Deterministic feedforward-based generation of large optical coherent-state superposition

Abstract

Large optical coherent-state superpositions are essential to advance quantum sensing, quantum repeaters and error-correction codes. We propose a deterministic feedforward protocol employing qubit-mode dispersive coupling, currently available in cavity quantum electrodynamics (QED). We show this single-mode protocol to outperform the advanced three-mode Gaussian-photon-number-resolving detector scheme both in terms of average fidelity and quantum non-Gaussian phase-space properties, and propose sensitivity to weak displacements of interference fringes as a feasible and conclusive witness of quantum interference. This approach combining QED with electro-optical feedforward is extendable to tailored states for applications and other platforms.

Figures (4)

• Figure 1: (a) Dispersive coupling protocol for even cat generation, based on qubit-light interaction and feedforward displacement conditioned on qubit measurement. (b) Gaussian-PNRD-based [${\rm GP}(2n)$] protocol, performing conditional displacement-PNRD over an optimized three-mode Gaussian state. Extensions of this basic scheme, including displacements in state preparation and feedforward PNRD, are not helpful to improve quality of the output state SuppMat.
• Figure 2: Fidelities $F^{(2n)}_{{\rm GP}}$ for $n=1,2,3$ and $F_{{\rm D}}$ as a function of the target cat amplitude $\alpha$. Dispersive interaction provides a deterministic scheme that outperforms ${\rm GP}(2n)$ in the limit of large $\alpha$; in SuppMat we discuss a probabilistic version of the protocol beating ${\rm GP}(2n)$ also for small $\alpha$. Moreover, the ${\rm GP}(2n)$ success probability for $\alpha\gg 1$ approaches $\approx 5\%, 0.19\%, 0.13\%$ for $n=1,2,3$SuppMat. The light-coloured lines represent fidelities after optical losses with transmissivity $\tau\le 1$, while the gray one refers to attenuation of the target cat $|{\psi_{\rm cat}}^{(+)}\rangle$.
• Figure 3: Oscillations of the Wigner function along the $y$ axis for the ${\rm GP}(2n)$ protocol, with $n=1,2$ (top row) and the dispersive protocol (bottom row) for different values of the cat amplitude $\alpha$. Boxes report the corresponding values of fidelity and Wigner-function minima.
• Figure 4: (a) Cavity QED implementation of dispersive interaction, consisting of a cavity containing a $\Lambda$-type atom, coupled to impinging free-space field with rate $\kappa_c$ and to scattering and loss modes with rate $\kappa_l$. The two atomic levels $|g\rangle$ and $|u\rangle$ are resonantly coupled to the cavity field with gain $\rm g$, and transition $|g\rangle \to |u\rangle$ is also associated with spontaneously decay rate $\gamma_{\rm se}$Rempe2019Ulrik2022Teja2023SuppMat. The system imperfections are conveniently described in terms of cooperativity $C$ and escape efficiency $\eta\le1$: ideal dispersive coupling is realized in the limits $C\gg1$ and $\eta=1$ (lossless cavity). (b) Fidelities $F_{{\rm D}}^{({\rm imp})}$ (top) of the imperfect dispersive coupling protocol and Wigner interference fringes $W_{\rm D}^{({\rm imp})}(0,y)$ with $\alpha=4$ (bottom) for different values of $C$ and $\eta$. The gray line corresponds to the ideal dispersive scheme; the dashed line refers to fidelity $F_{{\rm D}}^{({\rm pd})}$ in the presence of qubit phase damping of rate $\lambda=0.2$SuppMat. (c) Minimum resolvable displacements for ideal and imperfect dispersive schemes $\varepsilon_{\rm D}$ and $\varepsilon_{\rm D}^{({\rm imp})}$ as a function of $\alpha$: the ideal protocol approaches the target state sensitivity $\varepsilon_{\rm cat}^{(+)}$ for $\alpha\gg1$. The dashed line refers to $\varepsilon_{{\rm D}}^{({\rm pd})}$ in the case of qubit phase damping of rate $\lambda=0.2$SuppMat.