Academic Year 2010

It was recently shown how to implement a quantum probabilistic error correction protocol [1] in a solid state qubit [2,3] and in a photonic qubit [4]. In this talk it is shown how one can implement a quantum probabilistic error correction protocol in an open quantum system consisting of a single atom, with ground- and excited-state Zeeman structure, in a driven two-mode optical cavity. The ground-state superposition is manipulated and controlled through conditional measurements and external fields, which shield the coherence and correct quantum errors. Modeling an experimentally realistic situation demonstrates the robustness of the proposal for realization in the laboratory.

[1] Koashi M and Ueda M 1999 Reversing measurement and probabilistic quantum error correction Phys. Rev. Lett. 82 2598
[2] Korotkov A N and Jordan A N 2006 Undoing a weak quantum measurement of a solid-state qubit Phys. Rev. Lett. 97 166805
[3] Katz N et al 2008 Reversal of the weak measurement of a quantum state in a superconducting phase qubit Phys. Rev. Lett. 101 200401
[4] Kim Y-S, Cho Y-W, Ra Y-S and Kim Y-H 2009 Reversing the weak quantum measurement for a photonic qubit Opt. Express 17 11978–85

The consideration of an extended spin space aims to understand physical degrees of freedom from the most elementary, q-bit like ones, and unify symmetries, as does the Kaluza-Klein idea. We show that fermion and boson field equations, and the standard model in particular, can be equivalently formulated in an extended spin space, reproducing the required Lorentz, gauge and flavor symmetries. In turn, such an extended space puts restrictions on the interactions and the representations that match standard-model fields. Information on coupling constants is also provided.

Using a variational wave function constituted by the tensorial product of SU(2) and Weyl coherent states we determine the ground state and the phase transitions from the normal to the super-radiant regime of matter interacting with a one mode electromagnetic field described by the Dicke Hamiltonian. This Hamiltonain is invariant under a unitary transformation of the form U = exp ( i  \pi  \lambda) with \lambda denoting the excitation number operator. By restoring this Hamitlonian symmetry we get analytical expressions for the ground and first excited states of the Dicke model. The fitness between  the restored variational states and the numerical exact solutions is measured through the evaluation of the fidelity parameter; this is very close to $1$ except for the parameter values of the Dicke Hamiltonian close to the phase transition.

Quantum transients are temporary features of matter waves before they reach a stationary regime. Transients may arise after the preparation of an unstable initial state or due to a sudden interaction or a change in the boundary conditions. Examples are diffraction in time, buildup processes, decay, trapping, forerunners or pulse formation, as well as other phenomena recently scovered, such as the simultaneous arrival of a wave peak at arbitrarily distant observers. The interest on these transients is nowadays enhanced by new technological possibilities to control, manipulate and measure matter waves. We shall review the latest developments on the impact of quantum transients in ultracold atom physics, ranging from atom lasers to soliton dynamics.

We analyze, both from the numerical and experimental points of view, different configurations of elastic one-dimensional rods. A rod showing approximate translational symmetry is built by accumulating unit cells formed by small identical rods. A band spectrum emerges. The symmetry is then broken by introducing defects, producing localized states. Eventually, a disordered rod is achieved and the Anderson transition emerges. It is shown that the localization length is a linear function of the Brody repulsion parameter. The measurements are performed using an electromagnetic acoustic transducer which we have developed, and the calculations are carried out with the transfer matrix method.

The $H_3$ integrable system is $3D$ quantum system with rational potential related to the non-crystallographic root system $H_3$. It is isospectral to the $3D$ harmonic oscillator. It is shown that the gauge-rotated $H_3$ Hamiltonian, when written in terms of the variants of the Coxeter group $H_3$, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional variant subspaces in polynomials, they form the infinite flag with the characteristic vector $\vec \al\ =\ (1,2,3)$. A hidden algebra of the $H_3$ model is found. It is infinite-dimensional, finitely-generated algebra of differential operators possessing the finite-dimensional representations. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model in a space of invariants isospectral to the quantum $H_3$ rational model is defined.

