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Measures of quantum synchronization in continuous variable systems
A. Mari, A. Farace, N. Didier, V. Giovannetti, R. Fazio
TL;DR
The paper asks how complete and phase synchronization can be quantified for interacting continuous-variable quantum systems, where classical synchronization conditions encounter quantum constraints. It introduces measures extending both notions to the quantum domain, establishes bounds and resource dependence, examines entanglement connections, and applies them to opto-mechanical systems. The results show a universal quantum limit for complete synchronization, while phase synchronization can be unbounded in principle but is threshold-limited without resources such as squeezing.
Problem
Classical synchronization notions require extension to interacting continuous-variable quantum systems, where the uncertainty principle prevents exact complete synchronization.
Method
The paper introduces measures for complete and phase synchronization, analyzes their quantum bounds and relation to entanglement, and applies them to coupled opto-mechanical resonators.
Results
Quantum mechanics imposes a universal limit on complete synchronization, whereas phase synchronization is unbounded in principle but cannot be perfect for positive-P-function states and can surpass its weaker threshold with squeezing.
Takeaways & Limitations
High synchronization according to these measures does not require entanglement, and opto-mechanical arrays provide a setting for studying spontaneous synchronization near the quantum regime.
Takeaways & Limitations
The heuristic phase-synchronization bound is explicitly described as hand-waving, and the paper leaves the interplay with quantum correlations and extensions to other systems open.
Abstract
from arXiv · showhide
We introduce and characterize two different measures which quantify the level of synchronization of interacting continuous variable quantum systems. The two measures allow to extend to the quantum domain the notions of complete and phase synchronization. The Heisenberg principle sets a universal bound to complete synchronization. The measure of phase synchronization is in principle unbounded, however in the absence of quantum resources (e.g. squeezing) the synchronization level is bounded below a certain threshold. We elucidate some interesting connections between entanglement and synchronization and, finally, discuss an application based on quantum opto-mechanical systems.
SUPPLEMENTAL MATERIAL: HEURISTIC BOUND TO PHASE SYNCHRONIZATION
The heuristic bound connects phase-synchronization precision to amplitude synchronization under approximately circular limit cycles, phase-insensitive noise, and a beam-splitter-like interaction. The argument attributes synchronization to stable anti-symmetric-mode dynamics but is explicitly described as hand-waving.
- Assumptions: The conjectured bound assumes approximately circular limit cycles, phase-insensitive thermal or quantum noise, and an interaction Hint = −µ(a1a†2 + a2a†1).These assumptions are stated as often valid for optical or mechanical modes under the rotating wave approximation.
- Normal-mode dynamics: The interaction decomposes into symmetric and anti-symmetric normal modes, making the anti-symmetric mode rotate 2µ faster than the symmetric mode.The frequency difference follows after expressing the interaction in normal-mode form, up to a renormalization of bare frequencies.
- Normal-mode dynamics: The symmetric mode is stable in amplitude but retains a freely diffusing phase, corresponding to a zero Lyapunov exponent.This separates amplitude stabilization from phase freedom in the synchronized state.
- Normal-mode dynamics: Synchronization originates from negative eigenvalues of the new anti-symmetric-mode dynamical matrix when µ ≠ 0.With phase-insensitive noise, the diffusion matrix is proportional to the identity, enabling a steady-state analysis.
- Bound: The heuristic bound states that phase-synchronization precision may be limited by amplitude-synchronization precision in classical or quantum systems.The argument derives the bound from the steady-state anti-symmetric-mode variances.