🔬 atomic physics

Interaction-induced period shift of the Kohn oscillation in a parity-time-symmetric harmonic trap

This paper demonstrates that while interactions do not affect the Kohn oscillation frequency in a standard harmonic trap, they break the protection in a parity-time-symmetric gain-loss trap by coupling the modulated atom number to the cloud's width, thereby inducing a quadratic period shift with gain strength that persists until resonance with highly excited states occurs.

Original authors: Gaoqing Meng, Mingshu Zhao

Published 2026-10-05

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CC BY 4.0

Original authors: Gaoqing Meng, Mingshu Zhao

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

In the quiet world of ultra-cold atoms, physicists create clouds of matter so cold that the individual particles lose their separate identities and merge into a single, giant quantum wave. This state of matter, known as a Bose-Einstein condensate, behaves like a super-fluid that can be trapped and manipulated with magnetic or optical fields. When such a cloud is placed in a perfectly smooth, bowl-shaped trap, it has a remarkable property: if you nudge the entire cloud to one side and let it go, it swings back and forth like a pendulum. The speed of this swing depends only on the shape of the trap itself, not on how the atoms inside push or pull on one another. This stubborn independence, known as the Kohn oscillation, has long been a reliable standard for calibrating the traps used in experiments. However, the rules of this game change when the trap is no longer perfectly balanced. Imagine a trap where atoms are constantly being added to one side and removed from the other, a setup that mimics a system with a steady flow of energy in and out. For decades, theory suggested that even in this unbalanced, "gain-and-loss" environment, the atoms would still swing at the same steady pace, protected by a hidden symmetry. But a new study challenges this certainty, revealing that when the atoms interact with each other, this protection vanishes, and the rhythm of the swing begins to change.

The researchers, working with a theoretical model of these interacting atoms, set out to see what happens when the trap is tilted with this one-sided addition and removal of particles. They simulated a cloud of atoms in a harmonic trap, a standard bowl-shaped potential, but added a linear gradient where atoms are fed in on the right and drained on the left. In a world without interactions between atoms, the cloud would simply slide back and forth, its shape remaining rigid while the total number of atoms inside it swelled and shrank in a perfect rhythm. The researchers found that when they turned on the interactions—the forces the atoms exert on each other—this simple picture broke down. The changing number of atoms caused the cloud to expand and contract, a breathing motion that then pushed back against the center of the cloud. This feedback loop, driven by the constant gain and loss, subtly altered the timing of the swing. The period of the oscillation, the time it takes to complete one full back-and-forth cycle, began to grow longer as the strength of the gain and loss increased.

The team discovered that for small amounts of gain, this lengthening of the period follows a precise rule: it grows with the square of the gain strength. The amount of this shift is determined almost entirely by the size of the cloud itself. A larger cloud, which implies stronger interactions, experiences a more significant delay in its rhythm. The researchers built a simplified mathematical model using just five variables to track the cloud's position, speed, width, and internal correlations. This model successfully reproduced the behavior of the full, complex simulation, accurately predicting the period and the changing width of the cloud up to a certain point. It confirmed that the shift is not a random fluctuation but a direct consequence of the coupling between the atom number, the cloud's size, and its center of motion.

However, this rhythmic behavior does not last forever. As the researchers increased the gain strength further, they found a critical limit. The family of stable, repeating orbits that the cloud follows eventually hits a wall. At a specific threshold of gain, the oscillating cloud begins to resonate with the high-energy states of the trap, effectively coupling with excited states that are far beyond the main cloud. This resonance causes the family of orbits to fold back on itself, and the stable periodic motion can no longer be sustained. The researchers found that this critical point depends on both the size of the cloud and the amplitude of the swing; larger swings and larger clouds reach this limit at lower gain strengths. While the cloud becomes dynamically unstable just before this point, the instability does not destroy the oscillation entirely; the center of mass continues to swing, though the motion becomes slightly deformed.

The study also looked at how this phenomenon might be observed in real experiments, suggesting two potential platforms. One involves light traveling through a special glass waveguide where the refractive index creates a parabolic trap, and a linear gain-loss profile is imprinted using optical techniques. The other involves actual atoms in a magnetic trap, where a laser beam could be used to remove atoms from one side and a reservoir could feed them in from the other. In the atomic case, the researchers calculated that the required rate of atom loss at the turning points of the swing would be on the order of 100 per second. They noted that while the effect is theoretically robust, observing it requires extreme precision, as the shift in the period is tiny—on the order of one part in a thousand for the parameters they tested.

Ultimately, the work demonstrates that the "protection" of the Kohn oscillation is fragile. It holds only as long as the system remains non-interacting or perfectly balanced. Once interactions are introduced and the atom number is allowed to fluctuate by the environment, the atoms talk to each other in a way that changes the clock of their collective motion. The findings provide a clear, quantitative map of how this interaction-induced shift behaves, from the gentle quadratic growth at low gain to the sudden breakdown at the critical limit. By showing how a simple change in the environment can ripple through the internal structure of a quantum cloud to alter its fundamental rhythm, the study deepens the understanding of how open quantum systems behave when pushed beyond their ideal limits.

