arXiv:quant-ph/0511059 · Physical Review A

Phase sensitivity of a Mach-Zehnder interferometer

Luca Pezzè, Augusto Smerzi

Published Jan 3, 2006• 77 citations

Abstract

The best performance of a Mach-Zehnder interferometer is achieved with the input state $\ensuremath{\mid}{N}_{T}∕2+1⟩\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩+\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩\ensuremath{\mid}{N}_{T}∕2+1⟩$, ${N}_{T}$ being the total number of atoms or photons. This gives (i) a phase-shift error confidence ${C}_{68%}=2.67∕{N}_{T}$ with (ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling $\ensuremath{\sim}1∕{N}_{T}$ but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.

DOI: 10.1103/physreva.73.011801

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arXiv:quant-ph/0511059 · Physical Review A

Phase sensitivity of a Mach-Zehnder interferometer

Luca Pezzè, Augusto Smerzi

Published Jan 3, 2006• 77 citations

arXiv HTML PDF

Abstract

The best performance of a Mach-Zehnder interferometer is achieved with the input state

\ensuremath{\mid}{N}_{T}∕2+1⟩\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩+\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩\ensuremath{\mid}{N}_{T}∕2+1⟩

{N}_{T}

being the total number of atoms or photons. This gives (i) a phase-shift error confidence

{C}_{68%}=2.67∕{N}_{T}

with (ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling

\ensuremath{\sim}1∕{N}_{T}

but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.

DOI: 10.1103/physreva.73.011801

Related papers

The Phase-sensitivity of a Mach-Zehnder Interferometer for Coherent Light

Jong-Tae Shin, Heonoh Kim, Goo-Dong Park +1 more

We have studied the sensitivity of four different phase shift measurement schemes with a Mach-Zehnder interferometer. The input light is considered to be in a coherent state and the detectors are assumed to be ideal with the quantum efficiency of unity. It is shown by direct calculation of the operators corresponding to the measurement schemes that the uncertainty of the phase-shift measurement is limited to the classical one $\frac{1}{\sqrt{m}}$ (m is the average number of the photons in the input state) regardless of the phase-shift measurement schemes.We have studied the sensitivity of four different phase shift measurement schemes with a Mach-Zehnder interferometer. The input light is considered to be in a coherent state and the detectors are assumed to be ideal with the quantum efficiency of unity. It is shown by direct calculation of the operators corresponding to the measurement schemes that the uncertainty of the phase-shift measurement is limited to the classical one $\frac{1}{\sqrt{m}}$ (m is the average number of the photons in the input state) regardless of the phase-shift measurement schemes.

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Luca Pezzè, Augusto Smerzi

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Fidelity of quantum interferometers

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For a generic interferometer, the conditional probability density distribution $p(\ensuremath{\phi}\ensuremath{\mid}m)$, for the phase $\ensuremath{\phi}$ given measurement outcome $m$ will generally have multiple peaks. Therefore, the phase sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such as $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕\sqrt{N}$ (the standard limit), or $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕N$ (the Heisenberg limit). We propose an alternative measure of phase sensitivity---the fidelity of an interferometer---defined as the Shannon mutual information between the phase shift $\ensuremath{\phi}$ and the measurement outcomes $m$ . As an example application of interferometer fidelity, we consider a generic optical Mach-Zehnder interferometer, used as a sensor of a classical field. For the case where there exists no a priori information on the phase shift, we find the surprising result that maximally entangled state input leads to a lower fidelity than Fock state input, for the same photon number.For a generic interferometer, the conditional probability density distribution $p(\ensuremath{\phi}\ensuremath{\mid}m)$, for the phase $\ensuremath{\phi}$ given measurement outcome $m$ will generally have multiple peaks. Therefore, the phase sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such as $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕\sqrt{N}$ (the standard limit), or $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕N$ (the Heisenberg limit). We propose an alternative measure of phase sensitivity---the fidelity of an interferometer---defined as the Shannon mutual information between the phase shift $\ensuremath{\phi}$ and the measurement outcomes $m$. As an example application of interferometer fidelity, we consider a generic optical Mach-Zehnder interferometer, used as a sensor of a classical field. For the case where there exists no a priori information on the phase shift, we find the surprising result that maximally entangled state input leads to a lower fidelity than Fock state input, for the same photon number.

