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arXiv:2212.07708 · Symmetry

Sensitivity of Quantum-Enhanced Interferometers

Dariya Salykina, Farid Ya Khalili

Published Mar 22, 2023• 10 citations
arXivHTMLPDF

Abstract

We review various schemes of quantum-enhanced optical interferometers, both linear (SU(2)) and non-linear (SU(1,1)) ones, as well as hybrid SU(2)/SU(1,1) options, using the unified modular approach based on the Quantum Cramèr–Rao bound (QCRB), and taking into account the practical limitations pertinent to all real-world highly-sensitive interferometers. We focus on three important cases defined by the interferometer symmetry: (i) the asymmetric single-arm interferometer; (ii) the symmetric two-arm interferometer with the antisymmetric phase shifts in the arms; and (iii) the symmetric two-arm interferometer with the symmetric phase shifts in the arms. We show that while the optimal regimes for these cases differ significantly, their QCRBs asymptotically correspond to the same squeezing-enhanced shot noise limit (2), which first appeared in the pioneering work by C. Caves in 1981.We show also that in all considered cases the QCRB can be asymptotically saturated by the standard (direct or homodyne) detection schemes.

DOI
10.3390/sym15030774

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We review various schemes of quantum-enhanced optical interferometers, both linear (SU(2)) and non-linear (SU(1,1)) ones, as well as hybrid SU(2)/SU(1,1) options, using the unified modular approach based on the Quantum Cramèr–Rao bound (QCRB), and taking into account the practical limitations pertinent to all real-world highly-sensitive interferometers. We focus on three important cases defined by the interferometer symmetry: (i) the asymmetric single-arm interferometer; (ii) the symmetric two-arm interferometer with the antisymmetric phase shifts in the arms; and (iii) the symmetric two-arm interferometer with the symmetric phase shifts in the arms. We show that while the optimal regimes for these cases differ significantly, their QCRBs asymptotically correspond to the same squeezing-enhanced shot noise limit (2), which first appeared in the pioneering work by C. Caves in 1981.We show also that in all considered cases the QCRB can be asymptotically saturated by the standard (direct or homodyne) detection schemes.
Improving interferometric phase sensitivity is crucial for high-precision measurements in rapidly developing quantum technologies. The Mach–Zehnder interferometer (MZI) is a versatile tool for analyzing this phenomenon. By splitting and recombining a light beam using beam splitters, the MZI allows for precise phase sensitivity analysis using tools like the quantum Cramér–Rao bound (QCRB) and the quantum Fisher information (QFI). This paper analyzes the phase sensitivity of an MZI in various scenarios using different detection schemes and input states. We compare the single- and two-parameter quantum estimation and their associated QCRB for three phase-shift situations: in both arms; only in the upper arm (asymmetric); and in both arms symmetrically. We then investigate the phase sensitivity under three detection schemes: intensity difference; single-mode intensity; and balanced homodyne. Additionally, we explore the use of Perelomov and Barut–Girardello coherent states, two types of SU(1,1) coherent states, in all scenarios. Notably, we demonstrate that, under optimal conditions, all detection schemes can achieve the QCRB by utilizing SU(1,1) coherent states as input states.
We theoretically present the quantum Cramér-Rao bounds (QCRB) of an SU(1,1) interferometer for Gaussian states input with and without the internal photonic losses. The phase shifts in the single arm and in the double arms are studied and the corresponding analytical expressions of quantum Fisher information with Gaussian input states are presented. Different from the traditional Mach-Zehnder interferometer, the QCRB of single arm case is slightly higher or lower than that of double arms case depending on the input states. With a fixed mean photon number and for pure Gaussian state input, the optimal sensitivity is achieved with a squeezed vacuum input in one mode and the vacuum input in the other. We compare the QCRB with the standard quantum limit and Heisenberg limit. In the case of small internal losses the QCRB can beat the standard quantum limit.
