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Journal of Modern Optics

Critical phenomena in the temporal behaviour of phase in second harmonic generation

S. T. Gevorkyan, W. H. Maloyan

Published Aug 1, 1997• 9 citations
Publisher

Abstract

We investigate numerically the probability distribution density dynamics of phase in second harmonic generation in the region of instability. It is shown that in this system the quantum macroscopic Schrödinger-cat-like states of both modes can appear. The behaviour of the system moving through the critical point is also illustrated.
DOI
10.1080/09500349708230749

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For the process of intracavity second-harmonic generation in the region of unstable dynamics, the joint fluctuations of numbers of photons and phases of the fundamental mode and second harmonic are studied in positive P representation. It is shown that above the bifurcation point of the optical system the distribution functions of the phases of the fundamental mode and second harmonic have a two-component structure. The quantum trajectories of the phases of these modes are not localized in the components of their states, which demonstrates the presence of quantum-mechanical interference between these components. In both components of each state the same values of numbers of photons are realized. It is shown that in the case of coherent initial states of both modes the joint distribution functions of the numbers of photons and phases are in the oscillatory regime. With time, the modes pass from a two-component delocalized state to a one-component one and then back to the two-component state. The time of oscillation of the distribution function is of the order of the photon's lifetime in the cavity volume. Self-matching of phases of interacting modes is investigated above the bifurcation point.
We investigate quantum-mechanical counterpart of a classical instability in a phase space by the numerical method of quantum trajectories with moving basis. As an application the model of coupled two oscillators driven by a monochromatic force in the presence of dissipation (intracavity second harmonic generation) is analyzed. The system of interest is characterized by two bifurcations leading to ranges of instability: the Hopf bifurcation which connects a steady state dynamics of the oscillatory modes to a self-pulsing temporal dynamics and the bifurcation of the period-doubling. The both two regimes are analyzed on the framework of the semiclassical phase trajectories and the Wigner functions of the oscillatory modes in phase space.
We investigate the quantum evolution of the pump field in second-harmonic generation under strong pump depletion. Starting from a coherent state, the pump develops a nonclassical phase-space structure resembling a Schrödinger cat state. This behavior originates from phase instability induced by vacuum fluctuations of the harmonic mode. A rigorous quantum analysis has been performed for mean photon numbers up to ⟨n^⟩=100\langle \hat n \rangle = 100⟨n^⟩=100 in pump mode. For larger photon numbers, up to ⟨n^⟩=107\langle \hat n \rangle = 10^{7}⟨n^⟩=107, the dynamics have been analyzed using a classical trajectory method with sampled initial conditions that reproduces the main features of the quantum evolution. The results indicate that nonlinear frequency conversion can generate macroscopic superposition-like states of the pump field. Although the resulting state is not pure due to correlations with the second-harmonic wave, it remains non-classical with negative zones of Wigner function. These results indicate that strongly nonlinear frequency conversion can provide a scalable route toward macroscopic nonclassical states of light.
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In this, the first of two papers on the subject, we present an analysis of a non-linear optical system with a mode coupled to its second harmonic, exhibiting a rich variety of non-equilibrium steady-state transitions. Here we use a semiclassical analysis to examine the steady states and their stability, and thus predict the existence of various instabilities. We find that this system exhibits both first- and second-order phase transition-like behaviour, as well as having a hard mode transition. The application of this system as a switching device is also considered.