Sub-Planck spots of Schroedinger cats and quantum decoherence

Heisenberg's principle$^1$ states that the product of uncertainties of position and momentum should be no less than Planck's constant $\hbar$. This is usually taken to imply that phase space structures associated with sub-Planck ($\ll \hbar$) scales do not exist, or, at the very least, that they do not matter. I show that this deeply ingrained prejudice is false: Non-local "Schr\"odinger cat" states of quantum systems confined to phase space volume characterized by `the classical action' $A \gg \hbar$ develop spotty structure on scales corresponding to sub-Planck $a = \hbar^2 / A \ll \hbar$. Such structures arise especially quickly in quantum versions of classically chaotic systems (such as gases, modelled by chaotic scattering of molecules), that are driven into nonlocal Schr\"odinger cat -- like superpositions by the quantum manifestations of the exponential sensitivity to perturbations$^2$. Most importantly, these sub-Planck scales are physically significant: $a$ determines sensitivity of a quantum system (or of a quantum environment) to perturbations. Therefore sub-Planck $a$ controls the effectiveness of decoherence and einselection caused by the environment$^{3-8}$. It may also be relevant in setting limits on sensitivity of Schr\"odinger cats used as detectors.Comment: Published in Nature 412, 712-717 (2001

Topological Schr\"odinger cats: Non-local quantum superpositions of topological defects

Topological defects (such as monopoles, vortex lines, or domain walls) mark locations where disparate choices of a broken symmetry vacuum elsewhere in the system lead to irreconcilable differences. They are energetically costly (the energy density in their core reaches that of the prior symmetric vacuum) but topologically stable (the whole manifold would have to be rearranged to get rid of the defect). We show how, in a paradigmatic model of a quantum phase transition, a topological defect can be put in a non-local superposition, so that - in a region large compared to the size of its core - the order parameter of the system is "undecided" by being in a quantum superposition of conflicting choices of the broken symmetry. We demonstrate how to exhibit such a "Schr\"odinger kink" by devising a version of a double-slit experiment suitable for topological defects. Coherence detectable in such experiments will be suppressed as a consequence of interaction with the environment. We analyze environment-induced decoherence and discuss its role in symmetry breaking.Comment: 7 pages, 4 figure

Decoherence from a Chaotic Environment: An Upside Down "Oscillator" as a Model

Chaotic evolutions exhibit exponential sensitivity to initial conditions. This suggests that even very small perturbations resulting from weak coupling of a quantum chaotic environment to the position of a system whose state is a non-local superposition will lead to rapid decoherence. However, it is also known that quantum counterparts of classically chaotic systems lose exponential sensitivity to initial conditions, so this expectation of enhanced decoherence is by no means obvious. We analyze decoherence due to a "toy" quantum environment that is analytically solvable, yet displays the crucial phenomenon of exponential sensitivity to perturbations. We show that such an environment, with a single degree of freedom, can be far more effective at destroying quantum coherence than a heat bath with infinitely many degrees of freedom. This also means that the standard "quantum Brownian motion" model for a decohering environment may not be as universally applicable as it once was conjectured to be.Comment: RevTeX, 29 pages, 5 EPS figures. Substantially rewritten analysis, improved figures, additional references, and errors fixed. Final version (to appear in PRA

Fragility of a class of highly entangled states of many quantum-bits

We consider a Quantum Computer with n quantum-bits (`qubits'), where each qubit is coupled independently to an environment affecting the state in a dephasing or depolarizing way. For mixed states we suggest a quantification for the property of showing {\it quantum} uncertainty on the macroscopic level. We illustrate in which sense a large parameter can be seen as an indicator for large entanglement and give hypersurfaces enclosing the set of separable states. Using methods of the classical theory of maximum likelihood estimation we prove that this parameter is decreasing with 1/\sqrt{n} for all those states which have been exposed to the environment. Furthermore we consider a Quantum Computer with perfect 1-qubit gates and 2-qubit gates with depolarizing error and show that any state which can be obtained from a separable initial state lies inbetween a family of pairs of certain hypersurfaces parallel to those enclosing the separable ones.Comment: 9 Pages, RevTe

