Ameliorated phase sensitivity through intensity measurements in a Mach–Zehnder interferometer

The ultimate achievable phase sensitivity in a Mach–Zehnder interferometer is given by the Heisenberg limit of 1/⟨N⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\langle N\rangle $$\end{document}, with ⟨N⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle N \rangle $$\end{document} being the number of photons in the interferometer. However, the best-known phase sensitivity as obtained through intensity measurements is ≈2/⟨N⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 2/\langle N \rangle $$\end{document}. In this work, we provide examples of states that achieve improved phase sensitivity lesser than 2/⟨N⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2/\langle N \rangle $$\end{document}, albeit greater than 1/⟨N⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\langle N \rangle $$\end{document}. A modified scheme of the Mach–Zehnder interferometer that uses the first excited Fock state, single-mode squeezers, and an additional beam splitter is presented. It is shown that the modified scheme attains a phase sensitivity lesser than 2/⟨N⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2/\langle N \rangle $$\end{document}, through intensity measurements. The effect of attenuation and noise on phase sensitivity is considered. Phase sensitivity superior to the standard quantum limit is seen to persist for a finite range of attenuation and noise.