Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0211182 · Journal of Optics B Quantum and Semiclassical Optics

Approaching the Heisenberg limit with two-mode squeezed states

Ole Steuernagel, Stefan Scheel

Published Mar 1, 2004• 29 citations
arXivHTMLPDF

Abstract

Two mode squeezed states can be used to achieve Heisenberg limit scaling in interferometry: a phase shift of $δϕ\approx 2.76 / < N >$ can be resolved. The proposed scheme relies on balanced homodyne detection and can be implemented with current technology. The most important experimental imperfections are studied and their impact quantified.

DOI
10.1088/1464-4266/6/3/011

Related papers

Deterministic sub-Heisenberg limit phase sensitivity with the squeezed coherent state via balanced homodyne detection

Chenlu Li, Shaoqiu Ke, Pengxiang Ruan +5 more

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.

Physica Scripta•Feb 4, 2025•2 citations

Improving phase sensitivity of a hybrid interferometer with the two-mode squeezed coherent state

Tao Shao, Pengxiang Ruan, Minyang Zhang +4 more

Abstract We investigate the enhancement of phase sensitivity of a nonlinear-linear hybrid interferometer with the input of the two-mode squeezed coherent state (TMSCS). With the TMSCS produced by four-wave mixing, the quantum Cramér-Rao bounds (QCRB) can beat the Heisenberg limit (HL). Under the phase matched conditions, the optimal phase sensitivity with the balanced homodyne detection measurement can beat the HL and approach QCRB. The effects of internal and external losses on the measurement accuracy are also discussed. The results demonstrate that the scheme is robustness against to internal losses and this protocol can resist external detection loss which is up to 39%. Our results improve the performance of hybrid interferometers and this scheme can find important practical applications in quantum metrology.

Physica Scripta•Sep 18, 2024•5 citations

Heisenberg limit in phase measurements: the threshold detection approach

Dariya Salykina, V. S. Liamin, Valentin L. Gorshenin +2 more

The ultimate precision of phase estimation is limited by the Heisenberg scaling $Δϕ_0 = K/N$, where $K\sim1$ is a numerical prefactor and $N$ 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 $Δϕ_0$ and the width $δϕ$ of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that $δϕ$ strongly depends on $K$. 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 $δϕ=π/2$.

arXiv:2505.06714•May 10, 2025•0 citations

Heisenberg Scaling in a Continuous-Wave Interferometer

Hudson A. Loughlin, Melissa A. Guidry, J. Ding +7 more

Continuous-wave (CW) interferometry has stood at the frontier of precision measurement science since its inception, where it was used to search for the luminiferous ether, to the present day, where it forms the basis of interferometric gravitational-wave detection. Quantum theory predicts that this frontier can be expanded more rapidly by employing certain quantum resources, compared with the case of using only classical resources. In the quantum case, we can achieve ``Heisenberg scaling'', which manifests as a quadratic improvement over the best possible classical precision scaling. Although Heisenberg scaling has been demonstrated in pulsed operation, it has not been demonstrated for continuous operation. The challenge in doing so is two-fold: continuous measurements capable of Heisenberg scaling were previously unknown, and the requisite CW quantum states are fragile. Here we overcome these challenges and demonstrate the first CW interferometer exhibiting resource efficiency approaching Heisenberg scaling. Our scheme comprises a Mach-Zehnder interferometer illuminated with a pair of squeezed light sources, followed by a nonlinear estimator of the output homodyne record to estimate a differential phase modulation signal that drives the interferometer. We observe that this signal can be extracted with a precision that scales faster than what is allowed classically, and approaches the Heisenberg scaling limit.

arXiv:2509.25384•Sep 29, 2025•0 citations

The Super Phase Sensitivity of an SU(1,1) Interferometer with a Two-Mode Squeezed Coherent State via Balanced Homodyne and Intensity Detection

Changlan Xu, Lei Wang, Shaoqiu Ke +2 more

We propose a novel scheme that is used for the enhancement of phase sensitivity. The SU(1,1) interferometer with a two-mode squeezed coherent state input, using balanced homodyne detection (BHD) and intensity detection (ID), is shown. Our results demonstrate that the phase sensitivity achieved via BHD outperforms that of ID. The optimal phase sensitivity via BHD surpasses the Heisenberg limit (HL) and approaches the quantum Cramér–Rao bound. A larger photon number and parameter strength can make the phase sensitivity better. Furthermore, we show the effects of internal and external losses on phase sensitivity in detail. When external loss reaches 10%, the phase sensitivity can reach the HL. Next, we have a detailed discussion on the impact of the squeezing parameter and photon number on phase sensitivity, which shows that our scheme has better phase sensitivity and enhanced robustness. This interferometer system thus holds significant potential for applications in quantum precision measurement.

