Testing general relativity at the quantum level

It is shown that the effect of a gravitational field on a hydrogen atom is to admix states of opposite parity such as 2S _1/2 and 2P _1/2 The phase of this admixture is such as to produce circular polarization of the radiation emitted in transltions such as 2S _1/21S _1/2+ which arises from the interference between the gravity-induced amplitude and mat due to the weak neutral current. The predicted magnitude of the circular polarization, which could be sufficiently large to be detected in white dwarfs or in certain binary systems, varies from theory to theory. It is thus possible that a study of this effect could provide a feasible means of testing general relativity at the quantum level.This essay received the fourth award from the Gravity Research Foundation for the year 1979-Ed.     

On positive maps in quantum information

Using the Grothendieck approach to the tensor product of locally convex spaces, we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, a generalization of the idea of Choi matrices for genuine quantum systems will be presented.     

Classical decays in decoherent quantum maps

The linear entropy and the Loschmidt echo have proved to be of interest recently in the context of quantum information and of the quantum to classical transitions. We study the asymptotic long-time behavior of these quantities for open quantum maps and relate the decays to the eigenvalues of a coarse-grained superoperator. In specific ranges of coarse graining, and for chaotic maps, these decay rates are given by the Ruelle-Pollicott resonances of the classical map.     

Dephasing maps for quantum stochastic systems

Stochastic time evolution in a nonseparable and nonintegrable quantum system is manifested by rapid dephasing of gaussian wavepackets, whose topological distribution in the coordinate-momentum space defines its irregular regions, while wavepackets initiated in regular regions exhibit quasiperiodic evolution. A gradual transition from quasiperiodic to chaotic dynamics with increasing energy is observed.     

Measurement of the transverse spatial quantum state of light at the single-photon level

We present an experimental method to measure the transverse spatial quantum state of an optical field in coordinate space at the single-photon level. The continuous-variable measurements are made with a photon-counting, parity-inverting Sagnac interferometer based on all-reflecting optics. The technique provides a large numerical aperture without distorting the shape of the wavefront, does not introduce astigmatism, and allows for characterization of fully or partially coherent optical fields at the single-photon level. Measurements of the transverse spatial Wigner functions for highly attenuated coherent beams are presented and compared with theoretical predictions.     

Moment analysis and quantum effects in collision-induced absorption

In this work three moments of collison-induced absorption spectra which include lowest order quantum corrections are used to obtain information about the induced dipole moment and about the role of quantum effects in collision-induced absorption. The two experimental cases discussed are the induced translational band of a He-Ar mixture at 295 K and the induced fundamental band of a pH₂-Ar mixture at 160 K. In both cases the quantum corrections to a classical moment analysis were found to be significantly greater than the experimental error and cannot be ignored. Using several models for the dependence of the induced dipole moment on intermolecular separation, the form of the induced dipole is found for the small range of intermolecular separation (2·5 Å–3·2 Å) from which most of the absorption was found to arise.     

Quantum families of maps and quantum semigroups on finite quantum spaces

Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases we study quantum semigroups of maps preserving a fixed state and quantum commutants of given quantum families of maps.     

General quantum measurements: Local transition maps

On the basis of four physically motivated assumptions, it is shown that a general quantum measurement of commuting observables can be represented by a local transition map, a special type of positive linear map on a von Neumann algebra. In the case that the algebra is the bounded operators on a Hilbert space, these local transition maps share two properties of von Neumann-type measurements: they decrease matrix elements of states and increase their entropy. It is also shown that local transition maps have all the properties of a conditional expectation of a von Neumann algebra onto a subalgebra except that their range is not restricted to the subalgebra. The notion of locality arises from requiring that a quantum measurement may be treated classically when restricted to the commutative algebra generated by the measured observables. The formalism established applies to observables with arbitrary spectrum. In the case that the spectrum is continuous we have only incomplete measurements.     

Relaxation and localization in interacting quantum maps

Quantum relaxation is studied in coupled quantum baker's maps. The classical systems are exactly solvable Kolmogorov systems, for which the exponential decay to equilibrium is known. They model the fundamental processes of transport in classically chaotic phase space. The quantum systems, in the absence of global symmetry, show a marked saturation in the level of transport, as the suppression of diffusion in the quantum kicked rotor, and eigenfunction localization in the position basis. In the presence of a global symmetry we study another model that has classically an identical decay to equilibrium, but-quantally shows resonant transport, no saturation, and large fluctuations around equilibrium. We generalize the quantization to finite multibaker maps. As a byproduct we introduce some simple models of quantal tunneling between classically chaotic regions of phase space.     

