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arXiv:2011.06298 · Physical Review Applied

Nonlinear Charge- and Flux-Tunable Cavity Derived From an Embedded Cooper-Pair Transistor

B.L. Brock, Juliang Li, S. Kanhirathingal, B. Thyagarajan, William F. Braasch, M.P. Blencowe, A.J. Rimberg

Published Apr 5, 2021• 7 citations
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Abstract

We introduce the cavity-embedded Cooper-pair transistor (cCPT), a device that behaves as a highly nonlinear microwave cavity whose resonant frequency can be tuned both by charging a gate capacitor and by threading flux through a superconducting loop. We characterize this device and find excellent agreement between theory and experiment. A key difficulty in this characterization is the presence of frequency fluctuations comparable in scale to the cavity linewidth, which deform our measured resonance circles in accordance with recent theoretical predictions [Brock et al., Phys. Rev. Appl. 14, 054026 (2020)]. By measuring the power spectral density of these frequency fluctuations at carefully chosen points in parameter space, we find that they are primarily a result of the $1/f$ charge and flux noise common in solid-state devices. Notably, we also observe key signatures of frequency fluctuations induced by quantum fluctuations in the cavity field via the Kerr nonlinearity.

DOI
10.1103/physrevapplied.15.044009

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quant-pharXiv:2608.03440•Aug 5, 2026•0 citations
We introduce the cavity-embedded Cooper-pair transistor (cCPT), a device that behaves as a highly nonlinear microwave cavity whose resonant frequency can be tuned both by charging a gate capacitor and by threading flux through a superconducting loop. We characterize this device and find excellent agreement between theory and experiment. A key difficulty in this characterization is the presence of frequency fluctuations comparable in scale to the cavity linewidth, which deform our measured resonance circles in accordance with recent theoretical predictions [Brock et al., Phys. Rev. Appl. 14, 054026 (2020)]. By measuring the power spectral density of these frequency fluctuations at carefully chosen points in parameter space, we find that they are primarily a result of the 1/f1/f1/f charge and flux noise common in solid-state devices. Notably, we also observe key signatures of frequency fluctuations induced by quantum fluctuations in the cavity field via the Kerr nonlinearity.
We successfully demonstrate low-frequency noise suppression in the resonant-frequency fluctuations of a cavity-embedded Cooper-pair transistor (cCPT) driven at single-photon occupancy. In particular, we report a reduction in the resonant-frequency fluctuations caused by the internal charge noise over a bandwidth of approximately 1.4kHz1.4\phantom{\rule{0.2em}{0ex}}\text{kHz}1.4kHz when the cavity is driven at an average photon number n=10n=10n=10 and a bandwidth of 11 Hz for an average n=1n=1n=1. The gate-dependent tunability of the cCPT allows us to implement a feedback scheme derived from the Pound-Drever-Hall locking technique. This reduces fluctuations due to intrinsic charge noise, which otherwise interfere with the operation of the cCPT as a near quantum-limited electrometer. Our technique can be generalized to achieve frequency stabilization in tunable microwave resonators that play a vital role in today's quantum computing architectures, thereby moderating the limitations in detection caused by the intrinsic 1/f1/f1/f noise on such circuit devices. The work discusses the various aspects relating to the operation of a fully functional feedback loop down to the single-photon level.
Using an operator scattering approach, we analyze the quantum dynamics of an ultrasensitive electrometer—a Cooper pair transistor embedded in a quarter-wave microwave cavity (cCPT). While the cCPT is inherently a tunable, strongly nonlinear system affording a diverse range of functionalities, we restrict our present analysis to a necessary first investigation of its linear charge sensing capabilities, limiting to low pump powers corresponding to an average cavity photon number of ≲1. Assuming realizable cCPT parameters [B. L. Brock et al., Phys. Rev. Applied 15, 044009 (2021)], and not including noise from the subsequent amplifier chain, we predict the fundamental, photon shot noise-limited charge sensitivity to be 0.12μe/Hz when the pumped cavity has an average of one photon and the cCPT is operated close to charge degeneracy. This is to be compared with a first reported charge sensitivity value of 14μe/Hz in the single-photon regime [B. L. Brock et al., arXiv:2102.05362 (2021)].
