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Physical Review A

Six-photon de Broglie wavelength and phase-measurement sensitivity improving upon the standard quantum limit

B. Liu, Z. Y. Ou

Published Feb 23, 2010• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.81.023823

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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 cos6\ensuremathφ\mathrm{cos}6\ensuremath{\varphi}cos6\ensuremathφ dependence on the single-photon phase \ensuremathφ\ensuremath{\varphi}\ensuremathφ. 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.
The ultimate sensitivity of optical measurements is a key element of many recent works. Classically, it is mainly limited by the shot noise limit. However, a measurement setup that incorporates quantum mechanical principles can surpass the shot noise limit and reach the Heisenberg limit. Nevertheless, many of those experiments fail to break even the classical shot-noise limit. Following a recent proposal, we present here the results of optical phase measurements with a photon-number resolving detector using coherent states of up to 4200 photons on average. An additional scheme that can be implemented using standard single-photon detectors is also presented, and the results of the two schemes are compared. These measurements present deterministic single-shot sub-wavelength super-resolution up to 288 better than the optical wavelength. The results follow the classically limited sensitivity, up to 86 times better than the wavelength.
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.
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.
Quantum entanglement can help to increase the precision of optical phase measurements beyond the standard quantum limit (SQL) to the ultimate Heisenberg limit [1, 2]. However, the N-photon parity measurements required to achieve this optimal sensitivity are extremely difficult to realize with current photon detection technologies, requiring high-fidelity resolution of N + 1 different photon distributions between the output ports. Recent experimental demonstrations of precision beyond the SQL for two[3–5] and four[6–9] photons have therefore used only one or two photon-number detection patterns instead of parity measurements. Here we investigate the achievable phase sensitivity of the simple and efficient single interference fringe detection technique. We show that the maximally-entangled NOON[10] state does not achieve optimal phase sensitivity when N > 4, rather, the Holland-Burnett state[11] is optimal. We experimentally demonstrate this enhanced sensitivity using a single photon-counted fringe of the six-photon Holland-Burnett state. Specifically, our single-fringe six-photon measurement achieves a phase variance three times below the SQL.
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.
A measurement process is constructed to project an arbitrary two-mode NNN-photon state to a maximally entangled NNN-photon state. The result of this projection measurement shows a typical interference fringe with an NNN-photon de Broglie wavelength. For an experimental demonstration, this measurement process is applied to a four-photon superposition state from two perpendicularly oriented type-I parametric down-conversion processes. Generalization to the projection measurement of an arbitrary NNN-photon state can be easily made and may have wide applications in quantum information. As an example, we formulate it for precision phase measurement.