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Physical Review A

Phase measurement at the Heisenberg limit with three photons

Haibo Wang, Takayoshi Kobayashi

Published Feb 15, 2005• 41 citations
Publisher

Abstract

We describe an interferometer which is capable of measuring phase at a Heisenberg limit of $1∕3$. The proposed interferometer, which consumes three indistinguishable photons, consists of two beam splitters with two-third reflectivity and two mirrors. It is shown that the oscillation period of the quantum interference is three times shorter than in the classical case and the interference visibility is 100%. A de Broglie wavelength given by ${\ensuremath{\lambda}}_{0}∕3$ can then be observed in this scheme, where ${\ensuremath{\lambda}}_{0}$ is the wavelength of a single photon. Compared with similar phase measurements [Mitchell et al., Nature (London) 429, 161 (2004); Walter et al., ibid. 429, 158 (2004)] by multiphoton maximally entangled states (MES), a three-photon MES is not generated in our proposal.

DOI
10.1103/physreva.71.021802

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We describe an interferometer which is capable of measuring phase at a Heisenberg limit of 1∕31∕31∕3. The proposed interferometer, which consumes three indistinguishable photons, consists of two beam splitters with two-third reflectivity and two mirrors. It is shown that the oscillation period of the quantum interference is three times shorter than in the classical case and the interference visibility is 100%. A de Broglie wavelength given by \ensuremathλ0∕3{\ensuremath{\lambda}}_{0}∕3\ensuremathλ0​∕3 can then be observed in this scheme, where \ensuremathλ0{\ensuremath{\lambda}}_{0}\ensuremathλ0​ is the wavelength of a single photon. Compared with similar phase measurements [Mitchell et al., Nature (London) 429, 161 (2004); Walter et al., ibid. 429, 158 (2004)] by multiphoton maximally entangled states (MES), a three-photon MES is not generated in our proposal.
We describe an interferometer that is capable of measuring phase at a Heisenberg limit of one-third. The proposed interferometer, which consumes three indistinguishable photons, consists of two beam splitters with two-third reflectivity and two mirrors. It is shown that the oscillation period of the quantum interference is three times shorter than in the classical case and the interference visibility is 100%. A de Broglie wavelength given by λ 0 /3 can then be observed in this scheme, where λ 0 is the wavelength of a single photon. Compared with similar phase measurements by multiphoton maximally entangled states (MES), a three-photon MES is not generated in our proposal.
Using only linear optics and projection measurement, we propose a simple multiphoton interference scheme for the demonstration of six-photon de Broglie wavelength, which leads to a $\mathrm{cos}6\ensuremath{\varphi}$ dependence on the single-photon phase $\ensuremath{\varphi}$. Although the scheme does not reach the Heisenberg limit in sensitivity for phase measurement, it improves upon the standard quantum limit with super-resolved sensitivity in phase measurement. We discuss possible experimental implementation with three pairs of photons from parametric down-conversion.
Two schemes of projection measurement are realized experimentally to demonstrate the de Broglie wavelength of three photons without the need for a maximally entangled three-photon state (the NOON state). The first scheme is based on a proposal by [Wang and Kobayashi, Phys. Rev. A 71, 021802(R) (2005)] that utilizes a couple of asymmetric beam splitters while the second one applies the general method of NOON-state projection measurement to a three-photon case. Quantum interference of three photons is responsible for projecting out the unwanted states, leaving only the NOON-state contribution in these schemes of projection measurement. A detailed multimode analysis is made to account for imperfect situations in the experiments.
Two schemes of projection measurement are realized experimentally to demonstrate the de Broglie wavelength of three photons without the need for a maximally entangled three-photon state (the NOON state). The first scheme is based on the proposal by Wang and Kobayashi (Phys. Rev. A {\bf 71}, 021802) that utilizes a couple of asymmetric beam splitters while the second one applies the general method of NOON state projection measurement to three-photon case. Quantum interference of three photons is responsible for projecting out the unwanted states, leaving only the NOON state contribution in these schemes of projection measurement.
In quantum optics an ensemble of photons is treated as a Bose condensate which has a de Broglie wavelength given by ${\ensuremath{\lambda}}_{0}/N$ where ${\ensuremath{\lambda}}_{0}$ and NNN are the wavelength and average number of constituent photons, respectively. We describe an interferometer which is capable of measuring this de Broglie wavelength. We show specifically that a coherent state input has a de Broglie coherence length proportional to 1/N1/\sqrt{N}1/N​. Finally we show that the interferometer is Heisenberg limited in sensitivity ( $\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{min}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/N$) and thus forms a new class of interferometers operating at the fundamental quantum limit.
We study multiphoton quantum-interference effects at a beam splitter and its connection to the prospect of attaining interferometric phase-shift measurements with noise levels below the standard quantum limit. Specifically, we consider the mixing of the most classical states of light coherent states with the most nonclassical states of light number states at a 50 : 50 beam splitter. Multiphoton quantum-interference effects from mixing photon-number states of small photon numbers with coherent states of arbitrary amplitudes are dramatic even at the level of a single photon. For input vacuum and coherent states, the joint photon-number distribution after the beam splitter is unimodal, a product of Poisson distributions for each of the output modes but with the input of a single photon, the original distribution is symmetrically bifurcated into a bimodal distribution. With a two-photon-number state mixed with a coherent state, a trimodal distribution is obtained, etc. The obtained distributions are shown to be structured so as to be conducive for approaching Heisenberg-limited sensitivities in photon-number parity-based interferometry. We show that mixing a coherent state with even a single photon results in a significant reduction in noise over that of the shot-noise limit. Finally, based on the results of mixing coherent light with single photons, we consider the mixing coherent light with the squeezed vacuum and the squeezed one-photon states and find the latter yields higher sensitivity in phase-shift measurements for the same squeeze parameter owing to the absence of the vacuum state.