Photonics•Mar 23, 2026•0 citations

Coherent-light-boosted, sub-shot noise, quantum interferometry

William N. Plick, Jonathan P. Dowling, G. S. Agarwal

We present in this paper a new scheme for optical interferometry. We utilize coherent-beam-stimulated two-mode squeezed light, which interacts with a phase shifter and is then squeezed again before detection. Our theoretical device has the potential to reach far below the shot-noise limit in phase sensitivity. This new proposal avoids the pitfalls of other setups, such as difficulty in creating the required resource. Furthermore, our scheme requires no complicated detection protocol, relying instead only on simple intensity measurement. Also, bright, coherent sources 'boost' squeezed light, creating a very sensitive device. This hybrid scheme relies on no unknown components and can be constructed with current technology. In this paper, we present our analysis of this relatively straightforward device, using the operator propagation method. We derive the phase sensitivity and provide a simple numerical example of the power of our new proposal. Sensitivity to unknown phase shifts scales as a shot-noise-limited Mach–Zehnder interferometer, multiplied by a sub-Heisenberg contribution from the squeezed light.

arXiv:0911.5714•New Journal of Physics•Aug 6, 2010•165 citations
Two mode squeezed states can be used to achieve Heisenberg limit scaling in interferometry: a phase shift of δϕ≈2.76/<N>δϕ\approx 2.76 / < N >δϕ≈2.76/<N> can be resolved. The proposed scheme relies on balanced homodyne detection and can be implemented with current technology. The most important experimental imperfections are studied and their impact quantified.
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.
Abstract We investigate the enhancement of phase sensitivity of a nonlinear-linear hybrid interferometer with the input of the two-mode squeezed coherent state (TMSCS). With the TMSCS produced by four-wave mixing, the quantum Cramér-Rao bounds (QCRB) can beat the Heisenberg limit (HL). Under the phase matched conditions, the optimal phase sensitivity with the balanced homodyne detection measurement can beat the HL and approach QCRB. The effects of internal and external losses on the measurement accuracy are also discussed. The results demonstrate that the scheme is robustness against to internal losses and this protocol can resist external detection loss which is up to 39%. Our results improve the performance of hybrid interferometers and this scheme can find important practical applications in quantum metrology.
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.
Continuous-wave (CW) interferometry has stood at the frontier of precision measurement science since its inception, where it was used to search for the luminiferous ether, to the present day, where it forms the basis of interferometric gravitational-wave detection. Quantum theory predicts that this frontier can be expanded more rapidly by employing certain quantum resources, compared with the case of using only classical resources. In the quantum case, we can achieve ``Heisenberg scaling'', which manifests as a quadratic improvement over the best possible classical precision scaling. Although Heisenberg scaling has been demonstrated in pulsed operation, it has not been demonstrated for continuous operation. The challenge in doing so is two-fold: continuous measurements capable of Heisenberg scaling were previously unknown, and the requisite CW quantum states are fragile. Here we overcome these challenges and demonstrate the first CW interferometer exhibiting resource efficiency approaching Heisenberg scaling. Our scheme comprises a Mach-Zehnder interferometer illuminated with a pair of squeezed light sources, followed by a nonlinear estimator of the output homodyne record to estimate a differential phase modulation signal that drives the interferometer. We observe that this signal can be extracted with a precision that scales faster than what is allowed classically, and approaches the Heisenberg scaling limit.
We propose a novel scheme that is used for the enhancement of phase sensitivity. The SU(1,1) interferometer with a two-mode squeezed coherent state input, using balanced homodyne detection (BHD) and intensity detection (ID), is shown. Our results demonstrate that the phase sensitivity achieved via BHD outperforms that of ID. The optimal phase sensitivity via BHD surpasses the Heisenberg limit (HL) and approaches the quantum Cramér–Rao bound. A larger photon number and parameter strength can make the phase sensitivity better. Furthermore, we show the effects of internal and external losses on phase sensitivity in detail. When external loss reaches 10%, the phase sensitivity can reach the HL. Next, we have a detailed discussion on the impact of the squeezing parameter and photon number on phase sensitivity, which shows that our scheme has better phase sensitivity and enhanced robustness. This interferometer system thus holds significant potential for applications in quantum precision measurement.
We present in this paper a new scheme for optical interferometry. We utilize coherent-beam-stimulated two-mode squeezed light, which interacts with a phase shifter and is then squeezed again before detection. Our theoretical device has the potential to reach far below the shot-noise limit in phase sensitivity. This new proposal avoids the pitfalls of other setups, such as difficulty in creating the required resource. Furthermore, our scheme requires no complicated detection protocol, relying instead only on simple intensity measurement. Also, bright, coherent sources 'boost' squeezed light, creating a very sensitive device. This hybrid scheme relies on no unknown components and can be constructed with current technology. In this paper, we present our analysis of this relatively straightforward device, using the operator propagation method. We derive the phase sensitivity and provide a simple numerical example of the power of our new proposal. Sensitivity to unknown phase shifts scales as a shot-noise-limited Mach–Zehnder interferometer, multiplied by a sub-Heisenberg contribution from the squeezed light.