Non-Gaussian Mechanical Motion via Single and Multiphonon Subtraction from a Thermal State

Quantum optical measurement techniques offer a rich avenue for quantum control of mechanical oscillators via cavity optomechanics. In particular, a powerful yet little explored combination utilizes optical measurements to perform heralded non-Gaussian mechanical state preparation followed by tomography to determine the mechanical phase-space distribution. Here, we experimentally perform heralded single-phonon and multiphonon subtraction via photon counting to a laser-cooled mechanical thermal state with a Brillouin optomechanical system at room temperature and use optical heterodyne detection to measure the $s$-parametrized Wigner distribution of the non-Gaussian mechanical states generated. The techniques developed here advance the state of the art for optics-based tomography of mechanical states and will be useful for a broad range of applied and fundamental studies that utilize mechanical quantum-state engineering and tomography.

Figure 1

Multiphonon subtraction and tomography scheme and experimental schematic. (a) Optical pumping and heralding scheme. A pair of optical resonances spaced by the mechanical angular frequency ${\omega }_m$ is used to resonantly enhance the optomechanical interaction. A pump field drives a mode at ${\omega }_p$ creating an anti-Stokes signal at ${\omega }_{aS}$. An $n$-photon detection scheme is then used to herald $n$-phonon subtraction to the mechanical motion. (b) Experimental schematic. A tunable pump laser (1550 nm) drives an optical microresonator ($\mu \mathrm{RES}$) and the backscattered anti-Stokes signal is separated from the pump with an optical circulator (CIRC). The signal is subsequently split and detected via single-photon avalanche detectors (SPADs) to herald a single- or two-phonon subtraction operation. To characterize the mechanical state prepared, heterodyne detection is performed (BD, balanced detector; LO, local oscillator), and the signal recorded on an oscilloscope triggered by SPAD detection events. The two-phonon subtraction case is shown here that uses a two-photon-coincidence measurement.

Figure 2

Experimental $s$-parametrized Wigner functions $W_s$ (bottom row), with slices through $P_m=0$ (top row), for an (a) initial, (b) single-phonon subtracted, and (c) two-phonon subtracted mechanical thermal state. The phase-space distributions are plotted in units of the mechanical zero-point fluctuations $x_{\mathrm{ZP}}$ and are obtained through heterodyne detection of the optical anti-Stokes signal. The generation of non-Gaussianity from the originally Gaussian phase-space distribution is observed for single-phonon subtraction, which further grows upon two-phonon subtraction. The dash-dotted lines indicate the theoretically predicted maxima for the $W_s$ functions. The phase-space distribution of the optical vacuum contribution is shown in the inset in (a).

Figure 3

Dynamics and non-Gaussian distributions of the heralded mechanical states. (a) Time evolution of the heterodyne variance in units of optical vacuum noise about the heralding event for single- and two-phonon subtracted thermal states, plotted in green and red, respectively. The optical vacuum is plotted in gray and the variance of the initial thermal state is plotted in blue. Experimentally obtained variances are shown as solid lines, and the predictions of our theoretical model are shown as dott-dashed lines. The Poissonian experimental relative uncertainty is of order ${10}^{-3}$ and is not visible on this scale. The discrepancy between theory and experiment is attributed to filtering in postprocessing and dark counts from the SPADs. At the time of the heralding event, the ratio of the heterodyne variance to the optical vacuum noise increases by a factor of 1.94 and 2.94 relative to the initial thermal state for single- and two-phonon subtraction, respectively. (b) Marginal distributions for the $X_m$ quadrature of the mechanical oscillator as a function of time. (c) Mechanical quadrature probability distributions at the time of the heralding event, $t=t_0$, for the initial (blue), single-phonon subtracted (green), and two-phonon subtracted (red) thermal mechanical states.