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:0709.2143 · Physical Review Letters

Optimized Double-Well Quantum Interferometry with Gaussian Squeezed States

Y. P. Huang, M. G. Moore

Published Jun 27, 2008• 33 citations
arXivHTMLPDF

Abstract

A Mach-Zender interferometer with a Gaussian number-difference squeezed input state can exhibit sub-shot-noise phase resolution over a large phase interval. We derive the optimal level of squeezing for a given phase interval $\ensuremath{\Delta}{\ensuremath{\theta}}_{0}$ and particle number $N$. We then propose an adaptive measurement sequence in which the amount of squeezing is increased with each measurement. With this scheme, any phase on $(\ensuremath{-}\ensuremath{\Delta}{\ensuremath{\theta}}_{0},\ensuremath{\Delta}{\ensuremath{\theta}}_{0})$ can be measured with a precision of $3.5/N$, requiring only 2--4 measurements, provided only that $N\mathrm{tan}(\ensuremath{\Delta}{\ensuremath{\theta}}_{0})<{10}^{40}$. In a double-well Bose-Einstein condensate, the optimized input states can be created by adiabatic manipulation of the ground state.

DOI
10.1103/physrevlett.100.250406

Related papers

Sub-shot-noise phase sensitivity with a Bose-Einstein condensate Mach-Zehnder interferometer

Luca Pezzè, Lee A. Collins, Augusto Smerzi +2 more

Bose-Einstein condensates (BEC), with their coherence properties, have attracted wide interest for their possible application to ultraprecise interferometry and ultraweak force sensors. Since condensates, unlike photons, are interacting, they may permit the realization of specific quantum states needed as input of an interferometer to approach the Heisenberg limit, the supposed lower bound to precision phase measurements. To this end, we study the sensitivity to external weak perturbations of a representative matter-wave Mach-Zehnder interferometer whose input are two Bose-Einstein condensates created by splitting a single condensate in two parts. The interferometric phase sensitivity depends on the specific quantum state created with the two condensates, and, therefore, on the time scale of the splitting process. We identify three different regimes, characterized by a phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ scaling with the total number of condensate particles $N$ as (i) the standard quantum limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{1∕2}$, (ii) the sub shot-noise $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{3∕4}$, and the (iii) the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕N$. However, in a realistic dynamical BEC splitting, the $1∕N$ limit requires a long adiabaticity time scale, which is hardly reachable experimentally. On the other hand, the sub-shot-noise sensitivity $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{3∕4}$ can be reached in a realistic experimental setting. We also show that the $1∕{N}^{3∕4}$ scaling is a rigorous upper bound in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, while keeping constant all different parameters of the bosonic Mach-Zehnder interferometer.

arXiv:cond-mat/0508680•Physical Review A•Oct 26, 2005•68 citations

Optimal Gaussian squeezed states for atom interferometry in the presence of phase diffusion

Igor Tikhonenkov, M. G. Moore, Amichay Vardi

We optimize the signal-to-noise ratio of a Mach-Zehnder atom interferometer with Gaussian squeezed input states in the presence interactions. For weak interactions, our results coincide with those of Huang and Moore [Y. P. Huang and M. G. Moore, Phys. Rev. Lett. 100, 250406 (2008)], with an optimal initial number variance ${\ensuremath{\sigma}}_{o}\ensuremath{\propto}{N}^{1/3}$ and an optimal signal-to-noise ratio ${s}_{o}\ensuremath{\propto}{N}^{2/3}$ for the total atom number $N$. As the interaction strength $u$ increases past unity, phase diffusion becomes dominant, leading to a transition in the optimal squeezing from initial number squeezing to initial phase squeezing with ${\ensuremath{\sigma}}_{o}\ensuremath{\propto}\sqrt{\mathit{uN}}$ and ${s}_{o}\ensuremath{\propto}\sqrt{N/u}$ shot-noise scaling. The initial phase squeezing translates into hold-time number squeezing, which is less sensitive to interactions than coherent states and improves ${s}_{o}$ by a factor of $\sqrt{u}$.

