# Parametric instabilities in driven-dissipative Josephson circuits

Otten, Lisa; Hassler, Fabian (Thesis advisor); Kusminskiy, Silvia Viola (Thesis advisor)

*Aachen : RWTH Aachen University (2023)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2023

Abstract

A dc-biased Josephson junction coupled to one or more microwave resonators may display parametric resonance when the Josephson frequency matches an integer multiple of the resonance frequencies. In this work, we employ the Keldysh path-integral description to study qualitative and quantitative characteristics related to this phenomenon. We investigate degenerate, non-degenerate, and higher-order Josephson parametric driving, highlighting differences and similarities between the different processes. In particular, we focus on the dynamics close to parametric instabilities in order to understand the nonclassical properties of the radiation as well as the characteristics of the discrete time-symmetry breaking that can occur in the presence of parametric driving. We review the Keldysh path-integral description and show how the formalism relates to other well-known quantum and classical descriptions of open systems, such as the Lindblad master equation, the Langevin equation, or the Fokker-Planck equation. We demonstrate how a counting factor can be added to the path integral to gain access to the counting statistics of photons emitted by a driven-dissipative superconducting circuit. Based on this method, we study the counting statistics of a non-degenerate parametric oscillator below the threshold of instability and compare the results to the degenerate case. Furthermore, we analyze the second-order coherences of the radiation and discuss thermal effects on the counting statistics. We provide results for the driving strength at which the radiation demonstrates the strongest nonclassical features in so far that the Cauchy-Schwarz inequality is most strongly violated. Additionally, we study the impact of an asymmetry in the linewidth of the modes - a distinctive property of the non-degenerate resonance effect. Additionally, we investigate the stability of the vacuum state for higher-order parametric driving. We point out that - despite the absence of a classical instability threshold - quantum vacuum fluctuations or off-resonant frequency contributions of the Josephson drive can be employed to induce period multiplication. We focus on the period-tripling case as a textbook example and analyze the timescale and the conditions under which the time-symmetry breaking occurs. In particular, we show that for weak dissipation or strong driving, the system exhibits a non-equilibrium phase transition in the sense that the timescale over which the period-tripled state builds up can be arbitrarily separated from the timescale of the subsequent dephasing. Finally, we study the properties of the symmetry-broken states that emerge in the presence of the Josephson nonlinearity. Employing a rotating-wave approximation, we obtain an interesting sixfold symmetry in the phase of the period-tripled states which is weakly broken to a threefold symmetry in the presence of small dissipation and detuning. Beyond the rotating-wave approximation, off-resonant contributions of the parametric drive lead to a further reduction of this symmetry.

Institutions

- Department of Physics [130000]
- Theoretical Physics (condensed matter) Teaching and Research Area [135220]

### Identifier

- DOI: 10.18154/RWTH-2023-07132
- RWTH PUBLICATIONS: RWTH-2023-07132