We consider the decoherence of a finite quantum system under the coupling to an environment modeled by random matrix theory. In doing so, we assume that the Heisenberg time of the environment is of the same order as the decoherence time, as might be the case when only a few environmental degrees of freedom are involved in the coupling. The state of the central system is obtained from the unitary evolution of central system and environment after tracing out the latter. We consider decoherence measured by purity as well as entanglement decay measured by the concurrence. Analytic results are obtained in the linear response approximation. We show that the evolution of the central system becomes Markovian when the environmental Heisenberg time becomes small.

Over the past decade, digital holography has received a great deal of attention in the literature. Digital holography uses a digital camera in place of the films or plates of classical holography. The captured wave field is most often reconstructed numerically, most commonly using FFT-based reconstruction algorithms. The relationship of these numerical algorithms with the linear canonical transforms as a propagation model has explored recently, particularly from the perspective of sampling requirements. We present an overview of this work.

The structure of QCD at low energy is discussed, using a Hamiltonian justified by QCD at low energy. The Coulomb interaction term is approximated by an average potential and gluons are only taken into account via a static potential. We show that including 2 and 3 orbital states an analytic solution can be found, given by a SU(2) structure. Including more levels requires two successive unitary transformations, which leads again to an analytic solution for an arbitrary number of orbital levels. Future actions, like including gluons dynamically, are discussed.

The geometrical and topological structure of the chaotic set of a Hamiltonian scattering system with 3 degrees of freedom is presented. We start from a system with a symmetry and a corresponding conserved quantity which correspondingly can be reduced to a 2 degree of freedon system. Here the good knowledge of the structure of chaotic sets in 2 degree of freedom Hamiltonian systems is used to understand the higher dimensional chaotic set of the full 3 degree of freedom system. Finally the symmetry is broken and it is observed how the symmetry breaking modifies the chaotic set.

The vacuum (Casimir) energy in quantum field theory is a problem relevant both to new nanotechnology devices and to dark energy in cosmology. The crucial question is the dependence of the energy on the system geometry under study. Despite much progress since the first prediction of the Casimir effect in 1948 and its subsequent experimental verification in simple geometries, even the sign of the force in nontrivial situations is still a matter of controversy. Mathematically, vacuum energy fits squarely into the spectral theory of second-order self-adjoint elliptic linear differential operators. Specifically one promising approach is based on the small-t asymptotics of the cylinder kernel exp(-t sqrt(H)), H being the self-adjoint operator in question. In contrast with the well-studied heat kernel exp(-t H), the cylinder kernel depends in a non-local way on the geometry of the problem. We discuss some recent results by the Louisiana-Texas-Oklahoma collaboration on vacuum energy in model systems, including quantum graphs and two-dimensional cavities. The results may shed light on general questions, including the relationship between vacuum energy and periodic or closed classical orbits, and the ontribution to vacuum energy of boundaries, edges, and corners.

We propose a systematic group-theoretical procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n-qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.

Despite intensive studies of quantum dots in magnetic fields, the few-body dynamics of Coulomb interacting electrons WITHOUT exterior confinement in a magnetic field poses a challenge for theoretical methods. The ground state is an unusual state of infinite angular momentum and the dynamics of the system reveals a highly correlated dancing pattern which can be understood to some extend by using semiclassical methods. I present quantum mechanical and classical calculations of the time-dependent evolution of the system and compare them with results from the time-dependent variational principle, which build a bridge between the quantum and classical approach. The many-body results are obtained using high-performing Graphics Processing Units (GPUs).

