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Observation of Phase Space Dynamics of Inverted Harmonic Oscillator

This paper experimentally demonstrates the phase-space dynamics of an inverted harmonic oscillator using surface gravity water waves, revealing a distinct separatrix that governs whether wave packets are blocked or transmitted based on their energy relative to the potential barrier.

Original authors: Georgi Gary Rozenman, Maxim A. Efremov, Wolfgang P. Schleich, Lev Shemer, Ady Arie

Published 2026-07-22

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Georgi Gary Rozenman, Maxim A. Efremov, Wolfgang P. Schleich, Lev Shemer, Ady Arie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Wave Race: When Water Acts Like a Quantum Ghost

Imagine you are watching a race, but instead of runners, the competitors are ripples on a pond. In the world of physics, there is a famous character called the "harmonic oscillator." Think of it like a marble rolling back and forth in a smooth, U-shaped bowl. It's the ultimate playground for understanding how things vibrate, from atoms to guitar strings. But there is a spooky twin to this marble: the "inverted harmonic oscillator." Instead of a bowl, imagine a smooth, upside-down hill. If you place a marble on top, it doesn't stay put; it rolls down one side or the other, speeding up forever. In the strange world of quantum mechanics (the physics of the very small), this upside-down hill represents a barrier that particles can sometimes tunnel through or bounce off, depending on their energy.

Why do scientists care about this upside-down hill? Because it's a universal key. It helps explain how black holes might glow with "Hawking radiation," how lasers get their power, and even how electrons move through tiny computer chips. However, studying these quantum effects directly is incredibly hard because the particles are too small and the rules are too weird. So, physicists often look for a "analogue"—a bigger, easier-to-see system that follows the same mathematical rules. If you can make water waves behave exactly like quantum particles, you can watch the quantum drama play out in a bathtub. This is the stage where our story takes place: a team of scientists decided to turn a tank of water into a giant quantum simulator to see what happens when a wave tries to climb an invisible, upside-down hill.

The Experiment: Building an Upside-Down Hill in Water

In this study, the researchers built a 5-meter-long tank of water and set up a very specific, time-changing current. They didn't just push the water; they programmed a pump to create a flow that changed speed in a perfect parabolic curve over time. Think of it like a conveyor belt that starts slow, speeds up to a maximum, and then slows down again, but the "speed" of this belt is actually the speed of the water current itself.

This time-dependent current creates a "potential barrier" for the water waves. In the language of the experiment, the water waves are like quantum particles, and the changing current acts like that famous upside-down hill. The team launched tiny, perfectly shaped "Gaussian" wave packets (think of them as neat, bell-shaped ripples) into this current. They then watched what happened to these ripples as they encountered the barrier, measuring the height of the water and the speed of the wave at different points.

The Three Fates of the Wave

The most exciting part of the experiment was watching how the waves behaved based on their energy. The researchers found that the waves fell into three distinct categories, separated by a sharp boundary in the "phase space" (a map that plots a wave's position against its momentum).

The Transmitters (High Energy): When the wave packet had enough energy (specifically, an energy $E_0 > 0$ ), it was like a runner sprinting up a hill. It had enough momentum to roll right over the top of the barrier and continue on the other side. In the experiment, these waves were clearly transmitted through the system.
The Reflectors (Low Energy): When the wave packet had too little energy ( $E_0 < 0$ ), it was like a ball thrown at a wall that it couldn't climb. It hit the barrier, slowed down, and bounced back. The experiment showed these waves were blocked and reflected.
The Separatrix (The Edge Case): The most fascinating case was when the wave had exactly the right amount of energy to reach the very top of the hill ( $E_0 = 0$ ). This is the "separatrix," the razor-thin line between winning and losing. In this scenario, the wave didn't bounce back, nor did it zoom through. Instead, it crept toward the top of the barrier, slowing down more and more, stretching out, and fading away. It approached the peak but never quite seemed to get there in a finite time, eventually dissolving into the background noise.

What They Found and What They Ruled Out

The team measured the path of these waves with incredible precision. They tracked the "analog time coordinate" (where the wave was) and the "analog momentum" (how fast it was moving) and found that the waves followed the exact mathematical curves predicted by the theory of the inverted harmonic oscillator. They calculated the frequency of this "hill" to be approximately $30.14 \pm 0.40$ rad/s based on the wave's position and $30.55 \pm 0.21$ rad/s based on its momentum. These two independent measurements agreed with each other, confirming that their water tank was indeed acting as a perfect quantum simulator.

Crucially, the paper explicitly rules out a common quantum phenomenon in this specific setup. In many quantum scattering experiments, even if a particle has enough energy to go over a barrier, a tiny bit of it might still bounce back (quantum reflection). However, the authors state clearly that in their experiments with positive energy values, no quantum reflections occurred. They observed that the waves either went all the way through or bounced all the way back, with no "partial" reflections. They note that partial reflections only happen if the barrier is very small compared to the size of the wave, which was not the case in their setup.

