Probing the nonclassical dynamics of a quantum particle in a gravitational field

Abstract

In quantum mechanics, the time evolution of particles is given by the Schrödinger equation. It is valid in a nonrelativistic regime where the interactions with the particle can be modelled by a potential and quantised fields are not required. This has been verified in countless experiments when the interaction is of electromagnetic origin, but also corrections due to the quantised field are readily observed. When the interaction is due to gravity, then one cannot expect to see effects of the quantised field in current-technology Earth-bound experiments. However, this does not yet guarantee that in the accessible regime, the time evolution is accurately given by the Schrödinger equation. Here we propose to measure the effects of an asymmetric mass configuration on a quantum particle in an interferometer. For this setup we show that with parameters within experimental reach, one can be sensitive to possible deviations from the Schrödinger equation, beyond the already verified lowest-order regime. Performing this experiment will hence directly test the nonclassical behaviour of a quantum particle in the gravitational field.

Figures (2)

• Figure 1: Interferometric scheme to probe the dynamics of a quantum particle beyond the classical limit. A particle of mass $m$ is in superposition of the paths $| L \rangle$ and $| R \rangle$ with distance $\Delta x$. External masses $M_1$ and $M_2$ are at distances $d_1$ and $d_2$, respectively, from the midpoint of $\Delta x$. By choosing $M_1 / d_1^2 = M_2 / d_2^2$, the "classical" part of the time evolution cancels out while the reminder of the evolution leads to interference. Here a caesium atom with $m=133u$, $\Delta x = 10 \,\mathrm{cm}$ and two tungsten balls with $M_1 = 20 \,\mathrm{g}$, $M_2 = 40 \,\mathrm{g}$, $d_1 \approx 5.9 \,\mathrm{cm}$, $d_2 \approx 8.4 \,\mathrm{cm}$ are depicted in scale. Also shown are the potential energy $V(x)$ and the rescaled force $\Delta x V'(x)$.
• Figure 2: Spacetime diagram of the interferometric setup. At time $t=0$, a coherent momentum splitting by $\pm 10\hbar k$ is applied to the atom, yielding two arms at a relative velocity of $0.6 \,\mathrm{cm / s}$. After $1.67 \,\mathrm{s}$, another pulse eliminates the relative motion, and the arms are kept at a constant separation of $10\,\mathrm{cm}$ for $4 \,\mathrm{s}$. Then the pulse sequence is applied in reverse order to recombine the arms. For the proposed mass configuration, the phase shift would vanish exactly if the systematic effect of gravity were described by the Poisson bracket. The Schrödinger equation predicts a phase shift of $0.675 \,\mathrm{rad}$.