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Coherent scattering has recently attracted attention as a means of controlling the motion of levitated particles in three dimensions using a single optical cavity. In these systems, scattering of photons from the trapping field to a cavity mode has been used to cool all three modes of the centre-of-mass motion of levitated particles. The possibility of employing coherent scattering for more general quantum control has, however, not yet been discussed in the literature. Here, we present strategies for generating nonclassical correlations and for engineering interactions between motional modes of levitated particles using coherent scattering. We expand the theory developed by Gonzalez-Ballestero et al. to realize more general bilinear interactions in levitated optomechanics with coherent scattering. Going beyond the simple stationary picture, we introduce amplitude modulation as an important tool to modify the optomechanical interaction and discuss how it can be used to resonantly enhanced certain parts of the interaction, allowing, for example, strong one- and two-mode squeezing of motion. Our results thus show the potential of using coherent scattering for full quantum control of the motion of levitated particles.

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- 1. Motional Gaussian states and gates for a levitating particle Ondřej Černotík and Radim Filip Department of Optics, Palacký University Olomouc, 17. listopadu 1192/12, 77146 Olomouc, Czechia ondrej.cernotik@upol.cz Squeezing in one dimension Comparison to conventional optomechanics Convetional optomechanics relies on a dispersive shift of the cavity resonance, using tweezer only for trapping. Conventional opto- mechanics requires one cavity per motional mode for control. Full control over the motion needs also interactions between mechanical modes which are difficult to engineer. Coherent scattering Optomechanical interaction is mediated by tweezer photons scattered off the particle [1]. This coupling enables control of all three centre-of-mass modes using a single cavity mode [2,3]. Tunability of interactions intracavity intensity radial coupling axial coupling The coupling strength for axial and radial modes is controlled by positioning the particle within the cavity standing wave. The relative strength for the two radial modes is set by the polarization of the cavity and tweezer fields. Highlights Future plans Multidimensional motion Applications creating genuine three-mode entanglement novel states for quantum sensing below the standard quantum limit and testing quantum mechanics ultimate goal: full quantum control of particle motion Nonlinear motion higher-order optomechanical coupling [4] nonlinear potential with non-Gaussian tweezers [5] coupling to two-level systems [6] cavity-mediated interactions Radial and axial mode: Two radial modes: free oscillations and thermal noise cavity-mediated collective dissipation The radial and axial modes couple to orthogonal field quadratures which has important implications for the effective interactions. Interactions between mechanical modes can be induced by virtual photons (two radial modes), collective dissipation, or measurement-based feedback. This is not straightforward with standard dispersive optomechanics. References [1] C. Gonzalez-Ballestero et al., PRA, 100, 013805 (2019). [2] U. Delić et al., PRL 122, 123602 (2019). [3] D. Windey et al., PRL 122, 123601 (2019). [4] U. Delić et al., arXiv:1902.06605. [5] M. Šiler et al., PRL, 121, 230601 (2018). [6] G.P. Conangla et al., Nano Lett. 18, 3956 (2018). 10 2 10 1 100 Modulation depth 10 6 10 4 10 2 100 Squeezingdegree 100 101 SqueezedvarianceVsq 101 104 107 Thermal noise n 100 101 102 10 2 10 1 100 Sideband ratio / m 0.5 1.0 1.5 Basic Hamiltonian Equations of motion Adiabatic elimination of cavity dynamics Assuming a large detuning between the tweezer and cavity frequencies, , we get the effective mechanical dynamics with the effecitve rates 10 3 10 2 10 1 100 Modulation depth 10 3 10 2 10 1 Effectiveparameters |eff|/m,eff/m 0 250 500 750 1000 Time mt 10 6 10 3 100 Squeezingdegree 100 10 4 10 2 100 102 104 Initial temperature n0 10 1 100 101 102 SqueezedvarianceVsq 0 /2 3 /2 2 Modulation phase 10 1 100 Amplitude modulation of the tweezer Modulation of the tweezer field results in parametric amplification of the motion and modulation of the optomechanical coupling according to with modulation depth and phase . Adiabatic elimination of the cavity field now leads to the effective mechanical dynamics with , leading to a lower instability threshold and stronger squeezing. When we choose the detuning , the cavity field couples to the Bogoliubov mode via the Hamiltonian where . Combination of parametric and dissipative squeezing leads to stronger squeezing than either technique alone. parametric squeezing dissipative squeezing this approach opens the way towards full control of particle motion via coherent scattering strong mechanical squeezing possible with parametric amplification and coherent scattering prospect of interactions between all centre-of-mass modes , stable , unstable state-of-the-art parameters [2] state-of-the-art parameters [2] dissipative squeezing alone state-of-the-art parameters [2] (top) and enhanced coupling (bottom) 0 5 10 15 20 0.90 0.95 1.00 Squeezingdegree 1.0 1.2 1.4 0 100 200 300 400 500 Time mt 10 6 10 3 100 100 101 10 3 100 103 Initial temperature n0 10 1 100 101 102 103 SqueezedvarianceVsq state of the art unstable unstable stable