The role of entanglement in determining the non-classicality of a given interaction has gained significant traction over the last few years. In particular, as the basis for new experimental proposals to test the quantum nature of the gravitational field. Here we show that the rate of gravity mediated entanglement between two otherwise isolated optomechanical systems can be significantly increased by modulating the optomechanical coupling. This is most pronounced for low mass, high frequency systems – convenient for reaching the quantum regime – and can lead to improvements of several orders of magnitude, as well as a broadening of the measurement window. Nevertheless, significant obstacles still remain. In particular, we find that modulations increase decoherence effects at the same rate as the entanglement improvements. This adds to the growing evidence that the constraint on noise (acting on the position d.o.f) depends only on the particle mass, separation, and temperature of the environment and cannot be improved by novel quantum control. Finally, we highlight the close connection between the observation of quantum correlations and the limits of measurement precision derived via the Cramér-Rao Bound. An immediate consequence is that probing superpositions of the gravitational field places similar demands on detector sensitivity as entanglement verification.
Such an experiment, however, is expected to be exceedingly difficult – with the entanglement rate depending on the mass, separation and superposition size (or more generally, variance) that can be achieved during coherent evolution. The latter in particular poses a significant obstacle to optomechanical systems, which are often regarded as one of the most attractive platforms for tests of macroscopic quantum physics. These rely on the radiation pressure of a light field to drive the dynamics of a mechanical oscillator, the variance of which depends on that of the field as well as the strength of the optomechanical coupling. However, achieving a high photon number variance is typically difficult, and so the conventional approach is to simply increase the number of photons in a (possibly squeezed) coherent light field. This cannot be done without limit, as if the mechanical elements are driven too hard they risk colliding. As a result, predicted entanglement times are typically much longer than even the most optimistic noise timescales.
To address this problem, we show that if the optomechanical coupling strength, $k$, can be modulated close to the mechanical resonance, then the entanglement rate is significant enhanced – potentially by several orders of magnitude. This is because radiation pressure leads to a force on the mechanical elements proportional to the coupling, and so modulating $k$ is equivalent to resonantly driving the oscillator. As the force is also proportional to the photon number, fluctuations in the field are passed on to mechanics, but now enhanced by the increased displacement. This means that in ideal, zero noise conditions, state of the art systems could potentially generate appreciable entanglement on the order of seconds.
Unfortunately, however, we find that decoherence is also enhanced by a commensurate amount. This adds to a growing number of results – across multiple platforms – suggesting the existence of an unavoidable noise limit that depends only on the interaction term and the environment, i.e. it cannot be mitigated by local control or preparation of the mechanical states. In practice, this has severe implications for any attempts to probe gravity through entanglement tests. We highlight further difficulties by quantifying the extreme levels of control needed over the dynamics, which in the nonlinear optomechanical setting is reflected in both the degree that the mechanical frequencies of each oscillator must match as well as the measurement timing precision. Similar constraints should be expected in any experiment where entanglement is transferred from mechanical to ancilla degrees of freedom.
Finally, we compare our analysis against a variety of approximation methods, showing that these are often sufficient to obtain an accurate entanglement rate. In particular, by characterising the measurement sensitivity when two optomechanical systems are considered as a sensor-source pair, we find that the time needed to detect the quantum fluctuations in the source roughly coincides with that required to establish witnessable entanglement. This highlights the close connection to quantum metrology, and underlines the importance of improving sensor performance when attempting to access the quantum gravity regime.
 G. Amelino-Camelia, C. Lämmerzahl, F. Mercati, and G. M. Tino. ``Constraining the energy-momentum dispersion relation with planck-scale sensitivity using cold atoms''. Phys. Rev. Lett. 103, 171302 (2009).
 C. Anastopoulos and B. L. Hu. ``Probing a gravitational cat state''. Classical and Quantum Gravity 32, 165022 (2015).
 M. Carlesso, A. Bassi, M. Paternostro, and H. Ulbricht. ``Testing the gravitational field generated by a quantum superposition''. New Journal of Physics 21, 093052 (2019).
 A. Albrecht, A. Retzker, and M. B. Plenio. ``Testing quantum gravity by nanodiamond interferometry with nitrogen-vacancy centers''. Phys. Rev. A 90, 033834 (2014).
 S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, and G. Milburn. ``Spin entanglement witness for quantum gravity''. Phys. Rev. Lett. 119, 240401 (2017).
 C. Marletto and V. Vedral. ``Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity''. Phys. Rev. Lett. 119, 240402 (2017).
 C. Wan. ``Quantum superposition on nano-mechanical oscillator''. PhD thesis. Imperial College, London. (2017). url: https://spiral.imperial.ac.uk/handle/10044/1/74060.
 E. Chitambar, D. Leung, L. Mančinska, M. Ozols, and A. Winter. ``Everything you always wanted to know about locc (but were afraid to ask)''. Communications in Mathematical Physics 328, 303–326 (2014).
 N. Matsumoto, S. B. Cataño Lopez, M. Sugawara, S. Suzuki, N. Abe, K. Komori, Y. Michimura, Y. Aso, and K. Edamatsu. ``Demonstration of displacement sensing of a mg-scale pendulum for mm- and mg-scale gravity measurements''. Phys. Rev. Lett. 122, 071101 (2019).
 S. B. Cataño Lopez, J. G. Santiago-Condori, K. Edamatsu, and N. Matsumoto. ``High-$q$ milligram-scale monolithic pendulum for quantum-limited gravity measurements''. Phys. Rev. Lett. 124, 221102 (2020).
