N. Carlon ZambonPhotonics Laboratory, ETH Zürich, CH-8093 Zürich, SwitzerlandQuantum Center, ETH Zurich, CH-8093 Zürich, Switzerland M. RossiPresent address: Kavli Institute of Nanoscience, Department of Quantum Nanoscience, Delft University of Technology, 2628CJ Delft, The NetherlandsPhotonics Laboratory, ETH Zürich, CH-8093 Zürich, SwitzerlandQuantum Center, ETH Zurich, CH-8093 Zürich, Switzerland M. FrimmerPhotonics Laboratory, ETH Zürich, CH-8093 Zürich, SwitzerlandQuantum Center, ETH Zurich, CH-8093 Zürich, Switzerland L. NovotnyPhotonics Laboratory, ETH Zürich, CH-8093 Zürich, SwitzerlandQuantum Center, ETH Zurich, CH-8093 Zürich, Switzerland C. Gonzalez-BallesteroInstitute for Theoretical Physics, Vienna University of Technology (TU Wien), 1040 Vienna, Austria O. Romero-IsartICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels,Barcelona 08860, SpainICREA – Institucio Catalana de Recerca i Estudis Avancats, Barcelona 08010, Spain A. MilitaruPresent address: Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, AustriaPhotonics Laboratory, ETH Zürich, CH-8093 Zürich, SwitzerlandQuantum Center, ETH Zurich, CH-8093 Zürich, Switzerland

###### Abstract

We show how to entangle the motion of optically levitated nanoparticles in distant optical tweezers.The scheme consists in coupling the inelastically scattered light of each particle into transmission lines and directing it towards the other particle.The interference between this light and the background field introduces an effective coupling between the two particles while simultaneously reducing the effect of recoil heating.We analyze the system dynamics, showing that both transient and conditional entanglement between remote particles can be achieved under realistic experimental conditions.

Superposition states are one of the most fascinating manifestations of quantum mechanics. When dealing with two or more degrees of freedom, superpositions can produce strong correlations that make the joint state of the system non-separable, or entangled. Several works in the field of nanomechanics have prepared these quantum correlations between the motional degrees of freedom of two mechanical resonators, from individual atomsJost*etal.* (2009)to microbeamsRiedinger*etal.* (2018); Wollack*etal.* (2022), microscale drum resonatorsOckeloen-Korppi*etal.* (2018); Kotler*etal.* (2021); deLépinay*etal.* (2021), and acoustic modes of bulk resonatorsBienfait*etal.* (2020). Extending this capability to levitated optomechanicsMillen*etal.* (2020); Gonzalez-Ballestero*etal.* (2021)—i.e., generating motional entanglement between two optically levitated nanospheres in high vacuum—is a milestone in the field Rudolph*etal.* (2020); Chauhan*etal.* (2020); Brandão*etal.* (2021); Rudolph*etal.* (2023). On the one hand, entangled states would allow levitated nanoparticles to show quantum motional features without necessarily requiring the preparation of non-Gaussian states, a task which remains challenging despite recent proposalsMartinetz*etal.* (2020); Neumeier*etal.* (2024); Roda-Llordes*etal.* (2024). On the other hand, entangled states of two particles at controllable long distances could be used as probes to characterize yet unknown sources of decoherenceMiao*etal.* (2010), as well as for quantum-enhanced sensing and metrologyGiovannetti*etal.* (2004, 2004); Zhuang*etal.* (2018); Gessner*etal.* (2018, 2018); Xia*etal.* (2023).

Recent experiments have taken crucial steps towards entanglement in levitated optomechanics by showing mechanical ground-state cooling of levitated nanoparticles in free spaceTebbenjohanns*etal.* (2021); Magrini*etal.* (2021), as well as strong and controllable light-mediated interactions between two levitated nanoparticlesRieser*etal.* (2022); Reisenbauer*etal.* (2024). In these setups, however, the trapping laser’s shot noise induces a high degree of motional decoherence which prevents the generation of entanglement Rudolph*etal.* (2023). So far, proposals to address this issue have included trapping nanoparticles inside a high-finesse optical cavity to enhance the coupling-to-decoherence ratio Rudolph*etal.* (2023); Delić*etal.* (2019); Windey*etal.* (2019); Vijayan*etal.* (2024), non-optical coupling mechanismsRudolph*etal.* (2022); Winkler*etal.* (2024); Poddubny*etal.* (2024), and using squeezed light to reduce measurement backaction noise Rudolph*etal.* (2023); Gonzalez-Ballestero*etal.* (2023).

In this work we propose a method based on optical forces to generate entanglement between levitated nanoparticles across long distances (up to meter-scale) without a high-finesse optical cavity nor the use of squeezed light. We engineer long-range interactions by directional coupling of the light scattered off each nanoparticle into optical transmission lines within a closed loop configuration. Fine tuning of the accumulated phase in the transmission lines allows adjusting the effective coupling sign and strength, and to suppress the photon recoil. We derive the equations of motion for the system and provide analytical description of the system dynamics. Finally, we demonstrate the generation of both transient and conditional entanglement.

