Below-shot-noise capacity in phase estimation using nonlinear interferometers
Cristofero Oglialoro, Gerard J. Machado, Felix Farsch, Daniel F. Urrego, Alejandra A. Padilla, Raj B. Patel, Ian A. Walmsley, Markus Gräfe, Juan P. Torres, Enno Giese
quant-phphysics.opticsSubmitted Jan 7, 2026• Announced Jan 8, 2026• 0 citations
Over the past decade, several schemes for imaging and sensing based on nonlinear interferometers have been proposed and demonstrated experimentally. These interferometers exhibit two main advantages. First, they enable probing a sample at a chosen wavelength while detecting light at a different wavelength with high efficiency (bicolor quantum imaging and sensing with undetected light). Second, they can show quantum-enhanced sensitivities below the shot-noise limit, potentially reaching Heisenberg-limited precision in parameter estimation. Here, we compare three quantum-imaging configurations using only easily accessible intensity-based measurements for phase estimation: a Yurke-type SU(1,1) interferometer, a Mandel-type induced-coherence interferometer, and a hybrid scheme that continuously interpolates between them. While an ideal Yurke interferometer can exhibit Heisenberg scaling, this advantage is known to be fragile under realistic detection constraints and in the presence of loss. We demonstrate that differential intensity detection in the Mandel interferometer provides the highest and most robust phase sensitivity among the considered schemes, reaching but not surpassing the shot-noise limit, even in the presence of loss. Intensity measurements in a Yurke-type configuration can achieve genuine sub-shot-noise sensitivity under balanced losses and moderate gain; however, their performance degrades in realistic high-gain regimes. Consequently, in this regime, the Mandel configuration with differential detection outperforms the Yurke-type setup and constitutes the most robust approach for phase estimation.
Over the past decade, several schemes for imaging and sensing based on nonlinear interferometers have been proposed and demonstrated experimentally. These interferometers exhibit two main advantages. First, they enable probing a sample at a chosen wavelength while detecting light at a different wavelength with high efficiency (bicolor quantum imaging and sensing with undetected light). Second, they can show quantum-enhanced sensitivities below the shot-noise limit, potentially reaching Heisenberg-limited precision in parameter estimation. Here, we compare three quantum-imaging configurations using only easily accessible intensity-based measurements for phase estimation: a Yurke-type SU(1,1) interferometer, a Mandel-type induced-coherence interferometer, and a hybrid scheme that continuously interpolates between them. While an ideal Yurke interferometer can exhibit Heisenberg scaling, this advantage is known to be fragile under realistic detection constraints and in the presence of loss. We demonstrate that differential intensity detection in the Mandel interferometer provides the highest and most robust phase sensitivity among the considered schemes, reaching but not surpassing the shot-noise limit, even in the presence of loss. Intensity measurements in a Yurke-type configuration can achieve genuine sub-shot-noise sensitivity under balanced losses and moderate gain; however, their performance degrades in realistic high-gain regimes. Consequently, in this regime, the Mandel configuration with differential detection outperforms the Yurke-type setup and constitutes the most robust approach for phase estimation.
Bicolor quantum imaging can reach, in principle, phase supersensitivity approaching the Heisenberg limit using squeezed light from high-gain parametric down-conversion, but established quantum-imaging setups typically operate in the low-gain, spontaneous regime, where such scaling is inaccessible. Here, we show experimentally and theoretically that a symmetric nonlinear interferometer, even in the spontaneous regime, retains a phase-sensitivity advantage over configurations where entanglement is not exploited as a quantum-metrological resource, although the achievable phase sensitivity remains shot-noise-limited. This advantage appears as a shift of the optimal working point toward the dark fringe, the low-gain signature of the mechanism underlying high-gain supersensitivity. We derive design choices for phase-optimized nonlinear interferometers accordingly, favoring configurations that exploit entanglement as a metrological resource and specifying their optimal operating point.
We propose a high-precision phase estimation scheme in a hybrid interferometer by synergistically combining a Kerr nonlinear phase shifter and multi-photon subtraction operations. Using a coherent state and a vacuum state as input resources, we systematically evaluate the phase sensitivity via homodyne detection and analyze the quantum Fisher information as well as the quantum Cramér-Rao bound under both ideal and lossy conditions. Our results show that the joint integration of Kerr nonlinearity and multi-photon subtraction yields remarkable advantages over either technique used alone. The proposed scheme enables the phase sensitivity to surpass the standard quantum limit, exceed the conventional Heisenberg scaling (1/N), and approach the super-Heisenberg scaling (1/N2)-a direct consequence of Kerr nonlinearity. More precisely, the super-Heisenberg scaling $\propto $ 1/N2 is the ultimate precision limit permitted by the k=2 Kerr nonlinearity and does not violate the fundamental Heisenberg limit for linear phase accumulation. Even under moderate internal photon loss, the system maintains high precision and exhibits enhanced robustness to decoherence. The Kerr nonlinearity introduces an intensity-dependent phase shift proportional to the squared photon number, while multi-photon subtraction tailors non-Gaussian states to strengthen phase information extraction. Compared with existing schemes based on hybrid interferometers or SU(1,1) interferometers, our architecture achieves superior precision and stronger loss resilience. All components are experimentally accessible with current quantum optical technologies. This work provides a promising route for practical high-precision quantum metrology and quantum sensing.
