Abstract
We theoretically explore the role of mesoscopic fluctuations and noise on the spectral and temporal properties of systems of $\mathcal{PT}$-symmetric coupled gain-loss resonators operating near the exceptional point, where eigenvalues and eigenvectors coalesce. We show that the inevitable detuning in the frequencies of the uncoupled resonators leads to an unavoidable modification of the conditions for reaching the exceptional point, while, as this point is approached in ensembles of resonator pairs, statistical averaging significantly smears the spectral features. We also discuss how these fluctuations affect the sensitivity of sensors based on coupled $\mathcal{PT}$-symmetric resonators. Finally, we show that temporal fluctuations in the detuning and gain of these sensors lead to a quadratic growth of the optical power in time, thus implying that maintaining operation at the exceptional point over a long period can be rather challenging. Our theoretical analysis clarifies issues central to the realization of $\mathcal{PT}$-symmetric devices, and should facilitate future experimental work in the field.
Highlights
Pronounced sample-to-sample fluctuations constitute a hallmark of mesoscopic physics [1], where the finite number of degrees of freedom limits the self-averaging common to macroscopic systems
We have discussed the classical electrodynamics at exceptional point (EP) of PT -symmetric systems, where the spectrum can be real despite the presence of both loss and gain
We have emphasized mesoscopic fluctuations of classical origin, while we speculate that quantum optics and quantum fluctuations would experience dramatic enhancement near the EP
Summary
Pronounced sample-to-sample fluctuations constitute a hallmark of mesoscopic physics [1], where the finite number of degrees of freedom limits the self-averaging common to macroscopic systems. Schematic illustration of a PT -symmetric dimer formed by two identical (no frequency detuning) coupled optical resonators, but with opposite values of the gain–loss parameter G ≡g∕2κ. Perhaps the most notable characteristic of a PT -symmetric system is a PT -symmetry breaking transition that takes place around g∕2κ 1 In optical settings, this abrupt phase transition has been experimentally demonstrated in coupled waveguides and cavities, by measuring both the real and imaginary components of the eigenvalues, as well as by observing the evolution of the corresponding mode profile [23,24,25,26,27,28]. In order to analyze the influence of temporally fluctuating environments or sample-to-sample fluctuations associated with inevitable small variations in ωa and ωb, we shall in the following allow a small, but finite frequency detuning between the two coupled resonators. While we here focus on fluctuations in Δ, it is clear from Eq (2a) that fluctuations in G would have quite similar implications
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