated by a homogeneous bias field $F_{\mathrm{b}}$, leading to the total field $F=F_{\mathrm{ad}} + F_{\mathrm{b}}$. (b) Absolute value of total electric field $|F|$ near the chip surface vs coordinates $x,z$ for $y=0$. The bias field $F_{\mathrm{b}}=-30\:$V/cm fully compensates the adsorbate field in the vicinity of $x,y=0, z=15\:\mu$m. Arrows indicate the mechanical force $\bm{\nabla} (d_r|F|)$ acting on an atom in the Rydberg-Stark state $\ket{r}$ with static dipole moment $d_r>0$. (c) Schematic illustration of the Rydberg atom trapping scheme in one dimension: A pair of atomic Rydberg states $\ket{r}$ and $\ket{a}$ with static dipole moments $d_a\simeq -d_r$ are energy shifted by an electric field $F(z)$ in the opposite directions (left). The atom is irradiated by a microwave field of frequency $\omega$ detuned by $\Delta = \omega - \omega_{ra}$ from the unperturbed atomic resonance $\omega_{ra}$. In a spatially varying field $F(z)$, in the frame rotating with frequency $\omega$, the Rydberg levels cross at $z_0$ such that $\hbar \Delta + (d_r-d_a)|F(z_0)| =0$. The microwave field coupling the levels $\ket{r}$ and $\ket{a}$ with the Rabi frequency $\Omega$ lifts this degeneracy and results in avoided crossing of the two eigenstates $\ket{r_{\pm}}$ with energies $E_{\pm}$ split by $|\Omega|$ (right). The upper potential curve $E_+$ forms a trapping potential with the vibrational ground-state wavefunction $\chi_0^{(r)} (z)$ centered at $z_\mathrm{min} \simeq z_0$. (d) Spatial dependence of the trapping potential $E_+$ near the chip surface vs coordinates $x,z$ for $y=0$, for the static field as in panel (b) and atomic parameters as in Fig. \ref{['fig:Fnuz0']}. Gravity is neglected.
• Figure 2: (a) Total electric field $F(z)$ of Eq. (\ref{['eq:field']}) vs distance $z$ from the chip. The parameters are $F_0=37\:$V/cm, $\zeta=70\:\mu$m Kaiser2022 and $F_{\mathrm{b}} \simeq -30\:$V/cm. Inset: magnified view of the electric field in the linear regime $z\ll \zeta$. (b) Trap position $z_0$ and frequency $\nu$ (for $^{87}$Rb atom) as functions of the applied bias field $F_{\mathrm{b}}$ for fixed $\Delta/2\pi=-1.6\:$GHz, $\Omega/2\pi =30\:$MHz, and $|d_{r,a}|/h \simeq 400\:$MHz/(V/cm). (c) Microwave-dressed energy levels $E_{\pm}$ vs distance $z$ exhibiting avoided crossing at $z_0\simeq 10 \mu$m and $z_0 \simeq 25 \mu$m for $F_{\mathrm{b}}= -30\:$V/cm and $F_{\mathrm{b}}= -24\:$V/cm [vertical dashed lines in panel (b)], respectively. The upper eigenenergy $E_+$ forms a trapping potential approximated by a parabola.
• Figure 3: (a) Transition paths between the states $\ket{10,n}$ and $\ket{01,n}$ involving the intermediate states $\ket{00,n+1}$ and $\ket{11,n-1}$ containing one more or one less cavity photon. (b) Optimal detuning $\bar{\delta}_c$ vs the mean thermal photon number $\bar{n}_{\mathrm{th}}$. Inset shows the corresponding swap time $t_\mathrm{tr} \approx \frac{\pi}{2} \frac{\bar{\delta}_c}{g^2}$. (c) Illustration of oscillation dynamics between the states $\ket{01}$ and $\ket{10}$ for one full Rabi cycle $0<t<2t_{\mathrm{tr}}$ for $\bar{n}_{\mathrm{th}}=5$ as obtained from Monte Carlo simulations. (d) Population $p_{01}$ of the target state $\ket{01}$ at time $t_\mathrm{tr}$ vs $\bar{n}_{\mathrm{th}}$ as obtained analytically for $\kappa=0$ and $\gamma = 3 \times 10^{-4}g$, and numerically for $\kappa=10^{-3}g$ via the Monte Carlo simulations involving 5000 independent trajectories for each data point (error bars indicate one standard deviation). Inset shows the sum of populations of the atomic states, $p_\mathrm{tot} = p_{00}+p_{01}+p_{10}+p_{11}$.
• Figure 4: Fidelity of preparation of the two-atom entangled state $(\ket{10} - i \ket{01})/\sqrt{2}$ via the $\sqrt{\textsc{swap}}$ gate during time $t_\mathrm{tr}/2$ vs $\bar{n}_{\mathrm{th}}$, as obtained from Monte Carlo simulations averaged over $M=5000$ independent trajectories, with the parameters as in Fig. \ref{['fig:AtCavInt']}(c). Error bars correspond to one standard deviation and the dashed line is guide for the eye.
• Figure 5: Stark maps of suitable Rydberg state manifolds of an Rb atom in weak electric fields $F$ in the vicinity of (a) $48D$ and (b) $59D$ states. Thin lines with small arrows represent the microwave-field dressing of states $\ket{r}$ with $\ket{a}$ and $\ket{s}$ with $\ket{b}$ for trapping. The thicker line connecting states $\ket{r}$ and $\ket{s}$ represents either a microwave field driving the qubit transition with Rabi frequency $\Omega_q$ or the cavity field coupled to the qubit transition with strength $g$.