TLS-induced thermal nonlinearity in a micro-mechanical resonator

TL;DR

This work reveals a thermally driven, mixed reactive-dissipative nonlinearity in millikelvin quartz phononic crystal resonators arising from readout-power heating of a TLS ensemble. By marrying the standard tunneling model with a thermal-conductance description and solving self-consistently for TLS temperature, phonon occupancy, and resonance properties, the authors reproduce the observed power-dependent frequency shifts, linewidth changes, and hysteresis across wide dynamical ranges. They identify TLS relaxation damping as the primary factor limiting mechanical coherence to about $Q\sim 10^7$ and demonstrate discrete TLS signatures that can dominate the nonlinear response. The results yield a phase diagram predicting when TLS-induced nonlinearity is significant and suggest paths to suppress or harness these effects for sensing or TLS-based parametric amplification in quantum acoustics platforms.

Abstract

We present experimental evidence of a thermally-driven amplitude-frequency nonlinearity in a thin-film quartz phononic crystal resonator at millikelvin temperatures. The nonlinear response arises from the coupling of the mechanical mode to an ensemble of microscopic two-level system defects driven out of equilibrium by a microwave drive. In contrast to the conventional Duffing oscillator, the observed nonlinearity exhibits a mixed reactive-dissipative character. Notably, the reactive effect can manifest as either a softening or hardening of the mechanical resonance, depending on the ratio of thermal to phonon energy. By combining the standard TLS theory with a thermal conductance model, the measured power-dependent response is quantitatively reproduced and readout-enhanced relaxation damping from off-resonant TLSs is identified as the primary mechanism limiting mechanical coherence. Within this framework, we delineate the conditions under which similar systems will realize this nonlinearity.

Figures (9)