We theoretically analyze the phase sensitivity of the Induced-Coherence (Mandel-Type) Interferometer, including the case where the sensitivity is "boosted" into the bright input regime with coherent-light seeding. We find scaling which reaches below the shot noise limit, even when seeding the spatial mode which does not interact with the sample – or when seeding the undetected mode. It is a hybrid of a linear and a non-linear (Yurke-Type) interferometer, and aside from the supersensitivity, is distinguished from other systems by "preferring" an imbalance in the gains of the two non-linearities (with the second gain being optimal at low values), and non-monotonic behavior of the sensitivity as a function of the gain of the second non-linearity. Furthermore, the setup allows use of subtracted intensity measurements, instead of direct (additive) or homodyne measurements – a significant practical advantage. Bright, super-sensitive phase estimation of an object with different light fields for interaction and detection is possible, with various potential applications, especially in cases where the sample may be sensitive to light, or is most interesting in frequency domains outside what is easily detected, or when desiring bright-light phase estimation with sensitive/delicate detectors. We use an analysis in terms of general squeezing and discover that super-sensitivity occurs only in this case – that is, the effect is not present with the spontaneous-parametric-down-conversion approximation, which many previous analyses and experiments have focused on.
Quantum interferometric sensing plays a crucial role in a wide range of applications, including quantum metrology, quantum imaging, and quantum lithography, where minute phase shifts carry valuable physical information. The strength of quantum sensing lies in surpassing classical sensitivity limits, particularly through the use of quantum correlations and squeezing to suppress optical shot noise. Birefringence sensing is crucial for various applications, as it provides detailed information about the material's structure, stress, composition, and environmental conditions. We present an interferometric scheme for detecting unknown small birefringence beyond the shot-noise limit of sensitivity that leverages the hyper-squeezing (squeezed in both number-phase and polarization correlations) within a pair of polarized nonlinear SU(1,1) interferometers, coupled by the birefringence. Specifically, two pairs of crossed-polarization nonlinear media, both generate and measure two-mode quantum light that is squeezed and polarization-entangled. We present a complete theoretical analysis of the interferometer's sensitivity to small birefringence under realistic conditions of gain and internal loss, illuminating the potential for enhancement of the sensitivity by 3-15dB in practical, real-world experiments (the exact achievable enhancement is governed solely by the loss).
Quantum sensing using non-linear interferometers offers the possibility of bicolour imaging, using light that never interacted with the object of interest, and provides a way to achieve phase supersensitivity, i.e. a Heisenberg-type scaling of the phase uncertainty. Such a scaling behaviour is extremely susceptible to noise and only arises at specific phases that define the optimal working point of the device. While phase-shifting algorithms are to some degree robust against the deleterious effects induced by noise they extract an image by tuning the interferometer phase over a broad range, implying an operation beyond the working point. In our theoretical study, we investigate both the spontaneous and the high-gain regime of operation of a non-linear interferometer. In fact, in the spontaneous regime using a distillation technique and operating at the working point leads to a qualitatively similar behaviour. In the high-gain regime, however, typical distillation techniques inherently forbid a scaling better than the standard-quantum limit, as a consequence of the photon statistics of squeezed vacuum. In contrast, an operation at the working point still may lead to a sensitivity below shot noise, even in the presence of noise. Therefore, this procedure opens the perspective of bicolour imaging with a better than shot-noise phase uncertainty by working in the vicinity of the working point. Our results transfer quantum imaging distillation in a noisy environment to the high-gain regime with the ultimate goal of harnessing its full potential by combining bicolour imaging and phase supersensitivity.
We theoretically study the phase sensitivity of an SU(1,1) interferometer with a thermal state and a squeezed vacuum state as inputs and parity detection as the measurement. We find that the phase sensitivity can beat the shot-noise limit and approaches the Heisenberg limit, with increasing input photon number, in an ideal situation. We also consider the e ect of various noises, including photon loss, dark counts, and thermal photon noise. Our results show that the phase sensitivity is below the shot-noise limit with photon loss and dark counts, but cannot beat the shot-noise limit in the presence of thermal noise.