Model - A dielectric nanoparticle illuminated by a tightly focused laser experiences a restoring optical force. For small displacements, its motion is harmonic and imprints a position-dependent phase onto the scattered laser photons, which in turn generate recoilTebbenjohanns*etal.* (2019). Since scattering events occur randomly, recoil produces a fluctuating force acting on the nanoparticle. This form of optomechanical back-action represents the dominant fluctuating force in ultra-high-vacuum Jain*etal.* (2016).Hereafter, we focus on the nanoparticle motion along the optical axis $z$ and introduce the displacement operator $q=z/(\sqrt{2}z_{\text{zpf}})$ normalized to the zero point fluctuations $z_{\text{zpf}}=\sqrt{\hbar/(2m\Omega_{0}})$, where $m$ and $\Omega_{0}$ denote the nanoparticle mass and resonance frequency, respectively.While light-matter interactions in free space involve every mode of the electromagnetic continuum, it is possible to identify a collective mode that is solely responsible for the optomechanical interaction with $q$: the interacting modeMagrini*etal.* (2021); Militaru*etal.* (2022); Maurer*etal.* (2023).In the Heisenberg picture, the annihilation operators of the interacting mode before and after the interaction with the particle are denoted by $a_{\mathrm{in}}$ and $a_{\mathrm{out}}$, respectively. Crucially, the angular spectrum of the interacting mode associated with $q$ is strongly anisotropic: it predominantly propagates against the tweezer, see Fig.1(a).

Harnessing such directionality, we couple the back-scattered light, which carries information about the nanoparticle’s motion, from the output of one tweezer to the input of another. This can be accomplished using circulators that define a one-way loop in combination with transmission lines (e.g., optical fibers). Hereafter, we consider the case of two identical optical tweezers interconnected with identical optical transmission lines, as illustrated in Fig.1(b). Each transmission line introduces a phase lag $\theta$, and has a finite transmittance $\eta$ owing to finite collection efficiency and imperfect mode matching. Losses, in turn, allow independent, uncorrelated modes $a_{\mathrm{ext}}$ to leak in the loop. Inline optical switches allow to sever the loop and divert the backscattered light from each nanoparticle to separate hom*odyne receivers that are used for state initialization and tomography.

We now derive the equations of motion for the composite system. As the transmission-line loop can be regarded as a bad ring cavity, delayed interactions within a mechanical period can be neglected if $\Omega_{0}L/(\mathcal{F}c_{g})\ll 1$ where $L$, $c_{g}$ and $\mathcal{F}$ denote the loop length, optical group velocity and cavity finesse, respectively. For standard optical fibers, using $\mathcal{F}=(1-\eta^{2})/(1-\eta)^{2}$, $\Omega_{0}\approx 2\pi\times$100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$, and $\eta=0.5$, the approximation holds up to $L\sim 10\leavevmode\nobreak\ \mathrm{m}$sup . Each nanoparticle, labelled by the index $j=1,2$, obeys the Langevin equation

$\ddot{q}_{j}+\Omega_{0}^{2}q_{j}=\Omega_{0}\sqrt{2\Gamma_{\text{q}}}\left(a_{%\mathrm{in}}^{j}+a_{\mathrm{in}}^{j,\dagger}\right),$ | (1) |

where $\Gamma_{\text{q}}$ represents the decoherence rate due to quantum backaction.In addition to Eq.(1), we can write the input-output relationsGardinerandCollett (1985)

$a_{\mathrm{out}}^{j}=a_{\mathrm{in}}^{j}+\text{\rm i}\sqrt{2\Gamma_{\text{q}}}%q_{j},$ | (2) |

with $\text{\rm i}=\sqrt{-1}$ the imaginary unit. Equations(2) and(1) show that the amplitude quadrature of the interacting mode drives the motion, while the phase quadratures probes it. Given the loop geometry considered in Fig.1(b), the input-output fields must satisfy the closure relations

$a_{\mathrm{in}}^{j}=\left(\sqrt{\eta}a_{\mathrm{out}}^{3-j}+\sqrt{1-\eta}a_{%\mathrm{ext}}^{3-j}\right)\text{\rm e}^{\text{\rm i}\theta}.$ | (3) |

