Wigner Function of a Time-dependent Inverted Harmonic Oscillator

Seoktae Koh

Mar Sánchez–Córdova, Jasel Berra–Montiel, Alberto Molgado +1 more

Si-Cong Ji, Philipp Schüttelkopf, Nataliia Bazhan +6 more

When a quantum system passes through an unstable fixed point the local dynamics reduces to the inverted harmonic oscillator (IHO). It exponentially amplifies along one quadrature while squeezing the other, producing macroscopically extended quantum states from microscopic zero-point fluctuations. We realize this dynamics with a Bose-Einstein condensate on an AtomChip. Radio-frequency dressing flips the transverse harmonic confinement into an IHO. Through phase-space tomography we follow the full Wigner function of the evolving quantum state, observe sub-vacuum squeezing of 10.6(1.3) dB, and test coherent reversibility by time-reversing the IHO evolution. Matter-wave interference between the two daughter clouds confirms quantum coherence over timescales far beyond the initial expansion. Our experiment establishes ultra-cold atoms as a clean, controlled, many-body platform for unstable quantum dynamics opening a route to force sensing with time-reversal-based coherence certification and to analog studies of the amplification of quantum fluctuations in inflationary field dynamics.

Guangjie Guo, Zhongzhou Ren, Ju Guo-Xing +1 more

The time evolution of a time-dependent inverted harmonic oscillator (TDIHO) of arbitrary dimensions is investigated. Using the algebraic method, we obtain the exact orthogonal basis of solutions of a TDIHO in which the dimensionality d and angular momentum l appear as parameters, and also discuss its properties. With the wavepacket as the initial state, the general expressions of sojourn time of a TDIHO are given. The method is also applied to the inverted Caldirola–Kanai harmonic oscillator. The results show that the sojourn time appears as an increasing function of the dissipation parameter in arbitrary dimensions, but a decreasing function of the dimensionality.

Yuta Nasuda

Shangyun Wang, Songbai Chen, Jiliang Jing

Abstract We use out-of-time order correlators (OTOCs) to investigate the quantum instability and Ehrenfest time for an inverted harmonic oscillator (IHO). For initial states located in the stable manifolds of the IHO we find that the corresponding OTOC exhibits identical evolutionary characteristics to the saddle point before the Ehrenfest time. For initial states located in the unstable manifolds, the OTOCs still grow exponentially but the time to maintain exponential growth is related to the center position of its wave packet in phase space. Moreover, we use the Husimi Q function to visualize the quantum wave packets during exponential growth of the OTOCs. Our results show that quantum instability exists at arbitrary orbits in the IHO system, and the Ehrenfest time in the IHO system depends not only on the photon number of the initial system but also on the central positions of the initial states in phase space.