Universality of photon counting below a local bifurcation threshold

At a bifurcation point, a small change of a parameter causes a qualitative change in the system. Quantum fluctuations wash out this abrupt transition and enable the emission of quantized energy, which we term photons, below the classical bifurcation threshold. Close to the bifurcation point, the resulting photon counting statistics is determined by the instability. We propose a generic method to derive a characteristic function of photon counting close to a bifurcation threshold that only depends on the dynamics and the type of bifurcation, based on the universality of the Martin-Siggia-Rose action. We provide explicit expressions for the cusp catastrophe without conservation laws. Moreover, we propose an experimental setup using driven Josephson junctions that exhibits both a fold and a pitchfork bifurcation behavior close to a cusp catastrophe.

Figure 1

(a) Cusp shape in the canonical variables $(a,b)$. The fold bifurcation lines at $b=\pm {(a/3)}^{3/2}$ separate the region outside the cusp, where a single stable solution exists, from the region inside the cusp (gray) where two stable solutions exist. (b) Evolution of the fixed points as the systems passes to $a>0$ through the cusp point (0,0) along the vertical line indicated in (a). At the pitchfork bifurcation, the single stable fixed point splits into two stable (solid lines) and one unstable (dotted line) solution. (c) Evolution of the fixed points as the systems passes through the cusp along the horizontal line indicated in (a). At the first fold bifurcation, a second stable solution (solid line) and an unstable point (dotted line) become available, while the stability of the upper point remains unchanged. The classical switch between both solutions takes place at the second fold bifurcation, where the upper fixed point and the unstable fixed point merge, forcing the system onto the lower, stable fixed point.

Figure 2

Probability of large deviations of the photon current $I$ from the average current $\overline{I}$ in the vicinity of the pitchfork bifurcation (blue, solid line) and the fold bifurcation (orange, dashed line). The scaling is chosen such that both probabilities have identical Gaussian expansions around $I=\overline{I}$ as indicated by the thin, dotted line. Note that both probabilities are strongly asymmetric, leaning towards large current deviations.