how to compute the frequencies of the modes

[serpagano]

hello
I am using this package to verify our designs of JTWPA and I find it is very good.
However, I have difficulty in determining the frequencies of the idler modes.
The vector "modes" has the signal as first element but the other modes are labelled with even numbers, like (2,4,...8, -8,...,-4,-2)
Could someone explain how they are defined?
Many thanks
Sergio Pagano

[serpagano]

Solved.
see eq. 1 in referenced article X-parameter based design and simulation of Josephson traveling-wave parametric amplifiers for
quantum computing applications

[kpobrien]

I'm glad you found that reference! That's a good source of information.

To add a few examples, for a single-tone nonlinear simulation (eg. with hbnlsolve) the modes are the harmonics of the strong tone. For example, for the single pump JPA example, the pump mode frequencies can be obtained from the modes by multiplying by the pump frequency:

julia> using JosephsonCircuits;@variables wp;JosephsonCircuits.calcmodefreqs((wp,),[(1,), (3,), (5,), (7,), (9,), (11,), (13,), (15,)])
8-element Vector{Num}:
   wp
  3wp
  5wp
  7wp
  9wp
 11wp
 13wp
 15wp

For the simulations of the linearized system (eg. using hblinsolve) the modes are the pump harmonics that the signal is scattered by. For the JPA example, the signal and idler frequencies are given by the signal frequency plus the pump frequency times modes:

julia> using JosephsonCircuits;@variables ws wp;ws .+ JosephsonCircuits.calcmodefreqs((wp,),[(0,), (2,), (4,), (6,), (8,), (-8,), (-6,), (-4,), (-2,)])
9-element Vector{Num}:
        ws
  2wp + ws
  4wp + ws
  6wp + ws
  8wp + ws
 -8wp + ws
 -6wp + ws
 -4wp + ws
 -2wp + ws