We successfully demonstrate low-frequency noise suppression in the resonant frequency fluctuations of a cavity-embedded Cooper pair transistor (cCPT) driven at single-photon occupancy. In particular, we report a reduction in the resonant frequency fluctuations caused by the internal charge noise over a bandwidth of $\sim$1.4 kHz when the cavity is driven at an average photon number n=10n=10n=10, and a bandwidth of 11 Hz for average n=1n=1n=1. The gate-dependent tunability of the cCPT allows us to implement a feedback-scheme, derived from the Pound-Drever-Hall locking technique. This reduces fluctuations due to intrinsic charge-noise, which otherwise interferes with the cCPT's operation as a near quantum-limited electrometer. We believe our technique can be generalized to achieve frequency stabilization in tunable microwave resonators that play a vital role in today's quantum computing architecture, thereby moderating the limitations in detection caused by the intrinsic 1/f1/f1/f-noise on such circuit devices. The work discusses the various aspects relating to the operation of a fully functional feedback loop down to the single-photon level.
Using a field operator scattering approach, we analyze the quantum dynamics of an ultrasensitive electrometer - a Cooper pair transistor embedded in a quarter-wave microwave cavity (cCPT). While the cCPT is inherently a strongly nonlinear system, we restrict our present analysis to a necessary first investigation of its linear charge sensing capabilities, limiting to low pump powers corresponding to an average cavity photon number ≲1\lesssim 1≲1. Assuming realizable cCPT parameters similar to those obtained from a characterization of an actual experimental device (B. L. Brock et al., arXiv:2011.06298), we predict the fundamental, quantum noise-limited charge sensitivity to be 0.12 μe/Hz0.12\, \mu{\mathrm{e}}/\sqrt{\mathrm{Hz}}0.12μe/Hz​ when the pumped cavity has an average of one photon and the cCPT is operated close to charge degeneracy. This is to be compared with our first reported charge sensitivity value 14 μe/Hz14\, \mu{\mathrm{e}}/\sqrt{\mathrm{Hz}}14μe/Hz​ in the single-photon regime (B. L. Brock {\it et al}., arXiv:2102.05362)
Operating the cavity-embedded Cooper-pair transistor (CCPT) in the Kerr bistable regime, we demonstrate single-shot resolution between two charge states that are 0.09e0.09e0.09e apart. The measurement is performed with $94\mathrm{%}$ fidelity in a duration of around $3\phantom{\rule{0.2em}{0ex}}\text{\ensuremath{\mu}}\mathrm{s}$. This is done with a drive power that corresponds to only 20 intracavity photons on average in the high oscillation amplitude state of the CCPT, which is orders of magnitude smaller than that in rf single electron transistors. We find that the limiting factor for this mode of operation of the CCPT is the spontaneous fluctuation-induced switching between the two metastable oscillation amplitude states. We present empirical data on the variation of the switching dynamics with drive parameters and CCPT dc bias.
Resonance-frequency fluctuations can limit the sensitivity and stability of superconducting microwave cavities used for qubit readout, optomechanical displacement sensing, and magnetic flux detection. Here, we demonstrate the suppression of resonance-frequency fluctuations by locking a noisy nonlinear superconducting microwave cavity to a strong pump tone. The Kerr nonlinearity of this system, whereby the resonance frequency depends on the intracavity field amplitude, gives rise to an intrinsic feedback mechanism that enables passive stabilization without active external feedback. Using two-tone spectroscopy, we experimentally characterize the intrinsic nonlinear feedback mechanism and investigate its temporal stability through Allan deviation analysis. The frequency fluctuations of the locked cavity mode are reduced by nearly two orders of magnitude, reaching the 1/f1/f1/f noise floor, without continuous frequency tracking or active control. Kerr locking provides a general approach for self-stabilizing nonlinear microwave resonators by suppressing low-frequency cavity noise while preserving sensitivity to signals outside the locking bandwidth. This approach may benefit a broad range of systems, including SQUID-based resonators, optomechanical devices, and parametric amplifiers.