Impulsively Excited Gravitational Quantum States: Echoes and Time-Resolved Spectroscopy

We theoretically study an impulsively excited quantum bouncer (QB)—a particle bouncing off a surface in the presence of gravity. A pair of time-delayed pulsed excitations is shown to induce a wave-packet echo effect—a partial rephasing of the QB wave function appearing at twice the delay between pulses. In addition, an appropriately chosen observable [here, the population of the ground gravitational quantum state (GQS)] recorded as a function of the delay is shown to contain the transition frequencies between the GQSs, their populations, and partial phase information about the wave-packet quantum amplitudes. The wave-packet echo effect is a promising candidate method for precision studies of GQSs of ultracold neutrons, atoms, and antiatoms confined in closed gravitational traps.

Figure 1

Phase space analysis. $N=2\times {10}^4$ particles bounce on a surface, and kicked at $t=t_k=60$. Kick parameters: $a_k=0.5$ (e.g., for neutrons: $|\mathit{\mu }|=60.3\mathrm{neV}/\mathrm{T}$, $\widehat{\beta }\approx 0.8\mathrm{T}/\mathrm{m}$), ${\sigma }_k=0.5$. Initial distribution [light blue, (a)] parameters: ${\mu }_z=20.0$, ${\mu }_v=0.0$, ${\sigma }_z=4$, ${\sigma }_v=1/8$. (a) In blue—filamented phase space before the kick. (b) Shortly after the kick. (c) Close to echo event, at $t\approx 2t_k$. Arrows point at the tips (see the text). (d) Average position.

Figure 2

Echo induced by pulsed inhomogeneous magnetic field kick. The kick is applied at $t=t_k=60$. Initial state parameters: ${\mu }_z=20$, ${\sigma }_z=8$ [see Eq. (3)]. Excitation parameters: $a_k=|\mathit{\mu }|\widehat{\beta }/mg=0.5$ (e.g., for neutrons: $\widehat{\beta }\approx 0.8\mathrm{T}/\mathrm{m}$), and ${\sigma }_k=0.5$. The classical result [see Fig. 1] is added for comparison.

Figure 3

Schematic of a flow-through experimental setup. Bouncing (along $Z$ axis) particles propagate in $X$ direction. (a) First slit with a rough top surface used for preparation. (b) Interaction region. (c) Second slit used for detection.

Figure 4

Time-resolved GQS spectroscopy: kicks by pulsed inhomogeneous magnetic field. (a) $|c_1{|}^2(\tau )$, kicks’ parameters: $a_{k1}=2$, $a_{k2}=1$ (e.g., for neutron: $|\mathit{\mu }|=60.3\mathrm{neV}/\mathrm{T}$, ${\widehat{\beta }}_1\approx 2.4\mathrm{T}/\mathrm{m}$, ${\widehat{\beta }}_2\approx 1.2\mathrm{T}/\mathrm{m}$), ${\sigma }_{k1}={\sigma }_{k2}=0.2$. (b) Spectrum of $|c_1{|}^2(\tau )$. Peaks correspond to energy differences ${\mathcal{E}}_{i1}$ ($i=2,\dots ,6$). Theoretical energy differences [see Eq. (2)]: $z_{21}=1.750$, $z_{31}=3.182$, $z_{41}=4.449$, $z_{51}=5.606$, $z_{61}=6.684$.

Figure 5

Echo induced by surface shake. The first kick is applied at $t=0$, the delay of the second kick is $t_k=150$. The echo emerges at $t\approx 2t_k,3t_k$. Kicks’ parameters: $a_{k1}=1.5$, ${\sigma }_{k1}=0.1$, $a_{k2}=1.0$, ${\sigma }_{k2}=0.16$.

Figure 6

Time-resolved GQS spectroscopy: kicks by surface shake. (a) $|c_1{|}^2(\tau )$, kicks’ parameters: $a_{k1}=0.6$, $a_{k2}=0.1$, and ${\sigma }_{k1}={\sigma }_{k2}=0.2$. (b) Spectrum of $|c_1{|}^2(\tau )$ shown in panel (a). Peaks correspond to energy differences ${\mathcal{E}}_{i1}$ ($i=2,\dots ,6$). Theoretical differences [see Eq. (2)]: $z_{21}=1.750$, $z_{31}=3.182$, $z_{41}=4.449$, $z_{51}=5.606$, $z_{61}=6.684$, where $z_{i1}=z_i-z_1$.