Determining the state of a qubit on a time scale much shorter than its relaxation time is an essential requirement for quantum information processing. With the aid of a nondegenerate parametric amplifier, we demonstrate the continuous detection of quantum jumps of a transmon qubit with $90\mathrm{%}$ fidelity of state discrimination. Entirely fabricated by standard two-step optical-lithography techniques, this type of parametric amplifier consists of a dispersion-engineered Josephson-junction (JJ) array. By using long arrays, containing
103JJs, we can obtain amplification in multiple eigenmodes with frequencies below
10GHz, which is the typical range for qubit readout. Moreover, if a moderate flux tunability of each mode is introduced, employing superconducting-quantum-interference-device junctions, a single amplifier device could potentially cover the entire frequency band between 1 and 10 GHz.
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.