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arXiv:1103.2304 · Physical Review A

Formation of NOON states from Fock-state Bose-Einstein condensates

Hugo V. Cable, Franck Laloë, William J. Mullin

Published May 24, 2011• 13 citations
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Abstract

States of the form $|N\ensuremath{\rangle}{}_{a}|0\ensuremath{\rangle}{}_{b}+|0\ensuremath{\rangle}{}_{a}|N\ensuremath{\rangle}{}_{b}$, where $a$ and $b$ are single-particle states---i.e., NOON states---have been used for predicting violations of local realism (Greenberger-Horne-Zeilinger violations) and are valuable in metrology for precision measurements of phase at the Heisenberg limit. We show theoretically how the use of two Fock-state Bose-Einstein condensates as sources in a modified Mach-Zehnder interferometer can lead to creation of the NOON state in which $a$ and $b$ refer to arms of the interferometer and $N$ is a subset of the total number of particles in the two condensates. The modification of the interferometer involves making ``side'' measurements of a few particles near the sources. These measurements put the remaining particles in a superposition of two phase states, which are converted into NOON states by a beam splitter if the phase states are orthogonal. When they are not orthogonal, a ``feed-forward'' correction circuit is shown to convert them into proper form so that a NOON results. We apply the NOON to the measurement of phase. Here the NOON experiment is equivalent to one in which a large molecule passes through two slits. The NOON components can be recombined in a final beam splitter to show interference.

DOI
10.1103/physreva.83.053626

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States of the form ∣N\ensuremath⟩a∣0\ensuremath⟩b+∣0\ensuremath⟩a∣N\ensuremath⟩b|N\ensuremath{\rangle}{}_{a}|0\ensuremath{\rangle}{}_{b}+|0\ensuremath{\rangle}{}_{a}|N\ensuremath{\rangle}{}_{b}∣N\ensuremath⟩a​∣0\ensuremath⟩b​+∣0\ensuremath⟩a​∣N\ensuremath⟩b​, where aaa and bbb are single-particle states---i.e., NOON states---have been used for predicting violations of local realism (Greenberger-Horne-Zeilinger violations) and are valuable in metrology for precision measurements of phase at the Heisenberg limit. We show theoretically how the use of two Fock-state Bose-Einstein condensates as sources in a modified Mach-Zehnder interferometer can lead to creation of the NOON state in which aaa and bbb refer to arms of the interferometer and NNN is a subset of the total number of particles in the two condensates. The modification of the interferometer involves making side'' measurements of a few particles near the sources. These measurements put the remaining particles in a superposition of two phase states, which are converted into NOON states by a beam splitter if the phase states are orthogonal. When they are not orthogonal, a feed-forward'' correction circuit is shown to convert them into proper form so that a NOON results. We apply the NOON to the measurement of phase. Here the NOON experiment is equivalent to one in which a large molecule passes through two slits. The NOON components can be recombined in a final beam splitter to show interference.
NOON states (states of the form ∣N>a∣0>b+∣0>a∣N>b|N>_{a}|0>_{b}+|0>_{a}|N>_{b}∣N>a​∣0>b​+∣0>a​∣N>b​ where aaa and bbb are single particle states) have been used for predicting violations of local realism (Greenberger-Horne-Zeilinger violations) and are valuable in metrology for precision measurements of phase at the Heisenberg limit. We show theoretically how the use of two Fock state Bose-Einstein condensates as sources in a modified Mach-Zehnder interferometer can lead to the creation of the NOON state in which aaa and bbb refer to arms of the interferometer and NNN is a subset of the total number of particles in the two condensates. The modification of the interferometer involves making {}"side" measurements of a few particles near the sources. These measurements put the remaining particles in a superposition of two phase states, which are converted into NOON states by a beam splitter if the phase states are orthogonal. When they are not orthogonal, a {}"feedforward" correction circuit is shown to convert them into proper form so a NOON results. We apply the NOON to the measurement of phase. Here the NOON experiment is equivalent to one in which a large molecule passes through two slits. The NOON components can be recombined in a final beam splitter to show interference.
Quantum systems in Fock states do not have a phase. When two or more Bose-Einstein condensates are sent into interferometers, they nevertheless acquire a relative phase under the effect of quantum measurements. The usual explanation relies on spontaneous symmetry breaking, where phases are ascribed to all condensates and treated as unknown classical quantities. However, this image is not always sufficient: when all particles are measured, quantum mechanics predicts probabilities that are sometimes in contradiction with it, as illustrated by quantum violations of local realism. In this letter, we show that interferometers can be used to demonstrate a large variety of violations with an arbitrarily large number of particles. With two independent condensates, we find violations of the BCHSH inequalities, as well as new N-body Hardy impossibilities. With three condensates, we obtain new GHZ (Greenberger, Horne and Zeilinger) type contradictions.
Quantum systems in Fock states do not have a phase. When two or more Bose-Einstein condensates are sent into interferometers, they nevertheless acquire a relative phase under the effect of quantum measurements. The usual explanation relies on spontaneous symmetry breaking, where phases are ascribed to all condensates and treated as unknown classical quantities. However, this image is not always sufficient: when all particles are measured, quantum mechanics predicts probabilities that are sometimes in contradiction with it, as illustrated by quantum violations of local realism. In this Rapid communication, we show that interferometers can be used to demonstrate a large variety of violations with an arbitrarily large number of particles. With two independent condensates, we find violations of the Bell-Clauser-Horne-Shimony-Holt inequalities, as well as NNN-body Hardy impossibilities. With three condensates, we obtain Greenberger-Horne-Zeilinger-type contradictions.
In conventional Einstein-Rosen-Podolsky (EPR) experiments that violate local realism, particles are placed in very particular entangled states. We propose here to use two or three spinless Fock-state Bose-Einstein condensates as independent sources in interferometery experiments. While these states do not seem to be entangled, nevertheless we show that interferometers can be constructed that demonstrate a large variety of different violations local reality. We find violations of Bell inequalities, new Greenberger-Horne-Zeilinger (GHZ) contradictions, and Hardy impossibilities. These violations continue to arbitrarily large particle numbers. A necessary condition to observe the quantum effects is that all particles should be observed; if some are missed, the quantum effects disappear.
We discuss the prospects of performing Heisenberg-limited quantum sensing and metrology using a Mach-Zehnder interferometer with input states that are superpositions of twin-Fock states and where photon number parity measurements are made on one of the output beams of the interferometer. This study is motivated by the experimental challenge of producing twin-Fock states on opposite sides of a beam splitter. We focus on the use of the so-called pair coherent states for this purpose and discuss a possible mechanism for generating them. We also discuss the prospect of using other superstitions of twin-Fock states, for the purpose of interferometry.