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arXiv:2001.05797 · Physical Review A

Double-port measurements for robust quantum optical metrology

Wei Zhong, Lan Zhou, Yu‐Bo Sheng

Published Apr 21, 2021• 14 citations
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Abstract

It has been proposed and demonstrated that path-entangled Fock states (PEFSs) are robust against photon loss over NOON states [S. D. Huver et al., Phys. Rev. A 78, 063828 (2008)]. However, the demonstration was based on a measurement scheme which was yet to be implemented in experiments. In this work, we quantitatively illustrate the advantage of PEFSs over NOON states in the presence of photon losses by analytically calculating the quantum Fisher information. To realize such an advantage in practice, we then investigate the achievable sensitivities by employing three types of feasible measurements: parity, photon-number-resolving, and homodyne measurements. We here apply a double-port measurement strategy where the photons at each output port of the interferometer are simultaneously detected with the aforementioned types of measurements.

DOI
10.1103/physreva.103.042611

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It has been proposed and demonstrated that path-entangled Fock states (PEFSs) are robust against photon loss over NOON states [S. D. Huver et al., Phys. Rev. A 78, 063828 (2008)]. However, the demonstration was based on a measurement scheme which was yet to be implemented in experiments. In this work, we quantitatively illustrate the advantage of PEFSs over NOON states in the presence of photon losses by analytically calculating the quantum Fisher information. To realize such an advantage in practice, we then investigate the achievable sensitivities by employing three types of feasible measurements: parity, photon-number-resolving, and homodyne measurements. We here apply a double-port measurement strategy where the photons at each output port of the interferometer are simultaneously detected with the aforementioned types of measurements.
To acquire the best path-entangled photon Fock states for robust quantum-optical metrology with parity detection, we calculate phase information from a lossy interferometer by using twin entangled Fock states. We show that (a)(\text{a})(a) when loss is less than 50% twin entangled Fock states with large photon number difference give higher visibility, while when loss is higher than 50%, the ones with less photon number difference give higher visibility; and (b)(\text{b})(b) twin entangled Fock states with large photon number difference give sub--shot-noise limit sensitivity for phase detection in a lossy environment. This result provides a reference on what particular path-entangled Fock states are useful for real world metrology applications.
Abstract We theoretically investigate the phase sensitivity of a truncated SU(1,1) interferometer fed with a two-mode coherent state and employing double-port homodyne detection. On the one hand, we analytically demonstrate that the two-mode coherent state provides better phase sensitivity than the single-mode coherent state. In addition, we show that the double-port homodyne detection is a quasi-optimal measurement. For a bright coherent-state input, the sensitivity of this scheme saturates the phase-sensitivity bound determined by the quantum Fisher information. On the other hand, we quantitatively illustrate the advantage of double-port homodyne detection over the single-port scheme under ideal conditions and in the presence of photon loss, respectively. Furthermore, our analysis indicates that the scheme we propose is robust against photon loss.
Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] have argued that twin Fock states of equal photon number NNN injected at both input ports of a Mach-Zehnder interferometer lead to phase measurements with accuracies approaching the Heisenberg limit $\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{HL}}=1/(2N).$ However, the method of phase detection suggested by those authors, obtaining the difference of the photocurrents at the output ports of the interferometer, is not sensitive to the phase difference between the two interferometer paths; in fact, the photocurrent vanishes. In this paper we show that the use of parity measurements on just one of the output modes not only is sensitive to the phase difference but that the sensitivity approaches the Heisenberg limit for large N.N.N.
We propose a class of path-entangled photon Fock states for robust quantum optical metrology, imaging, and sensing in the presence of loss. We model propagation loss with beam splitters and derive a reduced density-matrix formalism from which we examine how photon loss affects coherence. It is shown that particular entangled number states, which contain a special superposition of photons in both arms of a Mach-Zehnder interferometer, are resilient to environmental decoherence. We demonstrate an order of magnitude greater visibility with loss than possible with path-entangled $\ensuremath{\mid}N,0⟩+\ensuremath{\mid}0,N⟩$ states. We also show that the effectiveness of a detection scheme is related to super-resolution visibility.
We study effects of phase fluctuations on phase sensitivity and visibility of a class of robust path-entangled photon Fock states (known as $m{m}^{\ensuremath{'}}$ states) as compared to the maximally path-entangled NOON states in the presence of realistic phase fluctuations such as turbulence noise. Our results demonstrate that the $m{m}^{\ensuremath{'}}$ states, which are more robust than the NOON state against photon loss, perform equally well when subject to such fluctuations. We derive the quantum Fisher information with the phase-fluctuation noise and show that the phase sensitivity with parity detection for both of the above states saturates the quantum Cram\'er-Rao bound in the presence of such noise, suggesting that parity detection is an optimal detection strategy.
In optical interferometry path-entangled states such as NOON states have shown to give quantum-enhanced precision measurements, but these states are notoriously fragile to particle losses, which typically collapse the quantum state and destroy the phase information. A class of inherently robust states that show the potential for great improvements over the alternatives are the entangled coherent states (ECSs). We show that these states allow substantial improvements over unentangled `classical' states and highly-entangled NOON states for a wide range of loss values. We then describe a measurement scheme that can be used to measure these states with a precision close to the theoretical bound given by the quantum Fisher information. \nWe then look at the quantum mechanisms that lead to precise measurements. In optical interferometry multi-mode entanglement is often assumed to be the driving force behind quantum enhanced measurements. Recent work has shown this assumption to be false, and here we show that when photon losses occur multi-mode entanglement is actually detrimental to obtaining quantum enhanced measurements. We specifically apply this idea to a superposition of coherent states, demonstrating that these states show a robustness to loss that allows them to significantly outperform their competitors in realistic systems. A practically viable measurement scheme is then presented that allows measurements close to the theoretical bound, even with loss. \nIn this thesis we also consider superpositions of spin coherent states and their application to quantum metrology. Compared to optical states, spin systems have a distinctly different process of decoherence known as non-Markovian dephasing, which has shown to give greatly improved robustness to loss. We see that spin cat states give an enhanced scaling over the shot noise limit, even with dephasing, whilst being realisable with current technology.