Two-Tone Optomechanical Instability and Its Fundamental Implications for Backaction-Evading Measurements

While quantum mechanics imposes a fundamental limit on the precision of interferometric measurements of mechanical motion due to measurement backaction, the nonlinear nature of the coupling also leads to parametric instabilities that place practical limits on the sensitivity by limiting the power in the interferometer. Such instabilities have been extensively studied in the context of gravitational wave detectors, and their presence has recently been reported in Advanced LIGO. Here, we observe experimentally and describe theoretically a new type of optomechanical instability that arises in two-tone backaction-evading (BAE) measurements, a protocol designed to overcome the standard quantum limit. We demonstrate the effect in the optical domain with a photonic crystal nanobeam cavity and in the microwave domain with a micromechanical oscillator coupled to a microwave resonator. In contrast to the well-known parametric oscillatory instability that occurs in single-tone, blue-detuned pumping, and results from a two-mode squeezing interaction between the optical and mechanical modes, the parametric instability in balanced two-tone optomechanics results from single-mode squeezing of the mechanical mode in the presence of small detuning errors in the two pump frequencies. Counterintuitively, the instability occurs even in the presence of perfectly balanced intracavity fields and can occur for both signs of detuning errors. We find excellent quantitative agreement with our theoretical predictions. Since the constraints on tuning accuracy become stricter with increasing probe power, the instability imposes a fundamental limitation on BAE measurements as well as other two-tone schemes, such as dissipative squeezing of optical and microwave fields or of mechanical motion. In addition to identifying a new limitation in two-tone BAE measurements, the results also introduce a new type of nonlinear dynamics in cavity optomechanics.

Popular Summary

Monitoring the motion of a macroscopic mechanical oscillator is central to sensing technologies such as gravitational-wave detectors and atomic force microscopes. The sensitivity of such efforts is ultimately limited by “quantum backaction,” a quantum effect in which the detector influences the measurement. To evade this backaction limit, researchers are exploring techniques available in optomechanical systems: The oscillator sits in an electromagnetic cavity in which changes in radiation pressure relay the oscillator movement. Here, we report on experiments that reveal a new optomechanical instability in these systems that can foil the measurements.

Backaction-evading measurements circumvent the quantum limit by controlling the acquired information, such as by measuring only the amplitude of the oscillator and keeping ignorant of its phase. This allows for unlimited measurement sensitivity at the cost of learning half of the information, and it can be realized by modulating the coupling between the oscillator and the cavity field. Aside from the technical challenges in performing such measurements, it was previously believed that dynamical effects arising from the optomechanical interaction posed no complications.

We perform backaction-evading measurements in two systems operating in different mechanical and electromagnetic domains. We find that tiny errors in the frequencies of the probing field and its modulation can unexpectedly conspire with the intrinsic optomechanical interaction itself, leading to strong self-oscillations that dominate over any measured signal. The more sensitivity one seeks to gain, the stricter the tuning accuracy requirement becomes.

This effect imposes a new fundamental limit on backaction-evading measurements and will require careful consideration in future quantum sensing and related experiments aiming to surpass the quantum limit.

Figure 1

Pumping scheme leading to two-tone instability. (a) Frequency-space representation of optical backaction-evading (BAE) measurement using two-tone pumping. An optomechanical system is pumped with two pumps that are placed on the lower and upper motional sideband of the cavity (shown in gray). Two detuning errors are introduced, due to imperfect knowledge of the mechanical oscillator frequency (${\mathrm{\Delta }}_m$), and due to imperfect symmetric spacing around the cavities’ resonance frequency ${\omega }_c$, as expressed by ${\mathrm{\Delta }}_c$. Also shown are the mechanical resonance at frequency ${\mathrm{\Omega }}_m$ and the scattered mechanical sidebands. The inset shows an optomechanical system: a mechanical oscillator (position coordinate $\widehat{x}$) that is the moving mirror of a Fabry-Perot cavity and that is coupled to the cavity mode by radiation pressure. (b) An equivalent system, in which the two-tone pumping is mapped to a Hamiltonian that exhibits the same dynamics as (a), consisting of a single continuous pump field applied at ${\omega }_c+{\mathrm{\Delta }}_c$ and with the mechanical oscillator frequency obeying the substitution ${\mathrm{\Omega }}_m\to -{\mathrm{\Delta }}_m$. Note that ${\mathrm{\Delta }}_c$ and ${\mathrm{\Delta }}_m$ have been exaggerated for clarity.

Figure 2

Experimental observation of two-tone instability. Panels (a) and (b) show two-tone BAE measurements in the presence of a small optical tuning error ${\mathrm{\Delta }}_c$ and a mechanical tuning error ${\mathrm{\Delta }}_m$. (a) BAE measurement in the optical domain with an optomechanical system based on a silicon photonic crystal nanobeam cavity (inset). A sequence of measurements is shown, where the mechanical sidebands are measured via quantum-limited heterodyne detection, normalized to the shot-noise level. Here, ${\omega }_{\mathrm{LO}}$ is the optical frequency of the heterodyne local oscillator laser. In the sequence, the mechanical “tuning error” ${\mathrm{\Delta }}_m$ is varied from a positive value toward zero, where ${\mathrm{\Delta }}_m=0$ ideally corresponds to a BAE measurement. Because of the cavity tuning error ${\mathrm{\Delta }}_c$, at ${\mathrm{\Delta }}_m/2\pi =0.04\mathrm{MHz}$ a strong increase in the total mechanical noise is observed, as well as narrowing of the sideband, instead of the expected decrease due to backaction cancellation. This is the onset of the two-tone instability. (b) BAE measurement in the microwave domain with a mechanically compliant capacitor coupled to a superconducting microwave resonator (inset). The procedure is the same as in (a). Measurements with both positive and negative values of ${\mathrm{\Delta }}_m$ are shown. The measurement at ${\mathrm{\Delta }}_m/2\pi =0.02\mathrm{kHz}$ occurs within the domain of instability (yellow curve), in which case the observed spectrum is distorted by saturation of the HEMT amplifier used in the detection chain. Note the logarithmic $y$ axis.