• Figure 1: A mixed reactive-dissipative thermal nonlinearity in a thin-film quartz PCR. (a) False-colored scanning electron micrograph of a PCR: a 1-µm-thick freestanding quartz beam (blue) patterned with aluminum electrodes (red) to excite the fundamental extensional mode (resonance frequency $f_r$) of the central block. A continuous microwave tone ($f\approx f_r$) is applied, and the reflected signal is analyzed to extract $S_{11}$. (b) Measured $|S_{11}|(f)$ for increasing drive power $P_s$ at select baseplate temperatures $T_0=25$-$100$ mK (Dataset #1). Traces were acquired by sweeping $f$ up (solid) and down (dotted) over a 10 kHz window centered at $f_r \approx 520.81$ MHz. With increasing $P_s$, the resonance dip becomes asymmetric and shifts to higher frequency. Curves are vertically offset in 5 dB steps for clarity; insets compare the lineshape at $P_s=-141$ dBm for $T_0=25$ and 50 mK. (c, e) The temperature-dependent internal quality factor $Q_i$ and resonance frequency shift $\Delta f_r$ measured in the low-power limit with $\overline{n}\approx 300$ (Dataset #2, teal disks). Black solid lines show fits to Eqs. \ref{['dfTLS']}–\ref{['Qimodel']}; in (c), resonant-TLS (red dashed) and relaxation-TLS (blue dashed) contributions are indicated. Resonant TLSs within the phononic bandgap are assumed to thermalize at an elevated temperature $T_0'=\sqrt{T_0^2+T_\mathrm{sat}^2}$ with $T_\mathrm{sat}=30$ mK; the red dotted line shows the ideal case $T_0'=T_0$. (d) Schematic of the coupling between external ($f, P_s$), internal ($\overline{n}, T$), and resonator ($f_r, Q$) parameters giving rise to reactive and dissipative nonlinearities (NL). (e) Right axis: the temperature coefficient of frequency, $TCF \equiv f_r^{-1} \textrm{d}\Delta f_r/\textrm{d}T$ (red solid line), and its high- and low-temperature approximations (Eqs. \ref{['TCFhighT']}, \ref{['TCFlowT']}, red dotted lines) as a function of $T_0$. The sign change at $T_c \approx 0.44 h f_{r,0}/k_B \approx 11$ mK marks a crossover from softening ($TCF<0$) to hardening ($TCF>0$).
• Figure 2: Nonlinear resonance frequency shift with power. (a) Resonance frequency $f_r$ versus applied probe power $P_s$ at temperatures $T_0 = 25, 50, 75, 100$ mK, extracted from the $S_{11}$ data in Fig. \ref{['fig1']}(b). $f_r$ is shown relative to its extrapolated zero-temperature value $f_{r,0} = 520.8083(1)$ MHz. Disk (square) markers correspond to upward (downward) frequency sweeps, with dashed (dotted) lines showing fits to the numerical model described in Appendix \ref{['app:frPssolver']}. A smooth step near $P_s \approx -140$ dBm is captured by including a discrete TLS at $546$ MHz; lighter solid lines show the model without this additional TLS. (b) The asymmetry and hysteresis in the magnitude and phase of $S_{11}$ at $P_s=-109$ dBm and $T_0=50$ mK are well-reproduced by the model.
• Figure 3: Power dependence of the $502.06$ MHz PCR probed at a fixed frequency $f=502.0657$ MHz and with a refrigerator temperature $T_0=25$ mK. (a) The average phonon occupancy $\overline{n}$, (b) effective TLS temperature $T$, (c) internal quality factor $Q_i$, (d) resonator--probe-tone detuning $f_r-f$, (e) magnitude of the reflection coefficient $|S_{11}(f)|$, (f) resonance frequency shift in number of linewidths, $\Delta \omega_r(T,T_0)/\kappa(T)=2\pi(f_r(T)-f_r(T_0))/\kappa(T)$, as a function of the probe power $P_s$. Teal and purple dots are data extracted from respectively ringdown (RD) and vector network analyzer (VNA) measurements. The black solid lines are fits to the numerical model. The dashed red and blue lines in (c) show the contributions from respectively $Q_\textrm{res}$ and $Q_\textrm{rel}$, corresponding to TLS resonant and relaxation damping processes, and the black dotted line indicates the value of $Q_e$. The two arrows in (d) indicate possible frequency jumps due to gradual saturation with $P_s$ of individual TLSs strongly-coupled to the resonator. The model parameters are summarized in Table \ref{['tableFitParams']}.
• Figure 4: Simulated impact of TLS loss tangent on the thermal nonlinearity strength. (a,b) The resonance frequency shift in number of linewidths $\Delta\omega_r(T,T_0)/\kappa=2\pi(f_r(T)-f_r(T_0))/\kappa(T)=-y$ and (c,d) the internal quality factor $Q_i$ as a function of the probe power $P_s$ at the sample and the TLS loss tangent $F\delta_0$ for two illustrative values of base temperature $T_0=25$ mK (a,c) and $300$ mK (b,d). In (a,b) the black dotted line shows the $y=y_c$ contour that delimits the resonance bistability region at high $P_s$. The three colored dashed lines represent contours of achieved phonon occupancies, $\overline{n}=1,n_s,n_h$ (respectively green, magenta and purple) and demarcate four qualitatively different regimes (see Supplemental Material S4). In (c,d) the black dash-dotted line tracks the maximum value of achieved $Q_i$ in the $P_s\hbox{-}F\delta_0$ space. (e) The achieved fractional detuning $y(T)=Q(T) x(T)$ and (f) the corresponding magnitude of the reflection coefficient $|S_{11}|$ as a function of the applied fractional detuning $y_0(T_0)=Q(T_0)x_0(T_0)$ for three illustrative points shown with the black, blue and red markers in (a-d) corresponding to $F\delta_0=10^{-6}$ and $P_s=-130, -118, -118$ dBm respectively. The responses for upward/downward linear sweeps of $y_0$ are distinguished using dark/light colors.
• Figure A1: Iterative solver for the TLS nonlinearity. (a) Overview of the numerical method: starting from an initial temperature $T_0$, the TLS saturation parameter $\alpha$ is updated iteratively to determine the total quality factor $Q(T_0)$ corresponding to an input power $P_s$. Given $Q(T_0)$, the resulting dissipated power is used to estimate the TLS temperature $T$ and corresponding detuning $x(T)$, which is refined through a second iteration loop. This process continues until convergence of $\alpha$ and $x$, yielding final values of $Q(T)$, $f_r(T)$, and $S_{11}$. (b) Detailed implementation of the $\alpha$- and $x$-iterations, shown as red and blue loops, respectively. Initialization and termination steps are indicated by dashed and double-lined arrows. Labels of the relevant equations for each step are shown in dark blue.