a, Schematic of the measurement circuit. b, A lateral displacement by dy (blue arrows) with an associated contraction and expansion by dz in a GaAs crystal generates a shear strain S_yz = 2dz/w at the upper surface that causes corresponding polarizations P_z and P_x = S_yz. The 2DEG free charge (+, −) will screen the P_x polarization charge in most areas but not under the QPC or its gates, resulting in a potential difference ε across the QPC. c, Reservoirs L and R are usually treated as having a fixed chemical potential, which is ordinarily a good approximation. Here, however, we cannot ignore the presence of the ohmic contacts (shown in a) that may themselves have a non-negligible impedance of several kilo-ohms. We assume that the metallic wires E and C connected to the crystal are held at a fixed potential difference and that the potentials of L and R are free to change as electrons tunnel through the QPC. The electron distribution functions, n_E and n_C, for E and C are Fermi distributions, whereas the corresponding distributions, n_L and n_R, for L and R are not. We treat the QPC as a tunnel barrier with coupling Ω_LR and the ohmic contacts as tunnel barriers with couplings Ω_EL and Ω_RC. d, The back-action associated with the measurement process naturally leads to the presence of a feedback loop. If a displacement dz of the crystal exists, piezoelectric transduction causes an energy difference between reservoirs L and R. The resulting flow of current, I, through the QPC can be used to obtain information regarding dz. Quantum statistical fluctuations in electron tunnelling invariably result in shot noise. Piezoelectric transduction leads to a fluctuating force, dF, that imparts a momentum kick to the lattice, affecting subsequent values of the displacement dz and closing the feedback loop. The energy needed to maintain the current I is provided by a bias voltage, V_CE.

a, Measurements of G_QPC versus gate voltage, V_g, at zero magnetic field, showing standard one-dimensional subbands. Inset, G_QPC after exposure of the sample to light, showing multiple conductance plateaux. b, Nonlinear differential conductance, G_QPC(V_dc, V_g). Each red trace corresponds to a different value of V_g (ΔV_g = 1 mV) and is plotted without offset. The vertical blue lines indicate the estimated root-mean-squared radio-frequency voltage, V_rf, applied to the QPC for noise measurements in c. c, Output power spectrum, P_n, of the RF-QPC for input power P_in = −78 dBm and G_QPC ≈ 0.5G₀. The frequency-dependent features are spontaneously generated and clearly exceed the noise floor of the cryogenic high-electron-mobility transistor (HEMT) amplifier, .

a, The reference modulation signal, P_s, is generated by applying an a.c. excitation to the QPC gates. b, Power dependence. The dashed lines are guides to the eye showing that P_s (red squares) scales as ~P_in whereas the integrated excess noise, (blue circles), scales as , the first signature of shot noise. c, Partition dependence. The reflected power spectra, P_n, for P_in = −88 dBm and G_QPC ≈ 0 (red), G_QPC ≈ 0.5G₀ (black) and G_QPC ≈ G₀ (green), showing that P_n is minimal for fully open or closed channels and maximal for half-open channels, the second signature of shot noise. d, Quantitative comparison of measured shot noise with theory. The measured integrated excess noise, (blue circles), and the calculated integrated noise power, ∫(2L/C_pZ₀)S_I(ω, ω₀) df (blue dashes), as functions of QPC differential conductance, G_QPC. We shift the calculated noise power downward by 3.9 dB but use no other fitting parameter. The suppression of the measured shot noise relative to the single-particle theory for G_QPC   0.5 has been observed by others and attributed to the many-body physics of the 0.7 structure³⁰. All data are from sample A.

a, The reflected power spectrum, P_n, from sample A reproduced with the inclusion of data from an addition input power excitation (P_in = −98 dBm). The diagram of the sample geometry includes the ohmic contacts (blue boxes), the QPC (yellow rectangles) and the linear dimensions. The calculated value for the excited three-dimensional acoustic mode (f_a = 510 kHz) falls almost exactly on the shot noise peak. As the power is increased (from red to black) the broadband shot noise decreases whereas the total increases, becoming concentrated at the resonant frequency. b, The shot noise spectrum for sample B, whose dimensions were chosen to provide ~1 MHz of bandwidth in which the shot noise is suppressed. The sample dimensions shown in the diagram correspond to a predicted resonant frequency (f_b = 1.07 MHz) remarkably close to the shot noise peak. The sharp features marked with asterisks are modulation noise rather than shot noise and scale as P_in, not . c, The reflected power spectra, P_n, for QPC 1 and QPC 2 (inset) in sample C when G_QPC ≈ 0 (red), G_QPC ≈ 0.5G₀ (black) and G_QPC ≈ G₀ (green), showing partition dependence for QPC 1 but no shot noise features for QPC 2. This confirms the existence of a back-action-mediated feedback loop between the shot noise and the normal vibrational mode. Without a bias across QPC 2, energy transferred to the Ey-1 mode cannot be returned to the current, thereby breaking the feedback loop. The sample dimensions shown in the diagram again correctly predict the location of the peak in the shot noise. d, Left axis: frequency-dependent Fano factor, , for sample A for P_in = −68 dBm, showing drastically super-Poissonian ( ≳100) and sub-Poissonian ( ) noise as a function of frequency (on a logarithmic scale). The red dotted line indicates Poissonian noise and the green dashed line gives the Fano factor expected for an uncoupled detector. Right axis: displacement dy versus frequency, showing the strongly non-thermal nature of the resonator dynamics. Inset, displacements dy and dz versus input power for sample A. The dashed line is a guide to the eye showing scaling as . In the diagram of the sample, blue arrows indicate the dy and dz deformations of the crystal in the Ey-1 mode.