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arXiv:2411.16334 · Physical Review A

Postselected amplification applied to the Mach-Zehnder interferometer for phase-shift measurement of optical coherent states

Jialin Li, Yazhi Niu, Lupei Qin, Xin-Qi Li

Published Apr 21, 2025• 5 citations
arXivHTMLPDF

Abstract

We propose a postselected amplification scheme for phase-shift measurement of optical coherent states when passing through the Mach-Zehnder interferometer (MZI). Different from the usual weak-value-amplification (WVA) formulation, the which-path states of the MZI cannot be described as subsystem states entangled with two optical coherent states. However, we obtain the same phase amplification result of the usual WVA in the Aharonov-Albert-Vaidman limit but, beyond this limit, the result is essentially different. We explicitly carry out the amplified phase shift, which can be extracted out from the field-quadrature measurement in the dark port of the MZI. We also evaluate the performance quality of the proposed scheme and analyze the technical advantages by considering possible errors in the quadrature measurement.

DOI
10.1103/physreva.111.042425

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We propose a postselected amplification scheme for phase-shift measurement of optical coherent states when passing through the Mach-Zehnder interferometer (MZI). Different from the usual weak-value-amplification (WVA) formulation, the which-path states of the MZI cannot be described as subsystem states entangled with two optical coherent states. However, we obtain the same phase amplification result of the usual WVA in the Aharonov-Albert-Vaidman limit but, beyond this limit, the result is essentially different. We explicitly carry out the amplified phase shift, which can be extracted out from the field-quadrature measurement in the dark port of the MZI. We also evaluate the performance quality of the proposed scheme and analyze the technical advantages by considering possible errors in the quadrature measurement.
Abstract We propose to embed the atomic magnetometer (AM) into an optical Mach–Zehnder interferometer (MZI). We analyze the effect of amplification of the Faraday rotation (FR) angle of the probe laser light, by properly postselecting the path-information state of the laser photons when passing through the MZI. In the presence of practical limitations, such as the polarization cross talk in the polarizing-beam-splitter, the amplified FR angle in the postselected photons can make the postselection scheme outperform the conventional measurement, thus further enhancing the sensitivity of the nowadays state-of-the-art optical AM.
The weak-value-amplification (WVA) technique has been extensively considered and debated in the field of quantum precision measurement, largely owing to the reduced Fisher information caused by the low probability of successful postselection. In this work we show that, rather than the transverse-spatial-wave-function meter typically considered, using the optical coherent state as a meter, the WVA measurement can outperform the conventional measurement not involving the strategy of postselection. We also show that the postselection procedure involved in the WVA scheme can make a mixture of coherent states work better than a pure coherent state with identical average photon numbers. This is in contrast to the main result proved in the absence of postselection. The postselection strategy can also result in the precision of Heisenberg scaling with the photon numbers without using expensive quantum resources. The present work may stimulate further interest in postselection strategy for quantum precision measurements.
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 1m\frac{1}{\sqrt{m}}m​1​ (m is the average number of the photons in the input state) regardless of the phase-shift measurement schemes.
Phase-shifting interferometry (PSI) promises excellent performance for quantitative phase measurement. However, the PSI is commonly realized by utilizing the Mach-Zehnder interferometer (MZI) fed with classical resources, and hence its phase sensitivity is hindered by the photon shot noise. In this paper, we propose to introduce specific phase-shifting algorithms into the well-established quantum-enhanced MZI, seeded by coherent and squeezed vacuum states and followed by single-mode intensity detection and difference intensity detection, to achieve quantitative phase measurement with sensitivity beyond that achieved by the corresponding Mach-Zehnder-type PSI with classical input. We show that when the average photon number of the coherent state is dominant, the optimal phase sensitivity of the Mach-Zehnder-type PSI with coherent plus squeezed vacuum input is better than the phase sensitivity of the Mach-Zehnder-type PSI with single coherent input under the same phase-sensing intensity condition in both of detection setups. We also show that for the typical MZI seeded with a high-power coherent state and a finitely squeezed vacuum state, the combination of intensity detection and phase-shifting algorithm can still improve the sensitivity of quantitative phase measurement. Furthermore, we derive the phase sensitivity of the fixed MZI as well as the quantum Cramér-Rao bound for two detection schemes. Finally, we explore the impacts of photon losses on the phase sensitivity enhancements of the Mach-Zehnder-type PSI and the fixed MZI in the high-intensity regime for each detection setup. The robustness to losses is compared between the Mach-Zehnder-type PSI and the fixed MZI. Our results may be helpful to promote the exploration of quantum-enhanced quantitative phase sensing.
A nonlinear phase shift is introduced to a Mach-Zehnder interferometer (MZI), and we present a scheme for enhancing the phase sensitivity. In our scheme, one input port of a standard MZI is injected with a coherent state and the other input port is injected with one mode of a two-mode squeezed-vacuum state. The final interference output of the MZI is detected with the method of active correlation output readout. Based on the optimal splitting ratio of beam splitters, the phase sensitivity can beat the standard quantum limit and approach the quantum Cram\'er-Rao bound. The effects of photon loss on phase sensitivity are discussed. Our scheme can also provide some estimates for units of ${\ensuremath{\chi}}^{(3)}$, due to the relation between the nonlinear phase shift and the susceptibility ${\ensuremath{\chi}}^{(3)}$ of the Kerr medium.