Quantum Systems Lose Synchronisation Via Newly Observed ‘phase Slips’

A thorough investigation into the breakdown of quantum synchronisation in self-sustained oscillators reveals a phenomenon analogous to phase locking in classical systems. Hans Christiansen and Jens Paaske at Niels Bohr Institute, in collaboration with University of Copenhagen, show how quantum phase slips degrade phase locking within continuous variable quantum systems, utilising a Keldysh path integral formulation of limit cycles. The work elucidates the impact of quantum noise on synchronisation and addresses non-Markovian effects, specifically within superconducting resonators coupled via a voltage biased double quantum dot, offering key insights into the behaviour of complex quantum systems.

Quantum phase slips define a synchronisation threshold in multi-photon oscillators

The phenomenon of synchronisation, where coupled oscillators adjust their rhythms to operate in harmony, is ubiquitous in physics and engineering. Classical oscillators, such as those found in mechanical clocks or electronic circuits, achieve synchronisation through the mutual exchange of energy and information. However, this synchronisation is fragile and susceptible to disruption by external noise, leading to phase slips, deviations from the coordinated rhythm. This research demonstrates that quantum oscillators, despite operating under fundamentally different principles, also exhibit a breakdown of synchronisation, but through a distinct mechanism: quantum phase slips. These slips arise from the inherent quantum fluctuations in the system’s phase, even in the absence of classical noise. For a five-photon limit cycle, representing a sustained oscillation involving five photons, the ratio of quantum phase slip diffusion to noise initially stands near unity, indicating minimal improvement in phase locking from coupling. This suggests that the initial coupling between oscillators does not significantly enhance their ability to maintain a stable, synchronised phase. However, this ratio diminishes to below 0.5 as the detuning parameter, ∆/D, increases beyond one. This threshold signifies the breakdown of synchronisation, mirroring the divergence observed in deterministic models of coupled oscillators, and was previously unattainable without accounting for quantum fluctuations. The detuning parameter, ∆/D, represents the difference in frequency between the oscillators normalised by the coupling strength, D; a larger value indicates a greater mismatch and weaker interaction.

Earlier methods struggled to accurately predict synchronization loss because they failed to incorporate the impact of these quantum phase slips on the limit cycle dynamics; even strong phase correlations cannot prevent the degradation of phase locking caused by these slips. Limit cycles describe self-sustained oscillations, and their dynamics are crucial for understanding synchronization. Traditional approaches often assume Markovian behaviour, meaning the system’s future state depends only on its present state, neglecting the influence of past states. This approximation breaks down in the presence of non-Markovian effects, which are particularly prominent in quantum systems. When the detuning parameter, ∆/D, exceeds one, the ratio of quantum phase slip diffusion to noise diminishes to below 0.5, mirroring the loss of synchronization seen in standard oscillator models. Simulations using superconducting resonators coupled via a voltage-biased double quantum dot revealed that the diffusion constant, σ2 −, decreases as the detuning increases, indicating a breakdown in stable phase relationships. Superconducting resonators, tiny circuits exhibiting quantum behaviour, are ideal platforms for studying these effects due to their sensitivity to quantum fluctuations. The voltage-biased double quantum dot acts as a tunable coupling element between the resonators. Furthermore, the phase diffusion rate vanishes when the coupling strength, D, approaches the relaxation rate, γ1, for all limit cycle radii. The relaxation rate, γ1, describes how quickly the system returns to equilibrium after a disturbance; when the coupling strength becomes comparable to the relaxation rate, the oscillators effectively lose their ability to exchange information and maintain synchronization.

Quantum disruptions to synchronisation and limitations of current modelling techniques

Technologies ranging from lasers to secure communication networks underpin synchronisation between oscillating systems; this research clarifies how quantum affects disrupt this delicate balance. The ability to precisely control and maintain synchronisation is essential for numerous applications, including high-precision measurements, signal processing, and quantum information processing. Understanding the mechanisms that limit synchronisation, particularly in quantum systems, is therefore crucial for advancing these technologies. The analysis relies on approximations within the Keldysh path integral, a complex mathematical technique for modelling quantum behaviour, and acknowledges that the influence of these approximations remains an open question. The Keldysh formalism allows for the treatment of non-equilibrium systems and the inclusion of quantum fluctuations, but it often requires simplifying approximations to make the calculations tractable. Assessing the validity and impact of these approximations is an ongoing area of research. Extending these findings to other continuous variable quantum systems presents a significant challenge, while the specific properties of the superconducting resonators used may not translate universally. Continuous variable quantum systems utilise properties like amplitude and phase, as opposed to discrete quantum bits, and their behaviour can vary significantly depending on the specific physical implementation.

Despite requiring further scrutiny, the approximations within the Keldysh path integral do not diminish the value of this work. Demonstrating quantum phase slips, sudden and random changes in the wave-like properties of superconducting resonators, establishes an important mechanism disrupting synchronisation. These phase slips are fundamentally different from classical phase slips, arising from the inherent uncertainty in the quantum phase. Understanding this breakdown is vital as it refines designs for sensitive devices like atomic clocks and improves the reliability of quantum communication protocols which depend on maintaining stable, coordinated oscillations. Atomic clocks, the most precise timekeeping devices, rely on the synchronisation of atomic oscillations, and any disruption to this synchronisation can lead to inaccuracies. Similarly, quantum communication protocols, such as quantum key distribution, require the precise coordination of quantum states between distant parties, making them vulnerable to phase slips.

Quantum phase slips, comparable to random jumps in a system’s wave properties, fundamentally limit how well continuous variable quantum systems can synchronise. Classical oscillators lose synchronisation due to similar, but distinct, phase slips caused by noise, whereas quantum systems exhibit this degradation even with strong correlations between their oscillating parts. This highlights the unique challenges posed by quantum synchronisation and the need for new theoretical frameworks to address these challenges. A Keldysh path integral, a complex mathematical method assessing all possible evolutionary routes, was used to model these limit cycles and reveal the impact of non-Markovian effects; these effects mean a system’s past influences its future behaviour. The Keldysh path integral effectively integrates over all possible trajectories of the system, allowing for the calculation of quantum observables and the identification of non-Markovian dynamics. This approach provides a powerful tool for understanding the complex behaviour of quantum oscillators and their susceptibility to phase slips.

The research demonstrated that quantum phase slips disrupt the synchronisation of continuous variable quantum systems, even when strong correlations exist between oscillating components. This breakdown differs from classical systems, where noise causes phase slips, and highlights unique challenges for achieving stable quantum oscillations. Understanding these slips is important because maintaining synchronisation is crucial for the reliable operation of sensitive devices such as atomic clocks and quantum communication protocols. The authors utilised a Keldysh path integral to model these limit cycles and analyse non-Markovian effects influencing synchronisation.