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arXiv:2606.05813 · Physical Review Applied

Strongly nonlinear regime of Josephson transmission lines revealed by two-tone spectroscopy

Alexander S. Averkin, A. A. Kopasov, I. Pologov, Aleksey N. Bolgar, Daria A. Kalacheva, Viktor B. Lubsanov, M. V. Fistul, Alexandre Karpov

Published May 26, 2026• 0 citations
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Abstract

We present experimental and theoretical studies of the off-resonant and strongly nonlinear regime of Josephson transmission lines (JTLs) with galvanically-coupled nonlinear elements. The transition from the weakly to the strongly nonlinear regime of a JTL induced by increasing the input power of the pump is probed via two-tone spectroscopy. Measurements of the phase of the transmission coefficient for a weak probe signal reveal a large increase and pronounced oscillations in the phase length variation as a function of the microwave power of the pump. Experimental observations are explained on the basis of the developed theoretical approach suitable for the description of the nonlinear response of strongly driven JTLs. Using the derived nonlinear wave equation, we show that the behavior of the phase length variation is associated with the oscillatory dependence of the Josephson inductances on the microwave power. It is demonstrated that the dissipation-induced propagation losses increase in the strongly nonlinear regime and also lead to smearing out the phase length oscillations. The developed theoretical analysis is in good agreement with experimental observations.

