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arXiv:2307.01723 · Physical Review Research

Phase sensitivity of spatially broadband high-gain SU(1,1) interferometers

D. Scharwald, T. Meier, Polina R. Sharapova

Published Nov 16, 2023• 13 citations
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Abstract

Nonlinear interferometers are promising tools for quantum metrology, as they are characterized by an improved phase sensitivity scaling compared to linear interferometers operating with classical light. However, the multimodeness of the light generated in these interferometers results in the destruction of their phase sensitivity, requiring advanced interferometric configurations for multimode light. Moreover, in contrast to the single-mode case, time-ordering effects play an important role for the high-gain regime in the multimode scenario and must be taken into account for a correct estimation of the phase sensitivity. In this paper, we present a theoretical description of spatially multimode $\mathrm{SU}(1,1)$ interferometers operating at low and high parametric gains. Our approach is based on a step-by-step solution of a system of integro-differential equations for each nonlinear interaction region. We focus on interferometers with diffraction compensation, where focusing elements such as a parabolic mirror are used to compensate for the divergence of the light. We investigate plane-wave and Gaussian pumping and show that for any parametric gain, there exists a region of phases for which the phase sensitivity surpasses the standard shot-noise scaling and discuss the regimes where it approaches the Heisenberg scale. Finally, we arrive at insightful analytical expressions for the phase sensitivity that are valid for both low and high parametric gain and demonstrate how it depends on the number of spatial modes of the system.

