Broadcasting quantum nonlinearity in hybrid systems

TL;DR

The paper introduces a hybrid approach to realize and broadcast nonlinear quantum gates from a nonlinear source to a linear target via pulsed light-mediated QND interactions. By leveraging a cubic (or higher) mechanical potential as the nonlinear resource and amplifying the effect with linear QND gates, it enables a nonlinear phase operation on a linear system (e.g., an atomic ensemble) that remains robust to the source’s initial state. The authors formalize the protocol with Heisenberg-picture input-output relations, define a nonlinear-variance metric ${\bm\sigma}(\lambda)$ to certify nonlinearity and non-Gaussianity, and evaluate two regimes (four-QND broadcasting and two-QND nonlinear squeezing generation) under realistic losses and heating. They demonstrate, via simulations and Wigner-function analyses, that nonlinearity broadcasting to atoms is feasible with current or near-future atom–optomechanical platforms, potentially enabling universal continuous-variable quantum processing beyond Gaussian capabilities. The work lays out practical optimization strategies and a versatile framework for deploying nonlinear processing across hybrid bosonic systems.

Abstract

Linear oscillators contribute to most branches of contemporary quantum science. They have already successfully served as quantum sensors and memories, found applications in quantum communication, and hold promise for cluster-state-based quantum computing. To master universal quantum processing with linear oscillators, an unconditional nonlinear operation is required. We propose such an operation using light-mediated interaction with another system that possesses a nonlinearity equivalent to more than a quadratic potential. Such a potential grants access to a nonlinear operation that can be broadcast to the target linear system. The nonlinear character of the operation can be verified by observing adequate negative values of the target system's Wigner function and the squeezing of the variance of a certain nonlinear combination of the quadratures below the thresholds attainable by Gaussian states. We explicitly evaluate an optically levitated mechanical oscillator as a flexible source of nonlinearity for a proof-of-principle demonstration of the nonlinearity broadcasting to linear systems, for example, mechanical oscillators or macroscopic atomic spin ensembles.

Figures (3)

• Figure 1: (a) Circuit diagram of the proposed protocol to broadcast nonlinearity from the source system (S, orange line) to the target system (T, red line). The protocol consists of five Hamiltonian transformations: four linear QND interactions between the systems (cyan) and a local nonlinear transformation of the source system (violet). The Hamiltonians corresponding to each step are written inside color boxes. (b) Circuit diagram of a simplified protocol with only two QND interactions. (c) A possible concrete implementation with a linear atomic cloud as the target system and a levitated nanoparticle as the nonlinearity source. Squeezed light pulse ($S$) mediates a linear QND interaction between the matter systems. Here, the nonlinearity is illustrated as a cubic function of the displacement of the nanoparticle. C --- circulators, HD --- homodyne detector. (d) Main figures of merit used to evaluate the broadcasting of the nonlinearity. Nonlinear variance ${\bm\sigma}(\lambda)$ (NLV, see \ref{['eq:nlv_definition']}) evaluated for the ground state $\lvert0\rangle$, a coherent state $\lvert \alpha = 1 + \mathrm{i}\rangle$, and for these two states after a cubic phase gate $\exp[ - \mathrm{i} \hat{x}^3 / 6]$. Filled regions of the NLV graph indicate areas reachable only by non-classical (yellow) states and quantum non-Gaussian (gray) states. The panels on the right show the contour plots of the Wigner functions of these states.
• Figure 2: Nonlinearity broadcasting to the atomic ensemble, visualized by the envelopes of the nonlinear variance ${\bm\sigma} (\lambda)$ curves. The nonlinear variance [see \ref{['eq:nlv_definition']}] of the final state of the atomic ensemble after (a) the full protocol, (b) the simplified protocol, see \ref{['fig:fig0-pdf']}. Areas reachable only by quantum non-Gaussian and non-classical states are indicated by, respectively, gray and light orange shading. In (a), thick lines show results of the protocol in the broadcasting regime. Dashed lines correspond to gain-symmetric QND gates, full lines to gain-asymmetric. Different colors correspond to different loss of the mediator $1 - \eta$. In the lossless case $\eta = 1$, both gain-symmetric and gain-asymmetric regimes correspond to the same configuration and hence perform identically. Thin lines show the optimized performance of the nonlinear squeezing generating regime. For all lines of this panel, the numerical parameters are: nonlinearity $\gamma = 0.07$, heating of the mechanics $\Gamma_\mathsf{m} = 10^{-4}\kappa$. The mechanics is initially in a thermal state with mean occupation $n = 0.45$ without squeezing. In (b), full and dashed lines, similarly, correspond to using gain-asymmetric and gain-symmetric QND gates in the simplified protocol. The mechanical oscillator is initially in a squeezed thermal state with mean occupation (before squeezing) $n = 0.45$, squeezing $7$ dB.
• Figure 3: Broadcasting of the nonlinearity witnessed by the Wigner function negativity and the nonlinear variance suppression. Wigner functions $2 \pi \; W (X,Y)$ of (a,b) the mechanical oscillator at $t = t_\mathsf{NL}$ (after the application of the nonlinearity) and (d, e) of the atomic system at the end of the protocol ($t = t_\mathsf{f}$). The graphs are made in the regime of (a, d) broadcasting the nonlinearity, (b, e) generation of nonlinear squeezing. In (c) the nonlinearity of the same strength is applied directly to the initial state of the mechanics. (f) Nonlinear variance ${\bm\sigma}(\lambda)$ evaluated for the corresponding state. Atoms are initially in vacuum, mechanics in a squeezed vacuum state with squeezing $r =0.1$. Numerical parameters: nonlinearity $\gamma = 0.15$, atomic decay rate $\zeta_\mathsf{a} = 10^{-3}$, mechanical decay rate $\zeta_\mathsf{m} = 10^{-6}$, mechanical bath occupation $n_\mathsf{th} = 10^3$, which results in the heating rate $\Gamma_\mathsf{m} = 10^{-3}$. The values defining the QND gains (see \ref{['eq:unitary_broadcast_io', 'eq:target_squeez']}) are $g = 1.26$ and $g_1 = 1.2$.