Optics and Spectroscopy

Optimized interferometry with Gaussian states

Stefano Olivares, Matteo G. A. Paris

Published Aug 1, 2007• 18 citations

Abstract

We revisit Mach—Zehnder interferometry using a suitable phase-space analysis and present a rigorous optimization of the sensitivity in realistic condition, i.e., for Gaussian input states, and taking into account nonunit quantum efficiency in the detection stage. The working regime of the interferometer is optimized at fixed input energy versus the squeezing phases and amplitudes as well as the distribution of squeezing in the two input signals. For ideal detection we find the known result that the squeezing resource allows to beat the shot-noise limit. For nonunit detection efficiency, we show that for fixed input energy one can always optimize the squeezing fraction in order to enhance the sensitivity with respect to the case of no squeezing, even in cases when one cannot go beyond the shot-noise limit.

DOI: 10.1134/s0030400x07080103

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Optics and Spectroscopy

Optimized interferometry with Gaussian states

Stefano Olivares, Matteo G. A. Paris

Published Aug 1, 2007• 18 citations

Publisher

Abstract

DOI: 10.1134/s0030400x07080103

Related papers

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We address in this work the phase sensitivity of a Mach-Zehnder interferometer with Gaussian input states. A squeezed-coherent plus squeezed vacuum input state allows us to unambiguously determine the optimal phase-matching conditions in order to maximize the quantum Fisher information. Realistic detection schemes are described and their performance compared with respect to the quantum Cram\'er-Rao bound. The core of this paper discusses in detail the most general Gaussian input state, without any a priori parameter restrictions. Prioritizing the maximization of various terms in the quantum Fisher information has the consequence of imposing the input phase-matching conditions. We discuss in detail when each scenario yields an optimal performance. Realistic detection scenarios are also considered and their performance compared to the theoretical optimum. The impact of the beam splitter types employed on the optimum phase-matching conditions is also discussed. We find a number of potentially interesting advantages of these states over the coherent plus squeezed vacuum input case.We address in this work the phase sensitivity of a Mach-Zehnder interferometer with Gaussian input states. A squeezed-coherent plus squeezed vacuum input state allows us to unambiguously determine the optimal phase-matching conditions in order to maximize the quantum Fisher information. Realistic detection schemes are described and their performance compared with respect to the quantum Cram\'er-Rao bound. The core of this paper discusses in detail the most general Gaussian input state, without any a priori parameter restrictions. Prioritizing the maximization of various terms in the quantum Fisher information has the consequence of imposing the input phase-matching conditions. We discuss in detail when each scenario yields an optimal performance. Realistic detection scenarios are also considered and their performance compared to the theoretical optimum. The impact of the beam splitter types employed on the optimum phase-matching conditions is also discussed. We find a number of potentially interesting advantages of these states over the coherent plus squeezed vacuum input case.

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We address the precision of optical interferometers fed by quantum and semiclassical Gaussian states involving passive and/or active elements, such as beam splitters, photodetectors, and optical parametric amplifiers. We first address the ultimate bounds to precision by discussing the behavior of the quantum Fisher information. We then consider photodetection at the output and calculate the sensitivity of the interferometers taking into account the nonunit quantum efficiency of the detectors. Our results show that in the ideal case of photon number detectors with unit quantum efficiency the best configuration is the symmetric one, namely, a passive (active) interferometer with a passive (active) detection stage: in this case one may achieve Heisenberg scaling of sensitivity by suitably optimizing over Gaussian states at the input. On the other hand, in the realistic case of detectors with nonunit quantum efficiency, the performances of the passive scheme are unavoidably degraded, whereas detectors involving optical parametric amplifiers allow us to fully compensate for the presence of loss in the detection stage, thus restoring the Heisenberg scaling.We address the precision of optical interferometers fed by quantum and semiclassical Gaussian states involving passive and/or active elements, such as beam splitters, photodetectors, and optical parametric amplifiers. We first address the ultimate bounds to precision by discussing the behavior of the quantum Fisher information. We then consider photodetection at the output and calculate the sensitivity of the interferometers taking into account the nonunit quantum efficiency of the detectors. Our results show that in the ideal case of photon number detectors with unit quantum efficiency the best configuration is the symmetric one, namely, a passive (active) interferometer with a passive (active) detection stage: in this case one may achieve Heisenberg scaling of sensitivity by suitably optimizing over Gaussian states at the input. On the other hand, in the realistic case of detectors with nonunit quantum efficiency, the performances of the passive scheme are unavoidably degraded, whereas detectors involving optical parametric amplifiers allow us to fully compensate for the presence of loss in the detection stage, thus restoring the Heisenberg scaling.

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Jiandong Zhang, Zijing Zhang, Hu, Jun-Yan +4 more

Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of $1/N$ with $N$ being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved with careful calculations of the quantum Fisher information regarding three kinds of common inputs: Gaussian states, squeezed number states, and Schrödinger cat states. The results suggest that shot-noise-style sensitivity limit is no longer available, and the ideal candidate is squeezed vacuum.Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of $1/N$ with $N$ being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved with careful calculations of the quantum Fisher information regarding three kinds of common inputs: Gaussian states, squeezed number states, and Schrödinger cat states. The results suggest that shot-noise-style sensitivity limit is no longer available, and the ideal candidate is squeezed vacuum.

arXiv:1906.10867Jun 26, 20191 citation

N

. As the interaction strength

u

increases past unity, phase diffusion becomes dominant, leading to a transition in the optimal squeezing from initial number squeezing to initial phase squeezing with ${\ensuremath{\sigma}}_{o}\ensuremath{\propto}\sqrt{\mathit{uN}}$ and ${s}_{o}\ensuremath{\propto}\sqrt{N/u}$ shot-noise scaling. The initial phase squeezing translates into hold-time number squeezing, which is less sensitive to interactions than coherent states and improves

{s}_{o}

by a factor of

\sqrt{u}

Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of

1/N

with

N

being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved with careful calculations of the quantum Fisher information regarding three kinds of common inputs: Gaussian states, squeezed number states, and Schrödinger cat states. The results suggest that shot-noise-style sensitivity limit is no longer available, and the ideal candidate is squeezed vacuum.