Cavity polariton blockade for non-local entangling gates with trapped atoms

TL;DR

The paper introduces a cavity-polariton blockade mechanism in cavity QED to realize non-local multi-qubit W-state preparation and non-local CZ and C2Z gates for an N-qubit register coupled to a single cavity mode. By engineering an effective blockade Hamiltonian between $|\, ext{D}_0 angle$ and a dressed $|ar{D}_1 angle$, the authors show how a global cavity drive and a global qubit pulse suffice to deterministically generate entanglement and perform gates without single-qubit addressing. They derive analytical expressions for W-state preparation and gate errors, demonstrating $\mathcal{O}(C^{-1/2})$ scaling with the single-particle cooperativity $C$, and provide optimal parameter regimes. The work analyzes realistic fidelities across neutral-atom, Rydberg-mominated microwave, and polar-molecule platforms, indicating practical routes toward scalable, non-local quantum information processing and sensing applications.

Abstract

We propose a scheme for realizing multi-qubit entangled W-state and non-local $CZ$ and $C_2Z$ gates via a cavity polariton blockade mechanism with a system of atomic qubits coupled to a common cavity mode. The polariton blockade is achieved by tuning the system, an $N-$qubit register, such that no two atoms are simultaneously excited to the qubit excited state, and there is an effective coupling only between the ground state and a singly-excited W state of the qubit register. The control step requires only an external drive of the cavity mode and a global qubit pulse and no individual qubit addressing. We analytically obtain the state preparation error for an $N-$qubit W state which scales as $\sqrt{(1-1/N)}/\sqrt{C}$ where $C$ is the single particle cooperativity. We additionally show the application of the polariton blockade mechanism in realizing a non-local $CZ$ and $C_2Z$ gate by using a different set of computational qubit states, and characterize the gate errors which scale as $\sim 1/\sqrt{C}$.

Figures (4)

