Quantitative calibration of a TWPA applied to an optomechanical platform

TL;DR

This work presents a thorough absolute calibration of a near-quantum-limited microwave optomechanical detector built around a Traveling Wave Parametric Amplifier (TWPA). By modeling Two-Level System (TLS)–driven absorption in both the superconducting cavity and the TWPA, the authors achieve quantitative extraction of the mechanical mode's phonon population across $4~\mathrm{mK}$ to $400~\mathrm{mK}$ with an uncertainty of $\pm 20\%$. The calibration hinges on a detailed TLS-based description of on-chip losses, enabling the conversion factor $\mathcal{M}$ to be used to obtain $n_{ph}$ from measured sideband signals via $A_{ph}=n_{ph}\,\mathcal{M}P_{in}$, and on calibrations of $g_0$, $\kappa_{ext}$, and $\kappa_{tot}$. The approach reveals that TLSs induce significant, power- and temperature-dependent losses in both the cavity and TWPA, and demonstrates robust absolute phonon measurements enabling quantitative microwave optomechanics and high-sensitivity quantum sensing applications.

Abstract

In the last decade, the microwave quantum electronics toolbox has been enriched with quantum-limited detection devices such as Traveling Wave Parametric Amplifiers (TWPAs). The extreme sensitivity they provide is not only mandatory for some physics applications within quantum information processing, but is also the key element that will determine the detection limit of quantum sensing setups. In the framework of microwave optomechanical systems, an unprecedented range of small motions and forces is accessible, for which a specific quantitative calibration becomes necessary. We report on near quantum-limited measurements performed with an aluminum drumhead mechanical device within the temperature range 4 mK - 400 mK. The whole setup is carefully calibrated, especially taking into account the power-dependence of microwave absorption in the superconducting optomechanical cavity. This effect is commonly attributed to Two-Level-Systems (TLSs) present in the metal oxide. We demonstrate that a similar feature exists in the TWPA, and can be phenomenologically fit with adapted expressions. If not taken into account, the error on the signal strength can be as large as a factor of about 2, which is unacceptable for quantitative experiments. The power and temperature dependence is studied over the full parameter range, leading to an absolute definition of phonon population (i.e. Brownian motion amplitude), with an uncertainty +- 20 % limited by sources of noise internal to the optomechanical element.

Figures (7)

• Figure 1: Low temperature microwave circuitry: after splitting the optomechanical pump signal into injection line and cancellation line, both signals are sent into the cryostat through attenuators and circulators. Both are recombined before the TWPA to avoid any saturation. Another circulator is implemented downstream to protect the TWPA from the noise generated by the cryogenic HEMT. The total attenuation in injection and gain in detection are calibrated within an uncertainty of about $\pm1$ dB. The different anchoring stages of the dilution unit are specified.
• Figure 2: Typical optomechanical data taken at $T = 20~$mK. a) Cavity measurement using a large probe tone ($1 \cdot 10^{-4}~$nW, and no pump); the resonance is fit with a Lorentzian function defined in Eq. (\ref{['S11']}); note the logarithmic scale. b) Raw mechanical peak obtained with the blue pumping scheme ($P_{in} = 2.5 \cdot 10^{-3}~$nW), and its Lorentz fit. Integrating this peak leads to the sideband area $A_{sdb}$ expressed in photons/s, and described by Eq. (\ref{['sidebandA']}), taking into account the proper gains and conversion factors. c) $\Gamma_{e\!f\!f}$ measurement using blue and red schemes (respectively anti-damping/damping processes) as a function of the on-chip applied power (corrected from injection attenuation). The absolute value of the slope directly gives $g_0$, the optomechanical coupling constant, through Eq. (\ref{['g0']}). d) Measured signal area for blue- and red- pumping configurations, with fits corresponding to Eq. (\ref{['sidebandA']}). In c) and d) the dashed vertical corresponds to the threshold for self-oscillation (see text).
• Figure 3: Main: Total damping rate of the cavity measured as a function of pump applied power (blue scheme, using a small probe tone, $5\cdot10^{-7}~$nW) at $T = 150$ mK. The fit corresponds to Eq. (\ref{['kappatot']}). The gray vertical corresponds to the inflection point defined by the critical power $P_{cav}^0$. A dramatic drop ($\sim 50$ %) in the total damping rate is observed while increasing the power (see text). Inset: temperature dependence demonstrating the characteristic $\tanh$ Eq. (\ref{['tanh']}) shape (see text).
• Figure 4: Main: background noise measurements as a function of cryogenic temperature. The $T_{cryo} \rightarrow 0~$K interpolation corresponds to the fundamental limit of the detection system, here about $3.5~$SQL. Note that the dashed line is only a guide: the temperature-dependence being due to both the last 50$~\Omega$ load of the setup, but also to the distributed losses within the TWPA itself, which we do not intend to model here. Inset: Typical result of a TWPA pump parameter tuning scan at $T = 20$ mK. The yellow part of the graph corresponds to the highest values of TWPA gain, while the red cross stands for the optimal set of parameters chosen (see text for further details), giving a gain of $18 \pm 1~$dB.
• Figure 5: Main: TWPA transmission around 5$~$GHz (with TWPA pump off) measured as a function of applied probe power, for various temperatures (and optomechanics pump off). Fitting functions correspond to Eqs. (\ref{['eq:fitTWPA']},\ref{['eq:fitTWPAT']}). The drop reaches about $40~\%$ at $20~$mK for low powers (see text). Inset: TWPA transmission at 150$~$mK (with TWPA pump off) measured with constant small probe power (about $3\cdot10^{-17}~$W), but as a function of the detuning of a large pump tone from the probe (optomechanics pump, about $2\cdot10^{-12}~$W). While detuning the pump, the losses reappear (for details, see text).