Solving Eq.(2) and(3) for the two input fields yieldssup

$a_{\mathrm{in}}^{j}=g_{\mathrm{L}}\left[\text{\rm i}\sqrt{2\Gamma_{\text{q}}}(%\alpha q_{j}+q_{3-j})+g_{\eta}(\alpha a_{\mathrm{ext}}^{j}+a_{\mathrm{ext}}^{3%-j})\right],$ | (4) |

where $\alpha(\theta)=\sqrt{\eta}\text{\rm e}^{\text{\rm i}\theta}$ denotes the transmission line transfer function, $g_{\mathrm{L}}(\theta)=\alpha/(1-\alpha^{2})$ the Airy function of the effective low-finesse resonator generated by the loop, and $g_{\mathrm{\eta}}^{2}=(1-\eta)/\eta$.The amplitude quadrature of $a_{\mathrm{in}}$ drives the nanoparticle’s motion. Equation(4) thus indicates that the loop effect is twofold. The first term, $\propto\alpha g_{\mathrm{L}}q_{j}$ renormalizes the trap stiffness depending on the round-trip phase $2\theta$. The second term, $\propto g_{\mathrm{L}}q_{3-j}$, introduces a coupling term originating from the modulation of the on-site optical force due to the interference between the tweezer and the interacting mode of the distant particle. For convenience, we write dynamics in the normal mode basis upon introducing the joint modes $q_{\pm}=(q_{1}\pm q_{2})/\sqrt{2}$ and $p_{\pm}=(p_{1}\pm p_{2})/\sqrt{2}$, associated with the nanoparticle common $(+)$ and relative $(-)$ motion. From Eqs.(1) and(4), we obtain

$\ddot{q}_{\pm}+\Omega_{\pm}^{2}q_{\pm}=\Omega_{0}\sqrt{4\Gamma_{\text{q}}}n_{%\pm},$ | (5) |

where

$\Omega_{\pm}^{2}=\Omega_{0}\left(\Omega_{0}\pm\frac{4\Gamma_{\text{q}}\sqrt{%\eta}\sin(\theta)}{1+\eta\mp 2\sqrt{\eta}\cos(\theta)}\right)$ | (6) |

defines the normal mode frequencies, and $n_{\pm}$ are two mutually uncorrelated and non-local fluctuations driving the joint modes. The derivation of Eq.(5) and the expressions for $n_{\pm}$ are provided insup .

In Fig.2(a) we plot $\Omega_{\pm}^{2}$ for $\Gamma_{\text{q}}/\Omega_{0}=1$ as a function of the phase $\theta$. For $\theta\in\{0,\pi\}$ we observe that the value of $\Omega_{+}^{2}$ ($\Omega_{-}^{2}$) is always larger (smaller) than the bare oscillator one $\Omega_{0}^{2}$. This fact is a manifestation of the normal mode splitting in coupled oscillators. Note that the splitting can exceed the frequency bare modes, which is the hallmark of ultra-strong coupling Forn-Díaz*etal.* (2019); Frisk Kockum*etal.* (2019); Marković*etal.* (2018). Consequently, $\Omega_{-}^{2}$ becomes negative and the anti-symmetric mode $q_{-}$ becomes unstable (shaded gray area)sup . Finally, we calculate the decoherence rates $\Gamma_{\pm}=2\Gamma_{\text{q}}\mathcal{N}_{\pm}^{2}$,which are proportional to the effective photon recoil rates that drive the joint modes. We obtain

$\mathcal{N}_{\pm}^{2}=\int_{\mathbb{R}}\differential t\,\langle n_{\pm}(t)n_{%\pm}(t^{\prime})\rangle=\frac{1}{2}\frac{1-\eta}{1+\eta\mp 2\sqrt{\eta}\cos(%\theta)}.$ | (7) |

Figure2(b) shows the dependence of $\Gamma_{\pm}$ on the phase $\theta$ and transmittance $\eta$. While the noise strength for the common mode ($+$) is peaked at $\theta=0$ and reaches a minimum at $\theta=\pi$, the trend is opposite for the relative mode ($-$). Importantly, it is possible to identify values of $\theta$ where both $\Gamma_{+}$ and $\Gamma_{-}$ are lower than the corresponding bare oscillator value (dashed black line). Thus, the loop effect is not just to redistribute photon recoil between the two particles but can actually reduce it overall. We can understand this fact by noticing that the loop introduces an effective low-finesse resonator (limited by $\eta$), which in turn suppresses the density of states into which the particle scatters. All panels in Fig.2 can be extended to the phase interval $[\pi,2\pi]$ by swapping the common and relative modes.

Dynamics- the expectation value and covariance matrix of the state vector $\mathbf{v}^{T}=(q_{+},p_{+},q_{-},p_{-})$ fully encode the state. Initially, we consider both nanoparticles to be in a low-occupation state with $\langle\mathbf{v}_{0}\rangle=0$ and covariance matrix $\bm{\Sigma}_{0}=\langle\overline{\mathbf{v}_{0}\mathbf{v}_{0}^{\mathrm{T}}}\rangle$, where the overline denotes symmetrization. State preparation can be accomplished by using the optical switches to reroute the scattered light from each particle to separate hom*odyne receivers, see Fig1(b). The measurement records are then used to stabilize the nanoparticle conditional state using feedback Mancini*etal.* (1998); Meng*etal.* (2020); IsaksenandAndersen (2023).