arXiv:1008.4114•Physical Review A•Oct 26, 2010•17 citations

Robust sub-shot-noise measurement via Rabi-Josephson oscillations in bimodal Bose-Einstein condensates

Igor Tikhonenkov, M. G. Moore, Amichay Vardi

Mach-Zehnder atom interferometry requires hold-time phase squeezing to attain readout accuracy below the standard quantum limit. This increases its sensitivity to phase diffusion, restoring shot-noise scaling of the optimal signal-to-noise ratio in the presence of interactions. The contradiction between the preparations required for readout accuracy and robustness to interactions is removed by monitoring Rabi-Josephson oscillations instead of relative-phase oscillations during signal acquisition. Optimizing the signal-to-noise ratio with a Gaussian squeezed input, we find that hold-time number squeezing satisfies both demands and that sub-shot-noise scaling is retained even for strong interactions.

arXiv:1110.1888•Physical Review A•Jun 23, 2011•2 citations

Optimized interferometry with Gaussian states

Stefano Olivares, Matteo G. A. Paris

We revisit Mach—Zehnder interferometry using a suitable phase-space analysis and present a rigorous optimization of the sensitivity in realistic condition, i.e., for Gaussian input states, and taking into account nonunit quantum efficiency in the detection stage. The working regime of the interferometer is optimized at fixed input energy versus the squeezing phases and amplitudes as well as the distribution of squeezing in the two input signals. For ideal detection we find the known result that the squeezing resource allows to beat the shot-noise limit. For nonunit detection efficiency, we show that for fixed input energy one can always optimize the squeezing fraction in order to enhance the sensitivity with respect to the case of no squeezing, even in cases when one cannot go beyond the shot-noise limit.

Optics and Spectroscopy•Aug 1, 2007•18 citations

Atom interferometer in a double-well potential

Karol Gietka, Jan Chwedeńczuk

We present a detailed study of an atom interferometer which can be realized in a double-well potential. We assume that the interferometric phase is imprinted in the presence of coherent tunneling between the wells. We calculate the ultimate bounds for the estimation sensitivity and show how they relate to the precision of the Mach-Zehnder interferometer. The interferometer presented here allows for sub-shot-noise sensitivity when fed with spin-squeezed states with either a reduced relative population imbalance or a reduced relative phase. We also calculate the precision of the estimation from the population imbalance and show that it overcomes the shot-noise limit when the entangled squeezed states are used at the input.

arXiv:1407.0858•Physical Review A•Dec 1, 2014•6 citations

Sub-shot-noise-limited phase estimation via single-mode inputs

Jiandong Zhang, Chenglong You, Shuai Wang

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.

