# Optical Bistability and the Jaynes-Cummings Model

• A

## Main Question or Discussion Point

I understand that otical bistability only occurs in a specific parameter regime defined by the bimodial leaf, but I have read that bistability originates from highly non linear dynamics of the system. As we are dealing with a qubit in a cavity that is being driven so things become non-trivial, the system is described by the Jaynes-Cummings Model with an additional drive term, so what terms in the equations or what in particular is non-linear that gives bistabilty?

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A. Neumaier
2019 Award
the system is described by the Jaynes-Cummings Model with an additional drive term, so what terms in the equations or what in particular is non-linear that gives bistabilty?
Please give a reference where ''the system'' is defined and the equations are stated.

Please give a reference where ''the system'' is defined and the equations are stated.
In the pape: Bistability effect in the extreme strong coupling regime of the Jaynes-Cummings model by A Dombi et.al
"Optical bistability is a benchmark of nonlinear lightmatter
interaction....Fluctuations are due to the dissipative processes, whereas the switching(Bistability) originates from a highly nonlinear dynamics." From the same paper, "the system : and the master equation: A. Neumaier
2019 Award
what terms in the equations or what in particular is non-linear that gives bistabilty?
The nonlinearity is in the term of H containing the factor g. For g=0, the system is decomposes into a harmonic oscillator and a 2-level system, which are both exactly solvable. The interaction creates the bistability.

• Raptor112
The nonlinearity is in the term of H containing the factor g. For g=0, the system is decomposes into a harmonic oscillator and a 2-level system, which are both exactly solvable. The interaction creates the bistability.
But with the interaction term it is still possible to diagonalise the Hamiltonian:

##H =\omega_c\bigg(\hat{a}^\dagger\hat{a} +\frac{1}{2}\bigg) + \frac{\omega_q}{2}\hat{\sigma}_z + g_{c}(\hat{a}\hat{\sigma}_+ + \hat{a}^\dagger\hat{\sigma}_-)##

has energy eigen values:

##E_{|n, \pm\rangle} = \omega_q\bigg(n+\frac{1}{2}\bigg) \pm\frac{\sqrt{\Delta^2 + 4g^2(n+1)}}{2}##

A. Neumaier
2019 Award
But with the interaction term it is still possible to diagonalise the Hamiltonian:
But only in the absence of interaction with the environment, which causes energy dissipation and hence activates the bistability under appropriate conditions. The resulting Lindblad equation (2) can no longer be diagonalized. The mean equation (4) becomes linear only if g=0 and inherits from the nonlinear Hamiltonian the nonlinear term ##S^*\alpha+\alpha^*S##.

f95toli
Gold Member
Note that they are talking about a driven system, the J-C Hamiltonian can give rise to very complicated dynamics once you add e.g. a coherent drive (Mollow triplets etc). This in combination with decoherence means that the behavior is far from non-trivial even if you are only using a "simple" Lindbladian to model the system. .

Note also that the J-C Hamiltonian is only valid in the RWA; if you are driving the system hard enough you need to use the full spin-boson model which is difficult to solve even with numerical methods.

with the environment, which causes energy dissipation and hence activates the bistability under appropriate conditions.
So interaction with the outside environment causes the bistability?

f95toli
Gold Member
So interaction with the outside environment causes the bistability?
Not sure there is a single "cause" here. I would say it is the very large value of g that is the "cause", with the understanding that the effects would only appear if you are driving the system hard enough and there is the right amount of dissipation.

But again, be very careful when working with the J-C Hamiltonian in this regime, you can easily go outside the regime where it is valid and if you solving it numerically you can easily get numerical artifacts if you space is not large enough

A. Neumaier
2019 Award
Bistability is present independent of the environment but becomes alive through the latter. This also holds classically: A classical particle in a double-well potential is a bistable system, but the effect of it is seen only if you add an external interaction.

Bistability is present independent of the environment but becomes alive through the latter. This also holds classically: A classical particle in a double-well potential is a bistable system, but the effect of it is seen only if you add an external interaction.
But I don't actually see whats non linear about the interaction term?

A. Neumaier