Starting from this initial condition, we compute the evolution of the covariance matrix $\mathbf{\Sigma}=\mathbf{\Sigma}^{c}+\mathbf{\Sigma}^{n}$, which is split into two terms associated with the coherent and incoherent dynamics, respectively. Since the joint modes in Eq.(5) are decoupled, each diagonal block of $\bm{\Sigma}$ evolves independently. The matrix exponential generating the flow of each subspace is $\mathbf{\Phi}_{\pm}(t)=\mathbf{S}[\sqrt{r_{\pm}}]\mathbf{R}[\phi_{\pm}]\mathbf%{S}[\sqrt{r_{\pm}}]^{-1}$, where $\mathbf{S}[\cdot]$ is a squeezing matrix with parameter $\sqrt{r_{\pm}}=\sqrt{\Omega_{0}/\Omega_{\pm}}$, and $\mathbf{R}[\phi_{\pm}]$ is a clockwise rotation by an angle $\phi_{\pm}=\Omega_{\pm}t$ in phase-space. The coherent terms yield

$\mathbf{\Sigma}^{c}_{\pm}(t)=\mathbf{\Phi}_{\pm}(t)\mathbf{\Sigma}_{0}\mathbf{%\Phi}_{\pm}(t)^{T}.$ | (8) |

For stable dynamics, i.e., $\Omega_{-}^{2}>0$, each element in Eqs.(8) oscillate at twice $\Omega_{\pm}$. In contrast, in the case of unstable dynamics, the covariance matrix elements get squeezed at a rate $\propto\exp(2\Omega_{\pm}t)$ Romero-Isart (2017). The incoherent contribution is

$\mathbf{\Sigma}^{n}_{\pm}(t)=\frac{r_{\pm}^{2}\Gamma_{\pm}}{\Omega_{\pm}}%\begin{pmatrix}\phi_{\pm}-\frac{\sin 2\phi_{\pm}}{2}&r_{\pm}^{-1}\sin^{2}\phi_%{\pm}\\r_{\pm}^{-1}\sin^{2}\phi_{\pm}&\frac{2\phi_{\pm}+\sin 2\phi_{\pm}}{2r_{\pm}^{2%}}\\\end{pmatrix},$ | (9) |

indicating a monotonous growth of the position and momentum variance at a rate $\Gamma_{\pm}$, and correlation oscillations at $2\Omega_{\pm}$sup . In the following, we use our knowledge of the time-dependent covariance matrix $\mathbf{\Sigma}(t)$ to demonstrate that the in-loop dynamics generates motional entanglement.

Transient entanglement - According to the Duan-Simon criterion, the motional state of the two particles is entangled if $\nu_{U}=\langle\Delta q^{2}_{+,U}\rangle+\langle\Delta p^{2}_{-,U}\rangle<1$ for some local symplectic transformation $U$ Simon (2000); Duan*etal.* (2000). We can generalize this statement by introducing $\nu_{\mathrm{min}}$, the minimum value of $\nu_{U}$ under all possible symplectic transformations $U$ Serafini*etal.* (2004); Weedbrook*etal.* (2012); Kotler*etal.* (2021). Figure3(a) shows the time evolution of $\nu_{\mathrm{min}}(t)$ for a loop efficiency $\eta=0.5$, a phase $\theta=2\pi/3$, and for some representative values of $\Gamma_{\text{q}}/\Omega_{0}$. In all curves $\Omega_{-}$ is imaginary, while the common mode (+) is stable. As a result, $\nu_{\mathrm{min}}$ oscillates at twice the stable mode frequency $\Omega_{+}$. The separability criterion is thus maximally violated at a time $t^{*}\sim\pi/(2\Omega_{+})$. Moreover, $\nu_{\text{min}}$ decreases with increasing $\Gamma_{\text{q}}$. This is because the correlations (two-mode squeezing) scale exponentially with $\Omega_{-}$, which in turn grows with $\Gamma_{\text{q}}$ according to Eq.(6). In contrast, the decoherence rate scales only linearly with $\Gamma_{\text{q}}$. Finally, the entanglement vanishes at large times as photon recoil eventually degrades the initial state purity.

Figure.3(b-e) shows the Wigner functions $\mathcal{W}_{j}\sim\text{exp}[-\mathbf{v}_{j}^{\mathrm{T}}\mathbf{\Sigma}%\mathbf{v}_{j}]$ of the initial state in the single particle basis $\mathcal{W}_{0}^{1,2}$, and after an interaction time $t^{*}$ in the joint mode basis $\mathcal{W}_{*}^{\pm}$, for $\Gamma_{\text{q}}/\Omega_{0}=2$. Position squeezing in Fig.3(b,c) is due to the departure from the weak measurement limit ($\eta\Gamma_{\text{q}}/\Omega_{0}\ll 1$) and the consequent break down of the rotating-wave approximation Meng*etal.* (2020). Moreover, we notice that the covariance ellipses (solid tangerine lines) associated with the Wigner functions $\mathcal{W}_{*}^{\pm}$ are anti-correlated and that the unstable mode (-) is squeezed $7.5\leavevmode\nobreak\ \mathrm{dB}$ below the zero point motion (dashed black lines) at an angle $\xi_{-}=\arg(1-r_{\pm}^{-1})$. These are both signatures of the emergence of large two-mode squeezing interactions in ultra-strongly coupled oscillators Marković*etal.* (2018); Kustura*etal.* (2022).

