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 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.
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 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.
We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.
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.