Figure 3

Domains of two-tone instability. The onset of the two-tone instability is given by the condition Eq. (6) and depends only on ${\tilde{\mathrm{\Delta }}}_m\equiv {\mathrm{\Delta }}_m/({\mathrm{\Gamma }}_m/2)$, ${\tilde{\mathrm{\Delta }}}_c\equiv {\mathrm{\Delta }}_c/(\kappa /2)$, and the cooperativity $\mathcal{C}$. Here we plot the domains of instability as a function of ${\tilde{\mathrm{\Delta }}}_m$ and ${\tilde{\mathrm{\Delta }}}_c$ for different cooperativities $\mathcal{C}$ (given as the contour labels). As $\mathcal{C}$ increases, the stable region in the vicinity of the origin ${\tilde{\mathrm{\Delta }}}_m={\tilde{\mathrm{\Delta }}}_c=0$ becomes smaller, reducing the range of ${\tilde{\mathrm{\Delta }}}_m$ and ${\tilde{\mathrm{\Delta }}}_c$ for which the system is stable.

Figure 4

Investigation of the two-tone instability in a circuit-electromechanical system. (a)–(c) Mapping of the total mechanical noise as a function of ${\tilde{\mathrm{\Delta }}}_m$ and ${\tilde{\mathrm{\Delta }}}_c$, for cooperativities $\mathcal{C}=\{3.5,7,14\}$, respectively. The total power $P$ in both mechanical sidebands in the output spectra is shown, normalized to the power $P_0$ at $({\tilde{\mathrm{\Delta }}}_m,{\tilde{\mathrm{\Delta }}}_c)=(-18,0)$ (no tuning error and far from the BAE regime). Since data points do not align on a regular grid (accounting for small changes in ${\tilde{\mathrm{\Delta }}}_c$ along the horizontal scan), the rasterization was implemented using nearest-neighbor (Voronoi) partitioning. Solid black lines are instability thresholds from Eq. (6). (d)–(f) Theory plots corresponding to (a)–(c). Gray areas are unstable regions predicted by theory. (g)–(i) Cross sections of (b) for ${\tilde{\mathrm{\Delta }}}_c=\{0,-0.07,-0.18\}$, respectively [nearest data points to horizontal dashed lines in (b)]. Solid black lines are theory predictions based on the full linear model.

Figure 5

Instability domains from eigenvalue analysis. Domains where an eigenvalue of the dynamical matrix Eq. (a2) has positive real part, signaling an instability. The blue domain, in the neighborhood of $-{\mathrm{\Delta }}_m={\mathrm{\Omega }}_m\approx {\mathrm{\Delta }}_c$, is the known parametric oscillatory instability. In this domain, the imaginary part of the eigenvalue is nonzero. In reality, the mechanical frequency is positive ${\mathrm{\Omega }}_m>0$, such that the lower right-hand part is unphysical. The red domain that occurs for $|{\mathrm{\Delta }}_m|\ll \kappa$ (enlargement shown in the inset) is the two-tone instability. In this domain ${\mathrm{\Delta }}_m{\mathrm{\Delta }}_c>0$, and the imaginary part of the eigenvalue is zero. The parameters used are $\kappa =1$, ${\mathrm{\Gamma }}_m={10}^{-2}$, and $\mathcal{C}=2$.

Figure 6

Vanishing of the effective mechanical frequency. The plot shows the difference between the measured effective mechanical frequency and pump modulation frequency as a function of the detunings. Near the instability domain (shaded in gray), the two frequencies become equal, which corresponds to vanishing of the effective mechanical frequency in the rotating frame, confirming the theoretical treatment. Measurement is the same as in Fig. 4.

Figure 7

Two-tone instability in the optical domain. (a) Two-tone BAE measurements (dark red and light red dots) scanning ${\mathrm{\Delta }}_m$ with ${\mathrm{\Delta }}_c$ kept approximately constant. The $({\tilde{\mathrm{\Delta }}}_m,{\tilde{\mathrm{\Delta }}}_c)$ coordinate of each measurement is plotted. The instability contours for the respective cooperativities, around $\mathcal{C}\sim 2.2–2.6$ (contour label), are indicated with the same color. While one measurement sequence ($\mathcal{C}=2.23$) remains near ${\tilde{\mathrm{\Delta }}}_c\approx 0$, the other ($\mathcal{C}=2.58$) encounters instability in the vicinity of its corresponding contour. The highlighted dot is the measurement right before instability. (b) Same as (a) with higher $\mathcal{C}\sim 4.1$. Here the stable region is smaller, highlighting the difficulty of achieving stable operation due to the inaccuracy in ${\tilde{\mathrm{\Delta }}}_c$. (c) Total noise power in the mechanical sidebands, normalized to 1 for large ${\mathrm{\Delta }}_m$, for the two measurements sequences in (a). The sudden increase in power for the unstable data is evident. The solid lines are theoretical fits using the full Langevin equations. See text for more details.