Observing and Verifying the Quantum Trajectory of a Mechanical Resonator

Continuous weak measurement allows localizing open quantum systems in state space and tracing out their quantum trajectory as they evolve in time. Efficient quantum measurement schemes have previously enabled recording quantum trajectories of microwave photon and qubit states. We apply these concepts to a macroscopic mechanical resonator, and we follow the quantum trajectory of its motional state conditioned on a continuous optical measurement record. Starting with a thermal mixture, we eventually obtain coherent states of 78% purity—comparable to a displaced thermal state of occupation 0.14. We introduce a retrodictive measurement protocol to directly verify state purity along the trajectory, and we furthermore observe state collapse and decoherence. This opens the door to measurement-based creation of advanced quantum states, as well as potential tests of gravitational decoherence models.

Figure 1

Measuring a mechanical quantum trajectory. (a) Experimental setup. The mechanical resonator is coupled to an optical cavity, driven resonantly by a probe laser (red). The motion is imprinted on the phase quadrature of the transmitted light, which is measured with a balanced homodyne detector. The photocurrent $I(t)$ is digitized and analyzed in postprocessing. (b) Example calibrated photocurrent, containing information about all mechanical modes coupled to the cavity. Inset shows one quadrature signal obtained by demodulating the photocurrent at ${\mathrm{\Omega }}_m$. (c) Sketch of a quantum trajectory in phase space, in terms of the first moment $\overrightarrow{\mathbf{r}}$ (red line) and conditional variance (dark gray area, standard deviation). The variance is reduced as information is gathered during the measurements. Averaging different realizations together leads to an unconditional, thermal state (light gray area), with variance $V_{\mathrm{bath}}$. (d)–(e) Measured single quantum trajectory $\overrightarrow{\mathbf{r}}(t)$ in terms of slowly varying quadratures, $\overrightarrow{X}(t)$ and $\overrightarrow{Y}(t)$. Insets illustrate predicted decay of the conditional variance as the conditional state collapses (gray shaded area, standard deviation).

Figure 2

Verification of the conditional state. (a) A single quantum trajectory, $\overrightarrow{\mathbf{r}}(t)$ (red line), calculated until time $t_0$ is compared with a retrodiction, $\overleftarrow{\mathbf{r}}(t)$ (blue line), backpropagated to the same time $t_0$. (b) An ensemble of relative state estimations at time $t_0$, as measured from the trajectory $\overrightarrow{\mathbf{r}}(t_0)$ (red circles) and verified by the retrodiction $\overleftarrow{\mathbf{r}}(t_0)$ (blue triangles). The prediction-retrodiction pairs are connected by a gray line. The gray box indicates the pair shown in Fig. 2. (c) Phase space distribution of the relative expectation values: $\overleftarrow{\mathbf{r}}(t_0)-\overrightarrow{\mathbf{r}}(t_0)$ (purple circles). The purple line corresponds to two standard deviations of the data (radius $2\sqrt{{\sigma }^2}$), as compared with the expected $\sqrt{{\sigma }^2}$ for a pure coherent state (black line, radius $2\sqrt{1}$).

Figure 3

Measurement-induced collapse and decoherence. (a) $X$ quadratures of relative trajectories $\overleftarrow{\mathbf{r}}(t_0)-\overrightarrow{\mathbf{r}}(t_0)$ (gray lines) shown during different time intervals. A few traces are highlighted in black for illustration purposes. (b) Relative variance (${\sigma }^2$, purple) of the ensemble of relative trajectories from Fig. 3. For comparison, the unconditional variance [calculated from $\overleftarrow{\mathbf{r}}(t)$ only] is shown in blue. The theoretical expectation ($V+V_E$) is shown in gray, and the variance of a pure coherent state is shown in black.