Phonon-Number-Sensitive Electromechanics

We use the strong intrinsic nonlinearity of a microwave superconducting qubit with a 4 GHz transition frequency to directly detect and control the energy of a micromechanical oscillator vibrating at 25 MHz. The qubit and the oscillator are coupled electrostatically at a rate of approximately $2\pi \times 22\mathrm{MHz}$. In this far off-resonant regime, the qubit frequency is shifted by 0.52 MHz per oscillator phonon, or about 14% of the 3.7 MHz qubit linewidth. The qubit behaves as a vibrational energy detector and from its line shape we extract the phonon number distribution of the oscillator. We manipulate this distribution by driving number state sensitive sideband transitions and creating profoundly nonthermal states. Finally, by driving the lower frequency sideband transition, we cool the oscillator and increase its ground state population up to $0.48\pm 0.13$, close to a factor of 8 above its value at thermal equilibrium. These results demonstrate a new class of electromechanics experiments that are a promising strategy for quantum nondemolition measurements and nonclassical state preparation.

Figure 1

(a) False-colored scanning electron micrograph (at an angle) of the micromechanical oscillator (blue) suspended above two electrodes and forming a vacuum gap capacitor. (b) Top view of the device showing the superconducting qubit electrodes (yellow and green), shunted by two Josephson junctions (JJ) in parallel. The dc bias line imposes a voltage $V_d$ onto the oscillator plate. (c) Equivalent circuit of the device.

Figure 2

(a) Low power spectroscopy of the qubit decoupled from the mechanical oscillator, at $V_d=0\mathrm{V}$. The solid line is a Lorentzian fit indicating an intrinsic qubit linewidth of about 3.7 MHz. (b) Principle of the ac-Stark shift. The bare qubit resonance is shifted by a fraction of its linewidth for each number state. For a mechanical thermal state, the dressed qubit line shape is the sum over all number states weighted by their population. (c) Spectroscopy of the qubit coupled to the mechanical oscillator (at thermal equilibrium). Inset: Phonon populations extracted with a fit assuming a thermal distribution (dashed line) or with a Bayesian-based deconvolution algorithm (full line).

Figure 3

(a) Energy level diagram for ground and excited states of the qubit ($g$ and $e$), dressed by phonon numbers $n-1$, $n$, and $n+1$, showing the number-sensitive qubit transition ${\tilde{\omega }}_q(n)$ as well as the number-sensitive blue and red sideband transitions ${\omega }_{B(R)}(n)$. (b) For a blue sideband drive centered around $n_{\text{drive}}$, blue sideband transitions at neighboring number states are also driven, at a smaller rate. (c) Qubit spectroscopy after a blue sideband drive centered at a few different $n_{\text{drive}}$. Dots are raw data, full lines are qubit line shapes expected from the reconstructed phonon number distribution. Vertical black arrows indicate the position of $n_{\text{drive}}$ mapped onto the spectroscopic frequency axis (that is, $\omega ={\omega }_q+2{\chi }_m{n}_{\text{drive}}$). (d) Reconstructed experimental phonon populations (full lines) and master equation simulation (dashed lines) based on independently measured parameters. The orange curves, corresponding to the reconstructed populations of the thermal state, are shown for reference. Confidence intervals on the reconstruction (lighter shade) are obtained from a nonparametric bootstrap [27]. Gray areas show where populations have been moved by the sideband drive.

Figure 4

Spectroscopy of the qubit after a red sideband drive of $150\mu \mathrm{s}$ (dark red) and 1.5 ms (black), centered around $n_{\text{drive}}\approx 8$. Dots are raw data, full lines are qubit line shapes expected from the reconstructed phonon number distribution. Inset: Reconstructed experimental phonon populations (log scale, with bootstrap confidence intervals in lighter shades). The dotted line is the distribution of a fit to a thermal state. The horizontal black arrow indicates the ground state population of about 0.48 (see text).