Enhanced Gravitational Entanglement via Modulated Optomechanics

A. Douglas K. Plato1, Dennis Rätzel2,3, and Chuanqi Wan

1Institut für Physik, Universität Rostock, Albert-Einstein-Straße 23, 18059 Rostock, Germany
2ZARM, University of Bremen, Am Fallturm 2, 28359 Bremen, Germany
3Institut für Physik, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany

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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.

One of the great mysteries of modern physics is how to reconcile quantum mechanics with the general theory of relativity. The prevailing assumption is that the gravitational field should somehow be quantised, though a number of alternative approaches exist and the fundamental nature of gravity still remains an open question. In the last few years, however, a potential route to resolving this issue has emerged from the field of quantum information. The idea is that certain types of correlations – for example, entanglement – cannot be created between two distinct subsystems if only local (quantum) operations and classical communication (LOCC) are allowed. This suggests that detecting gravity mediated entanglement between two macroscopic scale masses would indicate that the interaction is either quantised or that gravity acts non-locally in the macroscopic limit.

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.

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[2] Richard Howl, Nathan Cooper, and Lucia Hackermüller, "Gravitationally-induced entanglement in cold atoms", arXiv:2304.00734, (2023).

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