Resonant Phase Matching of Josephson Junction Traveling Wave Parametric Amplifiers

We propose a technique to overcome phase mismatch in Josephson-junction traveling wave parametric amplifiers in order to achieve high gain over a broad bandwidth. Using “resonant phase matching,” we design a compact superconducting device consisting of a transmission line with subwavelength resonant inclusions that simultaneously achieves a gain of 20 dB, an instantaneous bandwidth of 3 GHz, and a saturation power of $-98\mathrm{dBm}$. Such an amplifier is well suited to cryogenic broadband microwave measurements such as the multiplexed readout of quantum coherent circuits based on superconducting, semiconducting, or nanomechanical elements, as well as traditional astronomical detectors.

Figure 1

Resonantly phase-matched traveling wave parametric amplifier. (a) Signal photons are amplified through a nonlinear interaction with a strong pump as they propagate along the 2000 unit cell transmission line with a lattice period of $a=10\mu \mathrm{m}$. (b) In each unit cell a Josephson junction, a nonlinear inductor, is capacitively coupled to an $LC$ resonator. The circuit parameters are $C_j=329\mathrm{fF}$, $L=100\mathrm{pH}$, $C=39\mathrm{fF}$, $C_c=10\mathrm{fF}$, $C_r=7.036\mathrm{pF}$, $L_r=100\mathrm{pH}$, $I_0=3.29\mu \mathrm{A}$. (c) The $LC$ circuit opens a stop band (red line) in the dispersion relation of the TWPA (black dashed line) whose frequency depends on the circuit parameters. In the inset, we plot the pump frequency to phase match a pump current of $0.3I_0$ (blue), $0.5I_0$ (purple), and $0.7I_0$ (green), where $I_0$ is the junction critical current.

Figure 2

Gain of the RPM TWPA. (a) The gain as a function of signal frequency in dB with RPM (purple line) and without (black dashed line) for a pump current of $0.5I_0$ and a pump frequency of 5.97 GHz. (b) The phase mismatch with (purple line) and without (black dashed line) RPM. (c) The peak gain as a function of pump current without RPM (black dashed line) and with RPM for three different pump frequencies, which phase match the parametric amplification for pump currents of $0.3I_0$ (red line), $0.5I_0$ (purple line), and $0.7I_0$ (green line). (d) The phase mismatch as a function of pump current. The dots mark the pump current where the parametric amplification is perfectly phase matched.

Figure 3

Effect of pump depletion on dynamic range. (a) The gain as a function of input signal current (normalized to the pump current) for a small signal gain of 10, 15, and 20 dB obtained with a pump current of $0.5I_0$ and device lengths of 1150, 1530, and 1900 unit cells. The approximation for the gain depletion (dashed lines) from Eq. (4) is in excellent agreement with the result obtained by solving the full nonlinear dynamics (solid lines). (b) $P_{1\mathrm{dB}}$, the input signal power where the gain decreases by 1 dB, as a function of junction critical current $I_0$ with the pump current fixed at $0.5I_0$. The black dashed line corresponds to the device considered in this Letter, which has a gain compression point of $P_{1\mathrm{dB}}=-87$, $-93$, and $-98$ dBm, for a gain of 10, 15, and 20 dB for a junction critical current of $I_0=3.29\mu \mathrm{A}$.

Figure 4

Band structure of resonantly phase matched and periodically loaded TWPAs. The parametric amplifier without a resonant element [(a), black only], with a resonant element [(a), black and red], and with periodic loading (b): every 37th unit cell has a slightly different capacitance and inductance (green). (c) The wave vector as a function of frequency for the TWPA (black dashed line), RPM TWPA (red line), and TWPA with periodic loading (green lines). The yellow shaded region indicates the photonic band gap due to the periodic loading. The main difference between the photonic band gap and the resonator is the edge of the Brillouin zone. For the resonant element, the zone boundary is at $\pi /a$, while the other periodically loaded transmission line has zone boundary at $\pi /(37a)$, which is determined by the period of the loading. The effective momentum due to the periodic loading is close to $k_p$, which may phase match backward parametric amplification.