Random Walks of Light in Optical Resonators

We study random walks of light confined in a laser-driven optical micro-cavity. In this thesis, we analyse fluctuations in time-integrated observables, such as the light transmitted through the micro-cavity. In Chapter 2, we show that random walks in the time-integrated light intensity transmitted by an optical resonator follow Levy's arcsine laws. The arcsine laws state that, for a finite time trajectory of a stochastic observable, the time above the mean, the last time of crossing the mean, and the time to reach maximum deviation from the mean, all have the same cumulative distribution function which is an arcsine function. We establish that the laws are robust under a broad range of conditions including incoherent drives, non-equilibrium steady-states, Kerr nonlinearity, non-Gaussian noise and non-adiabatic driving protocols. Chapter 3 analyses the consequences of the arcsine laws for optical sensing. We show that the time-integrated transmitted intensity exhibits weak ergodicity breaking (WEB): unequal time and ensemble averages in the finite time limit. We show that this phenomenon arises from a separation of timescales: the observation time remains finite, but the time required by the integrated intensity to explore its phase space is infinite. This is evidenced by power-law tails in the first passage and return time distributions, indicating divergent means. We further investigate how WEB affects resonant optical sensing. We compare the sensing precision of schemes that rely on time versus ensemble averaging while using equal amounts of energy. We show that ensemble averaging significantly outperforms time averaging at short observation times when fluctuations are strong, but the advantage disappears as the observation time increases and ergodicity is recovered. In Chapter 4, we present a theoretical framework of stochastic thermodynamics for a linear optical cavity driven on resonance. We present expressions for thermodynamic quantities such as work and heat in terms of optical observables. We show that the steady-state intra-cavity field follows the equilibrium Boltzmann distribution, with the effective temperature given by the noise variance and the equilibration rate given by the dissipation rate. We formulate the first and second laws of thermodynamics for the optical cavity. We show that fluctuations in the work and heat obey fluctuation theorems. In particular, the work done on the optical cavity by a time-dependent laser power follows Crooks’ fluctuation theorem, relating non-equilibrium work distributions to equilibrium free energy differences. Chapter 5 demonstrates the breakdown of the arcsine laws due in an optical resonator filled with olive-oil. The oil's slow thermo-optical nonlinearity effectively imparts memory to random walks observed in the time-integrated transmitted intensity. This, in turn, results in significant deviation of the same three observables studied in Chapter 2 from the arcsine distribution. We characterize this behaviour by analysing the variance of the time-integrated intensity over time. The variance grows at different rates (fitted with power laws of varying exponents) in different time windows, which signifies a combination of anomalous diffusion and statistical aging. Through numerical simulations we show that the memory time of our system plays a crucial role in our observations.