Abstract Quantum states of light can enable sensing configurations with sensitivities beyond the shot-noise limit. In order to better take advantage of available quantum resources and obtain the maximum possible sensitivity, it is necessary to determine fundamental sensitivity limits for different possible configurations for a given sensing system. Here, due to their wide applicability, we focus on optical resonance sensors, which detect a change in a parameter of interest through a resonance shift. We compare their fundamental sensitivity limits set by the quantum Cramér–Rao bound (QCRB) based on the estimation of changes in transmission or phase of a probing bright two-mode squeezed state (bTMSS) of light. We show that the fundamental sensitivity results from an interplay between the QCRB and the transfer function of the system. As a result, for a resonance sensor with a Lorentzian lineshape a phase-based scheme outperforms a transmission-based one for most of the parameter space; however, this is not the case for lineshapes with steeper slopes, such as higher order Butterworth lineshapes. Furthermore, such an interplay results in conditions under which the phase-based scheme provides a higher sensitivity but a smaller degree of quantum enhancement than the transmission-based scheme. We also study the effect of losses external to the sensor on the degree of quantum enhancement and show that for certain conditions, probing with a classical state can provide a higher sensitivity than probing with a bTMSS. Finally, we discuss detection schemes, namely optimized intensity-difference and optimized homodyne detection, that can achieve the fundamental sensitivity limits even in the presence of external losses.
Abstract We theoretically study the phase sensitivities of two different phase-shift configurations in an SU(1,1) interferometer with coherent ⊗ squeezed vacuum states. According to quantum Cramér-Rao theorem, we analytically obtain the ultimate phase sensitivities for two types of phase shift accumulating in one- and two-arm. Compared with the case of one-arm phase shift, the model with phase shift encoding in both arms may provide a better sensitivity when the strength of squeezed vacuum state is large enough. Furthermore, we discuss the achievable sensitivities with the homodyne measurement by invoking of error-propagation formula. In addition, we study the effect of internal and outernal photon losses on the phase sensitivity of the SU(1,1) interferometer and find that the unbalanced interferometer is helpful to improve precision even with high external losses .
The ultimate precision of phase estimation is limited by the Heisenberg scaling Δϕ0=K/NΔϕ_0 = K/NΔϕ0​=K/N, where K∼1K\sim1K∼1 is a numerical prefactor and NNN is the mean number of photons interacting with the phase shifting object(s). However, achieving this fundamental limit often comes at the cost of an extremely narrow high-sensitive range, rendering schemes impractical. We analyze the precision limits of phase measurements in single- and two-arm optical interferometers with input Gaussian states. We consider two detection methods: conventional homodyne measurement and non-Gaussian threshold detection that saturates the quantum Cramér-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity Δϕ0Δϕ_0Δϕ0​ and the width δϕδϕδϕ of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that δϕδϕδϕ strongly depends on KKK. We derive an approximate analytic expression that describes this trade-off. We show also that the two-arm interferometer with antisymmetrically squeezed inputs exhibits exceptional performance, simultaneously achieving Heisenberg-limited sensitivity and a broad high-sensitivity range δϕ=π/2δϕ=π/2δϕ=π/2.
Homodyne detection is often used for interferometers based on nonlinear optical gain media. For the configuration of a seeded, "truncated SU(1,1)" interferometer Anderson, et al. [ Phys. Rev. A95, 063843 (2017)] showed how to optimize the homodyne detection scheme and demonstrated theoretically that it can saturate the quantum Cramer-Rao bound for phase estimation. In this work we extend those results by taking into account loss in the truncated SU(1,1) interferometer and determining the optimized homodyne detection scheme for phase measurement. Further, we build a truncated SU(1,1) interferometer and experimentally demonstrate that this optimized scheme achieves a reduction in noise level, corresponding to an enhanced potential phase sensitivity, compared to a typical homodyne detection scheme for a two-mode squeezed state. In doing so, we also demonstrate an improvement in the degree to which we can beat the standard quantum limit with this device.