Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Optica Acta International Journal of Optics

Multiple Bifurcations in Coherent n-photon Processes

Timothy Poston, Daniel F. Walls, P. Zoller

Published Dec 1, 1982• 3 citations
Publisher

Abstract

We consider a new class of bifurcations that may arise in multiphoton processes inside a coherently driven optical cavity involving more than one mode of the radiation field. For a non-saturable, non-linear medium, a single bifurcation point exists where the symmetric solution bifurcates into a number of non-symmetric solutions. The nature of the bifurcation ranges from a simple pitchfork bifurcation, in the case of four-wave mixing, to more complicated phenomena in higher-order processes. A saturable non-linear medium exhibits similar behaviour for low-input intensities; however, as the input intensity is increased the medium saturates, and at a second bifurcation point the symmetric branch regains its stability. The presence of fluctuations assures the accessibility of the symmetric branch. Thus, for example, for a two-photon absorbing medium we have the possibility of optical tristability involving one symmetric solution and two non-symmetric solutions.

DOI
10.1080/713820800

Related papers

Bifurcations and multistability in two-photon processes

Daniel F. Walls, C. V. Kunasz, P. D. Drummond +1 more

We consider an intracavity nonlinear medium interacting with two modes of the electromagnetic field via a two-photon process. The cavity is coherently driven at the frequencies of the two modes. We consider as the intracavity medium (a) a two-photon absorber, and (b) $N$ three-level atoms. These systems display a new class of bifurcations, and, in the case of the two-photon absorber, optical tristability is possible.

Physical Review A•Jul 1, 1981•22 citations

Periodic attractors in two-photon processes

Paul Mandel, Nicolas Pettiaux, Wang Kaige +2 more

We consider three nonlinear optical models that display a two-photon process and exhibit a Hopf bifurcation corresponding to a phase instability. The three models describe second-harmonic generation, two-photon optical bistability, and degenerate four-wave mixing in resonant cavities. In all three cases we consider only the resonant configuration and analyze the attractors which emerge when the driving field is increased beyond the Hopf bifurcation threshold. At a finite distance from threshold, we find a hysteresis domain that involves a pair of periodic attractors. For all three models, each of the two solutions has the same ``field portrait'' i.e., the same representation in a plane whose axes are the imaginary and the real parts of the electric field. This suggests that the scenario is generic for this class of systems. On the other hand, the models for two-photon optical bistability and degenerate four-wave mixing in resonant cavities exhibit additional nongeneric attractors when the driving field is further increased.

Physical Review A•Jan 1, 1991•16 citations

Bifurcation analysis of nonlinear coupled photonic nanocavities

Neil G. R. Broderick, Bernd Krauskopf, Soizic Terrien +4 more

Recently, various research groups have studied spontaneous symmetry breaking and bistable behaviour in both active and passively driven coupled photonic nanocavities[1]. This work is of interest both in itself and because it naturally extends to coupled nonlinear quantum systems described by the Bose-Hubbard model [2]. Here we present a theoretical study of the dynamics in such a system by considering the bifurcation behaviour as the different parameters are varied. In normalised units the model can be written as: dA/dt = i(δ τ + |A|2)A-A+(iκ + γ)τB+f (1) dB/dt = i(δ τ + |B|2)B-B+(iκ + γ)τA+f (2) where A and B are the slowly varying normalised amplitudes of the fields in each nano-cavity. Further, δ is the detuning from the cavity resonance, τ is the photon lifetime, κ and γ describe the linear coupling between the cavities (if γ = 0 and κ < 0 the model is identical to the Bose-Hubbard model [2] in the semiclassical regime), while f represents a coherent driving term. In deriving the above equations it is assumed that the cavities are identical, and so a focus is the switching between the symmetrical state [A(t) = B(t)] and asymmetrical states, which must come in pairs. Fig. 1(a) shows a typical boundary curve dividing the (δ, f) plane into the regions where the symmetric and asymmetric solutions exist. Further analysis shows that more complicated dynamics arise from the asymmetric state. Fig. 1(b) shows the field intensity for the symmetric and asymmetric steady-states as a function of f, where stable solutions are represented by solid and unstable ones by dashed curves. Hopf bifurcations, marked by H, cause the asymmetric solution to change stability. A typical bifurcating periodic solution is shown in Fig. 1(c) for f = 6.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 1, 2017•0 citations

