Force sensitivity of multilayer graphene optomechanical devices

Abstract

Mechanical resonators based on low-dimensional materials are promising for force and mass sensing experiments. The force sensitivity in these ultra-light resonators is often limited by the imprecision in the measurement of the vibrations, the fluctuations of the mechanical resonant frequency and the heating induced by the measurement. Here, we strongly couple multilayer graphene resonators to superconducting cavities in order to achieve a displacement sensitivity of 1.3 fm Hz(-1/2). This coupling also allows us to damp the resonator to an average phonon occupation of 7.2. Our best force sensitivity, 390 zN Hz(-1/2) with a bandwidth of 200 Hz, is achieved by balancing measurement imprecision, optomechanical damping, and measurement-induced heating. Our results hold promise for studying the quantum capacitance of graphene, its magnetization, and the electron and nuclear spins of molecules adsorbed on its surface.

Figure 1. Mechanical displacement and force sensitivity.

(a) Mechanical displacement spectrum S_z close to the mechanical resonance frequency ω_m/2π. The total displacement spectral density at ω_m is the sum of the displacement noise and the displacement imprecision . (b) Corresponding force sensitivity (dark grey). The individual components are the thermal force noise (dark yellow) and the imprecision force noise (turquoise), given by equations (1) and (2), respectively. The quantum back-action noise is neglected for simplicity. For the plots most of the parameters are those of device B, but we estimate the mass assuming that the graphene flake is a single layer. Further we choose n_add=0.5, T_bath=0.015 K, and n_p=2·10⁵ in a (see text).

Figure 2. Device and characterization.

(a) False-colour image of the device. The cavity is coloured in dark yellow. The graphene flake is clamped in between niobium support electrodes (grey) and cross-linked poly(methyl metracylate) (turquoise). The scale bar is 5 μm. (b) Schematic cross-section of the graphene resonator along the white dashed dotted line in a. (c) Schematic of the measurement circuit. The graphene mechanical resonator is coupled to the superconducting LC cavity through the capacitance C_m. The separation d between the suspended graphene flake and the cavity counter electrode is controlled by the constant voltage V_g. The cavity is pumped with a pump tone at ω_p and the output signal is amplified at 3 K. (d) Reflection coefficient |S₁₁|² and (e) reflected phase Δφ₁₁ of the superconducting cavity of device A at V_g=3.002 V. The dark yellow lines are fits to the data using κ_int/2π=950 kHz and κ_ext/2π=850 kHz using equation (7) (see Methods). (f) Driven vibration amplitude of the graphene resonator of device A as a function of drive frequency. The driving voltage is 22 nV and V_g=3.002 V. The dark yellow line is a lorentzian fit to the data. (g) Resonant frequency ω_c/2π of the superconducting cavity as a function of V_g. (h) Resonant frequency ω_m/2π of the graphene resonator as a function of V_g. The black line is the V_g dependence of ω_m expected from electrostatic softening (see Supplementary Note 1).

Figure 3. Effective mechanical energy decay rates extracted from ring-down measurements.

Mechanical dissipation rate measured on device A with the ring-down technique as a function of the number n_p of pump photons in the cavity at V_g=0 V and V_g=3.002 V, where n_p is proportional to the microwave power P_in applied at the input of the cryostat (see Supplementary Note 3). Red and blue data points correspond to red and blue detuned pumping, respectively. The measurements are well described by (red and blue lines) using g₀/2π=9.7 Hz in a and g₀/2π=42.6 Hz in b. The inset in b shows a ring-down measurement for n_p=1.4·10⁶. We plot the normalized vibration amplitude as a function of time t. The resonator is driven with a capacitive driving force for t<t₀. At t₀ the drive is switched off and the vibration amplitude decays freely (t>t₀). We fit the data with an exponential decay (black line) using with a decay rate . The vibration amplitude in ring-down measurements is larger than that in undriven displacement spectra, so that the motion in ring-down measurements can be resolved with lower n_p.

Figure 4. Thermal calibration and sideband cooling of fundamental mechanical mode with red-detuned pumping.

(a) Selected thermo-mechanical noise spectra for different temperatures and n_p=6·10⁴. (b) Plot of the measured mechanical mode temperature of device A, expressed in phonon occupation n_m, as a function of cryostat temperature at V_g=3.002 V where ω_m/2π=53.7 MHz and n_p=6·10⁴. On the right y-axis, we display the variance of the vibration amplitude 〈z²〉, which is obtained by integrating the thermal resonance, as is shown in a. The phonon occupation is quantified with (see Supplementary Note 3). The error bars are given by the standard deviation of 5 spectral measurements. (c) Mechanical displacement spectral density S_z measured for different pump photon number. The cryostat temperature is 15 mK. Note that the curves are not offset. (d) Displacement imprecision as a function of cavity pump photon population. The line is a fit of equation (3) with n_add=32. (e) Average phonon number n_m as a function of n_p. The error bars are given by the standard deviation of five spectral measurements.

Figure 5. Characterization of the force sensitivity.

(a) Force sensitivity as a function of cavity pump photon population measured when pumping the cavity on the red sideband. (b) Imprecision force noise (turquoise) and thermal force noise (dark yellow) versus n_p. The data in a,b are fitted to equations (2), (6). (c) Product of the bath temperature T_bath and the intrinsic mechanical decay rate as a function of cavity pump photon occupation. The line is a fit to the data. (d) Effective spectral mechanical line width and energy decay as a function of n_p. The data are fitted to with δΓ_noise/2π=8.7 kHz (red line). (e–h) Equivalent to (a–d) but for device B. The lowest value for the force sensitivity in e is . In e and f the data are fitted with n_add=22 and in h we use g₀/2π=7.3 Hz, κ/2π=2.5 MHz and δΓ_noise/2π=0.145 kHz. All the measurements on device A are performed at V_g=3.002 V and on device B at V_g=0 V. The cryostat temperature is 15 mK.