In this work the analysis of the equivalent rotations derived from the permutation inversion group formalism is revisited. We stress that  when dealing with the permutation inversion group formalism, the changes in the Euler angles are not required in order to determine the transformation that a given symmetry operation causes to the rotational functions. Indeed, as it is known,  the matrix elements involving the equivalent rotations are provided by a single Wigner's ${\bf D}^{(j)}{(R)}$ function. This analysis is used to propose a  symmetry projection approach to the rovibrational functions of methane, where the relevance of the anti-isomorphism between the permutations and the equivalent rotations is clearly manifested.  In our method the symmetry adapted functions are obtained by the diagonalization of a set of commuting operators, whose representation is given in terms of direct products of Wigner's D functions and vibrational matrix representations provided by a local scheme. The proposed approach is general and permits to obtain an orthonormal set of symmetry projected functions with good total angular momentum carrying the irreps of the molecular symmetry group in  systematic fashion.

Imposing the existence of a generalized symmetry for equations defined on a square lattice we obtain an integrability test. We use this test to prove the integrability and classify multilinear partial difference equations.

The umbral calculus is an old mathematical tool that began its first steps in the XVII century [1]. Since the second half of XIX [2, 3] it was systematically applied although it is in the second half of XX [4, 5] when a formal theory was established. Recently it has been used to provide discrete representations of canonical commutation relations, like [x, dx] = 1, [7]-[12]. This approach can be used to map equations and their solutions from a (continuous) framework to another (discrete) one. This umbral map preserves the point symmetries of the equations but, in general, new symmetries may appear originating a different behavior in some cases [13]-[14]. An umbral version of the orthogonal polynomials is presented. The umbral counterpart of the classical relations, that determine the polyniomials, is obtained [6].

In the probability representation of quantum mechanics, the are described by tomographic-probability-distribution functions as primary notions of the states alternatively to the wave unctions or density matrices. For continuous photon quadratures, the photon quantum states in the probability representation are identified with symplectic (or optical) tomograms directly measured by homodyne photon detectors. Since the tomograms are standard probability distributions, the notions of conventional probability theory like Shannon and Renyi entropies are associated with the quantum states as their new informational characteristics. The entropies are shown to obey new entropic uncertainty relations providing the integral bounds on the measured optical tomograms depending on the local oscillator phase. The bounds do not exist in classical domain of the electromagnetic-field states. In view of this, the tomographic-entropy bounds are suggested as a quantumness test to be checked in the experiments with homodyne detection of the photon states as well as a test of the accuracy of the experiments with homodyne detectors.

In this talk, I will introduce a new family of Hamiltonians with a deformed Kepler-Coulomb potential dependent on an indexing parameter $k$ which is related via coupling constant metamorphosis, to a family of superintegrable deformations of the harmonic oscillator given by Tremblay, Turbiner and Winternitz. I will discuss the superintegrability of the system for different choices of the parameter $k$ and also more generally, the coupling constant metamorphosis acting on potentials with a Kepler-Coulomb term. This is joint work with Pavel Winternitz.

Is it possible to trap a quantum particle in an open geometry? In this work, we deal with the boundary value problem of the Helmholtz equation within a waveguide with straight segments and a rectangular bending. The problem can be reduced to a one-dimensional matrix Schroedinger equation by means of conformal coordinates covering the interior of the guide. We use a corner-corrected WKB formalism to find the energies of the one-dimensional problem. It is shown that purely ondulatory effects are responsible for the presence of bound states at very low energies.

For physical systems where the Hamiltonian is no longer a constant of motion, like oscillators with time-dependent frequency or dissipative systems, a dynamical invariant of the so-called Ermakov type might still exist. This invariant depends on the classical position and velocity and an auxiliary variable that is proportional to position uncertainty. It is possible to construct this invariant via an algebraic method based on the classical Poisson bracket. A modified version for dissipative systems that also employs anti-Poisson brackets and is related to a description in an expanding coordinate system will be presented. The resulting invariant is identical with that obtained from a logarithmic nonlinear Schrödinger equation. The uncertainties of position and momentum, expressed in terms of the auxiliary variable, can be compared with the so-called moment-method for the description of the time-evolution of Bose-Einstein condensates, corresponding to a description in terms of the cubic nonlinear Gross-Pitaevskii equation. It will be shown how this method can be extended to also include dissipative damping effects into the Bose-Einstein dynamics.

