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arXiv:quant-ph/0612156

On the use of photonic N00N states for practical quantum interferometry

Gerald N. Gilbert, Michael Hamrick, Yaakov S. Weinstein

Published Dec 19, 2006• 2 citations
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Abstract

The performance of photonic $N00N$ states, propagating in an attenuating medium, is analyzed with respect to phase estimation. It is shown that, for $N00N$ states propagating through a lossy medium, the Heisenberg limit is never achieved. It is also shown that, for a given value of $N$, a signal comprised of an attenuated separable state of $N$ photons will actually produce a better phase estimate than will a signal comprised of an equally attenuated $N00N$ state, unless the transmittance of the medium is very high. This is a consequence of the need to utilize measurement operators appropriate to the different signal states. The result is that, for most practical applications in realistic scenarios with attenuation, the resolution of $N00N$ state-based phase estimation not only does not achieve the Heisenberg Limit, but is actually worse than the Standard Quantum Limit. It is demonstrated that this performance deficit becomes more pronounced as the number, $N$, of photons in the signal increases.

DOI
10.48550/arxiv.quant-ph/0612156

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The performance of photonic N00NN00NN00N states, propagating in an attenuating medium, is analyzed with respect to phase estimation. It is shown that, for N00NN00NN00N states propagating through a lossy medium, the Heisenberg limit is never achieved. It is also shown that, for a given value of NNN, a signal comprised of an attenuated separable state of NNN photons will actually produce a better phase estimate than will a signal comprised of an equally attenuated N00NN00NN00N state, unless the transmittance of the medium is very high. This is a consequence of the need to utilize measurement operators appropriate to the different signal states. The result is that, for most practical applications in realistic scenarios with attenuation, the resolution of N00NN00NN00N state-based phase estimation not only does not achieve the Heisenberg Limit, but is actually worse than the Standard Quantum Limit. It is demonstrated that this performance deficit becomes more pronounced as the number, NNN, of photons in the signal increases.
We study the phase estimation abilites of photonic N 00 N states, propagating in an attenuating medium, is analyzed. It is shown that N 00 N states of a given number of enangled photons N , never achieve the 1/ N Heisenberg limit if the propagation occurs through lossy medium. It is also shown that a signal comprised of an attenuated separable state of N photons will actually produce a better phase estimate than a signal comprised of an equally attenuated N 00 N state unless the transmittance of the medium is very high. Thus, for most practical applications in realistic scenarios with attenuation, the resolution of N 00 N state-based phase estimation not only does not achieve the Heisenberg Limit, but is actually worse than the 1/(square root of)N Standard Quantum Limit. This performance deficit becomes more pronounced as the number, N , of photons in the signal increases.
The phase estimation performance of photonic N00N states propagating in an attenuating medium is analyzed. It is shown that the Heisenberg limit is never achieved and that an attenuated separable state of N photons will actually produce a better phase estimate than an equally attenuated N00N state unless the transmittance of the medium is sufficiently high. Thus, for most practical applications with realistic attenuation, N00N-state-based phase estimation actually performs worse than the standard quantum limit. This performance deficit becomes more pronounced as the number of photons in the signal increases.
We analyze the phase estimation ability of photonic N00N states propagating in a lossy medium. In such a medium a N00N state of N enangled photons cannot achieve the maximum 1/N phase estimation resolution. In fact, unless the transmittance of the medium is extremely high, a signal comprised of an attenuated separable state of N photons will produce a better phase estimate than a comparable signal of an equally attenuated N00N state. Thus, for most practical applications the resolution provided photonic N00N states is actually worse than the 1/√N Standard Quantum Limit. This performance deficit becomes more pronounced as the number of photons in the signal increases.
We consider the viability of photonic N00N states for practical remote quantum sensing in realistic atmospheres. This is done by analyzing the phase estimation performance of N-photon N00N states propagating in an attenuating medium. It is shown that the 1/N Heisenberg limit is never achieved. Moreover, an attenuated N00N state will actually produce a worse phase estimate than an equally attenuated N-photon separable state unless the transmittance of the medium is sufficiently high. The transmittance threshold for which the N00N states perform better than separable states increases as the number of signal photons increases. The amount of photon attenuation that arises in realistic, turbulent atmospheres is generally far below this threshold even for low N N00N states. Photonic N00N states are thus not viable for use in practical remote quantum sensing.
The modern scientific method is critically dependent on precision measurements of physical parameters. A classic example is the measurement of the optical phase enabled by optical interferometry, where the error on the measured phase is conventionally bounded by the so-called Heisenberg limit. To achieve phase estimation at the Heisenberg limit, it has been common to consider protocols based on highly complex N00N states of light. However, despite decades of research and several experimental explorations, there has been no demonstration of deterministic phase estimation with N00N states reaching the Heisenberg limit or even surpassing the shot noise limit. Here we use a deterministic phase estimation scheme based on a source of Gaussian squeezed vacuum states and high-efficiency homodyne detection to obtain phase estimates with an extreme sensitivity that significantly surpasses the shot noise limit and even beats the conventional Heisenberg limit as well as the performance of a pure N00N state protocol. Using a high-efficiency setup with a total loss of about 11%, we achieve a Fisher information of 15.8(6) rad^{-2} per photon-a significant increase in performance compared to state of the art and beyond an ideal six photon N00N state scheme. This work represents an important achievement in quantum metrology, and it opens the door to future quantum sensing technologies for the interrogation of light-sensitive biological systems.
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.