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Optics Continuum

Squeezing-enhanced phase measurement with ternary-binning post-processing

jiaming Li, juntao he, Kui Liu, Jiangrui Gao

Published Jul 24, 2026• 0 citations
Publisher

Abstract

Quantum metrology can beat the shot-noise limit (SNL) with nonclassical resources, yet simultaneously realizing sub-Rayleigh resolution and sub-SNL sensitivity in a deployable interferometer remains challenging. Here we experimentally validate a deterministic phase-measurement protocol that combines single-port balanced homodyne detection in a Mach-Zehnder interferometer with ternary-binning post-processing. For a coherent probe with a mean photon number of about 67, ternary binning preserves sub-Rayleigh resolution while mitigating the near-zero-phase ill-conditioning of binary-binning readout, yielding a 7.15-fold resolution enhancement relative to the unprocessed interferometer. With 5.99 ± 0.13 dB squeezed-state injection into the vacuum port, the resolution enhancement increases to 13.69-fold, and sub-SNL sensitivity is achieved; the minimum sensitivity improves by a factor of 1.78 compared with the coherent probe. The protocol is deterministic and experimentally accessible, offering a practical route to multiparameter estimation and quantum-enhanced microscopy.

DOI
10.1364/optcon.608948

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Quantum metrology can beat the shot-noise limit (SNL) with nonclassical resources, yet simultaneously realizing sub-Rayleigh resolution and sub-SNL sensitivity in a deployable interferometer remains challenging. Here we experimentally validate a deterministic phase-measurement protocol that combines single-port balanced homodyne detection in a Mach-Zehnder interferometer with ternary-binning post-processing. For a coherent probe with a mean photon number of about 67, ternary binning preserves sub-Rayleigh resolution while mitigating the near-zero-phase ill-conditioning of binary-binning readout, yielding a 7.15-fold resolution enhancement relative to the unprocessed interferometer. With 5.99 ± 0.13 dB squeezed-state injection into the vacuum port, the resolution enhancement increases to 13.69-fold, and sub-SNL sensitivity is achieved; the minimum sensitivity improves by a factor of 1.78 compared with the coherent probe. The protocol is deterministic and experimentally accessible, offering a practical route to multiparameter estimation and quantum-enhanced microscopy.
Abstract We investigate the phase sensitivities of three schemes with the input of the squeezed coherent states based on the Mach–Zehnder interferometer (MZI). With employing the balanced homodyne detection, the phase sensitivities of these schemes can surpass the shot noise limit (SNL) and reach the sub-Heisenberg limit. By adjusting the combination of the squeezing parameter and the photon number of the coherent state, both the phase sensitivities with using the balanced homodyne detection and the quantum Cramer–Rao bound can be improved. In the regime of low squeezing parameters, the scheme with the two-mode squeezed coherent state outperforms the other two schemes. The phase sensitivity can only reach the sub-SNL when the photon number is much larger. These schemes exhibit strong robustness against internal losses of the MZI and photon detector losses. This work will play an important role in the field of quantum precision measurement.
An important part of optical sensing is interferometry, where direct or homodyne detection at the output of an interferometer yields information about small phase shifts. Classical interferometry operates at the shot-noise limit (SNL), caused by the photon structure of light and reached when an interferometer is fed with coherent light. The use of nonclassical light at the input of an interferometer enables overcoming the SNL. In particular, squeezed light is known to be less noisy than coherent light. An interferometer fed with strongly squeezed light can, in principle, provide the phase sensitivity at the Heisenberg limit, the ultimate limit allowed by the laws of physics. However, this supersensitive operation is strongly degraded by loss – both inside the interferometer and outside it, at the stage of detection. While the former (internal loss) seems to be inevitable, the latter (detection loss) can be overcome, and we demonstrate it in our recent experiment. The clue is to amplify light at the interferometer output using a phase-sensitive optical parametric amplifier. This device does not introduce additional noise, but it amplifies the fragile quantum signal and thus makes it tolerant to loss at the detection stage. In a proof-of-principle experiment, we build a polarization Mach-Zehnder interferometer and feed it with coherent light in one port and a squeezed vacuum in another port. At one output port, we amplify light with an optical parametric amplifier and measure a phase sensitivity of 13 mrad for 1500 photons in the interferometer. This result, achieved with 50% detection efficiency, overcomes the SNL by 3 dB. Moreover, by using stronger amplification, we can overcome the SNL under detection efficiency as low as 13%. Our results outline a way to use squeezed light for real-life sensing, where the detection efficiency cannot be very high, especially in spectral ranges like middle-infrared or terahertz.
In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.
Phase super-sensitivity is obtained when the sensitivity in a phase measurement goes beyond the quantum shot noise limit, whereas super-resolution is obtained when the interference fringes in an interferometer are narrower than half the input wavelength. Here we show experimentally that these two features can be simultaneously achieved using a relatively simple setup based on Gaussian states and homodyne measurement. Using 430 photons shared between a coherent- and a squeezed vacuum state, we demonstrate a 22-fold improvement in the phase resolution while we observe a 1.7-fold improvement in the sensitivity. In contrast to previous demonstrations of super-resolution and super-sensitivity, this approach is fully deterministic.
We theoretically study the phase estimation based on a Mach-Zehnder interferometer (MZI) with a two-mode squeezed coherent state. By maximizing the quantum Fisher information, we find that the quantum Cramér-Rao bounds (QCRB) can reach sub-Heisenberg limit under the phase-matched condition. The optimal phase sensitivity can reach the sub-shot noise limit (SNL) and approach the QCRB by employing the intensity difference detection. Meanwhile, compared with the MZI fed with a coherent plus a single-mode squeezed vacuum state, this scheme can have better performance by adjusting the squeezing parameter and the mean photon number. With the same parameter, our scheme shows more sensitive phase measurement than the SU(1,1) interferometer with a coherent plus a vacuum state. We also show that the phase sensitivity of our proposal can still reach the SNL when the loss of the photon is 36%. This scheme can provide potential applications in optical sensors.
Improvements in precision beyond the limit set by the vacuum state or zero-point fluctuations of the field are reported for two optical experiments. Both experiments employ the squeezed light from an optical parametric oscillator to reduce the level of fluctuations in one of two quadrature phase amplitudes below the shot-noise level (SNL). The first experiment follows the work of Caves1 on precision interferometry with squeezed states. An increase in signal-to-noise of 2.5 dB relative to the SNL is demonstrated for the detection of phase modulation in a Mach-Zehnder interferometer. In the second experiment amplitude modulation is encoded on a weak signal beam with an acoustooptic modulator. For the case of a squeezed signal beam compared with a beam in a coherent state, an improvement in sensitivity of 2.5 dB is achieved for the detection of amplitude changes of the field. The observed increases in sensitivity in both experiments are currently limited by simple linear losses in propagation and detection and not by the degree of available squeezing.