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arXiv:1609.06028 · Physical Review A

Quantifying the mesoscopic quantum coherence of approximate NOON states and spin-squeezed two-mode Bose-Einstein condensates

Bogdan Opanchuk, L. Rosales-Zárate, Run Yan Teh, M. D. Reid

Published Dec 29, 2016• 31 citations
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Abstract

We examine how to signify and quantify the mesoscopic quantum coherence of approximate two-mode NOON states and spin-squeezed two-mode Bose-Einstein condensates (BEC). We identify two criteria that verify a nonzero quantum coherence between states with quantum number different by $n$. These criteria negate certain mixtures of quantum states, thereby signifying a generalized $n$-scopic Schr\"odinger cat-type paradox. The first criterion is the correlation $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}\ensuremath{\ne}0$ (here $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}$ are the boson operators for each mode). The correlation manifests as interference fringes in $n$-particle detection probabilities and is also measurable via quadrature phase amplitude and spin-squeezing measurements. Measurement of $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}$ enables a quantification of the overall $n\text{th}$ order quantum coherence, thus providing an avenue for high efficiency verification of high-fidelity photonic NOON states. The second criterion is based on a quantification of the measurable spin-squeezing parameter ${\ensuremath{\xi}}_{N}$. We apply the criteria to theoretical models of NOON states in lossy interferometers and double-well trapped BECs. By analyzing existing BEC experiments, we demonstrate generalized atomic ``kitten'' states and atomic quantum coherence with $n⪆10$ atoms.

