Intrinsic decay rates and steady states of driven Josephson junction chains cavities

Abstract

Josephson junction (JJ) chains combine the coherence of superconductivity with the controllability of microwave-frequency circuits, making them a powerful platform for circuit quantum electrodynamics. In this work we consider a long JJ chain that effectively realizes a multi-mode cavity with nonlinear dispersion and additional multi-mode interactions. Individual modes appearing due to the finite size of the chain can be experimentally probed via microwave spectroscopy, both in equilibrium and in driven far-from-equilibrium settings. We study the role of multi-mode interactions in degrading internal coherence -- observable as excess linewidth -- in both equilibrium and driven regimes. Focusing on two-into-two mode scattering as the leading relaxation process, we classify the relevant scattering processes and derive their expected temperature- and frequency-scaling under equilibrium conditions. For experimentally relevant parameters, we show that the equilibrium decay rate is dominated by non-resonant processes, however weakly driving a particular set of modes out of equilibrium enhances resonant scattering, leading to observable signatures in the distribution function and linewidth. Finally, in the strong non-equilibrium regime we report a crossover to a qualitatively different non-equilibrium steady state.

Figures (15)

• Figure 1: (a) Sketch of system, consisting of a chain of JJs coupled to a transmission line. Small squares denote adjacent superconducting islands. Lines in different shades of magenta color illustrate the wave function of the first plasmonic modes with number $k=1,2,3$. The coupling to terminals $\kappa_\text{ex}$ and to the internal environment $\kappa_\text{i}$ are also shown. (b) Detail of the circuit, with $E_J$ the Josephson energy, $C_g$ the capacitance to ground, and $\theta_i$ the superconducting phase of each island. (c) Comparing exact, Eq. \ref{['eqn:exactdisp']}, and expanded, Eq. \ref{['eqn:mirlindisp']} dispersion relation of the chain shows agreement for low-lying modes.
• Figure 2: Introducing momentum unfolding: (a) plasmons modes labeled by a set of positive integers, with two-into-two collisions fulfilling the generalized momentum conservation $k\pm p\pm q_1\pm q_2=0$; (b) each mode is split into its positive and negative-momentum component, by introducing negative mode numbers with same frequency and occupation numbers as the positive ones. In this case, the constraint becomes $k+ p-q_1-q_2=0$. In what follows we adopt the unfolded notation as in (b).
• Figure 3: Schematic representation of large (i) and small (ii) momentum transfer decay of mode $k$. The only on-shell decay process involves momentum $p^*$ defined in the main text.
• Figure 4: Decay channel plot of mode $k=150$ assuming $k,p\to q_1,q_2$, and setting $T=0.1\,\text{K}$ and $\kappa^0/(2\pi)=5\,\text{MHz}$. The darker regions correspond to participating decay channels. The colored frames highlight two relevant and distinguished types of scattering processes: with large momentum transfer (i) (light and dark blue) and small momentum transfer (ii) (brown), respectively. Numerically, for the resonant processes of type (i) (see the main text), we sum $p$ over $-30\le p \le -1$, which includes $p^*$ up to $k\approx450$. More details on the selected channels in the other cases are discussed in the main text.
• Figure 5: Analysis of decays arising from process (i). (a) $\delta\kappa/2\pi,\delta\kappa^{(i),\text{off}}/2\pi,\delta\kappa^{(i),\text{on}}/2\pi$ as a function of mode number, for $T=0.002\,\text{K}$. The on-shell contributions strongly underestimate the total decay rate, although well-obeying the predicted power law $\sim k^4$ scaling with mode number. (b) The same quantities for fixed mode $k=180$, as a function of temperature. Above $T=0.005\,\text{K}$, small momentum transfer processes start to contribute to $\delta\kappa$, causing the deviation between the full decay rate and its estimate from large momentum transfer processes only. The contribution from off-resonant large momentum processes follows the expected $\sim T^2$ scaling, the contribution from resonant processes shows a weaker scaling although not exactly $\sim T$ (not all conditions for such scaling are fulfilled).