• Figure 1: (a) Schematic of atoms trapped inside a cavity and coupled to a common cavity mode, which is externally driven by a classical field $\eta(t)$. An additional global pulse addresses all the qubits. A multi-qubit entangled W state can be prepared with arbitrarily selected atoms atoms(in red) modeled as three-level systems (as shown in (b)) and a $C_2Z$ gate can be implemented with atoms (in yellow) modeled as four-level systems(as shown in (c)). (b,c) Level schematic for atoms implementing W state preparation and $CZ$ or $C_2Z$ gate. The $|1\rangle \leftrightarrow |e\rangle$ coupling is mediated by the cavity with coupling strength $g$. An additional (global) laser drive couples the states $|0\rangle$ and $|1\rangle$ with Rabi frequency $\Omega(t)$. The computational qubit states are highlighted in blue. (d) State population dynamics obtained numerically by simulating the dynamics under the full Hamiltonian in Eq. \ref{['eq::full_H']}, plotted for states $|\mathcal{D}_{0}\rangle, |\mathcal{D}_{1}\rangle$, and $|\mathcal{D}_{2}\rangle$ denoted by $P_{|\mathcal{D}_{0}\rangle}, P_{|\mathcal{D}_{1}\rangle}$ and $P_{|\mathcal{D}_{2}\rangle}$ respectively for a system with $N=2$. The populations $P_{|\mathcal{D}_{0}\rangle}$ (dashed lines), $P_{|\mathcal{D}_{1}\rangle}$ (solid lines), and $P_{|\mathcal{D}_{2}\rangle}$ (dash-dot lines) at each time add up to the trace of the reduced atomic density matrix (dotted lines) $\mathrm{Tr}(\rho_{\mathrm{symm}}) \leq 1$ where $\rho_{\mathrm{symm}}$ corresponds to the subspace spanned by states $\{|\mathcal{D}_{n}\rangle \forall n =0,1,\dots N\}$. (e) Infidelity ($1-F$) as a function of the total pulse duration $gT$ for W state preparation with $N=2$ for $C= 10^2, 10^6, 10^{10}$ and $\gamma/\kappa= 0.01, 0.1, 1, 10, 100$. The infidelity converges to the analytical estimate(dashed lines) $5.73\sqrt{1-1/N}/\sqrt{C}$ (See text Sec. \ref{['sec:fid_calculation']}) obtained in the limit $T \rightarrow \infty$. (f) Infidelity ($1-F$) as a function of single particle cooperativity for W state preparation with $N=50$, $CZ$ gate and $C_2Z$ gate. The dashed lines represent the analytically calculated errors, and numerical points obtained by simulating the dynamics with the full Hamiltonian (Eq. \ref{['eq::full_H']}) are plotted for $\gamma/\kappa = 0.01, 0.1, 1, 10, 100$ for a fixed pulse duration of $gT= 10^8$. (g,h) Time optimal pulses for implementing $CZ$ gate and $C_2Z$ gate from janduraTimeOptimalTwoThreeQubit2022everedHighfidelityParallelEntangling2023
• Figure 2: Level schematic overview of the blockade mechanism. (a) Eigenstates and eigenenergies of $\hat{H}^{(\Delta, \delta, g)}$ truncated to the subspace spanned by states in $n= 0,1,2$, $k=0,1$ (See text Sec. \ref{['subsec::cav_polaritons']}). (b) Couplings from $\hat{H}^{(\kappa, \gamma, \eta)}$ corresponding to the cavity drive with strength $\eta$ are denoted by red arrows. The blockade condition is achieved by setting $\epsilon_2^-=0$, which makes the cavity drive resonant to the $\ket{{2}_1{0}_e{0}_{\mathrm{ph}}} \leftrightarrow |p_2^-\rangle$ transition. (c) In the $n=0$ and $n=1$ subspaces, weak $\eta$ coupling shifts the respective states $\ket{{0}_1{0}_e{0}_{\mathrm{ph}}}$ and $\ket{{1}_1{0}_e{0}_{\mathrm{ph}}}$ in energy (red dashed lines) which also acquire linewidths to the first order in $\kappa, \gamma$. In the $n=2$ subspace, the states $\ket{{2}_1{0}_e{0}_{\mathrm{ph}}}$ and $|p_2^-\rangle$ are dressed by the $\eta$ interaction into new states $|\chi_{\pm}\rangle$ (red solid lines) with eigenvalues $\lambda_{\pm}$ (See text Sec. \ref{['subsec::shifts_H_eta']}). The couplings from $\hat{H}^{(\Omega)}$ are shown by blue dash-dot arrows. (d) The effective Hamiltonian restricted to the states $\ket{{0}_1{0}_e{0}_{\mathrm{ph}}}$ and $\overline{\ket{{1}_1{0}_e{0}_{\mathrm{ph}}}}$ (dressed state due to coupling to $n=2$ subspace via $\hat{H}^{(\Omega)}$) is obtained in the limit $|\Omega|\ll |\lambda_{\pm}|$(See text Sec. \ref{['subsec::H_Omega']}).
• Figure 3: (a) W state preparation error for $N=2$ as a function of the total operation time for $\kappa/g = 10^{-3}$, $\gamma/g = 10^{-3}$. The error due to the decay from $|e\rangle$ state, the error due to loss of photons and the error due to finite time (calculated in the limit $C \rightarrow \infty$) adds up to give the total error (dash-dot line). The dashed line is the analytical error given by $4.05/\sqrt{C}$ calculated in the limit $T \rightarrow \infty$. (a, inset) Final state population (in log-scale) in relevant states $\ket{{a}_1{b}_e{m}_{\mathrm{ph}}}$ as a function of the pulse duration $gT$ for the same parameters as in (a). The final state as $T \rightarrow \infty$ has non-vanishing components along the state $\ket{{0}_1{0}_e{0}_{\mathrm{ph}}}$ and $\ket{{2}_1{0}_e{0}_{\mathrm{ph}}}$ apart from the near-unity population in the target $\ket{{1}_1{0}_e{0}_{\mathrm{ph}}}$ state. (b) Final state populations (in log-scale) in the atomic symmetric Dicke states $|\mathcal{D}_{n}\rangle$ for $N=10$ and $N=2$ (inset) for $\kappa/g = 10^{-3}$, $\gamma/g = 10^{-3}$.
• Figure 4: Gate error for as a function of the total operation time for (a) $CZ$ gate and (b) $C_2Z$ gate for $C= 10^2, 10^4, 10^6, 10^8, 10^{10}$ and $\gamma/\kappa= 0.01, 0.1, 1, 10, 100$. The infidelity converges to the analytical estimate (dashed lines) $1- F \propto 1/\sqrt{C}$ (See text Sec. \ref{['subsec::CZ']}, \ref{['subsec::C2Z']}) obtained in the limit $T \rightarrow \infty$.