Quantum amplification of mechanical oscillator motion

Abstract

Detection of the weakest forces in nature is aided by increasingly sensitive measurements of the motion of mechanical oscillators. However, the attainable knowledge of an oscillator's motion is limited by quantum fluctuations that exist even if the oscillator is in its lowest possible energy state. We demonstrate a technique for amplifying coherent displacements of a mechanical oscillator with initial magnitudes well below these zero-point fluctuations. When applying two orthogonal squeezing interactions, one before and one after a small displacement, the displacement is amplified, ideally with no added quantum noise. We implemented this protocol with a trapped-ion mechanical oscillator and determined an increase by a factor of up to 7.3 (±0.3) in sensitivity to small displacements.

Fig. 1.. Conceptual illustration of the amplification protocol.

Each panel shows a Wigner function phase space distribution (not to scale) in a frame rotating at the oscillator frequency. A displacement ${\text{α}}_{\text{i}}$ of an initially squeezed ground state is amplified by subsequent reversed squeezing (“anti-squeezing”), resulting in a final coherent state with amplitude $G{\text{α}}_{\text{i}}$ with no added noise. Dashed red circles indicate the characteristic extent of the initial ground-state fluctuations.

Fig. 2.. Fock state population analysis.

(A to C) Histograms of Fock state populations extracted by fitting to BSB Rabi oscillations. Vertical bars are derived by fitting to an unconstrained population distribution. Solid blue lines are fits assuming parameterized functional forms of the ideal Fock state populations, yielding values of $r$, ${\text{α}}_{\text{i}}$, and ${\text{α}}_{\text{f}}$ (19). Insets show Wigner function illustrations of the corresponding motional states. (A) Initial squeezed motional ground state with $r=2.26\left(\pm 0.02\right)$. (B) After displacing this state by ${\text{α}}_{\text{i}}=0.200\left(\pm 0.002\right)$. (C) Final coherent state with amplitude ${\text{α}}_{\text{f}}=1.83\left(\pm 0.01\right)$, following the reversed squeezing operation. The initial displacement is amplified by $G={\text{α}}_{\text{f}}/{\text{α}}_{\text{i}}=9.17\left(\pm 0.09\right)$. (D) Squeezing parameter $r$ (black circles) as a function of the parametric drive duration. The solid line is a linear fit whose slope gives the parametric coupling strength $g$. (E) Measured gain (black circles) as a function of the squeezing parameter $r$. The solid line is the theoretical gain $G=\exp \left(r\right)$. Error bars denote SD.

Fig. 3.. Measurement sensitivity enhancement.

(A) Pulse sequence for displacement sensing protocol with PSRSB detection. (B) Population in $\left|\downarrow \right.〉$ as a function of the carrier $\text{π}/2$ pulse phase. Blue inverted triangles, data with no squeezing; red circles, data with amplification. Solid lines show sinusoidal fits to the data. (C) Contrast of the carrier phase scan, as shown in (B), as a function of the squeezing phase $\text{θ}$ for a fixed displacement. (D) Contrast as a function of the displacement amplitude $\left|{\text{α}}_{\text{i}}\right|$ for different initial squeezing pulse durations. Each data point is calculated from ~10⁴ experiments. The data shown in (B) and (C) have initial $\left|{\text{α}}_{\text{i}}\right|=0.0578\left(\pm 0.0006\right)$ and a squeezing duration of $t=8\text{μs}$ [nominally $r=2.54\left(\pm 0.03\right)$]. The solid black line in (D) is the maximum theoretical contrast without squeezing. Dashed lines in (C) and (D) are derived from a numerical model that includes motional decoherence. (E) Measurement sensitivity enhancement in the linear small-displacement regime as a function of the ideal gain $G=\exp \left(gt\right)$. For each squeezing duration, the enhancement is determined by dividing the slope of the contrast for $C\lesssim 0.25$ [obtained by fitting a straight line to data points in (D) with $C\lesssim 0.25$] by the slope of the 0 dB black line, which represents the standard quantum limit (19). Error bars denote SD.