The framework of third quantization – canonical quantization in theLiouville space -is discussed for open many-body bosonic systems, including some previous work by one of us on fermionic systems.References:T.H. Seligman and T. ProsenarXiv:1007.2921 T. Prosen2008 New J. Phys. 10 043026 

The Dirac (Moshinsky) oscillator is an elegant example of an exactly solvable quantum relativistic model. In this work we extend it by considering its coupling with an external (isospin) field and find the conditions that maintain the solvability. We then specialize in the 1+1 dimension case and evaluate the entanglement of the oscillator with the field. It is the shown that this model is related to the two-particle Tavis-Cummings model in quantum optics.

Overview of the rational potentials emerging in the Hamiltonian Reduction Method is given. All of them are the isospectral deformations of the harmonic oscillator with preservation of the property of its superintegrability. It is presented a list of discrete models defined on different lattices which are isospectral to the harmonic oscillator being also superintegrable. It is indicated a set of quantum canonical transformations which allow us to relate quantum (continuous) systems with discrete ones.

Seniority in the structure of nuclei refers to the number of nucleons that are not in pairs coupled to angular momentum J=0, and therefore it probes the most important two-body correlation within nuclei, “pairing”. Racah first introduced seniority in 1943 for the classification of complex atomic spectra and adapted it a few years later in the context of nuclear physics. Two key developments subsequent to Racah’s original idea are: the treatment of neutrons and protons and the treatment of nucleons in several non-degenerate orbits. The conditions for seniority conservation will be reviewed and compared to those necessary for the full integrability of a system of interacting particles. The more recent possibility of “partial” seniority will be explored, where most states are of mixed seniority but some remain pure. This explains the occurrence of nuclear seniority isomers, characterized by electromagnetic decay hindered by selection rules related to seniority. Finally, the relevance of seniority as a generic concept will be illustrated with an application to Bose-Einstein condensates.

The seminal paper of Aharonov and Bohm [Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959) 485-491 ] is at the origin of a very extensive literature in some of the more fundamental issues in physics. They claimed that electromagnetic fields can act at a distance on charged particles even if they are identically zero in the region of space where the particles propagate, that the fundamental electromagnetic quantities in quantum physics are not only the electromagnetic fields but also the circulations of the electromagnetic potentials; what gives them a real physical significance. They proposed two experiments to verify their theoretical conclusions. The magnetic Aharonov-Bohm effect, where an electron is influenced by a magnetic field that is zero in the region of space accessible to the electron, and the electric Aharonov-Bohm effect where an electron is affected by a time-dependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its tra jectory. The Aharonov-Bohm effects effect imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue after more than fifty years, on spite of the fact that its is discussed in most of the text books in quantum mechanics. The existence of the electric Ahronov-Bohm effect, that has not been confirmed experimentally, is a very controversial issue. In their 1959 paper Aharonov and Bohm proposed an Ansatz for the solution to the Schr ̈odinger equation in regions where there is a time-dependent electric potential that is constant in space. It consists in multiplying the free evolution by a phase given by the integral in time of the potential. The validity of this Ansatz predicts interference fringes between parts of a coherent electron beam that are sub jected to different potentials. In this talk we give the first rigorous proof that the exact solution to the Schr ̈odinger equation is given by the Aharonov-Bohm Ansatz, up to an error bound in norm that is uniform in time and that decays as a constant divided by the velocity. Our results give, for the first time, a rigorous proof that quantum mechanics predicts the existence of the electric Aharonov-Bohm effect, under conditions that we provide. We hope that our results will estimulate the experimental research on the electric Aharonov-Bohm effect.

One of the many fields to which Marcos Moshinsky made a fundamental contribution is that of accidental degeneracy and canonical transformations. Here we shall give a review of that field now usually referred to as "Superintegrability". We shall mainly concentrate on conceptual questions and on recent developments. These include the construction of infinite families of superintegrable systems with integrals of motion of arbitrary order, the relation between superintegrability and supersymmetry , and the construction of polynomial algebras of integrals of motion.