We conclude this section extracting the maximal logarithmic negativity $E_{N}=-10\text{min}[0,\log_{10}(\nu_{\text{min}})]$ as a function of $\theta$ and $\Gamma_{\text{q}}$. We show the results in Fig.4 for the case of $\eta=0.5$ [panel (a)] and $\eta=0.3$ [panel (b)]. Interestingly, even if larger negativities are reached within the unstable region in parameter space (solid red line), for $\eta=0.5$ the particles are entangled even in the stable portion of parameter space. Moreover, the negativity for a fixed $(\eta,\theta)$ first grows with $\Gamma_{\text{q}}$ but finally saturates for $\Gamma_{\text{q}}/\Omega_{0}\gg 1$. Indeed, in such a regime $\Omega_{\pm}^{2}\propto\pm\Gamma_{\text{q}}$ but as correlations grow $\propto\exp(\Omega_{-}t)$, the optimal interaction time $t^{*}\propto\Omega_{+}^{-1}$ decreases, resulting in a squeezing factor $\exp(2\Omega_{-}\Omega_{+}^{-1})$ independent of $\Gamma_{\text{q}}$.

To experimentally certify the generation of an entangled state of motion, one needs to reconstruct $\mathbf{\Sigma}$. As we show insup , this can be accomplished via an optimal retrodiction filter after subtracting the imprecision noise associated with the monitoring process WisemanandMilburn (2009); Wieczorek*etal.* (2015); Meng*etal.* (2020).

Stationary conditional state entanglement - In the previous sections we have considered a binary situation where all the back-scattered light was either circulating in the loop or was diverted into hom*odyne receivers. We now investigate an intermediate configuration where a fraction $\eta_{\text{m}}$ of the light coupled in the transmission line (with collection efficiency $\eta_{c}$) is used to measure the system. Simultaneously, a fraction $\eta=\eta_{c}(1-\eta_{\mathrm{m}})$ circulates in the loop. Our goal is to show that the conditional state of the system is also non-separable.

We denote with $a_{\text{m}}^{1,2}$ the optical fields at the measurement ports. In-loop correlations among the noise terms $a_{\mathrm{ext}}^{i}$ makes the analytic expression of $a_{\mathrm{m}}^{1,2}$ rather lengthy; we report it insup . The key feature is that combining the two fields with a beam splitter, as shown in Fig.5(a), results in output fields $a_{\mathrm{m}}^{\pm}=(a_{\mathrm{m}}^{1}\pm a_{\mathrm{m}}^{2})/\sqrt{2}$ that encode only information about the respective joint mode displacement $q_{\pm}$. A hom*odyne receiver with analyzer angle $\varphi$ measures the quadrature $Z_{\text{m}}^{\pm}=\cos\varphi X_{\text{m}}^{\pm}+\sin\varphi Y_{\text{m}}^{\pm}$, where $X_{\text{m}}^{\pm}$ and $Y_{\text{m}}^{\pm}$ denote respectively the amplitude and phase quadratures of the fields $a_{\mathrm{m}}^{\pm}$. Interestingly, we showed that fixing the analyzer angle $\tilde{\varphi}_{\pm}$ of each detector to

$\tan\tilde{\varphi}_{\pm}=\cot(\theta)\mp[\sqrt{\eta}\sin(\theta)]^{-1}$ | (10) |

allows collecting the maximal amount of information on $q_{\pm}$, with efficiency $\tilde{\eta}=\eta_{c}\eta_{\mathrm{m}}/(1-\eta)$. Correspondingly, the imprecision noise inherent to the measurement and the process noise (back-action) decorrelate, allowing to map our system onto the standard problem of continuous position measurements in optomechanics WisemanandMilburn (2009); Doherty*etal.* (2012). Drawing from its toolbox, we can readily write the conditional state covariance matrix $\mathbf{\Sigma}_{W}$ of the bipartite systemsup .

In Fig.5(b) we calculate $\nu_{\text{min}}$ for the conditional state covariance $\mathbf{\Sigma}_{W}$ as a function of the back-action rate $\Gamma_{\text{q}}$ and transmission line phase $\theta$. We assume a collection efficiency $\eta_{c}=0.5$, and measured fraction $\eta_{\mathrm{m}}=0.8$, yielding effectively a loop transmission $\eta=\eta_{c}(1-\eta_{\mathrm{m}})=0.1$. In some regions of the parameter space, especially in vicinity of the unstable region, $\nu_{\text{min}}$ reaches values that are comparable to those obtained in the transient dynamics, even for moderate ratios $\Gamma_{\text{q}}/\Omega_{0}\approx 1$. We therefore anticipate that the measurement outcomes at the optimal analyzer angles $\tilde{\varphi}_{\pm}$ can be processed using an optimal filter and controller to asymptotically stabilize the conditional state, thereby preparing an entangled steady state of motion of the two nanoparticles.

