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Research in Optical Sciences

Optimal multi-photon phase sensing with a single interference fringe

Guo‐Yong Xiang, Holger Friedrich Hofmann, Geoff J. Pryde

Published Jan 1, 2012• 0 citations
Publisher

Abstract

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.

DOI
10.1364/qim.2012.qm3b.2

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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.
Quantum entanglement can help to increase the precision of optical phase measurements beyond the shot noise limit (SNL) to the ultimate Heisenberg limit. 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 SNL 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" state does not achieve optimal phase sensitivity when N > 4, rather, we show that the Holland-Burnett state 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 SNL.
Path-entangled multi-photon states allow optical phase-sensing beyond the shot-noise limit, provided that an efficient parity measurement can be implemented. Realizing this experimentally is technologically demanding, as it requires coincident single-photon detection proportional to the number of photons involved, which represents a severe challenge for achieving a practical quantum advantage over classical methods. Here, we exploit advanced quantum state engineering based on superposing two photon-pair creation events to realize a new approach that bypasses this issue. In particular, optical phase shifts are probed with a two-photon quantum state whose information is subsequently effectively transferred to a single-photon state. Notably, without any multiphoton detection, we infer phase shifts by measuring the average intensity of the single-photon beam on a photodiode, in analogy to standard classical measurements. Importantly, our approach maintains the quantum advantage: twice as many interference fringes are observed for the same phase shift, corresponding to N = 2 path-entangled photons. Our results demonstrate that the advantages of quantum-enhanced phase sensing can be fully exploited in standard intensity measurements, paving the way toward resource-efficient and practical quantum optical metrology.
Quantum phase measurement with multiphoton twin-Fock states has been shown to be optimal for detecting equal numbers of photons at the output ports of a Mach–Zehnder interferometer (i.e., the so-called single-fringe detection), since the phase sensitivity can saturate the quantum Cramér–Rao lower bound at certain values of phase shift. Here we report a further step to achieve a global phase estimation at the Heisenberg limit by detecting the particle-number difference (i.e., the J ^ z measurement). We show the role of experimental imperfections on the ultimate estimation precision with the six-photon twin-Fock state of light. Our results show that both the precision and the sensing region of the J ^ z measurement are better than those of the single-fringe detection, due to combined contributions of the measurement outcomes. We numerically simulate the phase estimation protocol using an asymptotically unbiased maximum likelihood estimator.
One important problem in quantum optics is to resolve an extremely small change of phase shift. The complementarity between photon number and phase sets an ultimate limit, the so-called Heisenberg limit, on the phase measurement sensitivity. The precise phase estimation has many technological applications, such as optical gyroscopes, gravitational wave detection, quantum imaging and sensing. In this thesis I show that the utilization of the parity measurement in the optical interferometry is actually applicable to a wide range of quantum entangled input states. Comparison of the performance of the various quantum states then can be made within such a unified output measurement scheme. Based on such a universal detection scheme, we present a comparison of the phase sensitivity reduction for various quantum states of light in the presence of photon loss. I also provide a simple condition that could be used to check whether an arbitrary state can achieve the Heisenberg limit independently of the detection scheme. It implies that the fidelity between the two output states with zero phase and minimal detectable phase applied respectively should significantly different from unity as the minimal phase shift scaling as 1/N, whereas the average number of input photons N goes to infinity. Next I give several measures to characterize the which-way information in the interference experiment. We define a new distinguishability associated with the fidelity between two density matrices to measure the which-way information. We demonstrate that the changes of mutual entropy as well as the entanglement of formation of the whole system, i.e. the physical system plus the which-way detector can also be used to describe the which-way information. With such quantities, we show that as the fringe visibility of the interference pattern gets larger, the less which-way information is obtained. Finally, I show that coherent light coupled with photon number resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power. This scheme would have applications to laser radar, given the difficulty in making entangled states of light, as well as their susceptibility to atmospheric absorption.
While two-photon Hong-Ou-Mandel interference visibility has become a standard metric for single-photon sources, many optical quantum technologies require the generation and manipulation of larger photonic states. To date, efficiency limitations have prevented scaling quantum dot-based interference to the coalescence of more than two photons at a single beamsplitter. We overcome this limitation by combining a state-of-the-art quantum dot source with deterministic demultiplexing, enabling the direct observation of quantum interference fringes arising from up to four photons. We measure high mean interference contrasts of 93.0±0.1 %93.0 \pm 0.1~\%93.0±0.1 % for two photons, and 84.1±1.0 %84.1 \pm 1.0~\%84.1±1.0 % for four photons, with the complex fringe structure fully reproduced by a theoretical model. These results reveal the existence of "deep fringes" whose minima are unaffected by distinguishable photons, rendering the maximum contrast of four-photon interference highly sensitive to multi-photon emission but robust against photon distinguishability. We predict that these phenomena will extend to interference of larger numbers of photons, with relevance across a range of potential optical quantum technologies. A Fisher information analysis demonstrates that interference fringes from our source can exhibit phase sensitivity beyond the standard quantum limit, illustrating potential applications in quantum metrology.
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.