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arXiv:2202.04227

Feedback stabilization of the resonant frequency in tunable microwave cavities with single-photon occupancy

S. Kanhirathingal, B. Thyagarajan, B. L. Brock, Juliang Li, E. Jeffrey, M. P. Blencowe, J. Mutus, A. J. Rimberg

Published Feb 9, 2022• 0 citations
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Abstract

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 $n=10$, and a bandwidth of 11 Hz for average $n=1$. 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 $1/f$-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.

DOI
10.48550/arxiv.2202.04227

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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.
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.
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.
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)].
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.
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)
Microwave cavities are commonly used in many experiments, including optomechanics, magnetic field sensing, magnomechanics, and circuit quantum electrodynamics. Noise, such as variations in the magnetic field or mechanical vibrations, can cause fluctuations of the natural frequency of the cavity, creating challenges in operating them in experiments. To overcome these challenges, we demonstrate a dynamic feedback system implemented by the locking of a microwave drive to the noisy cavity. A homodyne-interferometer scheme monitors the cavity resonance fluctuations due to low-frequency noise, which is mitigated by frequency modulating the microwave generator. The feedback has a bandwidth of 400 Hz, with a reduction of cavity fluctuations by $85\mathrm{%}$ integrating up to a bandwidth of 2 kHz. Moreover, the cavity resonance frequency fluctuations are reduced by $73\mathrm{%}$. This scheme can be scaled to enable multitone experiments locked to the same feedback signal. As a demonstration, we apply the feedback to an optomechanical experiment and implement a cavity-locked multitone mechanical measurement. As low-frequency cavity frequency noise can be a limiting factor in many experiments, the multitone microwave locking technique presented here is expected to be relevant for a wide range of microwave-cavity experiments.