Fundamental sensitivity bounds for quantum enhanced optical resonance sensors based on transmission and phase estimation

Quantum states of light can enable sensing configurations with sensitivities beyond the shot-noise limit (SNL). In order to better take advantage of available quantum resources and obtain the maximum possible sensitivity, it is necessary to determine fundamental sensitivity limits for different possible configurations for a given sensing system. Here, due to their wide applicability, we focus on optical resonance sensors, which detect a change in a parameter of interest through a resonance shift. We compare their fundamental sensitivity limits set by the quantum Cramér-Rao bound (QCRB) based on the estimation of changes in transmission or phase of a probing bright two-mode squeezed state (bTMSS) of light. We show that the fundamental sensitivity results from an interplay between the QCRB and the transfer function of the system. As a result, for a resonance sensor with a Lorentzian lineshape a phase-based scheme outperforms a transmission-based one for most of the parameter space; however, this is not the case for lineshapes with steeper slopes, such as higher order Butterworth lineshapes. Furthermore, such an interplay results in conditions under which the phase-based scheme provides a higher sensitivity but a smaller degree of quantum enhancement than the transmission-based scheme. We also study the effect of losses external to the sensor on the degree of quantum enhancement and show that for certain conditions, probing with a classical state can provide a higher sensitivity than probing with a bTMSS. Finally, we discuss detection schemes, namely optimized intensity-difference and optimized homodyne detection, that can achieve the fundamental sensitivity limits even in the presence of external losses.

Enhanced on-chip phase measurement by inverse weak value amplification

Optical interferometry plays an essential role in precision metrology such as in gravitational wave detection, gyroscopes, and environmental sensing. Weak value amplification enables reaching the shot-noise-limit of sensitivity, which is difficult for most optical sensors, by amplifying the interferometric signal without amplifying certain technical noises. We implement a generalized form of weak value amplification on an integrated photonic platform with a multi-mode interferometer. Our results pave the way for a more sensitive, robust, and compact platform for measuring phase, which can be adapted to fields such as coherent communications and the quantum domain. In this work, we show a 7 dB signal enhancement in our weak value device over a standard Mach-Zehnder interferometer with equal detected optical power, as well as frequency measurements with 2 kHz sensitivity by adding a ring resonator.

Strategies for Positive Partial Transpose (PPT) States in Quantum Metrologies with Noise

Quantum metrology overcomes standard precision limits and has the potential to play a key role in quantum sensing. Quantum mechanics, through the Heisenberg uncertainty principle, imposes limits on the precision of measurements. Conventional bounds to the measurement precision such as the shot noise limit are not as fundamental as the Heisenberg limits, and can be beaten with quantum strategies that employ `quantum tricks’ such as squeezing and entanglement. Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered to be too weakly entangled for applications. Since no pure entanglement can be distilled from them, they are also called bound entangled states. We provide strategies, using which multipartite quantum states that have a positive partial transpose with respect to all bi-partitions of the particles can still outperform separable states in linear interferometers.

Versatile Super-Sensitive Metrology Using Induced Coherence

We theoretically analyze the phase sensitivity of the Induced-Coherence (Mandel-Type) Interferometer, including the case where the sensitivity is "boosted" into the bright input regime with coherent-light seeding. We find scaling which reaches below the shot noise limit, even when seeding the spatial mode which does not interact with the sample – or when seeding the undetected mode. It is a hybrid of a linear and a non-linear (Yurke-Type) interferometer, and aside from the supersensitivity, is distinguished from other systems by "preferring" an imbalance in the gains of the two non-linearities (with the second gain being optimal at low values), and non-monotonic behavior of the sensitivity as a function of the gain of the second non-linearity. Furthermore, the setup allows use of subtracted intensity measurements, instead of direct (additive) or homodyne measurements – a significant practical advantage. Bright, super-sensitive phase estimation of an object with different light fields for interaction and detection is possible, with various potential applications, especially in cases where the sample may be sensitive to light, or is most interesting in frequency domains outside what is easily detected, or when desiring bright-light phase estimation with sensitive/delicate detectors. We use an analysis in terms of general squeezing and discover that super-sensitivity occurs only in this case – that is, the effect is not present with the spontaneous-parametric-down-conversion approximation, which many previous analyses and experiments have focused on.

Overcoming detection loss and noise in squeezing-based optical sensing

Among the known resources of quantum metrology, one of the most practical and efficient is squeezing. Squeezed states of atoms and light improve the sensing of the phase, magnetic field, polarization, mechanical displacement. They promise to considerably increase signal-to-noise ratio in imaging and spectroscopy, and are already used in real-life gravitational-wave detectors. But despite being more robust than other states, they are still very fragile, which narrows the scope of their application. In particular, squeezed states are useless in measurements where the detection is inefficient or the noise is high. Here, we experimentally demonstrate a remedy against loss and noise: strong noiseless amplification before detection. This way, we achieve loss-tolerant operation of an interferometer fed with squeezed and coherent light. With only 50% detection efficiency and with noise exceeding the level of squeezed light more than 50 times, we overcome the shot-noise limit by 6 dB. Sub-shot-noise phase sensitivity survives up to 87% loss. Application of this technique to other types of optical sensing and imaging promises a full use of quantum resources in these fields.

Fundamental flow measurement capabilities of optical Doppler and time-of-flight principles

In order to understand the fundamental measurement capabilities of different flow velocity measurement principles based on Mie scattering, a fundamental equation of how to calculate the shot noise limit for a respective signal model is derived. The derivation is based on the well-known rules of uncertainty propagation and yields the Cramér–Rao bound without the necessity to calculate the Fisher information. The derived equation is next applied to compare the shot noise limit for Doppler and time-of-flight principles including laser Doppler anemometry (LDA), planar Doppler velocimetry (PDV), laser-two-focus velocimetry (L2F), particle tracking velocimetry (PTV) and particle image velocimetry (PIV). The comparison is performed for an identical mean laser power, while two cases are studied in detail: measuring on a single seeding particle as well as measuring on multiple seeding particles and averaging. LDA, L2F and PTV/PIV obey a similar shot noise limit. For the case of a measurement on multiple seeding particles, the minimal achievable measurement uncertainty is directly proportional to the absolute value of the measured velocity component and inversely proportional to the spatial resolution. The respective shot noise limit for PDV is almost independent of the measured flow velocity component and the spatial resolution. Since PDV is sensitive with respect to a different flow velocity component depending on the observation direction, a comparison with the other principles is only reasonable to a certain extent. However, all shot noise limits in case of measuring on multiple seeding particles show the expected inverse proportionality to the square root of the total number of detected photons and thus also to the square root of the measurement time. Considering a comparable spatiotemporal resolution, an identical mean light power and typical measurement configurations, the PDV shot noise limit is the largest. As a final result, it is derived that each measurement principle obeys an uncertainty principle between position and the respective component of the wave vector, which is in agreement with Heisenberg’s uncertainty principle. Therefore, a common basis is provided to assess the fundamental measurement capabilities of Doppler and time-of-flight measurement systems on the basis of what is possible within the quantum mechanical constraints. Graphic abstract