DOI
10.1103/gwdk-44lt

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We present experimental and theoretical studies of the off-resonant and strongly nonlinear regime of Josephson transmission lines (JTLs) with galvanically-coupled nonlinear elements. The transition from the weakly to the strongly nonlinear regime of a JTL induced by increasing the input power of the pump is probed via two-tone spectroscopy. Measurements of the phase of the transmission coefficient for a weak probe signal reveal a large increase and pronounced oscillations in the phase length variation as a function of the microwave power of the pump. Experimental observations are explained on the basis of the developed theoretical approach suitable for the description of the nonlinear response of strongly driven JTLs. Using the derived nonlinear wave equation, we show that the behavior of the phase length variation is associated with the oscillatory dependence of the Josephson inductances on the microwave power. It is demonstrated that the dissipation-induced propagation losses increase in the strongly nonlinear regime and also lead to smearing out the phase length oscillations. The developed theoretical analysis is in good agreement with experimental observations.
An electronic transmission line that contains an array of nonlinear elements (Josephson junctions) is studied theoretically. A continuous nonlinear wave equation describing the dynamics of the node flux along the transmission line is derived. It is shown that due to the nonlinearity of the system, a mixing process between four waves with different frequencies is possible. The mixing process can be utilized for amplification of weak signals due to the interaction with a strong pump wave. An analytical solution for the spatial evolution of the wave amplitudes is derived, and found to be in excellent agreement with the results of numerical computations. Simulations of realistic parameters show that the power gain can exceed 20 dB over a bandwidth of more than 2 GHz.
Atomic sized two-level systems (TLSs) in amorphous dielectrics are known as a major source of loss in superconducting devices. In addition, individual TLSs are known to induce large frequency shifts due to strong coupling to the devices. However, in the presence of a broad ensemble of TLSs these shifts are symmetrically canceled out and not observed in a typical single-tone spectroscopy experiment. We introduce a two-tone spectroscopy on the normal modes of a pair of coupled superconducting coplanar waveguide resonators to reveal this effect. Together with an appropriate saturation model this enables us to extract the average single-photon Rabi frequency of dominant TLSs to be ${\mathrm{\ensuremath{\Omega}}}_{0}/2\ensuremath{\pi}\ensuremath{\approx}79$ kHz. At high photon numbers we observe an enhanced frequency shift due to nonlinear kinetic inductance when using the two-tone method and estimate the value of the nonlinear coefficient as $K/2\ensuremath{\pi}\ensuremath{\approx}\ensuremath{-}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ Hz/photon. Furthermore, the lifetime of each resonance can be controlled (increased) by pumping of the other mode as demonstrated both experimentally and theoretically.
Josephson traveling-wave parametric amplifiers (JTWPAs) achieve high parametric gain through four-wave-mixing with near quantum-limited noise performance and extremely low energy consumption. This makes them suitable for the detection of ultra-low-power microwave signals in the lowest temperature stage of a dilution refrigerator. Distributed circuit representations of JTWPAs have been studied in the framework of classical theory. A nonlinear wave equation describing the flux along a nonlinear transmission line was derived in [1]. Dispersion engineering was introduced in [2] to increase the amplifier’s gain and bandwidth by means of resonant phase-matching (RPM). In the classical treatment [1], dielectric loss is considered for the formulation of the wave equation, however, a closed-form solution was only given when dissipation was neglected. Our work focuses on the effects of noise and dissipation in the nonlinear transmission line. Our model differs from [1],[2], where we derive the telegrapher’s equations taking into account the loss and noise contributions to construct the current-flux-relation. In the super-conducting state, the dissipation due to the imperfect electric insulation in the substrate needs to be taken into account. It is represented by a parallel conductance and an additional noise source describing temporal fluctuations [3]. The flux along the transmission line is expanded into three discrete modes, representing pump, signal, and idler contributions. The equations of motion of the signal and idler modes result in\begin{equation*}{\partial _z}{\tilde A_{\text{s}}} = - \left( {{\Gamma _{\text{s}}}/2 + {\text{i}}\Delta {k_{\text{T}}}/2} \right){\tilde A_{\text{s}}} + {\text{i}}{\kappa _{\text{s}}}\tilde A_{\text{i}}^{\ast} + {\tilde f_{\text{s}}}\left( {i_{\text{s}}^{{\text{noise}}}} \right),\quad {\partial _z}{\tilde A_{\text{i}}} = - \left( {{\Gamma _{\text{i}}}/2 + {\text{i}}\Delta {k_{\text{T}}}/2} \right){\tilde A_{\text{i}}} + {\text{i}}{\kappa _{\text{i}}}\tilde A_{\text{s}}^{\ast} + {\tilde f_{\text{i}}}\left( {i_{\text{i}}^{{\text{noise}}}} \right),\tag{1}\end{equation*}where Ãnare the mode amplitudes in a co-rotating frame, ∆kTis the total phase mismatch, κndescribe the coupling due to four-wave-mixing, Γnare the damping factors and f~n{\tilde f_n}f~​n​ represent thermal fluctuations on the transmission line, with n ∈ {s,i}. Pump-depletion was neglected when deriving the equations above. We define the signal power gain Gsas the ratio of output to input power. Assuming no initial idler contribution at the input of the device, we obtain\begin{equation*}{G_{\text{S}}} = {\left| {\cosh (gz) + \frac{{ - {\Gamma _{\text{s}}} + {\Gamma _{\text{i}}} - 2{\text{i}}\Delta {k_{\text{T}}}}}{{4g}}\sinh (gz)} \right|^2}{{\text{e}}^{ - \frac{1}{2}\left( {{\Gamma _{\text{s}}} + {\Gamma _{\text{i}}}} \right)z}},\tag{2}\end{equation*}where g is the gain factor, which depends on the coupling strength, the pump amplitude and the total phase mismatch. The gain spectrum of the JTWPA setup from [4] was calculated according to our model with good agreement to the results in the literature. The gain spectrum of the JTWPA is plotted in Fig. 1 (a) without RPM and in Fig. 1 (b) with RPM, both for the case with and without dissipation.
We study the power dependence of two-tone intermodulation arising in high-Tc superconducting microwave resonators as a function of each tone separately. In the regime where the intermodulation power varies as the square of the input power for equally strong tones, we observe unexpected behavior of the power dependence on the individual tones. We show that this behavior can be understood in terms of a nonlinear inductance which varies linearly with the magnitude of the input current. These findings provide a consistent phenomenological picture of the unusual power dependence and should place constraints on a microscopic description of nonlinear behavior in high-Tc films where the intermodulation products vary like the square of the input power.
High-kinetic-inductance (HKI) transmission lines provide a promising platform for compact nonlinear superconducting microwave devices. Existing theoretical descriptions are typically based on the slowly varying envelope approximation, which becomes inadequate for strongly nonlinear resonant structures with pronounced spatial field variations. Here, we develop a theoretical framework for single-frequency nonlinear wave propagation in HKI transmission lines by formulating the problem as a boundary value problem that retains the full spatial dependence of the electromagnetic fields. Applying the method to resonant transmission-line geometries, we demonstrate power-dependent resonance shifts, bistable transmission solutions, and strongly direction-dependent transport arising from asymmetric impedance barriers, yielding transmission contrasts of up to 93%93\%93% without magnetic bias fields. The framework is further extended to a three-port stub geometry, where nonlinear interference produces shifted anti-resonances and Duffing-like spectral distortions. Our approach provides a versatile tool for the analysis and design of strongly nonlinear superconducting microwave devices.
Josephson transmission lines (JTLs) are commonly used as signal interconnects in single-flux quantum (SFQ) circuits. They are also used, however, to transform arbitrary waveforms into SFQ pulses. Here we use a low-temperature electro-optic sampling system with subpicosecond resolution to observe the response of a JTL to a picosecond input pulse. Previously, this system has been used to directly observe SFQ pulses on niobium transmission lines. In our experiment, niobium metal semiconductor-metal (MSM) photodiodes were connected to 3, 4, 19, and 20-stage JTLs. The MSM photodiodes provided picosecond pulses to excite the JTL. The JTL bias and input pulse amplitude were varied to observe fluxon interaction and time delay of the JTL stages. Observed JTL behavior is dominated by dipolar oscillations radiated from the first stages of the JTL. By varying the values of line inductance and junction capacitance, thus changing the oscillation periods, these oscillations were traced to the coupling of the junction capacitance and line inductance. This is a new feature of the JTL that has thus far not been observed or incorporated into simulations. Our results could have significant implications on the limit of SFQ circuits.