Articles | Volume 15
21 Sep 2017
 | 21 Sep 2017

Markovian Dynamics of Josephson Parametric Amplification

Waldemar Kaiser, Michael Haider, Johannes A. Russer, Peter Russer, and Christian Jirauschek

Abstract. In this work, we derive the dynamics of the lossy DC pumped non-degenerate Josephson parametric amplifier (DCPJPA). The main element in a DCPJPA is the superconducting Josephson junction. The DC bias generates the AC Josephson current varying the nonlinear inductance of the junction. By this way the Josephson junction acts as the pump oscillator as well as the time varying reactance of the parametric amplifier. In quantum-limited amplification, losses and noise have an increased impact on the characteristics of an amplifier. We outline the classical model of the lossy DCPJPA and derive the available noise power spectral densities. A classical treatment is not capable of including properties like spontaneous emission which is mandatory in case of amplification at the quantum limit. Thus, we derive a quantum mechanical model of the lossy DCPJPA. Thermal losses are modeled by the quantum Langevin approach, by coupling the quantized system to a photon heat bath in thermodynamic equilibrium. The mode occupation in the bath follows the Bose-Einstein statistics. Based on the second quantization formalism, we derive the Heisenberg equations of motion of both resonator modes. We assume the dynamics of the system to follow the Markovian approximation, i.e. the system only depends on its actual state and is memory-free. We explicitly compute the time evolution of the contributions to the signal mode energy and give numeric examples based on different damping and coupling constants. Our analytic results show, that this model is capable of including thermal noise into the description of the DC pumped non-degenerate Josephson parametric amplifier.

Short summary
Recent advances in Josephson junction based parametric amplifiers (JPA) have reached quantum limited amplification. In the quantum regime, dissipation and thermal fluctuations play a significant role. We outline the classical treatment of the Josephson parametric amplifier and derive a quantum mechanical formalism of the lossy JPA. We investigate the influence of dissipation and fluctuations on the dynamics of the amplification mode and show limitations of the applied quantum Langevin approach.