We present a formalism for modelling parametric amplification by resonators subject to rate-limited nonlinearity of mixed reactive/dissipative character, with particular relevance tosuperconducting devices. The non-linearity is assumed to be characterised by a single state parameter, which responds to changes in the energy stored in the resonator with finite response time. We show how the operating point and small signal amplification behaviour of the pumped resonator can be calculated, characterised and optimised in terms of a set of three dimensionless parameters. The formalism is then illustrated with a simple, first-order, model nonlinearity and the implications for amplification via quasiparticle generation in a superconductor discussed. Throughout we describe how the parameters needed to characterise the device can be determined experimentally from steady-state measurements. A key result of this paper is that rate-limiting of a nonlinear mechanism does not preclude amplification, although it does limit the bandwidth over which it may be achieved.
The underlying nonlinear mechanisms behind the operation of travelling-wave parametric amplifiers (TWPAs) are important in determining their performance in terms of added noise, maximumgain, and bandwidth. We describe a method of characterising the underlying nonlinearity of a superconducting material in terms of its dissipative-reactive ratio and the response time of the underlying microscopic processes. We describe and calculate the different behaviour arising from the equilibrium supercurrent nonlinearity, which has low dissipation and fast response time, and the non-equilibrium heating nonlinearity, which has high dissipation and slow response time. We have fabricated TWPAs based on Al and Ti, and characterised their nonlinearities using our analysis. For both Al and Ti, the measured dissipative-reactive ratios and response times are quantitatively similar to predictions for the non-equilibrium heating nonlinearity. In line with this, we were able to obtain more than 20 dB of peak power gain, although only over a narrow bandwidth of a few kilohertz.
We have performed a quantum mechanical analysis of travelling-wave parametric amplifiers (TWPAs) in order to investigate five experimental phenomena related to their operations, namelythe effect of impedance mismatch, the presence of upper idler modes, the presence of quantum and thermal noise, the generation of squeezed states, and the preservation of pre-squeezed states during amplification. Our analysis uses momentum operators to describe the spatial evolution of quantised modes along a TWPA. We calculate the restriction placed on pump amplitude as well as amplifier gain as a result of impedance mismatch between a TWPA and its external system. We apply our analysis to upper idler modes and demonstrate that they will result in suppressed gain. We show that an ideal TWPA is indeed quantum-limited – i.e. it introduces a half-quantum of zero-point fluctuation which is the minimum possible noise contribution for a phase-preserving linear amplifier. We analyse the thermal noise associated with a TWPA by considering the effect of distributed sources along an amplifier transmission line. Our analysis predicts a doubling of thermal noise in the high gain limit as a result of wave-mixing between signal and idler modes. We study the operation of a TWPA in the presence of a DC bias current, and have shown that highly squeezed states can in principle be generated. However, amplifying a pre-squeezed state using a non-degenerate TWPA generally reduces the squeezing advantage.
We have developed a coupled-mode analysis framework for superconducting travelling-wave parametric amplifiers using the full Telegrapher’s equations to incorporate loss-relatedbehaviour. Our model provides an explanation of previous experimental observations regarding loss in amplifiers, advantages of concatenating amplifiers to achieve high gains, and signal gain saturation. This work can be used to guide the design of amplifiers in terms of the choice of material systems, transmission line geometry, operating conditions, and pump strength.
Thin-film superconducting transmission lines play important roles in many signal transmission and detection systems, including qubit coupling and read-out schemes, electron spin resonancesystems, parametric amplifiers, and various ultra high sensitivity detectors. Here we present a rigorous method for computing the electromagnetic behaviour of superconducting microstrip transmission lines and coplanar waveguides. Our method is based on conformal mapping, and is suitable for both homogeneous superconductors and proximity-coupled multilayers. We also present an effective conductivity approximation of multilayers, thereby allowing the multilayers to be analysed using existing electromagnetic design software. We compute the numerical results for Al-Ti bilayers and discuss the validity of our full computation and homogeneous approximation.