Quantum limits on optics-based precision measurements are described, contrasting single-mode, multi-mode, and continuous-time paradigms. The latter suggests that 1/N, where N is the measurement’s average photon number, is not the ultimate quantum limit on precision.
Optical techniques are widely used in many areas of science and technology to make accurate measurements and diagnostics, from microscopy, spectroscopy, chemical analysis, to gravitational wave detection and ranging. There are many reasons for this: light allows us to extract information in a remote and non-destructive way, it carries information in a massively parallel way, and perhaps more importantly, optical measurements can reach very high precision and sensitivity levels. It is therefore important to know what is the ultimate limit of sensitivity that can be possibly achieved in the estimation of a physical parameter p that is encoded in a light beam of fixed mean photon number N. This limit is imposed by the unavoidable quantum fluctuations of light and depends on the quantum state of light that conveys the information about p. When the light is in a coherent state, this limit is called ‘standard quantum limit’ and scales as 1/N1/2.
As one of the main pillars of quantum technologies, quantum metrology aims to improve measurement precision using techniques from quantum information. The two main strategies to achieve this are the preparation of nonclassical states and the design of optimized measurement observables. We discuss precision limits and optimal strategies in quantum metrology and sensing with a single mode of quantum continuous variables. We focus on the practically most relevant cases of estimating displacements and rotations and provide the sensitivities of the most important classes of states that includes Gaussian states and superpositions of Fock states or coherent states. Fundamental precision limits that are obtained from the quantum Fisher information are compared to the precision of a simple moment-based estimation strategy based on the data obtained from possibly sub-optimal measurement observables, including homodyne, photon number, parity and higher moments. Finally, we summarize some of the main experimental achievements and present emerging platforms for continuous-variable sensing. These results are of particular interest for experiments with quantum light, trapped ions, mechanical oscillators, and microwave resonators.
A fundamental aspect of quantum systems lies in their extreme sensitivity to external perturbations. This characteristic, which generally represents a considerable challenge for physicists seeking to isolate these systems in order to observe their quantum properties, can actually be transformed into an advantage. Indeed, this intrinsic sensitivity makes quantum systems ideal candidates for high-precision metrology. Quantum sensors exploit this susceptibility to measure physical quantities with precision difficult to match with classical approaches.
Optical and optomechanical systems are exploited for quantum technologies to perform highly precise measurements for fundamental physics as well as biological and engineering tasks. Theoretical studies using tools from statistics and quantum information can greatly aid studies of the sensing capabilities of quantum systems and experimental designs. These allow us to quantify the amount of information about a parameter encoded in a quantum state itself and examine how that can be extracted through measurements. In this thesis we show how mechanical squeezing and measurements beyond position can be utilised to improve the precision of wavepacket expansion measurements which can test collapse models of quantum mechanics. Particularly, that squeezing can compensate for the free-fall time which is often the most significant limiting factor in such experiments, and measurements of other quadratures can offer increased precision. We demonstrate that the use of additional optical fields to measure the displacement of a free mass in a radiation-pressure limited interferometer cannot surpass the ultimate precision of the single-mode interferometer. This work applies to the likes of laser-interferometric gravitational wave detectors. Finally, we venture into multi-parameter estimation: we calculate bounds for multi-mode Gaussian states used to estimate many phase shifts and optimise over the input states. In contrast to the—far less experimentally feasible—multi-mode generalisations of N00N states we see no significant improvement when using multi-mode Gaussian states for the task. Given their comparable performance in single-phase estimation this is a new limitation of Gaussian states appearing only at the multi-parameter level.
We consider estimating a small transverse displacement of an optical beam over a line-of-sight propagation path: a problem that has numerous important applications ranging from establishing a lasercom link, single-molecule tracking, guided munition, to atomic force microscopy. We establish the ultimate quantum limit of the accuracy of sensing a beam displacement, and quantify the classical-quantum gap. Further, using normal-mode decomposition of the Fresnel propagation kernel, and insights from recent work on entanglement-assisted sensing, we find a near-term realizable multi-spatio-temporal-mode continuous-variable entangled-state probe and a receiver design, which attains the quantum precision limit. We find a Heisenberg-limited sensitivity enhancement in terms of the number of entangled temporal modes, and a curious super-Heisenberg quantum enhanced scaling in terms of the number of entangled spatial modes permitted by the diffraction-limited beam propagation geometry.
Continuous monitoring of open quantum-optical systems offers a promising route towards quantum-enhanced sensing. In such sensing schemes, the ultimate precision limit is dictated by the quantum Cramér-Rao bound. However, reaching this fundamental bound with established continuous measurement techniques has remained an outstanding challenge. Here, we present an approach tailored to open quantum-optical sensors operating near dissipative critical points. The key insight of our approach is to exploit a synergy between the intrinsic critical dynamics and the unavoidable quantum measurement backaction of continuous homodyne detection. By carefully aligning the homodyne phase to the squeezing angle of the sensor state, quantum backaction can be evaded in the critical dynamics, thereby significantly boosting the attainable precision. We demonstrate the broad applicability of this approach across several sensor models and for both single- and multi-parameter estimation, showing that it consistently narrows the gap toward the ultimate precision bound.