  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 ultra-cold neutrons, atoms, and
anti-atoms confined in closed gravitational traps.

Quantum Physics;High Energy Physics - Phenomenology;Atomic Physics;Instrumentation and Detectors