Chirped-frequency excitation of gravitationally bound ultracold neutrons

Ultracold neutrons confined in the Earth’s gravitational field display quantized energy levels that have been observed for over a decade. In recent resonance spectroscopy experiments [T. Jenke et al., Nat. Phys. 7, 468 (2011)], the transition between two such gravitational quantum states was driven by the mechanical oscillation of the plates that confine the neutrons. Here we show that by applying a sinusoidal modulation with slowly varying frequency (chirp), the neutrons can be brought to higher excited states by climbing the energy levels one by one. The proposed experiment should make it possible to observe the quantum-classical transition that occurs at high neutron energies. Furthermore, it provides a technique to realize superpositions of gravitational quantum states, to be used for precision tests of gravity at short distances.

Figure 1

Bottom panel: Occupation probabilities of the different energy levels as a function of the drive frequency ${\omega }_d$. Top panel: Occupation probabilities as a function of the quantum state number at three values of the drive frequency (${\omega }_1=2\pi \times 338\mathrm{Hz}$, ${\omega }_2=2\pi \times 248\mathrm{Hz}$, ${\omega }_3=2\pi \times 136\mathrm{Hz}$). These frequencies are shown as dashed vertical lines in the bottom panel.

Figure 2

Wigner functions in the phase space $(x,p)$ at three different times corresponding to the three frequencies ${\omega }_{1,2,3}$ shown in Fig. 1. The red rectangle shows a phase-space area equal to $\hslash$. In the bottom panel, the dashed line shows the corresponding classical trajectory. Position and momentum are expressed in normalized units.

Figure 3

Average energy of the bouncer system as a function of the drive frequency. The green dashed line represents the classical formula. Inset: Average occupation number $⟨n⟩$ (thick blue line) and width of the level distribution $⟨n⟩\pm \mathrm{\Delta }n$ (thin green lines) as a function of the drive frequency.