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arXiv:1307.4770 · Physical Review A

Effects of phase fluctuations on phase sensitivity and visibility of path-entangled photon Fock states

Bhaskar Roy Bardhan, Kebei Jiang, Jonathan P. Dowling

Published Aug 29, 2013• 38 citations
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Abstract

We study effects of phase fluctuations on phase sensitivity and visibility of a class of robust path-entangled photon Fock states (known as $m{m}^{\ensuremath{'}}$ states) as compared to the maximally path-entangled NOON states in the presence of realistic phase fluctuations such as turbulence noise. Our results demonstrate that the $m{m}^{\ensuremath{'}}$ states, which are more robust than the NOON state against photon loss, perform equally well when subject to such fluctuations. We derive the quantum Fisher information with the phase-fluctuation noise and show that the phase sensitivity with parity detection for both of the above states saturates the quantum Cram\'er-Rao bound in the presence of such noise, suggesting that parity detection is an optimal detection strategy.

DOI
10.1103/physreva.88.023857

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We study effects of phase fluctuations on phase sensitivity and visibility of a class of robust path-entangled photon Fock states (known as mm\ensuremath′m{m}^{\ensuremath{'}}mm\ensuremath′ states) as compared to the maximally path-entangled NOON states in the presence of realistic phase fluctuations such as turbulence noise. Our results demonstrate that the mm\ensuremath′m{m}^{\ensuremath{'}}mm\ensuremath′ states, which are more robust than the NOON state against photon loss, perform equally well when subject to such fluctuations. We derive the quantum Fisher information with the phase-fluctuation noise and show that the phase sensitivity with parity detection for both of the above states saturates the quantum Cram'er-Rao bound in the presence of such noise, suggesting that parity detection is an optimal detection strategy.
To acquire the best path-entangled photon Fock states for robust quantum-optical metrology with parity detection, we calculate phase information from a lossy interferometer by using twin entangled Fock states. We show that (a)(\text{a})(a) when loss is less than 50% twin entangled Fock states with large photon number difference give higher visibility, while when loss is higher than 50%, the ones with less photon number difference give higher visibility; and (b)(\text{b})(b) twin entangled Fock states with large photon number difference give sub--shot-noise limit sensitivity for phase detection in a lossy environment. This result provides a reference on what particular path-entangled Fock states are useful for real world metrology applications.
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.
It has been proposed and demonstrated that path-entangled Fock states (PEFSs) are robust against photon loss over NOON states [S. D. Huver et al., Phys. Rev. A 78, 063828 (2008)]. However, the demonstration was based on a measurement scheme which was yet to be implemented in experiments. In this work, we quantitatively illustrate the advantage of PEFSs over NOON states in the presence of photon losses by analytically calculating the quantum Fisher information. To realize such an advantage in practice, we then investigate the achievable sensitivities by employing three types of feasible measurements: parity, photon-number-resolving, and homodyne measurements. We here apply a double-port measurement strategy where the photons at each output port of the interferometer are simultaneously detected with the aforementioned types of measurements.
The interference between coherent and squeezed vacuum light effectively produces path entangled N00N states with very high fidelities. We show that the phase sensitivity of the above interferometric scheme with parity detection saturates the quantum Cramer–Rao bound, which reaches the Heisenberg limit when the coherent and squeezed vacuum light are mixed in roughly equal proportions. For the same interferometric scheme, we draw a detailed comparison between parity detection and a symmetric-logarithmic-derivative-based detection scheme suggested by Ono and Hofmann.