We explore fundamental limits on transmission estimation with bright single and two-mode squeezed states of light. We show that their quantum Fisher information surpasses the one of coherent states and approaches the ultimate limit with Fock states.
Enhancing the precision of a measurement requires maximizing the information that can be gained about the quantity of interest from probing a system. For optical-based measurements, such an enhancement can be achieved through two approaches, increasing the number of photons used to interrogate the system and using quantum states of light to increase the amount of quantum Fisher information gained per photon. Here we consider the use of quantum states of light with a large number of photons, namely the bright single-mode squeezed state and the bright two-mode squeezed state, which take advantage of both of these approaches for the problem of transmission estimation. We show that, in the limit of large squeezing, these states approach the maximum possible quantum Fisher information per photon for transmission estimation that is achieved with the Fock state and the vacuum two-mode squeezed state. Since the bright states we consider can be generated at powers much higher than those of the quantum states that achieve the maximum quantum Fisher information per photon, they can achieve a much higher absolute precision as quantified by the quantum Cram\'er-Rao bound. We discuss the effects of losses external to the system on the precision of transmission estimation and identify simple measurement techniques that can saturate the quantum Cram\'er-Rao bound for the bright squeezed states even in the presence of such external losses.
The minimum possible uncertainty when estimating a parameter, in this case transmission, is bounded by the quantum Cramér-Rao bound (QCRB). This bound is independent of the measurement technique and as result can be used to quantify how close a measurement is to the optimal one for a given system and probing state, as an optimal measurement saturates the QCRB. The ability to perform transmission measurements through an optical system at the QCRB can lead to enhancements in the calibration of optimal states for interferometers, the characterization of high efficiency photodetectors, or the sensitivity of sensing devices based on transmission, such as plasmonic sensors or spectroscopy. We show theoretically that the ultimate sensitivity limit per photon in the estimation of transmission, achievable with Fock states or vacuum two-mode squeezed states (vTMSS), can be approached through the use of bright quantum states of light, such as bright single-mode squeezed states (bSMSS) or bright two-mode squeezed states (bTMSS). The use of bright quantum states has the advantage that they can be generated with a significantly larger number of photons, which leads to an absolute sensitivity that can be many orders of magnitude larger than the one achieved with Fock states or vTMSS. We identify a measurement strategy that saturates the QCRB for these bright quantum states and shown experimentally that transmission estimation at the QCRB can be achieved with bTMSS through optimized intensity difference measurements.
The field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical resources. Theoretically, the fundamental minimum uncertainty in the estimation of a parameter for a given probing state is bounded by the quantum Cramér-Rao bound. From a practical perspective, it is necessary to find physical measurements that can saturate this fundamental limit and to show experimentally that it is possible to perform measurements with the required precision to do so. Here we perform experiments that saturate the quantum Cramér-Rao bound for transmission estimation over a wide range of transmissions when probing the system under study with a continuous wave bright two-mode squeezed state. To properly take into account the imperfections in the generation of the quantum state, we extend our previous theoretical results to incorporate the measured properties of the generated quantum state. For our largest transmission level of 84%, we show a 62% reduction over the optimal classical protocol in the variance in transmission estimation when probing with a bright two-mode squeezed state with -8 dB of intensity-difference squeezing. Given that transmission estimation is an integral part of many sensing protocols, such as plasmonic sensing, spectroscopy, calibration of the quantum efficiency of detectors, etc., the results presented promise to have a significant impact on a number of applications in various fields of research.
Recently, there has been significant interest in multiple-parameter quantum estimation techniques that exploit quantum resources. In particular, the estimation of optical transmission is a crucial parameter in various scientific fields and industries. In optical-based sensing, precision can be enhanced through two approaches: increasing the number of photons that interact with the samples and utilizing quantum states of light. In this paper, we investigate multiple transmission estimation using bright two-mode squeezed states (bTMSSs), which combine the advantages of both bright light and quantum states. We calculate the precision bound for multiple transmission estimation by employing multiple copies of bTMSSs with experimentally feasible optimal measurement, determining both the quantum Cramer-Rao bound and the Cramer-Rao bound. Our results demonstrate that multiple copies of bTMSSs can achieve quantum-enhanced sensitivity for multiple transmissions compared to coherent states, and the ultimate limit of precision can be attained in regions with high nonlinear gain. Furthermore, as an application, we show the quantum-enhanced sensing in circular dichroism sensing using a pair of bTMSSs. Our strategy for multiple transmission estimation offers a practical platform for exploring real-world quantum sensing applications.
We consider an optical interferometer with coherent light in one input and a squeezed vacuum in another. Such an interferometer is known to beat the standard quantum limit of sensitivity to the difference of phase shifts in its arms. We find the ultimate limit of precision for such an interferometer by calculating the quantum Fisher information of the mixed quantum state in one of the interferometer's outputs about the difference phase. We show that, in the vicinity of the black fringe, this information is asymptotically close to the quantum Fisher information about this phase for the two-mode readout. We conclude that the single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and allows for the Heisenberg scaling of precision. We also show that the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.
Squeezed states of light are a crucial resource for improving the precision of parameter estimation [1], and this has made an impact in areas as wide ranging as biological imaging [2] and gravitational wave detection [3]. However, the optimal bounds of the Fisher information (FI) in continuous variable (CV) quantum metrology have, so far, only been abstractly defined and do not account for many important experimental parameters, such as the quantum efficiency and bandwidth of the detector [4]. We show that classical FI calculations can be used to compute these optimal bounds with more applicability to experimental measurement. This is demonstrated in the case of single parameter estimation of a modulated optical loss, using a bright amplitude squeezed state as a probe.