What is "system": some decoherence-theory arguments

We discuss the possibility of making the {\it initial} definitions of mutually different (possibly interacting, or even entangled) systems in the context of decoherence theory. We point out relativity of the concept of elementary physical system as well as point out complementarity of the different possible divisions of a composite system into "subsystems", thus eventually sharpening the issue of 'what is system'.Comment: 9 pages, no figure

Quantum Chaotic Environments, The Butterfly Effect, And Decoherence

We investigate the sensitivity of quantum systems that are chaotic in a classical limit, to small perturbations of their equations of motion. This sensitivity, originally studied in the context of defining quantum chaos, is relevant to decoherence in situations when the environment has a chaotic classical counterpart.Comment: 4 pages, 3 figure

The Chandra Fornax Survey - I: The Cluster Environment

We present the first results of a deep Chandra survey of the inner 1 degree of the Fornax cluster of galaxies. Ten 50 ksec pointings were obtained in a mosaic centered on the giant elliptical galaxy NGC 1399 at the nominal cluster center. Emission and temperature maps of Fornax are presented, and an initial study of 771 detected X-ray point sources is made. Regions as small as 100pc are resolved. The intra-cluster gas in Fornax exhibits a highly asymmetric morphology and temperature structure, dominated by a 180 kpc extended ``plume'' of low surface brightness, cool, ~1 keV) gas to the North-East of NGC 1399 with a sharper edge to the South West. The elliptical galaxy NGC 1404 also exhibits a cool halo of X-ray gas within the cluster, with a highly sharpened leading edge as it presumably falls into the cluster, and a cometary-like tail. We estimate that some ~200-400 point sources are physically associated with Fornax. Confirming earlier works, we find that the globular cluster population in NGC 1399 is highly X-ray active, extending to globulars which may in fact be intra-cluster systems. We have also found a remarkable correlation between the location of giant and dwarf cluster galaxies and the presence of X-ray counterparts, such that systems inhabiting regions of low gas density are more likely to show X-ray activity. Not only does this correlate with the asymmetry of the intra-cluster gas but also with the axis joining the center of Fornax to an infalling group 1 Mpc to the South-West. We suggest that Fornax may be experiencing an intergalactic ``headwind'' due to motion relative to the surrounding large-scale structure.Comment: 35 pages, 15 figures, submitted to ApJ. Most figures not included owing to severe compression degradation - we strongly recommend downloading the full resolution paper from http://www.astro.columbia.edu/~caleb/ms_highres.pdf (1.9Mb

Constraints on Scalar Asymmetric Dark Matter from Black Hole Formation in Neutron Stars

We consider possibly observable effects of asymmetric dark matter (ADM) in neutron stars. Since dark matter does not self-annihilate in the ADM scenario, dark matter accumulates in neutron stars, eventually reaching the Chandrasekhar limit and forming a black hole. We focus on the case of scalar ADM, where the constraints from Bose-Einstein condensation and subsequent black hole formation are most severe due to the absence of Fermi degeneracy pressure. We also note that in some portions of this constrained parameter space, non-trivial effects from Hawking radiation can modify our limits. We find that for scalar ADM with mass between 100 keV and 10^5 GeV, the constraint from pulsars in globular clusters on the scattering cross-section with neutrons ranges from \sigma_n < 10^{-45} cm^2 to 10^{-52} cm}^2. In particular, for scalar ADM with mass between 1 GeV and 1 TeV (in the case where black hole evaporation due to Hawking radiation is unimportant), the constraint on the scattering cross-section is below what is reachable with ton scale direct detection experiments.Comment: 24 pages, 5 figures; updated to match published versio