Shot-to-shot noise cancellation for parametric oscillators

Abstract

Powerful approaches to squeeze the motional state of a harmonic oscillator rely on the stepwise modulation of its resonance frequency. Such protocols can be limited by forces that vary slowly between experimental runs but are constant during a single experimental shot. Such shot-to-shot noise gives rise to a spread in experimental outcomes that masks the uncertainty intrinsic to quantum theory. Taking inspiration from spin-echo protocols, we propose a decoupling technique that, under ideal conditions, perfectly cancels shot-to-shot force noise in squeezing experiments based on parametric modulation. We implement the protocol using an optically levitated nanoparticle, where shot-to-shot force noise arises from slowly varying stray fields acting on the charge carried by the particle. Using our oscillator-echo protocol, we demonstrate shot-to-shot noise suppression to the measurement-backaction limit.

Figures (4)

• Figure 1: Experimental setup. A dielectric nanoparticle is optically trapped in a focused laser beam (optical axis $z$) in ultra-high vacuum (UHV). The phase of the light backscattered by the particle is compared to a local oscillator (LO) to measure the particle position along $z$. A feedback (FB) force is applied to cool the particle motion. The stiffness of the optical trap is modulated via the laser intensity with a pair of acousto-optic modulators (AOMs).
• Figure 2: State-space evolution of a harmonic oscillator in a potential with reduced eigenfrequency $\Omega/r$ for two different values of the force $f_a$ (black) and $f_b<f_a$ (red). The oscillator is initialized to a circular Gaussian state at the origin (dashed circles). The trajectories of the mean around the rotation centers $\mathbf{c}_{f_a,r}$ and $\mathbf{c}_{f_b,r}$ are shown as the solid ellipses. The covariances are illustrated as dashed ellipses. After evolution by a phase $\theta=\pi/2$ and $3\pi/2$, the state is maximally squeezed along the momentum axis. Averaging over many realizations including $f_a$ and $f_b$ leads to an additional contribution to the effective covariance, illustrated by the blue shading.
• Figure 3: State-space evolution during the oscillator-echo protocol. (a) Step (i): The oscillator frequency is instantaneously reduced from $\Omega$ to $\Omega/r'$ for a time corresponding to $\theta_1=\pi$ (see inset for frequency-jump sequence). The state's mean is displaced along the $Q$ axis due to rotation around the center point $\mathbf{c}_{f_0,r'}$, illustrated for two values $f_a$ (blue) and $f_b$ (red) of the force $f_0$. The state's covariance at the end of the step (solid circle) equals the one at the beginning (dashed) besides an increase due to white-noise heating (not illustrated). (b) Step (ii): The oscillator frequency is reduced further to $\Omega/r$ for any desired time (corresponding to the phase $\theta_2$), e.g., to prepare a squeezed state. When $r'$ is chosen correctly relative to $r$, the rotation points during the second step $\mathbf{c}_{f_a,r}$ and $\mathbf{c}_{f_b,r}$ fall at the position where the state's mean is located, such that no displacement of the mean occurs during this step. (c) Step (iii): The oscillator frequency is set back to $\Omega/r'$ for a time corresponding to $\theta_3=\pi$ as in step (i). The state is displaced back to the origin, irrespective of the value of the force $f_0$.
• Figure 4: (a) Oscillator frequency during the oscillator-echo protocol realized in panels (b--f) with $r=\sqrt{4}$ and $\Omega/2\pi=52$ kHz. The colored numbers indicate the sampling times at which the state was probed in (b--f). (b) Zoom of panel (c) around the origin of phase-space, showing the state at the beginning (red) and at the end (blue) of the protocol. The displacement $\textbf{d}$ of the state's mean by the protocol is shown as the solid arrow. The eigenvalues of the covariance matrix $V_P$ and $V_Q$ represent the main axes of the covariance ellipse (black dashed). (c) Evolution of the state throughout the oscillator-echo protocol. The color-coding of the datapoints corresponds to the sampling times indicated in panel (a). The small dashed ellipses (black) illustrate the covariances of the state at the different sampling times. The two purple dashed ellipses indicate the trajectories in phase-space as given by stray field values of $f_0\pm\sigma_{f_0}$. (d--f) Zoom of plot (c), showing the state at $t_5, t_6$, and $t_7$, respectively. (g) Measured magnitude of the phase-space displacement $|\mathbf{d}|$ as a function of $r'$ at the end of the protocol ($t_{11})$. (h) Measured determinant of the covariance matrix $V^\text{tot}=\det(\Sigma)$ at the end of the protocol ($t_{11})$. In (g) and (h), the squeezing ratio is $r=\sqrt{10}$ and the dashed gray lines are fits to Eqs. \ref{['eq:mean_echoPulse']} and \ref{['eq:cov_echoPulse']}. See main text for details.