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arXiv:0803.0155 · Journal of Modern Optics

Sub-shot-noise quantum optical interferometry: a comparison of entangled state performance within a unified measurement scheme

Yang Gao, Hwang Lee

Published Nov 10, 2008• 13 citations
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Abstract

Phase measurement using a lossless Mach–Zehnder interferometer with certain entangled N-photon states can lead to a phase sensitivity of the order of 1/N, the Heisenberg limit. However, previously considered output measurement schemes are different for different input states to achieve this limit. We show that it is possible to achieve this limit just by the parity measurement for all the commonly proposed entangled states. Based on the parity measurement scheme, the reductions of the phase sensitivity in the presence of photon loss are examined for the various input states.

DOI
10.1080/09500340802428298

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Phase measurement using a lossless Mach–Zehnder interferometer with certain entangled N-photon states can lead to a phase sensitivity of the order of 1/N, the Heisenberg limit. However, previously considered output measurement schemes are different for different input states to achieve this limit. We show that it is possible to achieve this limit just by the parity measurement for all the commonly proposed entangled states. Based on the parity measurement scheme, the reductions of the phase sensitivity in the presence of photon loss are examined for the various input states.
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.
It is known that the phase measurement using lossless Mach-Zehnder interferometer with certain entangled N-photon states can achieve a phase sensitivity of the order of 1/N, the Heisenberg limit. However, the output observable used is different for different input states to achieve the Heisenberg limit. In this paper, we show that it is possible to achieve this limit just by parity measurement for all the commonly proposed entangled states. Taking account of path absorption effect, the phase shifts with parity measurement for these states are obtained and indicate that the N00N state still remains to be the best candidate to perform phase detection if the transmittance of the medium is not too small and the number of photons is not very large.
Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] have argued that twin Fock states of equal photon number NNN injected at both input ports of a Mach-Zehnder interferometer lead to phase measurements with accuracies approaching the Heisenberg limit $\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{HL}}=1/(2N).$ However, the method of phase detection suggested by those authors, obtaining the difference of the photocurrents at the output ports of the interferometer, is not sensitive to the phase difference between the two interferometer paths; in fact, the photocurrent vanishes. In this paper we show that the use of parity measurements on just one of the output modes not only is sensitive to the phase difference but that the sensitivity approaches the Heisenberg limit for large N.N.N.
We investigate the utility of parity detection to achieve Heisenberg-limited phase estimation for optical interferometry. We consider the parity detection with several input states that have been shown to exhibit sub-shot-noise interferometry with their respective detection schemes. We show that with parity detection, all these states achieve the sub-shot-noise limited phase uncertainty. Thus making the parity detection a unified detection strategy for quantum optical metrology. We also consider quantum states that are a combination of a NOON state and a dual-Fock state, which gives a great deal of freedom in the preparation of the input state, and is found to surpass the shot-noise limit.
We investigate the phase sensitivity of the SU(1,1) interfereometer [SU(1,1)I] and the modified Mach–Zehnder interferometer (MMZI) with the entangled coherent states (ECS) as inputs. We consider the ideal case and the situations in which the photon losses are taken into account. We find that, under ideal conditions, the phase sensitivity of both the MMZI and the SU(1,1)I can beat the shot-noise limit (SNL) and approach the Heisenberg limit (HL). In the presence of photon losses, the ECS can beat the coherent and squeezed states as inputs in the SU(1,1)I, and the MMZI is more robust against internal photon losses than the SU(1,1)I.