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CRC Press eBooks

Phase Measurement at the Heisenberg Limit with Three Photons

Takayoshi Kobayashi

Published Oct 15, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781003618997-21

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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.
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 ${\ensuremath{\lambda}}_{0}∕3$ 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.
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 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.
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.
Summary form only given. Recently, Jacobson et al. ( see Phys. Rev. Lett., vol.74, p. 4835 (1995)) proposed the concept of "photonic de Broglie wave" in multiphoton states. They argued that the photonic de Broglie wavelength of an ensemble of photons with wavelength /spl lambda//sub 0/ and average number of photons N can be measured using a special interferometer with "effective beam splitters" that do not split the multiphoton states into constituent photons. Fonseca et al. see Phys. Rev. Lett., vol. 82, p. 2868 (1999)) measured the photonic de Broglie wavelength of a two-photon state ("biphoton") using a kind of Young's interferometer. We propose and demonstrate the measurement of photonic de Broglie wavelength of biphotons generated by parametric down-conversion utilizing the usual Mach-Zehnder interferometer.
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.