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Optica Acta International Journal of Optics

Non-equilibrium Transitions in Sub/Second Harmonic Generation

P. D. Drummond, K. J. McNeil, Daniel F. Walls

Published Mar 1, 1980• 290 citations
Publisher

Abstract

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.

DOI
10.1080/713820226

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Non-equilibrium Transitions in Sub/second Harmonic Generation

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Non-Gaussian phase transition and cascade of instabilities in the dissipative quantum Rabi model

Mingyu Kang, Yikang Zhang, Kenneth R. Brown +1 more

The open quantum Rabi model describes a two-level system coupled to a harmonic oscillator. A Gaussian phase transition for the nonequilibrium steady states has been predicted when the bosonic mode is soft and subject to damping. We show that oscillator dephasing is a relevant perturbation, which leads to a non-Gaussian phase transition and an intriguing cascade of instabilities for $k$-th order bosonic operators, as well as a jump in the steady-state qubit polarization. For the soft-mode limit, the equations of motion form a closed hierarchy and spectral properties can be efficiently studied. To this purpose, we establish a fruitful connection to non-Hermitian Hamiltonians. The results for the phase diagram, stability boundaries, and relevant observables are based on mean-field analysis, exact diagonalization, perturbation theory, and Keldysh field theory.

arXiv:2507.07092•Physical Review A•May 6, 2026•2 citations
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.
In this paper we present a quantum statistical analysis of a non-linear optical system with a mode coupled to its second harmonic, studied semiclassically in a previous paper. Quantum fluctuations are included via a Fokker-Planck equation for a generalized P representation. The spectrum of the light is shown to split into two peaks for critical values of the input fields. It is found that under certain conditions the non-classical effect of photon antibunching is possible in the steady state. In certain limits exact solutions for the generalized P representation are possible which allows an estimate of lifetimes in the metastable region to be calculated.
The simplest open, nonlinear optical system consists of a single bosonic mode with nonlinear self-interactions coupled to driving fields and reservoirs. In this chapter, we will examine a number of systems of this type using the methods that have been developed in the preceding chapters. Systems of this type can exhibit a variety of interesting physical effects. For example, in the case of a driven nonlinear oscillator, there can be more than one steady state (bistability), and quantum statistical effects can manifest themselves by affecting the stability of these states. Other quantum statistical effects have also been predicted in a variety of related systems – such as photon anti-bunching, squeezing and changes to spectra. In general, the size of these effects scales inversely with the size of the system. This ‘system size’ refers to a threshold photon number, a number of atoms, or some similar quantity. The nature of the steady states themselves can become less than straightforward. Nonequilibrium dissipative systems, of which the systems considered in this chapter are examples, have parameters that describe the energy input to the system, and their steady states depend on these parameters. When driven far from equilibrium, such systems can exhibit bifurcations in their steady states. In more complex cases, this eventually leads to periodic oscillations and chaos. Devices like this can be realized with multiple types of nonlinearity, ranging from nonlinear dielectrics through to cavity QED (with near-resonant atoms), cavity optomechanics (with nanomechanical oscillators) and circuit QED (with superconducting Josephson junctions). We will start with cases where all the interactions are with the reservoirs, which may be linear or nonlinear, while the cavity itself is harmonic in its response.
The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems. In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function. In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.
Abstract A scheme for active second harmonics generation is suggested. The system comprises N three-level atoms in ladder configuration, situated into a resonant cavity. The system generates the field whose frequency is twice the frequency of the pumping laser, and the field phase is locked to the phase of the pumping field. It is found that the system can lase in either superradiant or subradiant regime, depending on the number of atoms N . When N passes some critical value the transition from the super to subradiance occurs in a phase-transition-like manner. Stability study of the steady state supports this conclusion. For experimental realization of the super/subradiant second harmonics generation we propose semiconductor quantum well structures, superconducting quantum circuits, and evanescently coupled waveguides in which equally spaced levels relevant to this study exist.
The open quantum Rabi model describes a two-level system coupled to a harmonic oscillator. A Gaussian phase transition for the nonequilibrium steady states has been predicted when the bosonic mode is soft and subject to damping. We show that oscillator dephasing is a relevant perturbation, which leads to a non-Gaussian phase transition and an intriguing cascade of instabilities for kkk-th order bosonic operators, as well as a jump in the steady-state qubit polarization. For the soft-mode limit, the equations of motion form a closed hierarchy and spectral properties can be efficiently studied. To this purpose, we establish a fruitful connection to non-Hermitian Hamiltonians. The results for the phase diagram, stability boundaries, and relevant observables are based on mean-field analysis, exact diagonalization, perturbation theory, and Keldysh field theory.