Modeling of simple bandpass filters: bandwidth broadening of Josephson parametric devices due to non-Markovian coupling to dressed transmission-line modes

TL;DR

This work develops a non-Markovian framework for Josephson parametric devices by dressing transmission-line modes with representative filters. The authors derive a frequency-dependent self-energy Σ(Δ) and nonperturbative Hamiltonians for series-$LC$ and ladder-type couplings, then formulate generalized Langevin dynamics and input–output relations that reduce to the Markov limit when appropriate. They show that bandwidth broadening and improved gain–bandwidth products arise not only at resonant slope-matching but also in off-resonant regimes, driven by the structured self-energy. The approach provides design guidelines for bandwidth engineering in resonant JPDs using simple lumped-element filters, with potential impact on multiplexed readout and quantum-limited amplification in superconducting circuits.

Abstract

Josephson parametric devices are widely used in superconducting quantum computing research but suffer from an inherent gain-bandwidth trade-off. This limitation is partly overcome by coupling the device to its input/output transmission line via a bandpass filter, leading to wider bandwidth at undiminished gain. Here we perform a non-perturbative circuit analysis in terms of dressed transmission-line modes for representative resonant coupling circuits, going beyond the weak-coupling treatment. The strong frequency dependence of the resulting coupling coefficients implies that the Markov approximation commonly employed in cQED analysis is inadequate. By retaining the full frequency dependence of the coupling, we arrive at a non-Markovian form of the quantum Langevin equation with the frequency-dependent complex-valued self-energy of the coupling in place of a single damping parameter. We also consistently generalize the input-output relations and unitarity conditions. Using the exact self-energies of elementary filter networks -- a series- and parallel-LC circuit and a simple representative bandpass filter consisting of their combination -- we calculate the generalized parametric gain factors. Compared with their Markovian counterpart, these gain profiles are strongly modified. We find bandwidth broadening not only in the established parameter regime, where the self-energy of the coupling is in resonance with the device and its real part has unity slope, but also within off-resonant parameter regimes where the real part of the self-energy is large. Our results offer insight for the bandwidth engineering of Josephson parametric devices using simple coupling networks.

Figures (12)