The Mystery of the Fading Wave

When the researchers looked at the "edge case" wave ( $E_0 = 0$ ), they saw something visually striking. As the wave packet approached the top of the barrier, it didn't just stop; it stretched out and became incredibly thin. The amplitude (the height of the wave) dropped so low that it disappeared into the noise of the measuring equipment before it could reach the very peak. The authors suggest this isn't just a measurement error. They propose that the rapid stretching of the water surface near the top of the barrier dilutes both the wave and the random background ripples, effectively reducing the noise in that specific area. It's as if the wave was so stretched out that it became invisible, a perfect demonstration of the "separatrix" dynamics.

Why This Matters

This experiment didn't just watch water waves; it proved that we can use simple, classical water tanks to model complex, quasi-infinite quantum systems. By creating this "inverted harmonic oscillator" with a water pump, the team has opened a door to studying things that are usually impossible to see directly. They suggest that this setup could be used to probe the "quantum reflection" regime (if they make the barrier smaller), measure the "logarithmic phase singularity" (a weird mathematical point that mimics the edge of a black hole), or even control wave packets to bring them to a complete stop.

The paper concludes that this water-based platform is a powerful new tool. It bridges the gap between the abstract math of quantum mechanics and the tangible world of fluid dynamics, showing that sometimes, to understand the universe's smallest secrets, you just need a really good wave maker and a tank of water.

Technical Summary: Observation of Phase Space Dynamics of Inverted Harmonic Oscillator

Problem and Motivation
The inverted harmonic oscillator (IHO) serves as a fundamental model in quantum mechanics, characterizing parabolic potential barriers rather than binding potentials. It is central to understanding quantum scattering, tunneling through barriers, and phenomena in quantum optics such as laser gain and difference-frequency resonance. Furthermore, the IHO shares significant phase-space features with black hole horizons, specifically the existence of a separatrix that translates into a logarithmic phase singularity associated with Hawking radiation. While the IHO is theoretically well-established, direct experimental observation of its phase-space dynamics, particularly the distinction between transmission, reflection, and motion along the separatrix, has been challenging. This study addresses the need to experimentally realize an IHO to map these dynamics and observe the "stopping power" of the potential barrier.

Methodology
The authors experimentally realized an IHO using classical surface-gravity water waves. The system exploits the mathematical analogy between the Schrödinger equation for a quantum particle and the wave equation governing paraxial optical beams and water waves, with a crucial interchange of space and time roles.

Physical Setup: Experiments were conducted in a 5-meter long water tank. Surface gravity waves were generated by a computer-controlled wave maker at one end and measured by capacitance-type wave gauges.
Realization of the Potential: Instead of a spatial potential barrier, the IHO potential was realized through a spatially homogeneous but time-dependent water current. The current velocity followed a quadratic time dependence, $U(t) = U_0 + c(t - t_p)^2$ , creating an effective inverted parabolic potential $-\Omega^2\tau^2$ in the co-moving frame.
Wave Packets: Gaussian wave packets were launched with varying initial conditions to simulate different energy states relative to the barrier maximum.
Data Reconstruction: The complex amplitude envelope $A(\tau, \xi)$ of the wave was reconstructed from surface elevation measurements using the Hilbert transform. From this, the expectation values of the analog time coordinate $\langle \tau \rangle$ and analog momentum $\langle p \rangle$ were calculated to map the phase-space trajectories.

Key Contributions and Results
The study successfully mapped the phase-space dynamics of Gaussian wave packets interacting with an IHO, identifying three distinct regimes based on the classical energy $E_0 = p_0^2 - \Omega^2\tau_0^2$ :

Transmission ( $E_0 > 0$ ): Wave packets with energy above the barrier maximum were transmitted through the potential.
Reflection ( $E_0 < 0$ ): Wave packets with energy below the barrier maximum were reflected (blocked).
Separatrix Motion ( $E_0 = 0$ ): Wave packets with energy exactly at the barrier maximum propagated along the separatrix. In this marginal case, the wave packet exhibited significant spreading, causing its amplitude to decay below the noise level of the gauges before reaching the potential maximum. The authors observed a reduction in ambient noise near the barrier top, attributed to the rapid stretching of the surface field diluting both the packet and background fluctuations.

The experimental trajectories for $\langle \tau \rangle$ and $\langle p \rangle$ were found to be in quantitative agreement with analytical hyperbolic predictions derived from the Schrödinger-like equation. Two independent methods of extracting the IHO frequency $\Omega$ (from coordinate trajectories and momentum trajectories) yielded consistent mean values ( $\bar{\Omega} \approx 30.14 \pm 0.40$ and $\bar{\Omega} \approx 30.55 \pm 0.21$ rad/s, respectively), confirming the validity of the analogy.

Significance
The paper claims that this work provides a clear experimental observation of the separatrix in phase-space dynamics, a feature that separates transmitted and reflected states. By demonstrating the stopping power of the IHO and the specific behavior of wave packets at zero energy, the study validates the IHO as a paradigmatic model for quasi-infinite space phenomena.

The authors position this realization as a platform for future investigations into:

The quantum-reflection regime where the barrier length is comparable to the wave packet width.
The direct measurement of the logarithmic phase singularity at the separatrix, which defines the analog Hawking spectrum.
The implementation of time-dependent control to bring wave packets to rest.

The study emphasizes that this water-wave platform, along with potential extensions to optical, acoustic, and matter-wave systems, offers a cross-platform avenue for studying parabolic-barrier physics relevant to quantum scattering, gravitational analogs, and non-equilibrium dynamics.