Conclusion- We proposed a schemeto entangle levitated nanoparticles held in optical tweezers at meter-scale distances solely harnessing optical forces. Importantly, our scheme does not rely on high-finesse cavities or the injection of squeezed light.The nanoparticles must be trapped in ultra-high-vacuum where their motion is predominantly driven by photon recoil, an already demonstrated regimeJain*etal.* (2016). Coupling the backscattered light into the loop is equivalent to maximizing the detection efficiencyTebbenjohanns*etal.* (2019), where values higher than 30% have been achievedMagrini*etal.* (2021). Finally, the phase acquired in the loop can be stabilized by extracting and monitoring a small fraction of the circulating light.

Generalizing our results to an asymmetric configuration, e.g. by setting uneven transmission line phases, will feature a rich parameter space characterized by non-reciprocal interactions and vacuum noise correlations Rudolph*etal.* (2023).Such entangled states may enhance the force-gradient sensing capabilities of our platform Rudolph*etal.* (2022), with applications in searches of new physics Afek*etal.* (2022). Moreover, entangling the motion of massive objects at large distances is a promising prospect for testing quantum mechanicsGhirardi*etal.* (1986); Bassi*etal.* (2013); ArndtandHornberger (2014); Gonzalez-Ballestero*etal.* (2021) or to perform locality loophole-free Bell tests with levitated objects Ralph*etal.* (2000); Thearle*etal.* (2018), a task that is significantly more challenging as it requires non-Gaussian operationsBanaszekandWódkiewicz (1998); Stobińska*etal.* (2007).

###### Acknowledgements.

## I Acknowledgements

This research has been supported by the European Research Council (ERC) under the grant agreement No. [951234] (Q-Xtreme ERC-2020-SyG), the Swiss SERI Quantum Initiative (grant no. UeM019-2), and the Swiss National Science Foundation (grant no. 51NF40-160591). N.C-Z. thanks for support through an ETH Fellowship (grant no. 222-1 FEL-30). The research was funded in part by the Austrian Science Fund (FWF) [10.55776/COE1]. For Open Access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission.

