Quantum Physics in the Inverted Harmonic Oscillator

Unstable fixed points play a central role in physics, governing the amplification of fluctuations across fields as diverse as quantum Hall transport, black-hole horizons, and cosmological reheating. The inverted harmonic oscillator (IHO) is the canonical, exactly solvable model of such instabilities, simultaneously serving as the generator of squeezing in quantum optics and as a universal scattering barrier. Here, we experimentally study the evolution of wave packet placed on top of an inverted harmonic oscillator potential using a dilute Bose-Einstein Condensate on an AtomChip. By rapidly transforming a harmonic confinement into an inverted potential via radio frequency dressing, we observe phase-space dilation of the atomic state up to \textbf{10.6 dB squeezing}, reconstruct its Wigner function through full tomography, and demonstrate coherent evolution by interference and substantial restoration of the initial coherent state. Our results establish (dilute) ultracold atoms as a pristine platform for accessing the physics of unstable quantum dynamics, bridging fundamental connections to horizons, quantum field instabilities while opening pathways for quantum-enhanced sensing and tests of macroscopic coherence.