DOI
10.1103/physrevresearch.5.043158

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Nonlinear interferometers are promising tools for quantum metrology, as they are characterized by an improved phase sensitivity scaling compared to linear interferometers operating with classical light. However, the multimodeness of the light generated in these interferometers results in the destruction of their phase sensitivity, requiring advanced interferometric configurations for multimode light. Moreover, in contrast to the single-mode case, time-ordering effects play an important role for the high-gain regime in the multimode scenario and must be taken into account for a correct estimation of the phase sensitivity. In this paper, we present a theoretical description of spatially multimode SU(1,1)\mathrm{SU}(1,1)SU(1,1) interferometers operating at low and high parametric gains. Our approach is based on a step-by-step solution of a system of integro-differential equations for each nonlinear interaction region. We focus on interferometers with diffraction compensation, where focusing elements such as a parabolic mirror are used to compensate for the divergence of the light. We investigate plane-wave and Gaussian pumping and show that for any parametric gain, there exists a region of phases for which the phase sensitivity surpasses the standard shot-noise scaling and discuss the regimes where it approaches the Heisenberg scale. Finally, we arrive at insightful analytical expressions for the phase sensitivity that are valid for both low and high parametric gain and demonstrate how it depends on the number of spatial modes of the system.
An SU(1,1){\rm SU(1,1)}SU(1,1) interferometer uses a sequence of two optical parametric amplifiers for achieving sub-shot-noise sensitivity to a phase shift introduced in between. We present the first realization of a wide-field SU(1,1){\rm SU(1,1)}SU(1,1) interferometer, where the use of a focusing element enables spatially multimode operation within a broad angle. Over this angle, the interference phase is found to be flat. This property is important for the high sensitivity to the phase front disturbance. Further, −4.3±0.7-4.3\pm0.7−4.3±0.7 dB quadrature squeezing, an essential requirement to the high sensitivity, is experimentally demonstrated for plane-wave modes inside the interferometer. Such an interferometer is useful not only for quantum metrology, but also in remote sensing, enhanced sub-shot-noise imaging, and quantum information processing.
Multimode quantum light has promising applications in many areas of physics, such as quantum communications and quantum computing. However, its multimode nature also makes it challenging to measure its properties. Recently [I. Barakat , ], a technique for the simultaneous measurement of squeezing of multiple broadband modes based on a phase-sensitive amplification approach was experimentally implemented using a setup that effectively corresponds to an SU(1,1) interferometer. Here, we aim to provide a complete theoretical analysis of the modal structure of (generally unbalanced) SU(1,1) interferometers and a detailed theoretical formal derivation of the framework for this technique. Utilizing the joint Schmidt decomposition of the transfer functions, we investigate the shape and phase profiles of the modes of the SU(1,1) interferometer and its components [parametric down-conversion (PDC) sections] for different parametric gain regimes. We discover a complicated interplay between the PDC modes and the modes of the entire interferometer, and analyze it by using their overlap coefficients as a similarity measure. Finally, we develop a rigorous processing method for the aforementioned multimode squeezing measurement technique and discuss necessary approximations to make this method experimentally feasible.
An SU(1,1) interferometer uses a sequence of two optical parametric amplifiers for achieving sub-shot-noise sensitivity to a phase shift introduced in between. We present the first realization of a wide-field SU(1,1) interferometer, where the use of a focusing element enables spatially multimode operation within a broad angle. Over this angle, the interference phase is found to be flat. This property is important for the high sensitivity to the phase front disturbance. Further, -4.3 +/- 0.7 dB quadrature squeezing, an essential requirement to the high sensitivity, is experimentally demonstrated for plane-wave modes inside the interferometer. Such an interferometer is useful not only for quantum metrology, but also in remote sensing, enhanced sub-shot-noise imaging, and quantum information processing. (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
We provide a general theoretical derivation of the phase sensitivity achieved by SU(1,1) interferometers under homodyne detection. The general expressions obtained accommodate arbitrary input states and include internal and external losses. In this systematic review, both full SU(1,1) interferometers with two parametric amplifiers and the truncated interferometers with only one parametric amplifier are examined. We investigate scenarios involving both single-output ports and joint homodyne detection, and consider parametric amplifiers with equal gains or with a boosted gain second amplifier. Our analytical formulation provides physical insight and understanding of the improvements in the sensitivity, which are shown to originate from noise reduction and/or signal amplification, depending on the configurations and practical implementations. Surprisingly, the configuration with single-output mode detection and parametric amplifiers with equal gains exhibits the highest robustness to very high internal losses. We finally apply this framework to a ubiquitous ∣α,0⟩|α,0\rangle∣α,0⟩ input two-mode coherent probe state. This approach permits the comparison of different strategies and the optimization of the interferometer performance in the presence of losses. In particular, we determine which amplification and detection configurations provide the best performance, depending on the level of losses. This exemplifies how this general analytical approach provides a powerful tool to design quantum-enhanced interferometers and achieve optimal sensitivity with selected probe states and homodyne detection.
Multimode quantum light has promising applications in many areas of physics, such as quantum communications and quantum computing. However, its multimode nature also makes it challenging to measure its properties. Recently [I. Barakat et al., Optica Quantum 3, 36 (2025)], a technique for the simultaneous measurement of squeezing of multiple broadband modes based on a phase-sensitive amplification approach was experimentally implemented using a setup that effectively corresponds to an SU(1,1) interferometer. Here, we aim to provide a complete theoretical analysis of the modal structure of (generally unbalanced) SU(1,1) interferometers and a detailed theoretical formal derivation of the framework for this technique. Utilizing the joint Schmidt decomposition of the transfer functions, we investigate the shape and phase profiles of the modes of the SU(1,1) interferometer and its components [parametric down-conversion (PDC) sections] for different parametric gain regimes. We discover a complicated interplay between the PDC modes and the modes of the entire interferometer, and analyze it by using their overlap coefficients as a similarity measure. Finally, we develop a rigorous processing method for the aforementioned multimode squeezing measurement technique and discuss necessary approximations to make this method experimentally feasible.
Nonlinear SU(1,1) interferometers are fruitful and promising tools for spectral engineering and precise measurements with phase sensitivity below the classical bound. Such interferometers have been successfully realized in bulk and fiber-based configurations. However, rapidly developing integrated technologies provide higher efficiencies, smaller footprints, and pave the way to quantum-enhanced on-chip interferometry. In this work, we theoretically realised an integrated architecture of the multimode SU(1,1) interferometer which can be applied to various integrated platforms. The presented interferometer includes a polarization converter between two photon sources and utilizes a continuous-wave (CW) pump. Based on the potassium titanyl phosphate (KTP) platform, we show that this configuration results in almost perfect destructive interference at the output and supersensitivity regions below the classical limit. In addition, we discuss the fundamental difference between single-mode and highly multimode SU(1,1) interferometers in the properties of phase sensitivity and its limits. Finally, we explore how to improve the phase sensitivity by filtering the output radiation and using different seeding states in different modes with various detection strategies.