Nonlinear dynamics of a class-A two-photon laser with injected signal

Javier F. Urchueguía, V. Espinosa, Eugenio Roldán +1 more

We performed a comprehensive study of the temporal dynamics of a two-photon laser with resonant injected signal in the high-Q cavity limit (class-A laser). The stationary solutions as well as their stability are completely characterized. In particular it is shown that tristable behaviour is possible, in contrast with two-photon optical bistability. It is also shown that there are two independent Hopf bifurcations, the sub- or supercritical character of which is analytically determined. The dynamics of the system far from the bifurcation points is classified through a numerical study.

Journal of Modern Optics•Nov 1, 1996•7 citations

Nonlinear Mixing and Multi photon Processes in the Field of Two Pumps

G. S. Agarwal, N. Nayak

A very large number of physical phenomena in the resonant interaction of atomic-molecular systems with the radiation fields can be studied by treating the atomic system as a two level system and by examining its response to the applied fields. Various perturbative results are well known. A number of recent experimental works require the knowledge of the nonperturbative response functions, when both the applied fields are strong. These include a variety of experiments on instabilities(1) in lasers involving the switching from a single mode to for example two mode operation. Similarly recent nonlinear mixing experiments, where not only four wave mixing but also higher mixing in saturating fields has been studied. Pump probe experiments have also been repeated with probes whose intensity could be large. The purpose of this paper is to formulate the general theory and use it to understand the experiments. The formulation we present is also well suited for studying the effect of collisions. Details of various collisional induced coherences in fourwave and six wave mixing etc. can be studied.

International Meeting on Instabilities and Dynamics of Lasers and Nonlinear Optical Systems•Jan 1, 1985•0 citations

Criteria for multiphoton optical bistability in a high-Q ring cavity

Shenping Li, Yizhou Zhang

The static criterion for multiphoton optical bistability in a high-Q ring cavity is analytically derived for a homogeneously broadened atomic system when the input coherent field is in perfect resonance with the atoms and with the cavity. It is shown how the static threshold condition, the bistable regime, the critical point, and the threshold input field amplitude for the optical bistability, change with the variables of the photon number involved in the transition and the system parameter C/sub n/. Criteria for the stability of the stationary solutions are derived and discussed. The results show that in the input-output curves, all parts of the branches with negative slope are unstable, and some parts of the branches with positive slope may be unstable. Considering the stability of the stationary solutions, complete criteria for optical bistability are discussed.>