## References

- Jost
*etal.*(2009)J.D.Jost, J.P.Home, J.M.Amini, D.Hanneke, R.Ozeri, C.Langer, J.J.Bollinger, D.Leibfried, andD.J.Wineland,Nature459,683 (2009). - Riedinger
*etal.*(2018)R.Riedinger, A.Wallucks, I.Marinković, C.Löschnauer, M.Aspelmeyer, S.Hong, andS.Gröblacher,Nature556,473 (2018). - Wollack
*etal.*(2022)E.A.Wollack, A.Y.Cleland, R.G.Gruenke, Z.Wang, P.Arrangoiz-Arriola, andA.H.Safavi-Naeini,Nature604,463 (2022). - Ockeloen-Korppi
*etal.*(2018)C.F.Ockeloen-Korppi, E.Damskägg, J.-M.Pirkkalainen, M.Asjad, A.A.Clerk, F.Massel, M.J.Woolley, andM.A.Sillanpää,Nature556,478 (2018). - Kotler
*etal.*(2021)S.Kotler, G.A.Peterson, E.Shojaee, F.Lecocq, K.Cicak, A.Kwiatkowski, S.Geller, S.Glancy, E.Knill, R.W.Simmonds, J.Aumentado, andJ.D.Teufel,Science372,622 (2021),arXiv:2004.05515 . - deLépinay
*etal.*(2021)L.M.deLépinay, C.F.Ockeloen-Korppi, M.J.Woolley, andM.A.Sillanpää,Science372,625 (2021). - Bienfait
*etal.*(2020)A.Bienfait, Y.P.Zhong, H.-S.Chang, M.-H.Chou, C.R.Conner, É.Dumur, J.Grebel, G.A.Peairs, R.G.Povey, K.J.Satzinger, andA.N.Cleland,Phys. Rev. X10,021055 (2020). - Millen
*etal.*(2020)J.Millen, T.S.Monteiro, R.Pettit, andA.N.Vamivakas,Rep. Prog. Phys.83,026401 (2020). - Gonzalez-Ballestero
*etal.*(2021)C.Gonzalez-Ballestero, M.Aspelmeyer, L.Novotny, R.Quidant, andO.Romero-Isart,Science374 (2021),10.1126/science.abg3027,arXiv:2111.05215 . - Rudolph
*etal.*(2020)H.Rudolph, K.Hornberger, andB.A.Stickler,Physical Review A101,011804 (2020),arXiv:2001.05769 . - Chauhan
*etal.*(2020)A.K.Chauhan, O.Černotík, andR.Filip,New Journal of Physics22,123021 (2020),arXiv:2006.03342 . - Brandão
*etal.*(2021)I.Brandão, D.Tandeitnik, andT.Guerreiro,Quantum Science and Technology6,045013 (2021),arXiv:2102.08969 . - Rudolph
*etal.*(2023)H.Rudolph, U.Delić, K.Hornberger, andB.A.Stickler,arXiv (2023),10.48550/arXiv.2306.11893,2306.11893 . - Martinetz
*etal.*(2020)L.Martinetz, K.Hornberger, J.Millen, M.S.Kim, andB.A.Stickler,npj Quantum Inf.6,1 (2020). - Neumeier
*etal.*(2024)L.Neumeier, M.A.Ciampini, O.Romero-Isart, M.Aspelmeyer, andN.Kiesel,Proc. Natl. Acad. Sci. U.S.A.121,e2306953121 (2024). - Roda-Llordes
*etal.*(2024)M.Roda-Llordes, A.Riera-Campeny, D.Candoli, P.T.Grochowski, andO.Romero-Isart,Phys. Rev. Lett.132,023601 (2024). - Miao
*etal.*(2010)H.Miao, S.Danilishin, H.Müller-Ebhardt, H.Rehbein, K.Somiya, andY.Chen,Phys. Rev. A81,012114 (2010). - Giovannetti
*etal.*(2004)V.Giovannetti, S.Lloyd, andL.Maccone,Science306,1330 (2004). - Zhuang
*etal.*(2018)Q.Zhuang, Z.Zhang, andJ.H.Shapiro,Physical Review A97,032329 (2018),arXiv:1711.10459 . - Gessner
*etal.*(2018)M.Gessner, L.Pezzè, andA.Smerzi,Physical Review Letters121,130503 (2018),arXiv:1806.05665 . - Xia
*etal.*(2023)Y.Xia, A.R.Agrawal, C.M.Pluchar, A.J.Brady, Z.Liu, Q.Zhuang, D.J.Wilson, andZ.Zhang,Nature Photonics17,470 (2023),arXiv:2210.16180 . - Tebbenjohanns
*etal.*(2021)F.Tebbenjohanns, M.L.Mattana, M.Rossi, M.Frimmer, andL.Novotny,Nature595,378 (2021). - Magrini
*etal.*(2021)L.Magrini, P.Rosenzweig, C.Bach, A.Deutschmann-Olek, S.G.Hofer, S.Hong, N.Kiesel, A.Kugi, andM.Aspelmeyer,Nature595,373 (2021). - Rieser
*etal.*(2022)J.Rieser, M.A.Ciampini, H.Rudolph, N.Kiesel, K.Hornberger, B.A.Stickler, M.Aspelmeyer, andU.Delić,Science377,987 (2022). - Reisenbauer
*etal.*(2024)M.Reisenbauer, H.Rudolph, L.Egyed, K.Hornberger, A.V.Zasedatelev, M.Abuzarli, B.A.Stickler, andU.Delić,Nat. Phys.,1 (2024). - Delić
*etal.*(2019)U.Delić, M.Reisenbauer, D.Grass, N.Kiesel, V.Vuletić, andM.Aspelmeyer,Physical Review Letters122,123602 (2019),arXiv:1812.09358 . - Windey
*etal.*(2019)D.Windey, C.Gonzalez-Ballestero, P.Maurer, L.Novotny, O.Romero-Isart, andR.Reimann,Physical Review Letters122,1 (2019),arXiv:1812.09176 . - Vijayan
*etal.*(2024)J.Vijayan, J.Piotrowski, C.Gonzalez-Ballestero, K.Weber, O.Romero-Isart, andL.Novotny,Nature Physics (2024),10.1038/s41567-024-02405-3,arXiv:2308.14721 . - Rudolph
*etal.*(2022)H.Rudolph, U.Delić, M.Aspelmeyer, K.Hornberger, andB.A.Stickler,Physical Review Letters129,193602 (2022),arXiv:2204.13684 . - Winkler
