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arXiv:0806.0457 · Physical Review A

Criteria for generalized macroscopic and mesoscopic quantum coherence

Eric G. Cavalcanti, M. D. Reid

Published Jun 18, 2008• 33 citations
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Abstract

We consider macroscopic, mesoscopic, and ``$S$-scopic'' quantum superpositions of eigenstates of an observable and develop some signatures for their existence. We define the extent, or size $S$ of a superposition, with respect to an observable $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$, as being the range of outcomes of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ predicted by that superposition. Such superpositions are referred to as generalized $S$-scopic superpositions to distinguish them from the extreme superpositions that superpose only the two states that have a difference $S$ in their prediction for the observable. We also consider generalized $S$-scopic superpositions of coherent states. We explore the constraints that are placed on the statistics if we suppose a system to be described by mixtures of superpositions that are restricted in size. In this way we arrive at experimental criteria that are sufficient to deduce the existence of a generalized $S$-scopic superposition. The signatures developed are useful where one is able to demonstrate a degree of squeezing. We also discuss how the signatures enable a new type of Einstein-Podolsky-Rosen gedanken experiment.

DOI
10.1103/physreva.77.062108

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We consider macroscopic, mesoscopic, and ``SSS-scopic'' quantum superpositions of eigenstates of an observable and develop some signatures for their existence. We define the extent, or size SSS of a superposition, with respect to an observable x\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}x\ifmmode^\elseˆ\fi, as being the range of outcomes of x\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}x\ifmmode^\elseˆ\fi predicted by that superposition. Such superpositions are referred to as generalized SSS-scopic superpositions to distinguish them from the extreme superpositions that superpose only the two states that have a difference SSS in their prediction for the observable. We also consider generalized SSS-scopic superpositions of coherent states. We explore the constraints that are placed on the statistics if we suppose a system to be described by mixtures of superpositions that are restricted in size. In this way we arrive at experimental criteria that are sufficient to deduce the existence of a generalized SSS-scopic superposition. The signatures developed are useful where one is able to demonstrate a degree of squeezing. We also discuss how the signatures enable a new type of Einstein-Podolsky-Rosen gedanken experiment.
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.
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 $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}\ensuremath{\ne}0$ (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 nnn-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 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 ${\ensuremath{\xi}}_{N}$. 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⪆10n⪆10n⪆10 atoms.
In recent years, there has been an increased interest in the generation of superposition of coherent states with opposite phases, the so-called photonic Schrodinger-cat states. These experiments are very challenging and so far, cats involving small photon numbers only have been implemented. Here, we propose to consider two-mode squeezed states as examples of a Schrodinger-cat-like state. In particular, we are interested in several criteria aiming to identify quantum states that are macroscopic superpositions in a more general sense. We show how these criteria can be extended to continuous variable entangled states. We apply them to various squeezed states, argue that two-mode squeezed vacuum states belong to a class of general Schrodinger-cat states and compare the size of states obtained in several experiments. Our results not only promote two-mode squeezed states for exploring quantum effects at the macroscopic level but also provide direct measures to evaluate their usefulness for quantum metrology.
We consider fundamental limits on the detectable size of macroscopic quantum superpositions. We argue that a full quantum mechanical treatment of system plus measurement device is required, and that a (classical) reference frame for phase or direction needs to be established to certify the quantum state. When taking the size of such a classical reference frame into account, we show that to reliably distinguish a quantum superposition state from an incoherent mixture requires a measurement device that is quadratically bigger than the superposition state. Whereas for moderate system sizes such as generated in previous experiments this is not a stringent restriction, for macroscopic superpositions of the size of a cat the required effort quickly becomes intractable, requiring measurement devices of the size of the Earth. We illustrate our results using macroscopic superposition states of photons, spins, and position. Finally, we also show how this limitation can be circumvented by dealing with superpositions in relative degrees of freedom.