IEEE Journal of Quantum Electronics•Jan 1, 1993•0 citations
We consider a new class of bifurcations that may arise in multiphoton processes inside a coherently driven optical cavity involving more than one mode of the radiation field. For a non-saturable, non-linear medium, a single bifurcation point exists where the symmetric solution bifurcates into a number of non-symmetric solutions. The nature of the bifurcation ranges from a simple pitchfork bifurcation, in the case of four-wave mixing, to more complicated phenomena in higher-order processes. A saturable non-linear medium exhibits similar behaviour for low-input intensities; however, as the input intensity is increased the medium saturates, and at a second bifurcation point the symmetric branch regains its stability. The presence of fluctuations assures the accessibility of the symmetric branch. Thus, for example, for a two-photon absorbing medium we have the possibility of optical tristability involving one symmetric solution and two non-symmetric solutions.
We consider an intracavity nonlinear medium interacting with two modes of the electromagnetic field via a two-photon process. The cavity is coherently driven at the frequencies of the two modes. We consider as the intracavity medium (a) a two-photon absorber, and (b) NNN three-level atoms. These systems display a new class of bifurcations, and, in the case of the two-photon absorber, optical tristability is possible.
We consider three nonlinear optical models that display a two-photon process and exhibit a Hopf bifurcation corresponding to a phase instability. The three models describe second-harmonic generation, two-photon optical bistability, and degenerate four-wave mixing in resonant cavities. In all three cases we consider only the resonant configuration and analyze the attractors which emerge when the driving field is increased beyond the Hopf bifurcation threshold. At a finite distance from threshold, we find a hysteresis domain that involves a pair of periodic attractors. For all three models, each of the two solutions has the same ``field portrait'' i.e., the same representation in a plane whose axes are the imaginary and the real parts of the electric field. This suggests that the scenario is generic for this class of systems. On the other hand, the models for two-photon optical bistability and degenerate four-wave mixing in resonant cavities exhibit additional nongeneric attractors when the driving field is further increased.
Recently, various research groups have studied spontaneous symmetry breaking and bistable behaviour in both active and passively driven coupled photonic nanocavities[1]. This work is of interest both in itself and because it naturally extends to coupled nonlinear quantum systems described by the Bose-Hubbard model [2]. Here we present a theoretical study of the dynamics in such a system by considering the bifurcation behaviour as the different parameters are varied. In normalised units the model can be written as: dA/dt = i(δ τ + |A|2)A-A+(iκ + γ)τB+f (1) dB/dt = i(δ τ + |B|2)B-B+(iκ + γ)τA+f (2) where A and B are the slowly varying normalised amplitudes of the fields in each nano-cavity. Further, δ is the detuning from the cavity resonance, τ is the photon lifetime, κ and γ describe the linear coupling between the cavities (if γ = 0 and κ < 0 the model is identical to the Bose-Hubbard model [2] in the semiclassical regime), while f represents a coherent driving term. In deriving the above equations it is assumed that the cavities are identical, and so a focus is the switching between the symmetrical state [A(t) = B(t)] and asymmetrical states, which must come in pairs. Fig. 1(a) shows a typical boundary curve dividing the (δ, f) plane into the regions where the symmetric and asymmetric solutions exist. Further analysis shows that more complicated dynamics arise from the asymmetric state. Fig. 1(b) shows the field intensity for the symmetric and asymmetric steady-states as a function of f, where stable solutions are represented by solid and unstable ones by dashed curves. Hopf bifurcations, marked by H, cause the asymmetric solution to change stability. A typical bifurcating periodic solution is shown in Fig. 1(c) for f = 6.
We performed a comprehensive study of the temporal dynamics of a two-photon laser with resonant injected signal in the high-Q cavity limit (class-A laser). The stationary solutions as well as their stability are completely characterized. In particular it is shown that tristable behaviour is possible, in contrast with two-photon optical bistability. It is also shown that there are two independent Hopf bifurcations, the sub- or supercritical character of which is analytically determined. The dynamics of the system far from the bifurcation points is classified through a numerical study.
A very large number of physical phenomena in the resonant interaction of atomic-molecular systems with the radiation fields can be studied by treating the atomic system as a two level system and by examining its response to the applied fields. Various perturbative results are well known. A number of recent experimental works require the knowledge of the nonperturbative response functions, when both the applied fields are strong. These include a variety of experiments on instabilities(1) in lasers involving the switching from a single mode to for example two mode operation. Similarly recent nonlinear mixing experiments, where not only four wave mixing but also higher mixing in saturating fields has been studied. Pump probe experiments have also been repeated with probes whose intensity could be large. The purpose of this paper is to formulate the general theory and use it to understand the experiments. The formulation we present is also well suited for studying the effect of collisions. Details of various collisional induced coherences in fourwave and six wave mixing etc. can be studied.
The static criterion for multiphoton optical bistability in a high-Q ring cavity is analytically derived for a homogeneously broadened atomic system when the input coherent field is in perfect resonance with the atoms and with the cavity. It is shown how the static threshold condition, the bistable regime, the critical point, and the threshold input field amplitude for the optical bistability, change with the variables of the photon number involved in the transition and the system parameter C/sub n/. Criteria for the stability of the stationary solutions are derived and discussed. The results show that in the input-output curves, all parts of the branches with negative slope are unstable, and some parts of the branches with positive slope may be unstable. Considering the stability of the stationary solutions, complete criteria for optical bistability are discussed.>