Quantum sensing using non-linear interferometers (NLIs) 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 (WP) 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 WP. In our theoretical study, we investigate both the spontaneous and the high-gain regime of operation of an NLI. In fact, in the spontaneous regime using a distillation technique and operating at the WP 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 WP 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 WP. 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.
Quantum sensing using non-linear interferometers (NLIs) 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 (WP) 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 WP. In our theoretical study, we investigate both the spontaneous and the high-gain regime of operation of an NLI. In fact, in the spontaneous regime using a distillation technique and operating at the WP 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 WP 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 WP. 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.
Abstract Quantum sensing using non-linear interferometers (NLIs) 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 (WP) 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 WP. In our theoretical study, we investigate both the spontaneous and the high-gain regime of operation of an NLI. In fact, in the spontaneous regime using a distillation technique and operating at the WP 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 WP 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 WP. 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.
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.
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.
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.
A nonlinear interferometer uses nonlinear elements as beam splitters to split and to recombine optical waves for interference. As a result, the interference fringe size has a nonlinear dependence on the intensity of the field for phase sensing and leads to an enhanced phase signal. In this paper, a practical scheme of nonlinear interferometry for precision phase measurement is analyzed with parametric amplifiers as the nonlinear beam splitters. It is found that the signal due to phase shift is enhanced by a factor of the amplification gain as compared to a linear interferometer with the same phase-sensing light intensity while the quantum noise is kept at the vacuum level, thus, effectively increasing the signal-to-noise ratio (SNR) beyond the standard quantum limit. Furthermore, the scheme is not as sensitive to the detection loss as the linear scheme with a squeezed state for noise reduction. However, losses inside the interferometer limit the enhancement factor in SNR. We apply the concept to a Michelson interferometer but with parametric amplifiers involved for gravitational-wave detection. We find that effective power is increased by the gain of the amplifiers without actually increasing the cycling power inside the interferometer. Furthermore, the full benefits with squeezed input and variational output or the combination of a quantum nondemolition interferometer for sensitivity beyond the standard quantum limit apply here with even better results. Such a nonlinear interferometer will find wide applications in precision measurements.
The classically defined minimum uncertainty of the optical phase is known as the standard quantum limit or shot-noise limit (SNL), originating in the uncertainty principle of quantum mechanics. Based on the SNL, the phase sensitivity is inversely proportional to K, where K is the number of interfering photons or statistically measured events. Thus, using a high-power laser is advantageous to enhance sensitivity due to the K gain in the signal-to-noise ratio. In a typical interferometer, however, the resolution remains in the diffraction limit of the K = 1 case unless the interfering photons are resolved as in quantum sensing. Here, a projection measurement method in quantum sensing is adapted for classical sensing to achieve an additional K gain in the resolution. To understand the projection measurements, several types of conventional interferometers based on N-wave interference are coherently analyzed as a classical reference and numerically compared with the proposed method. As a result, the Kth-order intensity product applied to the N-wave spectrometer exceeds the diffraction limit in classical sensing and the Heisenberg limit in quantum sensing, where the classical N-slit system inherently satisfies the Heisenberg limit of π/N in resolution.