An Adiabatic Theorem without a Gap Condition

The basic adiabatic theorems of classical and quantum mechanics are over-viewed and an adiabatic theorem in quantum mechanics without a gap condition is described.Comment: Talk at QMath 7, Prague, 1998. 10 pages, 7 figure

The expression pattern of MUC1 (EMA) is related to tumour characteristics and clinical outcome of invasive ductal breast carcinoma

Aims: To clarify MUC1 patterns in invasive ductal breast carcinoma and to relate them to clinicopathological parameters, coexpression of other biological markers and prognosis. Methods and results: Samples from 243 consecutive patients with primary ductal carcinoma were incorporated into tissue microarrays (TMAs). Slides were stained for MUC1, oestrogen receptor (ER), progesterone receptor (PR), Her2/neu, p53 and cyclin D1. Apical membrane MUC1 expression was associated with smaller tumours (P = 0.001), lower tumour grades (P < 0.001), PR positivity (P = 0.003) and increased overall survival (OS; P = 0.030). Diffuse cytoplasmic MUC1 expression was associated with cyclin D1 positivity (P = 0.009) and increased relapse-free survival (RFS; P = 0.034). Negativity for MUC1 was associated with ER negativity (P = 0.004), PR negativity (P = 0.001) and cyclin D1 negativity (P = 0.009). In stepwise multivariate analysis MUC1 negativity was an independent predictor of both RFS [hazard ratio (HR) 3.5, 95% confidence interval (CI) 1.5, 8.5; P = 0.005] and OS (HR 14.7, 9 5% Cl 4.9, 44. 1; P < 0.001). Conclusions: The expression pattern of MUC1 in invasive ductal breast carcinoma is related to tumour characteristics and clinical outcome. In addition, negative MUC1 expression is an independent risk factor for poor RFS and OS, besides 'classical' prognostic indicators

Remarks on the method of comparison equations (generalized WKB method) and the generalized Ermakov-Pinney equation

The connection between the method of comparison equations (generalized WKB method) and the Ermakov-Pinney equation is established. A perturbative scheme of solution of the generalized Ermakov-Pinney equation is developed and is applied to the construction of perturbative series for second-order differential equations with and without turning points.Comment: The collective of the authors is enlarged and the calculations in Sec. 3 are correcte

Hyperasymptotic solutions for certain partial differential equations

We present the hyperasymptotic expansions for a certain group of solutions of the heat equation. We extend this result to a more general case of linear PDEs with constant coefficients. The generalisation is based on the method of Borel summability, which allows us to find integral representations of solutions for such PDEs.Comment: 17 page

Longitudinal Atomic Beam Spin Echo Experiments: A possible way to study Parity Violation in Hydrogen

We discuss the propagation of hydrogen atoms in static electric and magnetic fields in a longitudinal atomic beam spin echo (lABSE) apparatus. Depending on the choice of the external fields the atoms may acquire both dynamical and geometrical quantum mechanical phases. As an example of the former, we show first in-beam spin rotation measurements on atomic hydrogen, which are in excellent agreement with theory. Additional calculations of the behaviour of the metastable 2S states of hydrogen reveal that the geometrical phases may exhibit the signature of parity-(P-)violation. This invites for possible future lABSE experiments, focusing on P-violating geometrical phases in the lightest of all atoms.Comment: 6 pages, 4 figure

Ray and wave chaos in asymmetric resonant optical cavities

Optical resonators are essential components of lasers and other wavelength-sensitive optical devices. A resonator is characterized by a set of modes, each with a resonant frequency omega and resonance width Delta omega=1/tau, where tau is the lifetime of a photon in the mode. In a cylindrical or spherical dielectric resonator, extremely long-lived resonances are due to `whispering gallery' modes in which light circulates around the perimeter trapped by total internal reflection. These resonators emit light isotropically. Recently, a new category of asymmetric resonant cavities (ARCs) has been proposed in which substantial shape deformation leads to partially chaotic ray dynamics. This has been predicted to give rise to a universal, frequency-independent broadening of the whispering-gallery resonances, and highly anisotropic emission. Here we present solutions of the wave equation for ARCs which confirm many aspects of the earlier ray-optics model, but also reveal interesting frequency-dependent effects characteristic of quantum chaos. For small deformations the lifetime is controlled by evanescent leakage, the optical analogue of quantum tunneling. We find that the lifetime is much shortened by a process known as `chaos-assisted tunneling'. In contrast, for large deformations (~10%) some resonances are found to have longer lifetimes than predicted by the ray chaos model due to `dynamical localization'.Comment: 4 pages RevTeX with 7 Postscript figure

High-fidelity quantum driving

The ability to accurately control a quantum system is a fundamental requirement in many areas of modern science such as quantum information processing and the coherent manipulation of molecular systems. It is usually necessary to realize these quantum manipulations in the shortest possible time in order to minimize decoherence, and with a large stability against fluctuations of the control parameters. While optimizing a protocol for speed leads to a natural lower bound in the form of the quantum speed limit rooted in the Heisenberg uncertainty principle, stability against parameter variations typically requires adiabatic following of the system. The ultimate goal in quantum control is to prepare a desired state with 100% fidelity. Here we experimentally implement optimal control schemes that achieve nearly perfect fidelity for a two-level quantum system realized with Bose-Einstein condensates in optical lattices. By suitably tailoring the time-dependence of the system's parameters, we transform an initial quantum state into a desired final state through a short-cut protocol reaching the maximum speed compatible with the laws of quantum mechanics. In the opposite limit we implement the recently proposed transitionless superadiabatic protocols, in which the system perfectly follows the instantaneous adiabatic ground state. We demonstrate that superadiabatic protocols are extremely robust against parameter variations, making them useful for practical applications.Comment: 17 pages, 4 figure

Atomic Physics: Neutral atoms put in charge

An elegant experiment shows that atoms subjected to a pair of laser beams can behave like electrons in a magnetic field, as demonstrated by the appearance of quantized vortices in a neutral superfluid

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

Geometric quantum computation with NMR

The experimental realisation of the basic constituents of quantum information processing devices, namely fault-tolerant quantum logic gates, requires conditional quantum dynamics, in which one subsystem undergoes a coherent evolution that depends on the quantum state of another subsystem. In particular, the subsystem may acquire a conditional phase shift. Here we consider a novel scenario in which this phase is of geometric rather than dynamical origin. As the conditional geometric (Berry) phase depends only on the geometry of the path executed it is resilient to certain types of errors, and offers the potential of an intrinsically fault-tolerant way of performing quantum gates. Nuclear Magnetic Resonance (NMR) has already been used to demonstrate both simple quantum information processing and Berry's phase. Here we report an NMR experiment which implements a conditional Berry phase, and thus a controlled phase shift gate. This constitutes the first elementary geometric quantum computation.Comment: Minor additions at request of referees. 4 pages revtex including 2 figures (1 eps). Nature in pres