Interference Patterns Control Chaotic Movement Within Quantum Oscillators

Umair Abdul Halim at UPM Serdang and colleagues demonstrate that the extent of chaos directly correlates with the temporal coherence of interfering oscillator modes. Their analysis reveals that near resonance, sustained interference creates expansive chaotic regions, while sharp frequency detuning restricts chaotic behaviour to smaller areas. The research provides a coherence parameter as a key set of tools for understanding chaotic transport in similar low-dimensional Bohmian systems, potentially advancing the design and control of quantum systems.

Coherence parameter defines transition to extended chaotic motion through sustained interference

The dimensionless coherence parameter, χ, now accurately predicts the extent of chaotic motion, improving upon prior methods reliant on incommensurate frequency ratios that masked results with dephasing. Traditionally, the identification of chaotic behaviour in quantum systems has been challenging due to the inherent probabilistic nature of quantum mechanics and the difficulty in defining classical trajectories. Bohmian mechanics offers a deterministic interpretation, describing particle motion via trajectories guided by the wavefunction. However, even within this framework, quantifying chaos in systems like the anisotropic harmonic oscillator has proven difficult. Previous approaches often focused on analysing the frequency ratios between the excited states, assuming that incommensurate frequencies would naturally lead to chaotic trajectories. These methods, however, failed to adequately account for the crucial role of quantum interference and the resulting phase structure of the wavefunction, often obscuring the true extent of chaotic motion through dephasing effects. The coherence parameter, χ, circumvents this issue by directly quantifying the temporal coherence of the interfering modes, providing a more accurate measure of the underlying chaotic dynamics. This parameter is fundamentally linked to the lifetime of the interference pattern, reflecting how long the superposition of states maintains its oscillatory behaviour.

Sustained interference, generating long-lived phase structures, occurred when trajectories were repeatedly stretched and folded, most noticeably with slower beating frequencies between oscillator modes. The phase of the wavefunction, which dictates the Bohmian velocity field, exhibits complex structures, including regions of constructive and destructive interference. When the frequency detuning between the modes is small, indicating sustained interference, these phase structures become more intricate and spatially extended. This leads to a greater degree of trajectory stretching and folding, characteristic of chaotic dynamics. The slower beating frequencies allow for more complete exploration of phase space, resulting in trajectories that wander throughout the system. Conversely, rapid detuning disrupts the interference pattern, leading to a loss of synchronisation and a breakdown in coherent phase evolution. This results in chaotic dynamics confined to smaller areas of the system, as trajectories are unable to fully explore the available phase space. The analysis of Lyapunov exponents, a measure of trajectory divergence, confirmed that higher values of χ corresponded to more spatially extended chaotic regions, while lower values maintained localisation. Specifically, a positive Lyapunov exponent indicates chaotic behaviour, with the magnitude of the exponent reflecting the rate of divergence of nearby trajectories. The team observed a clear correlation between χ and the spatial distribution of these positive Lyapunov exponents, demonstrating that higher coherence values lead to a broader distribution and, therefore, more extensive chaos.

Currently, these calculations assume idealised conditions and do not yet account for external disturbances or the complexities of many-body systems, limiting their immediate application to real-world quantum simulations. The model employed is a simplified representation of a quantum system, focusing on a two-dimensional anisotropic harmonic oscillator with only three energy states: the ground state and the first two excited states. This simplification allows for a clear and tractable analysis of the underlying physics, but it neglects several factors that could influence the behaviour of real-world quantum systems. External disturbances, such as electromagnetic fields or collisions with other particles, can introduce decoherence and disrupt the interference pattern, reducing the coherence parameter and suppressing chaotic motion. Furthermore, the model does not account for interactions between multiple particles, which are prevalent in many physical systems. Incorporating these complexities would require significantly more computational resources and analytical effort. For a long time, scientists have sought reliable indicators of chaos within quantum systems, vital for understanding phenomena ranging from molecular interactions to material behaviour. This work builds upon this foundation by offering a new diagnostic, linking the persistence of quantum interference to the extent of chaotic motion, a major advancement over previous reliance on potentially misleading frequency ratios. Understanding the interplay between coherence and chaos is crucial for controlling and manipulating quantum systems, with potential applications in areas such as quantum computing and quantum materials.

The team demonstrated this coherence parameter within a specific, simplified model, a two-dimensional harmonic oscillator with just three energy states, and are now investigating its limitations. The chosen system, while simple, exhibits the key features necessary for observing Bohmian chaos, namely a non-integrable Hamiltonian and the presence of interference effects. The anisotropy of the harmonic oscillator, meaning that the potential energy is different in the x and y directions, is crucial for generating the necessary non-integrability. Further work will focus on extending the model to incorporate more realistic conditions, such as external disturbances and many-body interactions, to broaden its applicability. This includes exploring the effects of dissipation and noise on the coherence parameter and investigating how it behaves in systems with a larger number of interacting particles. A refined understanding of low-dimensional quantum systems is offered by establishing a clear link between the persistence of quantum interference and the scale of chaotic movement. The spatial extent of chaos within a harmonic oscillator is dictated by temporal coherence, the duration for which quantum states interfere with each other, rather than energy levels. Quantifying this relationship with a dimensionless coherence parameter, physicists have created a new set of tools for analysing chaotic behaviour, comparing the rate of interference ‘beating’ with particle transport, and opens questions regarding its applicability to more complex quantum scenarios. The implications extend to understanding transport phenomena in various quantum systems, potentially influencing the design of more efficient quantum devices and materials.

The research demonstrated that the extent of chaotic movement in a two-dimensional harmonic oscillator is governed by the temporal coherence of quantum interference. This means the duration of interference between quantum states, rather than energy levels, dictates how widely chaotic motion spreads. By introducing a coherence parameter that compares the rate of interference with particle transport, researchers have provided a new method for analysing chaotic behaviour in low-dimensional quantum systems. The authors are currently investigating the limitations of this parameter and exploring its behaviour in more complex scenarios with multiple interacting particles.