arXiv:quant-ph/0602123Physical Review ANov 15, 200616 citations

Swap-test interferometry with biased qubit noise

Ondřej Černotík, Iivari Pietikäinen, Shruti Puri +2 more

The Mach-Zehnder interferometer is a powerful device for detecting small phase shifts between two light beams. Simple input states, such as coherent states or single photons, can reach the standard quantum limit of phase estimation, while more complex states can be used to reach Heisenberg scaling; the latter, however, require challenging preparation and measurement strategies. The quest for highly sensitive phase estimation therefore calls for interferometers with nonlinear devices which would make the preparation of these complex states more efficient. Here, we show that the Heisenberg scaling can be recovered with simple input states (including Fock and coherent states) when the linear mirrors in the interferometer are replaced with controlled-swap gates and measurements on auxiliary qubits. These swap tests project the input Fock and coherent states onto NOON and entangled coherent states, respectively, and allow optimal or near-optimal measurements, leading to improved sensitivity to small phase shifts in one of the interferometer arms. We analyze auxiliary qubit errors in detail, showing that biasing the qubit towards phase flips offers a great advantage, and perform thorough numerical simulations of a possible implementation in circuit quantum electrodynamics with an auxiliary Kerr-cat qubit. Our results thus present a viable approach to phase estimation approaching Heisenberg-limited sensitivity and demonstrate potential advantages of using biased-noise qubits in quantum metrology. Published by the American Physical Society 2024The Mach-Zehnder interferometer is a powerful device for detecting small phase shifts between two light beams. Simple input states, such as coherent states or single photons, can reach the standard quantum limit of phase estimation, while more complex states can be used to reach Heisenberg scaling; the latter, however, require challenging preparation and measurement strategies. The quest for highly sensitive phase estimation therefore calls for interferometers with nonlinear devices which would make the preparation of these complex states more efficient. Here, we show that the Heisenberg scaling can be recovered with simple input states (including Fock and coherent states) when the linear mirrors in the interferometer are replaced with controlled-swap gates and measurements on auxiliary qubits. These swap tests project the input Fock and coherent states onto NOON and entangled coherent states, respectively, and allow optimal or near-optimal measurements, leading to improved sensitivity to small phase shifts in one of the interferometer arms. We analyze auxiliary qubit errors in detail, showing that biasing the qubit towards phase flips offers a great advantage, and perform thorough numerical simulations of a possible implementation in circuit quantum electrodynamics with an auxiliary Kerr-cat qubit. Our results thus present a viable approach to phase estimation approaching Heisenberg-limited sensitivity and demonstrate potential advantages of using biased-noise qubits in quantum metrology. Published by the American Physical Society 2024

arXiv:2112.02568Physical Review ResearchJul 17, 20243 citations

The best performance of a Mach-Zehnder interferometer is achieved with the input state

\ensuremath{\mid}{N}_{T}∕2+1⟩\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩+\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩\ensuremath{\mid}{N}_{T}∕2+1⟩

{N}_{T}

being the total number of atoms or photons. This gives (i) a phase-shift error confidence

{C}_{68%}=2.67∕{N}_{T}

with (ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling

\ensuremath{\sim}1∕{N}_{T}

but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.

\frac{1}{\sqrt{m}}

(m is the average number of the photons in the input state) regardless of the phase-shift measurement schemes.

{N}_{T}

is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter

r

and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.

For a generic interferometer, the conditional probability density distribution $p(\ensuremath{\phi}\ensuremath{\mid}m)$, for the phase $\ensuremath{\phi}$ given measurement outcome

m

will generally have multiple peaks. Therefore, the phase sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such as $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕\sqrt{N}$ (the standard limit), or $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕N$ (the Heisenberg limit). We propose an alternative measure of phase sensitivity---the fidelity of an interferometer---defined as the Shannon mutual information between the phase shift $\ensuremath{\phi}$ and the measurement outcomes

m

. As an example application of interferometer fidelity, we consider a generic optical Mach-Zehnder interferometer, used as a sensor of a classical field. For the case where there exists no a priori information on the phase shift, we find the surprising result that maximally entangled state input leads to a lower fidelity than Fock state input, for the same photon number.