Spectral properties of dissipative chaotic quantum maps

I examine spectral properties of a dissipative chaotic quantum map with the help of a recently discovered semiclassical trace formula. I show that in the presence of a small amount of dissipation the traces of any finite power of the propagator of the reduced density matrix, and traces of its classical counterpart, the Frobenius-Perron operator, are identical in the limit of variant Planck's over 2pi -->0. Numerically I find that even for finite variant Planck's over 2pi the agreement can be very good. This holds in particular if the classical phase space contains a strange attractor, as long as one stays clear of bifurcations. Traces of the quantum propagator for iterations of the map agree well with the corresponding traces of the Frobenius-Perron operator if the classical dynamics is dominated by a strong point attractor. (c) 1999 American Institute of Physics.     

Composition maps in self-assembled alloy quantum dots

Nanoscale variations in composition arising from the competition between chemical mixing effects and elastic relaxation can substantially influence the electronic and optical properties of self-assembled alloy quantum dots. Using a combination of finite element and quadratic programming optimization methods, we have developed an efficient technique to compute the equilibrium composition profiles in strained quantum dots. We find that the composition profiles depend strongly on the morphological features such as the slopes and curvatures of their surfaces and the presence of corners and edges as well as the ratio of the strain and chemical mixing energy densities. More generally, our approach provides a means to quantitatively model the interplay among the composition variations, the temperature, the strain, and the shapes of small-scale lattice-mismatched structures.     

On completely positive maps in generalized quantum dynamics

Several authors have hypothesized that completely positive maps should provide the means for generalizing quantum dynamics. In a critical analysis of that proposal, we show that such maps are incompatible with the standard phenomenological theory of spin relaxation and that the theoretical argument which has been offered as justification for the hypothesis is fallacious.     

The classical skeleton of open quantum chaotic maps

We have studied two complementary decoherence measures, purity and fidelity, for a generic diffusive noise in two different chaotic systems (the baker map and the cat map). For both quantities, we have found classical structures in quantum mechanics-the scar functions-that are specially stable when subjected to environmental perturbations. We show that these quantum states constructed on classical invariants are the most robust significant quantum distributions in generic dissipative maps.     

Repeatable procedures and maps in open quantum dynamics

Examples of repeatable procedures and maps are found in the open quantum dynamics of one qubit that interacts with another qubit. They show that a mathematical map that is repeatable can be made by a physical procedure that is not.     

Maps and inverse maps in open quantum dynamics

Two kinds of maps that describe evolution of states of a subsystem coming from dynamics described by a unitary operator for a larger system, maps defined for fixed mean values and maps defined for fixed correlations, are found to be quite different for the same unitary dynamics in the same situation in the larger system. An affine form is used for both kinds of maps to find necessary and sufficient conditions for inverse maps. All the different maps with the same homogeneous part in their affine forms have inverses if and only if the homogeneous part does. Some of these maps are completely positive; others are not, but the homogeneous part is always completely positive. The conditions for an inverse are the same for maps that are not completely positive as for maps that are. For maps defined for fixed mean values, the homogeneous part depends only on the unitary operator for the dynamics of the larger system, not on any state or mean values or correlations. Necessary and sufficient conditions for an inverse are stated several different ways: in terms of the maps of matrices, basis matrices, density matrices, or mean values. The inverse maps are generally not tied to the dynamics the way the maps forward are. A trace-preserving completely positive map that is unital cannot have an inverse that is obtained from any dynamics described by any unitary operator for any states of a larger system.     

The structure of states and maps in quantum theory

The structure of statistical state spaces in the classical and quantum theories are compared in an interesting and novel manner. Quantum state spaces and maps on them have rich convex structures arising from the superposition principle and consequent entanglement. Communication channels (physical processes) in the quantum scheme of things are in one-to-one correspondence with completely positive maps. Positive maps which are not completely positive do not correspond to physical processes. Nevertheless they prove to be invaluable mathematical tools in establishing or witnessing entanglement of mixed states. We consider some of the recent developments in our understanding of the convex structure of states and maps in quantum theory, particularly in the context of quantum information theory.     

Measuring the energy level repulsion in quantum dots

Energy level repulsion is one of the remnants of classical chaos in quantum mechanics. Measurements of the distribution of nearest neighbor spacings in quantum dots reveal, in contrast to other classically chaotic systems, deviations from the predictions made by random matrix theory. Here, we survey possible contributions to these deviations from experimental peculiarities present in measurements on quantum dots, and discuss the methods to eliminate or reduce such distortions.