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arXiv:1509.02118 · Physical Review A

Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators

Wim Casteels, Florent Storme, Alexandre Le Boité, Cristiano Ciuti

Published Mar 10, 2016• 93 citations

Abstract

We explore theoretically the physics of dynamic hysteresis for driven-dissipative nonlinear photonic resonators. In the regime where the semiclassical mean-field theory predicts bistability, the exact steady-state density matrix is known to be unique, being a statistical mixture of two states; in particular, no static hysteresis cycle of the excited population occurs as a function of the driving intensity. Here, we predict that in the quantum regime a dynamic hysteresis with a rich phenomenology does appear when sweeping the driving amplitude in a finite time. The hysteresis area as a function of the sweep time reveals a double power-law decay, with a behavior qualitatively different from the mean-field predictions. The dynamic hysteresis power-law in the slow sweep limit defines a characteristic time, which depends dramatically on the size of the nonlinearity and on the frequency detuning between the driving and the resonator. In the strong nonlinearity regime, the characteristic time oscillates as a function of the intrinsic system parameters due to multiphotonic resonances. We show that the dynamic hysteresis for the considered class of driven-dissipative systems is due to a nonadiabatic response region with connections to the Kibble-Zurek mechanism for quenched phase transitions. We also consider the case of two coupled driven-dissipative nonlinear resonators, showing that dynamic hysteresis and power-law behavior occur also in the presence of correlations between resonators. Our theoretical predictions can be explored in a broad variety of physical systems, e.g., circuit QED superconducting resonators and semiconductor optical microcavities.

DOI: 10.1103/physreva.93.033824

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We experimentally explore the dynamic optical hysteresis of a semiconductor microcavity as a function of the sweep time. The hysteresis area exhibits a double power law decay due to the shot noise of the driving laser, which triggers switching between metastable states. Upon increasing the average photon number and approaching the thermodynamic limit, the double power law evolves into a single power law. This algebraic behavior characterizes a dissipative phase transition. Our findings are in good agreement with theoretical predictions, and the present experimental approach is promising for the exploration of critical phenomena in photonic lattices.

We experimentally explore the dynamical optical hysteresis of a semiconductor microcavity as a function of the sweep time. The hysteresis area exhibits a double power law decay due to the influence of fluctuations, which trigger switching between metastable states. Upon increasing the average photon number and approaching the thermodynamic limit, the double power law evolves into a single power law. This algebraic behavior characterizes a dissipative phase transition. Our findings are in good agreement with theoretical predictions for a single mode resonator influenced by quantum fluctuations, and the present experimental approach is promising for exploring critical phenomena in photonic lattices.

We introduce a two-level atom into a cavity as an ancilla to control the hysteresis of the system where the cavity is described by a driven-dissipative nonlinear Kerr model. We find that the dynamic optical hysteresis can be modulated by changing the atom-cavity coupling strength, and the nonlinearity induced by atom-cavity coupling weakens the Kerr nonlinearity in the weak coupling regime. This leads to the change in the critical point corresponding to the decrease in the hysteresis area. The physics behind the modulation is closely related to the effect of the atom-cavity coupling on the discontinuity of the first nonzero Liouvillian eigenvalue. A relative deviation of the metastable state and target steady state is defined to characterize the relation between the hysteresis and adiabaticity of the system. Taking only two Liouvillian eigenvectors into account in the composite system, we derive an expression for the dependence of the hysteresis area on the sweeping speed and discuss its feature in the slow sweeping limit.

Dissipative phase transitions are characteristic features in open quantum systems. Key signatures are the dynamical switching between different states in the vicinity of the phase transition and the appearance of hysteresis. Here, we experimentally study dynamic sweeps across a first order dissipative phase transition in a multi-mode driven-dissipative system. In contrast to previous studies, we perform sweeps of the dissipation strength instead of the driving strength. We extract exponents for the scaling of the hysteresis area in dependence of the sweep time and study the g^{(2)}(0) g ( 2 ) ( 0 ) correlations, which show non-trivial behavior. By changing the temperature of the system we investigate the importance of coherently pumping the system. We compare our results to numerical calculations done for a single mode variant of the system, and find surprisingly good agreement. Furthermore, we identify and discuss the differences between a scan of the dissipation strength and a scan of the driving strength.

Driven-dissipative systems have recently attracted great attention due to the existence of novel physical phenomena with no analog in the equilibrium case. The Keldysh path-integral theory is a powerful tool to investigate these systems. However, it has still been a challenge to study strong nonlinear effects implemented by recent experiments, since in this case the photon number is few and quantum fluctuations play a crucial role in the dynamics of the system. Here we develop an approach for deriving exact steady states of driven-dissipative systems by introducing the Keldysh partition function in the Fock-state basis and then mapping the standard saddle-point equations into Keldysh-Heisenberg equations. We take the strong Kerr nonlinear resonators with and without the nonlinear driving as two examples to illustrate our method. It is found that, in the absence of the nonlinear driving, the exact steady state obtained does not exhibit bistability and agrees well with the complex

P

-representation solution. While in the presence of the nonlinear driving, the multiphoton resonance effects are revealed and are consistent with the qualitative analysis. Our method provides an intuitive way to explore a variety of driven-dissipative systems especially with strong correlations.

The response of nonlinear resonators to multifrequency driving reveals rich dynamics beyond conventional single-tone theory. We study a Duffing resonator under bichromatic excitation and identify a competition between the two drives, governed by their detuning and relative amplitudes. In the slow-beating regime, where the tones are closely spaced, the secondary drive acts as a modulation that induces dynamical phase transitions between coexisting stationary states. We introduce the cycle-averaged amplitude as an order parameter and map the resulting phase diagram as a function of the drive detuning and amplitude ratio, capturing the pronounced asymmetry observed for blue versus red detuning in experiment. We devise a model to link the onset of these transitions to the resonance properties around the nonlinear stationary mode of the system. Our results provide a framework for controlling driven nonlinear systems, enabling state manipulation, and sensing in nanomechanical, optical, and superconducting circuit platforms.