Fundamental quantum limits to waveform detection

Ever since the inception of gravitational-wave detectors, limits imposed by quantum mechanics to the detection of time-varying signals have been a subject of intense research and debate. Drawing insights from quantum information theory, quantum detection theory, and quantum measurement theory, here we prove lower error bounds for waveform detection via a quantum system, settling the long-standing problem. In the case of optomechanical force detection, we derive analytic expressions for the bounds in some cases of interest and discuss how the limits can be approached using quantum control techniques.

Figure 1

Quantum-circuit diagrams for the waveform detection problem. The quantum system is modeled as a pure state $|\psi \rangle$ with unitary evolution ($U_0$ or $U_1$) under each hypothesis (${\mathcal{H}}_0$ or ${\mathcal{H}}_1$) in a large enough Hilbert space for a given classical waveform $x\left(t\right)$, which perturbs the evolution under ${\mathcal{H}}_1$. If $x\left(t\right)$ is stochastic, the final quantum state under ${\mathcal{H}}_1$ is mixed. Measurements are modeled as a positive-operator-valued measure (POVM) $E\left[y\right]$ at the final time through the principle of deferred measurement.

Figure 2

A cavity optomechanical force detector. An optical cavity with a moving mirror is pumped on-resonance with an input field $A_{\mathrm{in}}$, and the output field $A_{\mathrm{out}}$ is measured to infer whether a force $x\left(t\right)$ has perturbed the motion of the mirror.

Figure 3

An integrated QNC-Kennedy receiver. The output field $A_{\mathrm{out}}$ from the optomechanical force detector in Fig. 2 is displaced by $-\mathcal{A}$ and then passed through an optical setup that removes the measurement back action noise. The dashed arrows represent a red-detuned optical cavity mode that mimics a negative-mass oscillator and interacts with the optical probe field via a beam splitter (BS) and a two-mode optical parametric amplifier (OPA). Details of how this setup works can be found in Refs. [17, 18]. If the field $A_{\mathrm{in}}$ is in a coherent state, the final output field should be in a vacuum state under the null hypothesis ${\mathcal{H}}_0$. Any photon detected at the output indicates that ${\mathcal{H}}_1$ must be true.