Quantum motion of a neutron in a waveguide in the gravitational field

We study theoretically the quantum motion of a neutron in a horizontal waveguide in the gravitational field of the Earth. The waveguide in question is equipped with a mirror below and a rough surface absorber above. We show that such a system acts as a quantum filter, i.e. it effectively absorbs quantum states with sufficiently high transversal energy but transmits low-energy states. The states transmitted are determined mainly by the potential well formed by the gravitational field of the Earth and the mirror. The formalism developed for quantum motion in an absorbing waveguide is applied to the description of the recent experiment on the observation of the quantum states of neutrons in the Earth’s gravitational field.

Figure 1

Schematic view of the experiment. From left to the right: the vertical bold lines indicate the upper and lower plates of the input collimator (1); the solid arrows correspond to classical neutron trajectories (2) between the input collimator and the entry slit between a mirror [(3), empty rectangle below] and a scatterer [(4), black rectangle above]. The dotted horizontal arrows illustrate the quantum motion of neutrons above a mirror (5), and the black box represents the neutron detector (6).

Figure 2

Neutron count rate at the exit of the waveguide as a function of absorber position. Circles correspond to the experimental data, solid line corresponds to the theoretical fit.

Figure 3

Lifetime of the first three gravitational states as a function of absorber position in a typical arrangement of our experiment.

Figure 4

Energy of the first three gravitational states as a function of absorber position in an arrangement typical for our experiment.

Figure 5

The relative neutron flux as a function of absorber position for different absorber diffuseness in an arrangement typical for our experiment. $F^{*}$ is the flux calculated at absorber position $H=45\mu \mathrm{m}$ and diffuseness $\rho =1\mu \mathrm{m}$.

Figure 6

The relative neutron flux in the presence of, and in the absence of, gravity. $F^{*}$ is the flux with gravity calculated for absorber position $H=45\mu \mathrm{m}$, $|\mathrm{Im}a|=2\mu \mathrm{m}$, and ${\tau }^{\mathrm{pass}}=0.02\mathrm{s}$.

Figure 7

The relative neutron flux in the direct and inverse geometry experiment. $F^{*}$ is the flux in the direct geometry case, calculated for absorber position $H=45\mu \mathrm{m}$, $|\mathrm{Im}a|=2\mu \mathrm{m}$, and ${\tau }^{\mathrm{pass}}=0.02\mathrm{s}$.

Figure 8

The relative neutron flux as a function of the slit height in the time-dependent model. $F^{*}$ is the flux calculated for absorber position $H=40\mu \mathrm{m}$, $b=1\mu \mathrm{m}$, $|\mathrm{Im}a|=0.1\mu \mathrm{m}$, and ${\tau }^{\mathrm{pass}}=0.02\mathrm{s}$.

Figure 9

Repopulation coefficient of the ground state above the 2nd mirror as a function of the relative shift of the two mirrors in $\mu \mathrm{m}$.

Figure 10

The relative neutron flux as a function of slit height for two values of the bottom mirror shift. $F^{*}$ is the flux calculated at absorber position $H=35\mu \mathrm{m}$ and diffuseness $\rho =1\mu \mathrm{m}$.