Towards the Fundamental Quantum Limit of Linear Measurements of Classical Signals

The quantum Cramér-Rao bound (QCRB) sets a fundamental limit for the measurement of classical signals with detectors operating in the quantum regime. Using linear-response theory and the Heisenberg uncertainty relation, we derive a general condition for achieving such a fundamental limit. When applied to classical displacement measurements with a test mass, this condition leads to an explicit connection between the QCRB and the standard quantum limit that arises from a tradeoff between the measurement imprecision and quantum backaction; the QCRB can be viewed as an outcome of a quantum nondemolition measurement with the backaction evaded. Additionally, we show that the test mass is more a resource for improving measurement sensitivity than a victim of the quantum backaction, which suggests a new approach to enhancing the sensitivity of a broad class of sensors. We illustrate these points with laser interferometric gravitational-wave detectors.

Figure 1

A schematic for a quantum measurement of a classical signal using a linear detector. One degree of freedom of the detector is singled out as the input port, for which the observable, $\widehat{F}$, is coupled to the signal, $x$, and another one as the output port with its observable, $\widehat{Z}$, projectively measured by the observer. The detector is a quantum interface between two classical domains.

Figure 2

Illustration of the single-shot measurement with the detector in a pure, Gaussian, squeezed state (the noise ellipse represents its Wigner function). The optimal observable $\widehat{Z}$ to achieve the QCRB is neither the conjugate variable of $\widehat{F}$ (along the horizontal axis), which contains the largest signal, nor the one having the minimum noise (parallel to the semiminor axis of the noise ellipse). Instead, it is the one uncorrelated with $\widehat{F}$ and $\tan \theta =\sin (2\varphi )\sinh (2r)/[\cosh (2r)+\cos (2\varphi )\sinh (2r)]$, in which $r$ and $\varphi$ are the squeezing factor and angle, as derived in Ref. [5] using Eq. (2).

Figure 3

Schematic diagram of a LIGO-like interferometer (left). Two approximately equivalent physical pictures for the detection principle are shown (right).

Figure 4

The top row shows the QCRB (solid curve) for a LIGO-type GW detector with detuning frequency $\mathrm{\Delta }=0$ (left) and $\mathrm{\Delta }/(2\pi )=400\mathrm{Hz}$ (right), and various sensitivity curves for comparison: dashed curve, constant phase quadrature readout; dash-dotted curve, readout quadrature optimized to maximize sensitivity at each frequency; and dotted curve, the SQL $\sqrt{4\hslash /(M{\omega }^2)}$. The bottom row shows the ratio to the QCRB for selected curves. Other relevant parameters are $M=40\mathrm{kg}$, $P_{\text{cav}}=800\mathrm{kW}$, $L_{\text{arm}}=4\mathrm{km}$, $\gamma /(2\pi )\approx 100\mathrm{Hz}$, and laser frequency ${\omega }_0/(2\pi )\approx 3\times 10^{14}\mathrm{Hz}$.