 M. Rademacher, J. Millen, and Y. L. Li. ``Quantum sensing with nanoparticles for gravimetry: when bigger is better''. Advanced Optical Technologies 9, 227–239 (2020).
 C. Montoya, E. Alejandro, W. Eom, D. Grass, N. Clarisse, A. Witherspoon, and A. A. Geraci. ``Scanning force sensing at micrometer distances from a conductive surface with nanospheres in an optical lattice''. Applied optics 61, 3486–3493 (2022).
 S. Qvarfort, A. D. K. Plato, D. E. Bruschi, F. Schneiter, D. Braun, A. Serafini, and D. Rätzel. ``Optimal estimation of time-dependent gravitational fields with quantum optomechanical systems''. Phys. Rev. Research 3, 013159 (2021).
 A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen. ``Mechanical Squeezing via Parametric Amplification and Weak Measurement''. Physical Review Letters 107, 213603 (2011).
 J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker. ``Cavity cooling a single charged levitated nanosphere''. Physical Review Letters 114, 123602 (2015).
 P. Z. G. Fonseca, E. B. Aranas, J. Millen, T. S. Monteiro, and P. F. Barker. ``Nonlinear dynamics and strong cavity cooling of levitated nanoparticles''. Physical Review Letters 117, 173602 (2016).
 E. B. Aranas, P. Z. G. Fonseca, P. F. Barker, and T. S. Monteiro. ``Split-sideband spectroscopy in slowly modulated optomechanics''. New Journal of Physics 18, 113021 (2016).
 D. Kafri, J. M. Taylor, and G. J. Milburn. ``A classical channel model for gravitational decoherence''. New Journal of Physics 16, 065020 (2014).
 C. Anastopoulos and B. L. Hu. ``Problems with the newton-schrödinger equations''. New Journal of Physics 16, 085007 (2014).
 S. Qvarfort, A. Serafini, A. Xuereb, D. Braun, D. Rätzel, and D. E. Bruschi. ``Time-evolution of nonlinear optomechanical systems: Interplay of mechanical squeezing and non-gaussianity''. Journal of Physics A: Mathematical and Theoretical 53, 075304 (2020).
 M. T. Naseem, A. Xuereb, and Ö. E. Müstecaplıoğlu. ``Thermodynamic consistency of the optomechanical master equation''. Physical Review A 98, 052123 (2018).
 Á. Rivas, A. D. K. Plato, S. F. Huelga, and M. B. Plenio. ``Markovian master equations: a critical study''. New Journal of Physics 12, 113032 (2010).
 S. L. Adler, A. Bassi, and E. Ippoliti. ``Towards quantum superpositions of a mirror: an exact open systems analysis — calculational details''. Journal of Physics A: Mathematical and General 38, 2715–2727 (2005).
 J. Chiaverini, S. J. Smullin, A. A. Geraci, D. M. Weld, and A. Kapitulnik. ``New experimental constraints on non-newtonian forces below 100 $\mu$ m''. Physical Review Letters 90, 151101 (2003).
 T. W. van de Kamp, R. J. Marshman, S. Bose, and A. Mazumdar. ``Quantum gravity witness via entanglement of masses: Casimir screening''. Physical Review A 102, 062807 (2020).
 O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac. ``Optically levitating dielectrics in the quantum regime: Theory and protocols''. Physical Review A 83, 013803 (2011).
 J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vamivakas. ``Optomechanics with levitated particles''. Reports on Progress in Physics 83, 026401 (2020).
 C. M. DeWitt and D. Rickles. ``The role of gravitation in physics: report from the 1957 chapel hill conference''. Volume 5. epubli. (2011).
 F. Schneiter, S. Qvarfort, A. Serafini, A. Xuereb, D. Braun, D. Rätzel, and D. E. Bruschi. ``Optimal estimation with quantum optomechanical systems in the nonlinear regime''. Phys. Rev. A 101, 033834 (2020).
 S. Ast, M. Mehmet, and R. Schnabel. ``High-bandwidth squeezed light at 1550 nm from a compact monolithic ppktp cavity''. Opt. Express 21, 13572–13579 (2013).
 J. Z. Bernád, L. Diósi, and T. Geszti. ``Quest for quantum superpositions of a mirror: high and moderately low temperatures''. Physical review letters 97, 250404 (2006).
 S. Rijavec, M. Carlesso, A. Bassi, V. Vedral, and C. Marletto. ``Decoherence effects in non-classicality tests of gravity''. New Journal of Physics 23, 043040 (2021).
 S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert. ``Observation of non-markovian micromechanical brownian motion''. Nature Communications 6, 7606 (2015).
 M. Ludwig, K. Hammerer, and F. Marquardt. ``Entanglement of mechanical oscillators coupled to a nonequilibrium environment''. Phys. Rev. A 82, 012333 (2010).
 A. Datta and H. Miao. ``Signatures of the quantum nature of gravity in the differential motion of two masses''. Quantum Science and Technology 6, 045014 (2021).
 B. Dakić, V. Vedral, and Č. Brukner. ``Necessary and sufficient condition for nonzero quantum discord''. Physical review letters 105, 190502 (2010).
 B. L. Hu, J. P. Paz, and Y. Zhang. ``Quantum brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise''. Phys. Rev. D 45, 2843–2861 (1992).
 Daisuke Miki, Nobuyuki Matsumoto, Akira Matsumura, Tomoya Shichijo, Yuuki Sugiyama, Kazuhiro Yamamoto, and Naoki Yamamoto, "Generating quantum entanglement between macroscopic objects with continuous measurement and feedback control", Physical Review A 107 3, 032410 (2023).
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