Macroscopic Quantum Mechanics in a Classical Spacetime

We apply the many-particle Schrödinger-Newton equation, which describes the coevolution of a many-particle quantum wave function and a classical space-time geometry, to macroscopic mechanical objects. By averaging over motions of the objects’ internal degrees of freedom, we obtain an effective Schrödinger-Newton equation for their centers of mass, which can be monitored and manipulated at quantum levels by state-of-the-art optomechanics experiments. For a single macroscopic object moving quantum mechanically within a harmonic potential well, its quantum uncertainty is found to evolve at a frequency different from its classical eigenfrequency—with a difference that depends on the internal structure of the object—and can be observable using current technology. For several objects, the Schrödinger-Newton equation predicts semiclassical motions just like Newtonian physics, yet quantum uncertainty cannot be transferred from one object to another.

Figure 1

Left: according to standard quantum mechanics, both the vector ($⟨x⟩$, $⟨p⟩$) and the uncertainty ellipse of a Gaussian state for the c.m. of a macroscopic object rotate clockwise in phase space, at the same frequency $\omega ={\omega }_{\mathrm{c}.\mathrm{m}.}$. Right: according to the c.m. Schrödinger-Newton equation (2), ($⟨x⟩$, $⟨p⟩$) still rotates at ${\omega }_{\mathrm{c}.\mathrm{m}.}$, but the uncertainty ellipse rotates at ${\omega }_{\mathrm{q}}\equiv ({\omega }_{\mathrm{c}.\mathrm{m}.}^2+{\omega }_{\mathrm{SN}}^2{)}^{1/2}>{\omega }_{\mathrm{c}.\mathrm{m}.}$.