• Figure 1: A (nonlinear) lumped element resonator (e.g. a JPA) coupled to a transmission line of impedance $Z_0$ via various coupling networks (red): (a) a series-$LC$ circuit, (b) a parallel-$LC$ circuit, (c) a simple ladder filter, (d,e) extended ladder filters and (f) an extended parallel-$LC$ train filter.
• Figure 2: Self-energy $\Sigma_E(\Delta)$ vs $\omega = \omega_s + \Delta$ of series-$LC$ coupling circuit for $Z_s/Z_0=0.8$, $C_c/C_s = 1.0$ and various values of $L_c/L_s$. All cases belong to the parameter regime $4\beta/\alpha > 1$, with $\beta/\alpha = 32 L_c/(25 L_s)$. Grey solid and black dotted lines show the resonant approximations $\Sigma_{E,\text{res}}(\Delta)$ and $\Sigma_{E,\text{res},0}(\Delta)$ of Eq. \ref{['eq:selfenergy_seriescoupled_resapprox']} and Eq. \ref{['eqapp:selfenergy_seriescoupled_resapprox_exact']}, respectively. The vertical lines show the (approximate) resonance position $\omega_\Sigma$ (solid), Eq. \ref{['eq:wres_selfenergy_imag_seriescoupled']}, and the bare coupling frequency $\omega_c = 1/\sqrt{C_c L_c}$ (dashed). The black dashed line has slope $-1$, which represents the so-called 'slope-matching' condition Eq. \ref{['eq:resonantcondition_bandwidthbroadening']} (see Sec. \ref{['sec:results_bandwidthbroadening']}).
• Figure 3: Self-energy $\Sigma_E(\Delta)$ vs $\omega = \omega_s + \Delta$ for the simple ladder filter with $Z_s/Z_0=1.0$, $C_g/C_s=0.5$, $L_g/L_s=2.0$, $C_c/C_s = 5.0$ and various values of $L_c/L_s$. All cases belong to the parameter regime $4\alpha/\beta > 1$, with $\alpha/\beta = 10 L_s/(3 L_c)$. Grey solid lines show the resonant approximation $\Sigma_{E,\text{res}}(\Delta)$ of Eq. \ref{['eq:selfenergy_parallelbeforeseriescoupled_resapprox']}. The vertical lines show the (approximate) resonance position $\omega_\Sigma$ (solid), Eq. \ref{['eq:wres_selfenergy_imag_parallelbeforeseriescoupled']}, which here coincides with the bare coupling frequency $\omega_c = 1/\sqrt{L_c C_c}$.
• Figure 4: Pump-parameter space $(\delta_p, \epsilon_p)$ showing stable and unstable parameter regions for trivial state $A_s=0$ of undriven resonator ($B_{\text{in}} = 0$). It is stable for pump strength $\epsilon_p$ below the pump threshold $\epsilon_{p,\text{th}}$ (bold solid and dashed lines), but for $|\epsilon| \geq \epsilon_{p,\text{th}}$ it turns unstable, while parametric oscillations with finite amplitude $A_s \neq 0$ set in. The solid and dashed bold lines show $\epsilon_{p,\text{th}}$ from Eq. \ref{['eq:h_thresh__wdependent']} and from the corresponding Markov approximation, Eq. \ref{['eq:h_thresh__Markov']}, respectively. Thin dash-dotted and dotted lines show the underlying $\delta_p$-dependence of $\Sigma(\Delta=0)=\Sigma_E(\Delta=0)$ for $\Gamma_I=0$: either calculated for series-$LC$ coupling with $Z_s/Z_0 = 2$, $C_c/C_s = 10$, $L_c/L_s = 0.4$ from Eqs. \ref{['eq:imSig_seriescoupled']} and \ref{['eqapp:reSig1_seriescoupled']}--\ref{['eqapp:reSig2_seriescoupled']} (yellow, resulting $\epsilon_{p,\text{th}}$ in red), or held fixed at value $\Sigma(\Delta=0)[\delta_p=0] \approx (-0.36 - 0.34\text{i})\omega_s$ (gray, resulting $\epsilon_{p,\text{th}}$ in black).
• Figure 5: Pump-parameter space $(\delta_{p,s}, \epsilon_p)$ showing stable and unstable parameter regions for trivial state $A_s = A_a =0$ of undriven resonator ($B_{\text{in}} = 0$) coupled to the TL via the simple ladder filter. It is stable for pump strength $\epsilon_p$ below the pump threshold $\epsilon_{p,\text{th}}$ (red solid line), where all eigenvalues of Eq. \ref{['eq:dynamicalmatrix_filter']} have negative real parts. For the outer two lobes of the instability region (shaded), the threshold $\epsilon_{p,\text{th}}$ (where the real part of one or more eigenvalues turns zero) is equivalent to the value $\tilde{\epsilon}_{p,\text{th}}$ (bold magenta) defined by the condition $|\det(\boldsymbol{\mathcal{A}})| = |D(0)| = 0$. The red dashed line shows $\epsilon_{p,\text{th}}$ according to the Markov approximation for the threshold, obtained by setting $\text{Re}\Sigma_E(0) = 0$ in Eq. \ref{['eq:dynamicalmatrix_filter']}. Thin dash-dotted and dotted lines show the underlying $\delta_{p,s}$-dependence of $\Sigma(\Delta=0)$ (yellow) and $\Sigma_E(\Delta=0)$ (teal), calculated for parameters $Z_s/Z_0 = 0.7$, $C_g/C_s = 120$, $L_g/L_s = 0.002$, $C_c/C_s = 1.12$, $L_c/L_s = 4.0$ and $\Gamma_I=0$ from Eqs. \ref{['eq:imSig_filtercoupled']}, Eqs. \ref{['eqapp:reSig1_parallelbeforeseriescoupled']}--\ref{['eqapp:reSig2_parallelbeforeseriescoupled']}, and \ref{['eqF:selfenergy_tot']}.