Optics Express•Oct 24, 2022•2 citations
A Mach-Zender interferometer with a Gaussian number-difference squeezed input state can exhibit sub-shot-noise phase resolution over a large phase interval. We derive the optimal level of squeezing for a given phase interval \ensuremathΔ\ensuremathθ0\ensuremath{\Delta}{\ensuremath{\theta}}_{0}\ensuremathΔ\ensuremathθ0​ and particle number NNN. We then propose an adaptive measurement sequence in which the amount of squeezing is increased with each measurement. With this scheme, any phase on (\ensuremath−\ensuremathΔ\ensuremathθ0,\ensuremathΔ\ensuremathθ0)(\ensuremath{-}\ensuremath{\Delta}{\ensuremath{\theta}}_{0},\ensuremath{\Delta}{\ensuremath{\theta}}_{0})(\ensuremath−\ensuremathΔ\ensuremathθ0​,\ensuremathΔ\ensuremathθ0​) can be measured with a precision of 3.5/N3.5/N3.5/N, requiring only 2--4 measurements, provided only that Ntan(\ensuremathΔ\ensuremathθ0)<1040N\mathrm{tan}(\ensuremath{\Delta}{\ensuremath{\theta}}_{0})<{10}^{40}Ntan(\ensuremathΔ\ensuremathθ0​)<1040. In a double-well Bose-Einstein condensate, the optimized input states can be created by adiabatic manipulation of the ground state.
Bose-Einstein condensates (BEC), with their coherence properties, have attracted wide interest for their possible application to ultraprecise interferometry and ultraweak force sensors. Since condensates, unlike photons, are interacting, they may permit the realization of specific quantum states needed as input of an interferometer to approach the Heisenberg limit, the supposed lower bound to precision phase measurements. To this end, we study the sensitivity to external weak perturbations of a representative matter-wave Mach-Zehnder interferometer whose input are two Bose-Einstein condensates created by splitting a single condensate in two parts. The interferometric phase sensitivity depends on the specific quantum state created with the two condensates, and, therefore, on the time scale of the splitting process. We identify three different regimes, characterized by a phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ scaling with the total number of condensate particles NNN as (i) the standard quantum limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{1∕2}$, (ii) the sub shot-noise $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{3∕4}$, and the (iii) the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕N$. However, in a realistic dynamical BEC splitting, the 1∕N1∕N1∕N limit requires a long adiabaticity time scale, which is hardly reachable experimentally. On the other hand, the sub-shot-noise sensitivity $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{3∕4}$ can be reached in a realistic experimental setting. We also show that the 1∕N3∕41∕{N}^{3∕4}1∕N3∕4 scaling is a rigorous upper bound in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, while keeping constant all different parameters of the bosonic Mach-Zehnder interferometer.
We optimize the signal-to-noise ratio of a Mach-Zehnder atom interferometer with Gaussian squeezed input states in the presence interactions. For weak interactions, our results coincide with those of Huang and Moore [Y. P. Huang and M. G. Moore, Phys. Rev. Lett. 100, 250406 (2008)], with an optimal initial number variance ${\ensuremath{\sigma}}_{o}\ensuremath{\propto}{N}^{1/3}$ and an optimal signal-to-noise ratio ${s}_{o}\ensuremath{\propto}{N}^{2/3}$ for the total atom number NNN. As the interaction strength uuu increases past unity, phase diffusion becomes dominant, leading to a transition in the optimal squeezing from initial number squeezing to initial phase squeezing with ${\ensuremath{\sigma}}_{o}\ensuremath{\propto}\sqrt{\mathit{uN}}$ and ${s}_{o}\ensuremath{\propto}\sqrt{N/u}$ shot-noise scaling. The initial phase squeezing translates into hold-time number squeezing, which is less sensitive to interactions than coherent states and improves so{s}_{o}so​ by a factor of u\sqrt{u}u​.
Mach-Zehnder atom interferometry requires hold-time phase squeezing to attain readout accuracy below the standard quantum limit. This increases its sensitivity to phase diffusion, restoring shot-noise scaling of the optimal signal-to-noise ratio in the presence of interactions. The contradiction between the preparations required for readout accuracy and robustness to interactions is removed by monitoring Rabi-Josephson oscillations instead of relative-phase oscillations during signal acquisition. Optimizing the signal-to-noise ratio with a Gaussian squeezed input, we find that hold-time number squeezing satisfies both demands and that sub-shot-noise scaling is retained even for strong interactions.
We revisit Mach—Zehnder interferometry using a suitable phase-space analysis and present a rigorous optimization of the sensitivity in realistic condition, i.e., for Gaussian input states, and taking into account nonunit quantum efficiency in the detection stage. The working regime of the interferometer is optimized at fixed input energy versus the squeezing phases and amplitudes as well as the distribution of squeezing in the two input signals. For ideal detection we find the known result that the squeezing resource allows to beat the shot-noise limit. For nonunit detection efficiency, we show that for fixed input energy one can always optimize the squeezing fraction in order to enhance the sensitivity with respect to the case of no squeezing, even in cases when one cannot go beyond the shot-noise limit.
We present a detailed study of an atom interferometer which can be realized in a double-well potential. We assume that the interferometric phase is imprinted in the presence of coherent tunneling between the wells. We calculate the ultimate bounds for the estimation sensitivity and show how they relate to the precision of the Mach-Zehnder interferometer. The interferometer presented here allows for sub-shot-noise sensitivity when fed with spin-squeezed states with either a reduced relative population imbalance or a reduced relative phase. We also calculate the precision of the estimation from the population imbalance and show that it overcomes the shot-noise limit when the entangled squeezed states are used at the input.
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.