*etal.*(2024)K.Winkler, A.V.Zasedatelev, B.A.Stickler, U.Delić, A.Deutschmann-Olek, andM.Aspelmeyer,arXiv (2024),10.48550/arXiv.2408.07492,2408.07492 . - Poddubny
*etal.*(2024)A.N.Poddubny, K.Winkler, B.A.Stickler, U.Delić, M.Aspelmeyer, andA.V.Zasedatelev,arXiv (2024),10.48550/arXiv.2408.06251,2408.06251 . - Gonzalez-Ballestero
*etal.*(2023)C.Gonzalez-Ballestero, J.A.Zielińska, M.Rossi, A.Militaru, M.Frimmer, L.Novotny, P.Maurer, andO.Romero-Isart,PRX Quantum4,030331 (2023). - Tebbenjohanns
*etal.*(2019)F.Tebbenjohanns, M.Frimmer, andL.Novotny,Physical Review A100,043821 (2019),arXiv:1907.12838 . - Jain
*etal.*(2016)V.Jain, J.Gieseler, C.Moritz, C.Dellago, R.Quidant, andL.Novotny,Physical Review Letters116,243601 (2016). - Militaru
*etal.*(2022)A.Militaru, M.Rossi, F.Tebbenjohanns, O.Romero-Isart, M.Frimmer, andL.Novotny,Phys. Rev. Lett.129,053602 (2022). - Maurer
*etal.*(2023)P.Maurer, C.Gonzalez-Ballestero, andO.Romero-Isart,Physical Review A108,033714 (2023). - (37)“See supplemental material for a derivation of the equations of motion in the joint mode basis, a detailed description of the transient evolution, of the in-loop position measurements, and of the protocol for the entanglement verification.”.
- GardinerandCollett (1985)C.W.GardinerandM.J.Collett,Phys. Rev. A31,3761 (1985).
- Forn-Díaz
*etal.*(2019)P.Forn-Díaz, L.Lamata, E.Rico, J.Kono, andE.Solano,Reviews of Modern Physics91,025005 (2019). - Frisk Kockum
*etal.*(2019)A.Frisk Kockum, A.Miranowicz, S.De Liberato, S.Savasta, andF.Nori,Nature Reviews Physics1,19 (2019). - Marković
*etal.*(2018)D.Marković, S.Jezouin, Q.Ficheux, S.Fedortchenko, S.Felicetti, T.Coudreau, P.Milman, Z.Leghtas, andB.Huard,Physical Review Letters121,040505 (2018),arXiv:1804.08705 . - Mancini
*etal.*(1998)S.Mancini, D.Vitali, andP.Tombesi,Physical Review Letters80,688 (1998),arXiv:9802034 [quant-ph] . - Meng
*etal.*(2020)C.Meng, G.A.Brawley, J.S.Bennett, M.R.Vanner, andW.P.Bowen,Physical Review Letters125,043604 (2020). - IsaksenandAndersen (2023)F.W.IsaksenandU.L.Andersen,Physical Review A107,023512 (2023),arXiv:2207.07785 .
- Romero-Isart (2017)O.Romero-Isart,New Journal of Physics19,123029 (2017).
- Simon (2000)R.Simon,Physical Review Letters84,2726 (2000).
- Duan
*etal.*(2000)L.-M.Duan, G.Giedke, J.I.Cirac, andP.Zoller,Physical Review Letters84,2722 (2000). - Serafini
*etal.*(2004)A.Serafini, F.Illuminati, andS.D.Siena,Journal of Physics B: Atomic, Molecular and Optical Physics37,L21 (2004),arXiv:0307073 [quant-ph] . - Weedbrook
*etal.*(2012)C.Weedbrook, S.Pirandola, R.García-Patrón, N.J.Cerf, T.C.Ralph, J.H.Shapiro, andS.Lloyd,Reviews of Modern Physics84,621 (2012),arXiv:1110.3234 . - Kustura
*etal.*(2022)K.Kustura, C.Gonzalez-Ballestero, A.D. L.R.Sommer, N.Meyer, R.Quidant, andO.Romero-Isart,Physical Review Letters128,143601 (2022),arXiv:2112.01144 . - WisemanandMilburn (2009)H.M.WisemanandG.J.Milburn,
*Quantum Measurement and Control*(Cambridge University Press,2009). - Wieczorek
*etal.*(2015)W.Wieczorek, S.G.Hofer, J.Hoelscher-Obermaier, R.Riedinger, K.Hammerer, andM.Aspelmeyer,Physical Review Letters114,223601 (2015),arXiv:1505.01060 . - Doherty
*etal.*(2012)A.C.Doherty, A.Szorkovszky, G.I.Harris, andW.P.Bowen,Philos. Trans. Royal Soc. A370,5338 (2012). - Afek
*etal.*(2022)G.Afek, D.Carney, andD.C.Moore,Phys. Rev. Lett.128,101301 (2022). - Ghirardi
*etal.*(1986)G.C.Ghirardi, A.Rimini, andT.Weber,Physical Review D34,470 (1986). - Bassi
*etal.*(2013)A.Bassi, K.Lochan, S.Satin, T.P.Singh, andH.Ulbricht,Reviews of Modern Physics85,471 (2013),arXiv:1204.4325 . - ArndtandHornberger (2014)M.ArndtandK.Hornberger,Nature Physics10,271 (2014),arXiv:1410.0270 .
- Ralph
*etal.*(2000)T.C.Ralph, W.J.Munro, andR.E.S.Polkinghorne,Physical Review Letters85,2035 (2000). - Thearle
*etal.*(2018)O.Thearle, J.Janousek, S.Armstrong, S.Hosseini, M.Schünemann (Mraz), S.Assad, T.Symul, M.R.James, E.Huntington, T.C.Ralph, andP.K.Lam,Physical Review Letters120,040406 (2018),arXiv:1801.03194. - BanaszekandWódkiewicz (1998)K.BanaszekandK.Wódkiewicz,Phys. Rev. A58,4345 (1998).
- Stobińska
*etal.*(2007)M.Stobińska, H.Jeong, andT.C.Ralph,Phys. Rev. A75,052105 (2007).