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arXiv:1310.0443 · Physical Review A

Quantum-enhanced phase estimation with an amplified Bell state

Jaspreet Sahota, Daniel F. V. James

Published Dec 10, 2013• 17 citations
arXivHTMLPDF

Abstract

We propose a phase estimation protocol for optical interferometry that employs a probe state (containing on average $\overline{n}$ photons) obtained by squeezing each mode, separately, of a single photon path entangled Bell state. This scheme involves a Mach-Zehnder type interferometer for which each mode is squeezed after the first beam splitter. Information about the differential phase is extracted using a parity detection and the resulting measurement signal is super-resolving and supersensitive, with a minimum phase uncertainty $\ensuremath{\Delta}\ensuremath{\varphi}=2/(\overline{n}+1)$. This probe state can be generated with current technologies where $\overline{n}$ is in the order of many thousands of photons.

DOI
10.1103/physreva.88.063820

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arXiv:1611.01498•Physical Review A•Oct 16, 2017•8 citations
We propose a phase estimation protocol for optical interferometry that employs a probe state (containing on average n‾\overline{n}n photons) obtained by squeezing each mode, separately, of a single photon path entangled Bell state. This scheme involves a Mach-Zehnder type interferometer for which each mode is squeezed after the first beam splitter. Information about the differential phase is extracted using a parity detection and the resulting measurement signal is super-resolving and supersensitive, with a minimum phase uncertainty \ensuremathΔ\ensuremathφ=2/(n‾+1)\ensuremath{\Delta}\ensuremath{\varphi}=2/(\overline{n}+1)\ensuremathΔ\ensuremathφ=2/(n+1). This probe state can be generated with current technologies where n‾\overline{n}n is in the order of many thousands of photons.
Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
We investigate the sensitivity and resolution of phase estimation in a Mach–Zehnder interferometer with photon-added two-mode squeezed vacuum states (PA-TMSVS) with a photon-number parity measurement scheme. Our results indicate that, when the phase to be estimated is small, the PA-TMSVS offers an improved phase resolution over both the two-mode squeezed vacuum and the photon-subtracted two-mode squeezed vacuum for a given initial squeezing level. In addition, we study the behavior of the quantum Fisher information of the PA-TMSVS, and show that the optimum phase uncertainty predicted by the corresponding Cramer–Rao bound can be achieved with a parity measurement.
We explore optical quantum engineering of phase-parameterized continuous-variable (CV) probe states to exploit nonclassical light to solve the problem of precise phase estimation. The optical interferometer consists of a single beam splitter (BS) with tunable transmittance and reflectance, and two single-mode squeezed vacuum states (SMSVs). The reference SMSV state is mixed with a weakly squeezed state carrying an unknown phase at the beam splitter to form an output hybrid entangled state. Then, in the measurement mode, the number of photons is measured to generate the target CV state parameterized by the unknown phase. Using the CV states, we propose a sub-Heisenberg metrology protocol in which the quantum Cramer-Rao (QCR) boundary is saturated by intensity measurement. The advantage of quantum engineering of CV probe states for ultra-precise phase estimation of unknown phase is due solely to the nonclassical photonic properties of the measurement induced CV states of definite parity and is independent of the mode entanglement.
We explore optical quantum engineering of phase-parameterized continuous-variable (CV) probe states to exploit the nonclassical light to solve the problem of precise phase estimation. The optical interferometer consists of a single beam splitter (BS) with tunable transmittance and reflectance, and two single-mode squeezed vacuum states (SMSVs). The reference SMSV state is mixed with a weakly squeezed state carrying an unknown phase at the beam splitter to form an output hybrid entangled state. Then, in the measurement mode, the number of photons is measured to generate the target CV state parameterized by the unknown phase. By using a non-unitary encoding of the probe state with an unknown phase, we propose a sub-Heisenberg metrology protocol, in which the quantum Cramer-Rao (QCR) boundary is saturated by the intensity measurement. The advantage of quantum engineering of CV probe states for ultra-precise phase estimation of an unknown phase is due solely to the nonclassical photonic properties of the measurement-induced CV states of definite parity, and the protocol execution is independent of the mode entanglement.
We theoretically investigate the quantum enhanced metrology using two-mode squeezed twin-Fock states and parity detection. Our results indicate that, for a given initial squeezing parameter, compared with the two-mode squeezed vacuum state, both phase sensitivity and resolution can be enhanced when the two-mode squeezed twin-Fock state is considered as an input state of a Mach–Zehnder interferometer. Within a constraint on the total photon number, although the two-mode squeezed vacuum state gives the better phase sensitivity when the phase shift φ to be estimated approaches to zero, the phase sensitivity offered by these non-Gaussian entangled Gaussian states is relatively stable with respect to the phase shift itself. When the phase shift slightly deviates from φ = 0, the phase sensitivity can be still enhanced by the two-mode squeezed twin-Fock state over a broad range of the total mean photon number where the phase uncertainty is still below the quantum standard noise limit. Finally, we numerically prove that the quantum Cramér–Rao bound can be approached with the parity detection.
The supersensitivity attained in quantum phase estimation is known to be compromised in the presence of decoherence. This is particularly patent at blind spots---phase values at which sensitivity is totally lost. One remedy is to use a precisely known reference phase to shift the operation point to a less vulnerable phase value. Since this is not always feasible, we present here an alternative approach based on combining the probe with an ancillary degree of freedom containing adjustable parameters to create an entangled quantum state of higher dimension. We validate this concept by simulating a configuration of a Mach-Zehnder interferometer with a two-photon probe and a polarization ancilla of adjustable parameters, entangled at a polarizing beam splitter. At the interferometer output, the photons are measured after an adjustable unitary transformation in the polarization subspace. Through calculation of the Fisher information and simulation of an estimation procedure, we show that optimizing the adjustable polarization parameters using an adaptive measurement process provides globally supersensitive unbiased phase estimates for a range of decoherence levels, without prior information or a reference phase.