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arXiv:0708.2809 · Journal of Physics B Atomic Molecular and Optical Physics

Quantum enhancement ofN-photon phase sensitivity by interferometric addition of down-converted photon pairs to weak coherent light

Takafumi Ono, Holger F Hofmann

Published Apr 22, 2008• 11 citations
arXivHTMLPDF

Abstract

It is shown that the addition of down-converted photon pairs to coherent laser light enhances the N -photon phase sensitivity due to the quantum interference between components of the same total photon number. Since most of the photons originate from the coherent laser light, this method of obtaining non-classical N -photon states is much more efficient than methods based entirely on parametrically down-converted photons. Specifically, it is possible to achieve an optimal phase sensitivity of about δϕ 2 = 1/ N 3/2 , equal to the geometric mean of the standard quantum limit and the Heisenberg limit, when the average number of down-converted photons contributing to the N -photon state approaches .

DOI
10.1088/0953-4075/41/9/095502

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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.

Research in Optical Sciences•Jan 1, 2012•0 citations
It is shown that the addition of down-converted photon pairs to coherent laser light enhances the N -photon phase sensitivity due to the quantum interference between components of the same total photon number. Since most of the photons originate from the coherent laser light, this method of obtaining non-classical N -photon states is much more efficient than methods based entirely on parametrically down-converted photons. Specifically, it is possible to achieve an optimal phase sensitivity of about δϕ 2 = 1/ N 3/2 , equal to the geometric mean of the standard quantum limit and the Heisenberg limit, when the average number of down-converted photons contributing to the N -photon state approaches .
We show that the quantum interference between down-converted photon pairs and photons from coherent laser light can produce a maximally path entangled NNN-photon output component with a fidelity greater than 90% for arbitrarily high photon numbers. A simple beam splitter operation can thus transform the two-photon coherence of down-converted light into an almost optimal NNN-photon coherence.
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.
We investigate phase estimation in an SU(1,1) interferometer employing mmm-photon-added even and odd coherent states as nonclassical input resources. The phase sensitivity is evaluated through intensity detection and the error propagation method, while the ultimate precision limit is determined from the quantum Cramér-Rao bound with the quantum Fisher information serving as the relevant metrological quantity. Our results demonstrate that photon addition significantly enhances the phase sensitivity, increases the quantum Fisher information, and reduces the quantum Cramér-Rao bound, leading to a clear improvement over the corresponding even and odd coherent states. Furthermore, the achievable sensitivity exceeds the standard quantum limit and gradually approaches the Heisenberg scaling with increasing photon-addition number. We also find that the mmm-photon-added even coherent states exhibit a modest advantage over their odd counterparts. As mmm increases, however, this distinction becomes progressively weaker, suggesting that photon addition diminishes the role of the initial parity of the coherent state in determining the interferometric performance.
Quantum metrology promises greater sensitivity for optical phase measurements than could ever be achieved classically. Here, we present a theory of the phase sensitivity for the general case where the detection probability is given by an N photon interference fringe. We find that the phase sensitivity has a complicated dependence on both the intrinsic efficiency of detection η and the interference fringe visibility V . Most importantly, the phase that gives maximum phase sensitivity is in general not the same as the phase at which the slope of the interference fringe is a maximum, as has previously been assumed. We determine the parameter range where quantum enhanced sensitivity can be achieved. In order to illustrate these theoretical results, we perform a four-photon experiment with η=3/4 and V =82±6% (an extension of our previous work (Nagata et al 2007 Science 316 726)) and find a phase sensitivity 1.3 times greater than the standard quantum limit at a phase different to that which gives maximum slope of the interference fringe.
Two-path interferometry with coherent states and a squeezed vacuum can achieve phase sensitivities close to the Heisenberg limit when the average photon number of the squeezed vacuum is close to the average photon number of the coherent light. Here, we investigate the phase sensitivity of such states in the presence of photon losses. It is shown that the Cramer-Rao bound of phase sensitivity can be achieved experimentally by using a weak local oscillator and photon counting in the output. The phase sensitivity is then given by the Fisher information FFF of the state. In the limit of high squeezing, the ratio (F−−N)/N2(F--N)/{N}^{2}(F−−N)/N2 of Fisher information above shot noise to the square of the average photon number NNN depends only on the average number of photons lost, nloss{n}_{\mathrm{loss}}nloss​, and the fraction of squeezed vacuum photons $\ensuremath{\mu}$. For $\ensuremath{\mu}=1/2$, the effect of losses is given by (F−−N)/N2=1/(1+2nloss)(F--N)/{N}^{2}=1/(1+2{n}_{\mathrm{loss}})(F−−N)/N2=1/(1+2nloss​). The possibility of increasing the robustness against losses by lowering the squeezing fraction $\ensuremath{\mu}$ is considered, and an optimized result is derived. However, the improvements are rather small, with a maximal improvement by a factor of 2 at high losses.
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.