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Optical and Quantum Sensing and Precision Metrology II

Estimation of the average of arbitrary unknown phase delays with Heisenberg-scaling precision

Danilo Triggiani, Vincenzo Tamma

Published Mar 3, 2022• 1 citation
Publisher

Abstract

We show an estimation scheme which reaches the Heisenberg-scaling sensitivity in the estimation of the average of the optical phases along the two arms of a Mach-Zehnder interferometer, by using a single squeezed vacuum state and homodyne detection at a single output port. We show that, in order to achieve this quantum advantage, it is required only a classical prior knowledge about the two phases, namely obtainable with a classical estimation strategy with shot-noise limited precision.

DOI
10.1117/12.2618306

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We show an estimation scheme which reaches the Heisenberg-scaling sensitivity in the estimation of the average of the optical phases along the two arms of a Mach-Zehnder interferometer, by using a single squeezed vacuum state and homodyne detection at a single output port. We show that, in order to achieve this quantum advantage, it is required only a classical prior knowledge about the two phases, namely obtainable with a classical estimation strategy with shot-noise limited precision.
Achieving the ultimate quantum precision in the estimation of multiple physical parameters simultaneously is a challenge in quantum metrology due to fundamental limitations and experimental challenges in harnessing the necessary quantum resources. We propose an experimentally feasible scheme to reach Heisenberg-scaling precision in the simultaneous estimation of two unknown phase parameters in a Mach-Zehnder interferometer by using squeezed and coherent states of light as input and homodyne detections at the outputs. Our results open a new research paradigm in harnessing the full quantum advantage of multiparameter estimation in distributed quantum sensing and quantum information processing, as enabled by multiphoton quantum interference with scalable experimental resources.
We address the problem of distributed quantum metrology with a single squeezed-vacuum source by using the formalism of quantum mechanics in phase space. In particular, we demonstrate Heisenberg-limited sensitivity in the measurement of the average of two arbitrary phase shifts in the arms of a Mach–Zehnder interferometer. We obtain exact results for the measurement probability at the interferometer output for any value of the phases, which give us insight into the emergence of Heisenberg-limited sensitivity for periodical values of the phases.
Quantum sensing and quantum metrology propose schemes for the estimation of physical properties, such as lengths, time intervals, and temperatures, achieving enhanced levels of precision beyond the possibilities of classical strategies. However, such an enhanced sensitivity usually comes at a price: the use of probes in highly fragile states, the need to adaptively optimise the estimation schemes to the value of the unknown property we want to estimate, and the limited working range, are some examples of challenges which prevent quantum sensing protocols to be practical for applications. This work reviews two feasible estimation schemes which address these challenges, employing easily realisable resources, i.e., squeezed light, and achieve the desired quantum enhancement of the precision, namely the Heisenberg-scaling sensitivity. In more detail, it is here shown how to overcome, in the estimation of any parameter affecting in a distributed manner multiple components of an arbitrary M-channel linear optical network, the need to iteratively optimise the network. In particular, we show that this is possible with a single-step adaptation of the network based only on a prior knowledge of the parameter achievable through a “classical” shot-noise limited estimation strategy. Furthermore, homodyne measurements with only one detector allow us to achieve Heisenberg-limited estimation of the parameter. We further demonstrate that one can avoid the use of any auxiliary network at the price of simultaneously employing multiple detectors.
We propose a metrological strategy reaching Heisenberg-scaling precision in the estimation of functions of any fixed number ppp of arbitrary parameters encoded in a generic MMM-channel linear network. This scheme is experimentally feasible since it only employs a single-mode squeezed vacuum and homodyne detection on a single output channel. Two auxiliary linear networks are required and their role is twofold: to refocus the signal into a single channel after the interaction with the interferometer, and to fix the function of the parameters to be estimated according to the linear network analyzed. Although the refocusing requires some knowledge on the parameters, we show that the required precision on the prior measurement is achievable with a classic measurement. We conclude by discussing two paradigmatic schemes in which the choice of the auxiliary stages allows us to change the function of the unknown parameter to estimate.
We show that the phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1/{N}_{T}$, where NT{N}_{T}NT​ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter rrr and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.
We propose a quantum metrological protocol based on a Mach-Zehnder interferometer with a squeezed vacuum input state and an antisqueezing operation at one of its output channels. A simple and intuitive geometrical picture of the state evolution is provided by the marginal Wigner functions of the state at each interferometer output channel. The protocol allows us to detect the values of the sum $\ensuremath{\beta}=\frac{1}{2}({\ensuremath{\varphi}}_{1}+{\ensuremath{\varphi}}_{2})+{\ensuremath{\theta}}_{\mathrm{in}}\ensuremath{-}{\ensuremath{\theta}}_{\mathrm{out}}$, of the relative phase ${\ensuremath{\theta}}_{\mathrm{in}}\ensuremath{-}{\ensuremath{\theta}}_{\mathrm{out}}$ between the two squeezers, and of the average of the phase delays ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ in the two arms of the interferometer. The detection sensitivity scales at the Heisenberg limit and, remarkably, is robust not only to detector inefficiencies but also to any photon losses occurring before the antisqueezing operation. Interestingly, we demonstrate that in the latter case an increase of sensitivity can even occur by increasing the losses in a suitable range.