Motional entanglement of remote optically levitated nanoparticles (2024)

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

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 atomsJostetal. (2009)to microbeamsRiedingeretal. (2018); Wollacketal. (2022), microscale drum resonatorsOckeloen-Korppietal. (2018); Kotleretal. (2021); deLépinayetal. (2021), and acoustic modes of bulk resonatorsBienfaitetal. (2020). Extending this capability to levitated optomechanicsMillenetal. (2020); Gonzalez-Ballesteroetal. (2021)—i.e., generating motional entanglement between two optically levitated nanospheres in high vacuum—is a milestone in the field Rudolphetal. (2020); Chauhanetal. (2020); Brandãoetal. (2021); Rudolphetal. (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 proposalsMartinetzetal. (2020); Neumeieretal. (2024); Roda-Llordesetal. (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 decoherenceMiaoetal. (2010), as well as for quantum-enhanced sensing and metrologyGiovannettietal. (2004, 2004); Zhuangetal. (2018); Gessneretal. (2018, 2018); Xiaetal. (2023).

Recent experiments have taken crucial steps towards entanglement in levitated optomechanics by showing mechanical ground-state cooling of levitated nanoparticles in free spaceTebbenjohannsetal. (2021); Magrinietal. (2021), as well as strong and controllable light-mediated interactions between two levitated nanoparticlesRieseretal. (2022); Reisenbaueretal. (2024). In these setups, however, the trapping laser’s shot noise induces a high degree of motional decoherence which prevents the generation of entanglement Rudolphetal. (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 Rudolphetal. (2023); Delićetal. (2019); Windeyetal. (2019); Vijayanetal. (2024), non-optical coupling mechanismsRudolphetal. (2022); Winkleretal. (2024); Poddubnyetal. (2024), and using squeezed light to reduce measurement backaction noise Rudolphetal. (2023); Gonzalez-Ballesteroetal. (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 recoilTebbenjohannsetal. (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 Jainetal. (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 modeMagrinietal. (2021); Militaruetal. (2022); Maureretal. (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.

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 Mancinietal. (1998); Mengetal. (2020); IsaksenandAndersen (2023).

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); Duanetal. (2000). We can generalize this statement by introducing $\nu_{\mathrm{min}}$, the minimum value of $\nu_{U}$ under all possible symplectic transformations $U$ Serafinietal. (2004); Weedbrooketal. (2012); Kotleretal. (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 $\mathcal{W}_{j}\sim\text{exp}[-\mathbf{v}_{j}^{\mathrm{T}}\mathbf{\Sigma}$ 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 Mengetal. (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 Markovićetal. (2018); Kusturaetal. (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); Wieczoreketal. (2015); Mengetal. (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.

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 regimeJainetal. (2016). Coupling the backscattered light into the loop is equivalent to maximizing the detection efficiencyTebbenjohannsetal. (2019), where values higher than 30% have been achievedMagrinietal. (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 Rudolphetal. (2023).Such entangled states may enhance the force-gradient sensing capabilities of our platform Rudolphetal. (2022), with applications in searches of new physics Afeketal. (2022). Moreover, entangling the motion of massive objects at large distances is a promising prospect for testing quantum mechanicsGhirardietal. (1986); Bassietal. (2013); ArndtandHornberger (2014); Gonzalez-Ballesteroetal. (2021) or to perform locality loophole-free Bell tests with levitated objects Ralphetal. (2000); Thearleetal. (2018), a task that is significantly more challenging as it requires non-Gaussian operationsBanaszekandWódkiewicz (1998); Stobińskaetal. (2007).

Acknowledgements.