Enhancing Gravitational Interaction between Quantum Systems by a Massive Mediator

In 1957 Feynman suggested that the quantum or classical character of gravity may be assessed by testing the gravitational interaction due to source masses in superposition. However, in all proposed experimental realizations using matter-wave interferometry, the extreme weakness of this interaction requires pure initial states with extreme squeezing to achieve measurable effects of nonclassical interaction for reasonable experiment durations. In practice, the systems that can be prepared in such nonclassical states are limited to small masses, which in turn limits the strength of their interaction. Here we address this key challenge—the weakness of gravitational interaction—by using a massive body as an amplifying mediator of gravitational interaction between two test systems. Our analysis shows that this results in an effective interaction between the two test systems that grows with the mass of the mediator, is independent of its initial state and, therefore, its temperature. This greatly reduces the requirement on the mass and degree of delocalization of the test systems and, while still highly challenging, brings experiments on gravitational source masses a step closer to reality.

Figure 1

Setup: A test particle (system $A$) of radius $r$ and mass $m_a$ is subject to a double-well potential with wells separated by a distance $d_0$ and behaves as a two-level system with states $|L⟩$ and $|R⟩$, which are stationary for the duration of the experiment provided that $d_0$ is large enough to make any tunneling negligible. A massive oscillator (system $C$) with frequency $\omega$, radius $R$, and mass $m_c$ has its equilibrium position at a distance $d$ from the center of the double-well potential and acts as a mediator between the test mass and an ancillary qubit (system $B$) that has states $|0⟩$ and $|1⟩$ and bare energy splitting ${\omega }_0$. The mediator is weakly coupled to the test mass through gravitational interaction with energy ${\widehat{V}}_a(\widehat{X})$ depending on the position of the oscillator, and strongly coupled to the ancillary system with a much stronger interaction energy ${\widehat{V}}_b(\widehat{X})$ of a nature other than gravitational, e.g., Casimir force. The direct interaction between systems $A$ and $B$ is negligible.

Figure 2

System dynamics: In (a) we show the evolution in phase space of the four components of the mediator state correlated with each of the four states of the TLTM and the AQ. Here, $\{⟨\widehat{x}⟩,⟨\widehat{p}⟩\}$ are dimensionless position and momentum quadratures of the redefined oscillator, with shifted frequency and displaced equilibrium position. (b) The evolution of the entanglement, as quantified by the logarithmic negativity, between the TLTM and the AQ in continuous lines and between the TLTM and the mediator in dashed lines, for different temperatures of the mediator. Here, $g_a=1/48\tilde{\omega }$ and $g_b=\tilde{\omega }$. Continuous lines in (c) display the entanglement between the TLTM and the AQ for different values of the coupling $g_b$ expressed in units of $\tilde{\omega }$. For this simulation we initialize the mediator in a thermal state with mean phonon occupation ${⟨n⟩}_0=10$, and set $g_a$ as in (b). Dots indicate the value of the entanglement at the decoupling times $t_n$. Dashed lines correspond to the evolution of entanglement between two generic qubits governed by $\widehat{H}=(2g_a{g}_b/\tilde{\omega }){\widehat{\sigma }}_z{\widehat{\sigma }}_z$, which at times $t_n$ has a unitary-evolution operator equivalent to that of the full-system Hamiltonian, see Eq. (4).