Circuit quantum electrodynamic model of a resonantly phase-matched Josephson traveling wave parametric amplifier

In recent years Josephson-junction-based parametric amplifiers have been developed for single-photon-level signals in the microwave spectral region, where they found applications in quantum information processing. Typically, Josephson traveling wave parametric amplifiers (JTWPA) have been studied based on classical circuit models. Although this approach shows good agreement with experimental results, it describes the system behavior in the framework of a classical theory, neglecting the quantum nature of the device. A quantum mechanical treatment of the JTWPA was only given in a few recent papers, where [1] derives a Hamiltonian for a chain of unit cells without resonant phase-matching (RPM), and [2] solves the dynamics of the system with and without RPM, also partly considering noise and squeezed state generation. However, our approach differs from [2] in that we give an explicit solution to the resulting nonlinear wave equation for narrow-band signals, starting from discrete chain Hamiltonians with and without RPM. We focus on describing the dynamics of the device in a noiseless quantum setting and present solutions for the governing nonlinear wave equation in Fig. 1. The starting point of this work is the amplifier design as given in [3]. The mathematical description relies on the Hamiltonian of a discrete chain of unit cells, which was derived in [1], and which we extended by a RPM circuit [2]. As the size ∆l of a single chain element is relatively small compared to the wavelength, the chain of discrete Josephson elements can be considered in terms of a continuous transmission line. Using the continuum approximation, the discrete annihilation operators â_n are replaced by â (x, t) and the equation of motion for the case without RPM can be written as (Equation presented) where ω₀ is the natural resonance frequency, K is the Kerr nonlinearity constant, C is the shunt capacitance, and C_J is the Josephson capacitance. The form of the resulting wave equation resembles the classical one in [3]. In addition to the approximations used in [3, 4], we expand the flux operator into a set of modes and assume a strong classical pump field [2]. Finally, we obtain a closed analytic solution of the four-wave-mixing process. Comparing the results in Fig. 1 (a) and (c) to the results of the quantum mechanical model from [2] and to the classical model in [4] shows good agreement. The same holds for the phase mismatch in Fig. 1 (b) and (d).