Symbolic-Numeric Co-Simulation of Josephson Traveling-Wave Parametric Amplifiers

Analytic solutions fail to accurately predict the gain and added quantum noise in traveling-wave parametric amplifiers for large signal amplitudes. Thus, the underlying system Hamiltonian needs to be extended to include more and more modes in order to describe effects such as spectral leakage. For a sufficiently complex Hamiltonian, the procedure of deriving the Heisenberg equations of motion for each mode is tedious or even prohibitive. Therefore, we present a symbolic-numeric co-simulation approach, where the equations of motion are automatically derived from a given Hamiltonian using a modern computer algebra system. The resulting equations of motion can be solved numerically. We verified our approach using existing analytic and numeric solutions from the literature and found good agreement. Additionally, we simulate an amplifier from the literature, where the pump mode experiences substrate losses beyond the stiff pump approximation.