We show, how the concept of doorways states carries beyond the typical applica- tions and the usual concepts of doorways. We further increase the scale on which it may occur to large classical wave systems. Specically we analyze the seismic response of sedimentary basins covered by water-logged clays, a rather common sit- uation for urban sites. A model is introduced in which the doorway state is a plane wave propagating in the interface between the sediments and the clay. This wave is produced by the coupling of a Rayleigh and an evanescent SP-wave. This in turn leads to a strong resonant response in the soft clays near the surface of the basin. Our model calculations are compared with measurements during Mexico City earthquakes, showing quite good agreement. This not only provides a transparent explanation of catastrophic resonant seismic response in certain basins but at the same time constitutes up to this date the largest scale example of doorway mecha- nisms in wave scattering. Furthermore the doorway state itself has interesting and rather unusual characteristics.

We study the fidelity decay in the $k$-body embedded ensembles of random matricesfor many bosons distributed over two single-particle states, where time-reversal symmetry holds, i.e., $k$-body Embedded Ensemble of Random Matrices for bosons with $\beta=1$. The un-perturbed Hamiltonian consists of the diagonal elements of a fixed member of the ensemble, while the perturbation contains all the off-diagonal elements. This situation mimics the typical mean-field basis used in several calculations. We find that the ensemble-averaged fidelity as well as the fidelity of typical members with respect to an initial random state displays the typical revival of period 1 Heisenberg time of the harmonic oscillator, and a freeze in the fidelity decay. The scaling properties are obtained with respect to the number of bosons and the perturbation strength. For certain members of the ensemble we show that the revival occurs at earlier fractional times of the Heisenberg time, which is related to certain specific $k$ body interactions. The case $k=2$ is presented in detail.

One of the most important science contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis à vis the standard Schrödinger equation being the only one potential model with energy spectra equally spaced. At present, in modern physics, spatially-dependent mass Schrödinger equation (SDMSE) appears in the study of the dynamics of charge carriers (electrons and/or holes) in semiconductor heterostructures and/or inhomogeneous crystals. Consequently, the search of exact and quasi-exact solutions to the SDMSE for various potentials with a specific position-dependent mass distribution (PDMD) has attracted considerable attention during the past few years. Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the harmonic oscillator potential model for the SDMSE. To that purpose, we propose an algorithm based on the point canonical transformation method to convert a general second order differential equation, of Sturm-Liouville type, into a Schrödinger-like standard equation. In that case, depending on the choice of PDMD, the eigenfunctions are not more proportional to Hermite polynomials and the energy spectra is not more equally spaced. For example, for the PDMD m(x)=m0(α2x2+1)-2 the wavefunctions are given in terms of Jacobi polynomials. Besides, by considering the standard Schrodinger equation with the harmonic oscillator potential model we obtain those pairs PDMD-effective potential, allowing harmonic oscillator eigenvalues. i.e. isospectral partners.

The ability of metal adsorption to transfer charge to the surface of single molecular carbon sheets is explored in this paper. Though other metals are considered we basically will deal with Lithium. We concentrate on fairly small sheets and examined the minimum threshold size of a molecular surface needed to separate metals. From our quantum chemical calculations we deduce that a molecular surface of six benzene rings is needed for Lithium dimers to be separated. We further observe symmetry breaking, when two lithium atoms are adsorbed right opposite to each other on the two sides of the sheet.

We examine the evolution in phase space of an N -point signal, produced and sensed at finite arrays transverse to a planar waveguide, within the framework of the finite quantization of geometric optics. We use the Kravchuk coherent states provided by the finite oscillator model to evince the nonlinear transfor- mations that elliptic-profile waveguides produce on phase space by means of the SO(3) Wigner function.

We present advancements toward understanding the chaotic saddle in four dimensional Hamiltonian Maps. Our approach was to use cross section data and examine the Rainbow Singularities, which characterize the complementary set of the saddle. An analytic model has been proposed to explain the shape of these singularities.