DOI
10.1103/physreva.94.062125

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We examine how to signify and quantify the mesoscopic quantum coherence of approximate two-mode NOON states and spin-squeezed two-mode Bose-Einstein condensates (BEC). We identify two criteria that verify a nonzero quantum coherence between states with quantum number different by nnn. These criteria negate certain mixtures of quantum states, thereby signifying a generalized nnn-scopic Schr"odinger cat-type paradox. The first criterion is the correlation \ensuremath⟨a\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\finb\ifmmode^\elseˆ\fin\ensuremath⟩\ensuremath≠0\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}\ensuremath{\ne}0\ensuremath⟨a\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\finb\ifmmode^\elseˆ\fi​n\ensuremath⟩\ensuremath=0 (here a\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}a\ifmmode^\elseˆ\fi and b\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}b\ifmmode^\elseˆ\fi​ are the boson operators for each mode). The correlation manifests as interference fringes in nnn-particle detection probabilities and is also measurable via quadrature phase amplitude and spin-squeezing measurements. Measurement of \ensuremath⟨a\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\finb\ifmmode^\elseˆ\fin\ensuremath⟩\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}\ensuremath⟨a\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\finb\ifmmode^\elseˆ\fi​n\ensuremath⟩ enables a quantification of the overall nthn\text{th}nth order quantum coherence, thus providing an avenue for high efficiency verification of high-fidelity photonic NOON states. The second criterion is based on a quantification of the measurable spin-squeezing parameter \ensuremathξN{\ensuremath{\xi}}_{N}\ensuremathξN​. We apply the criteria to theoretical models of NOON states in lossy interferometers and double-well trapped BECs. By analyzing existing BEC experiments, we demonstrate generalized atomic ``kitten'' states and atomic quantum coherence with n⪆10n⪆10n⪆10 atoms.
According to classical theory, a system with two or more macroscopically distinct states available to it is in one of those states at all times. Quantum mechanics gives a different interpretation where the system can be in a superposition of such states. In this paper, we derive criteria in the form of inequalities to detect this effect, referred to as mesoscopic quantum coherence, where the states are (at least) mesoscopically distinct. Such criteria are also signatures of a mesoscopic Schr\"odinger cat paradox. We extend the treatment to consider definitions and criteria for mesoscopically entangled states, which are a subset of the states exhibiting mesoscopic quantum coherence. It is shown how the criteria (referred to as type I criteria) can be applied to detect the mesoscopic entanglement of systems prepared in NOON states, Greenberger-Horne-Zeilinger states, and entangled cat states involving coherent states. Here, a larger system CCC is entangled with a second system SSS, which can be small. The proposed definition of mesoscopic (macroscopic) entanglement involves the Schmidt decomposition, and can be applied to systems in a superposition of many states, only some pairs of which are mesoscopically (macroscopically) distinguishable. We prove that a higher-order Hillery-Zubairy entanglement criterion will detect mesoscopic entanglement of this type, and use this to demonstrate the mesoscopic entanglement of the ground state of a two-mode Bose-Einstein condensate. Finally, we explain how a subset of the type I criteria (called type II criteria) are EPR steering inequalities allowing realization of a mesoscopic or macroscopic version of the Einstein-Podolsky-Rosen paradox for macroscopic superposition states. Where the two systems CCC and SSS are spatially separated, we then use results from the literature to point out it is not possible to complete quantum mechanics using hidden-variable theories compatible with the assumption of locality between the two systems. The violation of certain Bell inequalities is sufficient to prove this result, which gives a stricter signature of the Schr\"odinger cat paradox and of mesoscopic quantum coherence. We call such signatures type III criteria.
Quantum weak values arise when the mean outcome of a weak measurement made on certain preselected and postselected quantum systems goes beyond the eigenvalue range for a quantum observable. Here, we propose how to determine quantum weak values for superpositions of states with a macroscopically or mesoscopically distinct mode number, that might be realized as two-mode Bose-Einstein condensate or photonic NOON states. Specifically, we give a model for a weak measurement of the Schwinger spin of a two-mode NOON state, for arbitrary NNN. The weak measurement arises from a nondestructive measurement of the two-mode occupation number difference, which for atomic NOON states might be realized via phase contrast imaging and the ac Stark effect using an optical meter prepared in a coherent state. The meter-system coupling results in an entangled cat-state. By subsequently evolving the system under the action of a nonlinear Josephson Hamiltonian, we show how postselection leads to quantum weak values, for arbitrary NNN. Since the weak measurement can be shown to be minimally invasive, the weak values provide a useful strategy for a Leggett-Garg test of NNN-scopic realism.
Using criteria based on superselection rules that apply to massive bosons but not to photons, we analyze the quantum correlations between the two condensate modes of a Bose-Einstein condensate interferometer [described in Egorov et al., Phys. Rev. A 84, 021605 (2011)]. We show that the observation of interference fringes can only be explained by Einstein-Podolsky-Rosen steering correlations between the two modes. Moreover, the size of the two-mode correlation linked to the fringe visibility will place a lower bound on the mean number of atoms of the (pure) two-mode steerable state. In order to determine the two-mode correlations, we develop a multimode theory describing the dynamics of the condensate atoms and the thermal fraction through the interferometer sequence, in agreement with the experimentally measured fringe visibility. We thus present experimental evidence for two-mode entangled states genuinely involving 40 000 87Rb^{87}\mathrm{Rb}87Rb atoms, and for two-way steerability between two groups of 20 000 indistinguishable atoms.
We show how one can prepare and detect entanglement and Einstein-Podolsky-Rosen (EPR) steering between two distinguishable groups (modes) of atoms in a Bose-Einstein condensate (BEC) atom interferometer. Our paper extends previous work that developed criteria for two-mode entanglement and EPR steering based on the reduced variances of two spins defined in a plane. Observation of planar spin squeezing will imply entanglement, and sufficient planar spin squeezing implies EPR steering, between the two groups of atoms. By using a two-mode dynamical model to describe BEC interferometry experiments, we show that the two-mode entanglement and EPR steering criteria are predicted to be satisfied for realistic parameters. The reported observation of spin squeezing in these parameter regimes suggests it is very likely that the criteria can be used to infer an EPR steering between mesoscopic groups of atoms, provided the total atom number can be determined to sub-Poissonian uncertainty. The criteria also apply to a photonic Mach-Zehnder interferometer. Finally, we give a method based on the amount of planar spin squeezing to determine a lower bound on the number of particles that genuinely comprise the two-mode EPR steerable state---the so-called two-mode EPR steering depth.
Evidence for Bell's nonlocality is so far mainly restricted to microscopic systems, where the elements of reality that are negated predetermine results of measurements to within one spin unit. Any observed nonlocal effect (or lack of classical predetermination) is then limited to no more than the difference of a single photon or electron being detected or not (at a given detector). In this paper, we analyze experiments that report the Einstein-Podolsky-Rosen (EPR) steering form of nonlocality for mesoscopic photonic or Bose-Einstein condensate systems. Using an EPR steering parameter, we show how the EPR nonlocalities involved can be quantified for four-mode states, to give evidence of EPR-nonlocal effects corresponding to a two-mode number difference of 10^{5} photons, or of several tens of atoms (at a given site). Applying to experiments, we also show how the variance criterion of Duan, Giedke, Cirac and Zoller for EPR entanglement can be used to determine a lower bound on the number of particles in a pure two-mode EPR-entangled or steerable state.
We construct Einstein-Podolsky-Rosen (EPR) steering signatures for the nonlocality of the entangled superposition state described by $\frac{1}{\sqrt{2}}{|N\ensuremath{\rangle}|0\ensuremath{\rangle}+|0\ensuremath{\rangle}|N\ensuremath{\rangle}}$, called the two-mode NOON state. The signatures are a violation of an EPR steering inequality based on an uncertainty relation. The violation confirms an EPR steering between the two modes and involves certification of an intermode correlation for number, as well as quadrature phase amplitude measurements. We also explain how the signatures certify an NthN\mathrm{th}Nth order quantum coherence, so the system (for larger NNN) can be signified to be in a superposition of states distinct by a mesoscopic value of quantum number in each mode. Finally, we examine the limitations imposed for lossy scenarios, discussing how experimental realizations may be possible for N=2,3N=2,3N=2,3.