Arcsine Laws of Light

We demonstrate that the time-integrated light intensity transmitted by a coherently driven resonator obeys Lévy’s arcsine laws—a cornerstone of extreme value statistics. We show that convergence to the arcsine distribution is algebraic, universal, and independent of nonequilibrium behavior due to nonconservative forces or nonadiabatic driving. We furthermore verify, numerically, that the arcsine laws hold in the presence of frequency noise and in Kerr-nonlinear resonators supporting non-Gaussian states. The arcsine laws imply a weak ergodicity breaking which can be leveraged to enhance the precision of resonant optical sensors with zero energy cost, as shown in our companion manuscript [V. G. Ramesh et al., companion paper, Phys. Rev. Res. (2024).]. Finally, we discuss perspectives for probing the possible breakdown of the arcsine laws in systems with memory.

Figure 1

(a) Inset: We use a laser-driven single-mode cavity to test the arcsine laws and study their implications for optical processes. The cavity length and hence the laser-cavity detuning $\mathrm{\Delta }$ are periodically modulated. Main panel: Single-shot and averaged intensity transmitted through the cavity shown as a black curve and a red curve, respectively, both as a function of $\mathrm{\Delta }$ referenced to the loss rate $\mathrm{\Gamma }$. (b) Experimental setup for measuring the cavity transmission while adding noise to the laser amplitude and phase using electro-optic modulators EOM-A and EOM-P, respectively. MO means microscope objective and PD means photodetector. (c)–(e) Color and white arrows illustrate the force magnitude $|\overrightarrow{F}/\mathrm{\Gamma }|$ and direction, respectively, exerted on the light field $\alpha ={\alpha }_R+i{\alpha }_I$. $A/\sqrt{\mathrm{\Gamma }}=7$ for all calculations, and (c) $\mathrm{\Delta }/\mathrm{\Gamma }=-2$, (d) $\mathrm{\Delta }/\mathrm{\Gamma }=0$, and (e) $\mathrm{\Delta }/\mathrm{\Gamma }=2$. The dynamics of $\alpha$ in (c) and (e) resemble Brownian motion in a stirred fluid.

Figure 2

Sample trajectory of the time-integrated transmitted intensity ${\mathcal{I}}_{n\tau }$, with the growing expectation value $\mathbb{E}[{\mathcal{I}}_{n\tau }]$ subtracted. The shaded area, dashed blue line, and solid red line, indicate the fractional times $T_{+}$, $T_0$, and $T_m$, respectively, to which the arcsine laws apply. The green horizontal line indicates the time average of this trajectory.

Figure 3

(a) Probability distribution and (b) cumulative distribution for $T_{j=+,0,m}$ extracted from trajectories of the time-integrated intensity as shown in Fig. 2. Black curves in (a) and (b) are analytical predictions of Eqs. (1a) and (1b), respectively.

Figure 4

Mean squared residuals between the arcsine distribution $\mathcal{C}(T_{+})$ [Eq. (1b)] and the finite time distribution ${\mathcal{C}}_{n\tau }(T_j)$ as the transmitted intensity is integrated over an increasing number of cycles. Purple stars correspond to experiments. All other data are calculated numerically, using ${10}^4$ simulations with distinct noise realizations per data point. Each symbol corresponds to a different protocol. Lines are power laws with exponent $-1$ fitted to the data. Blue squares, green stars, and orange diamonds correspond to periodic modulations of $A/\sqrt{\mathrm{\Gamma }}$ between 5 and 12 while $\mathrm{\Delta }/\mathrm{\Gamma }$ is fixed at 0, 3, and 10 respectively. Black circles correspond to a periodic modulation of $\mathrm{\Delta }/\mathrm{\Gamma }$ between $-10$ and 10 while $A/\sqrt{\mathrm{\Gamma }}=7$. $D=2\sqrt{\mathrm{\Gamma }}$ and $\tau =50/\mathrm{\Gamma }$ in all simulations; both choices do not alter the results.