Technical Summary: Interaction-induced period shift of the Kohn oscillation in a parity-time-symmetric harmonic trap

Problem Statement
In a standard harmonic trap, the center of mass (COM) of a Bose–Einstein condensate (BEC) oscillates at the trap frequency, a phenomenon known as the Kohn oscillation. This frequency is protected against interactions due to the harmonic potential theorem, provided the interaction depends only on relative coordinates. Previous work established that introducing a parity-time (PT) symmetric gain-loss gradient (an imaginary potential linear in position) to a non-interacting system preserves this frequency; the cloud oscillates rigidly while its atom number is periodically modulated.

However, the presence of interactions breaks this protection. The gain and loss mechanisms modulate the total atom number ( $N$ ), which alters the interaction energy. This modulation drives changes in the cloud's width ( $S$ ), which in turn feeds back onto the COM dynamics via the time derivative of the width ( $\dot{S}$ ) and the position-momentum covariance ( $C$ ). The central questions addressed are: Does this interaction-induced feedback shift the oscillation period? By how much? And up to what gain strength do these periodic orbits persist?

Methodology
The authors investigate this problem using the one-dimensional Gross–Pitaevskii (GP) equation with an imaginary potential term $i\gamma x$ :
$i\partial_t\psi = \left[ -\frac{1}{2}\partial_x^2 + \frac{1}{2}x^2 + i\gamma x + g|\psi|^2 \right]\psi$
where $\gamma$ represents the gain-loss strength and $g$ the interaction strength.

Numerical Continuation: Periodic orbits are computed by numerically continuing the known Kohn orbits (at $\gamma=0$ ) to finite $\gamma$ . The method utilizes the PT-symmetry of the system to define symmetric orbits that cross the trap center at specific phases. Floquet multipliers are calculated to assess linear stability.
Moment Model: To gain analytical insight, a five-variable moment model is derived from the GP equation. This model tracks the atom number ( $N$ ), COM position ( $X$ ), momentum ( $P$ ), squared width ( $S$ ), and covariance ( $C$ ). The model assumes a fixed density profile shape (scaling ansatz) to close the hierarchy of moment equations, incorporating quantum pressure and interaction terms.
Perturbative Analysis: Analytical expansions are performed for small $\gamma$ to derive the leading-order shift in the period, utilizing the Non-Hermitian Ehrenfest theorem and the coupling between the dipole mode and the breathing mode.

Key Results

Period Shift Mechanism: The period $T$ increases quadratically with the gain strength $\gamma$ for small $\gamma$ . The shift arises because the gain modulates $N$ , which drives the width $S$ . Since $S$ responds to $N$ with a phase lag (due to the breathing mode resonance), the term $2\gamma S$ in the COM equation of motion acts against the restoring force.
Analytical Prediction: In the Thomas–Fermi (TF) regime, the relative period shift is given by:
$\frac{T}{2\pi} - 1 \approx 0.79 g^{2/3} \gamma^2$
This coefficient depends solely on the cloud size (via the ground-state width $\sigma_0$ ) and is independent of the oscillation amplitude.
Model Accuracy: The five-variable moment model reproduces the period and width of the cloud with high accuracy (within a few percent) up to the critical gain, provided the cloud profile remains approximately symmetric. The model captures the quadratic shift exactly in the small- $\gamma$ limit.
Critical Gain and Stability: Each family of periodic orbits terminates at a critical gain $\gamma_c$ $γ_{c}$ . At this point:
- The orbit resonates with highly excited states of the trap (e.g., levels $n \approx 475$ and $559$ for the parameters used).
- The family of solutions "turns back" (a fold bifurcation), meaning two orbits with the same parameters exist just below $\gamma_c$ .
- Dynamical instability (Floquet multipliers leaving the unit circle) often occurs shortly before or at $\gamma_c$ , though the instability deforms the cloud without immediately destroying the oscillation.
Scaling Laws: The critical gain scales as $\gamma_c \propto (A \cdot R_{TF})^{-1/2}$ , where $A$ is the oscillation amplitude and $R_{TF}$ is the Thomas–Fermi radius. Specifically, $\gamma_c \sqrt{A R_{TF}} \approx 0.280$ across the computed families.

Significance and Claims
The paper claims to demonstrate that the "protection" of the Kohn frequency in PT-symmetric traps is fragile in the presence of interactions. The interaction-induced coupling between the atom number modulation and the cloud width breaks the rigid oscillation, leading to a measurable period shift.

The authors emphasize that:

The shift is a second-order effect in $\gamma$ driven by the interplay of gain, loss, and interactions.
The five-variable moment model provides a robust, parameter-free description of the dynamics up to the critical point, validating the physical mechanism of the period shift.
The existence of a critical gain where orbits resonate with high-energy trap states limits the observable range of these periodic solutions.
The results are relevant for experimental realization in optical waveguides (fluid of light) and atomic condensates with engineered loss, provided the gain is coherent and the trap remains harmonic over the large excursions of the cloud.

The work does not claim to solve the full nonlinear dynamics beyond the critical gain or to predict chaotic behavior, focusing instead on the properties of the periodic orbits emerging from the Kohn limit.

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