Lowest Eigenstates Research Articles

High-Spin Polaron in Lightly DopedCuO2Planes

We derive and investigate numerically a minimal yet detailed spin-polaron model that describes lightly doped CuO2 layers. The low-energy physics of a hole is studied by total-spin-resolved exact diagonalization on clusters of up to 32 CuO2 unit cells, revealing features missed by previous studies. In particular, spin-polaron states with total spin 3/2 are the lowest eigenstates in some regions of the Brillouin zone. In these regions, and also at other points, the quasiparticle weight is identically zero indicating orthogonal states to those represented in the one electron Green's function. This highlights the importance of the proper treatment of spin fluctuations in the many-body background.

Superconducting phase qubit based on the Josephson oscillator with strong anharmonicity

We propose a superconducting phase qubit on the basis of the radio-frequency SQUID with the screening parameter value $\beta_L = (2\pi/\Phi_0)LI_c \approx 1$, biased by a half flux quantum $\Phi_e=\Phi_0/2$. Significant anharmonicity ($> 30%$) can be achieved in this system due to the interplay of the cosine Josephson potential and the parabolic magnetic-energy potential that ultimately leads to the quartic polynomial shape of the well. The two lowest eigenstates in this global minimum perfectly suit for the qubit which is insensitive to the charge variable, biased in the optimal point and allows an efficient dispersive readout. Moreover, the transition frequency in this qubit can be tuned within an appreciable range allowing variable qubit-qubit coupling.

Diquark and triquark correlations in the deconfined phase of QCD

We use the non-perturbative Q\bar Q potential at finite temperatures derived in the Field Correlator Method to obtain binding energies for the lowest eigenstates in the Q\bar Q and QQQ systems (Q=c,b). The three--quark problem is solved by the hyperspherical method. The solution provides an estimate of the melting temperature and the radii for the different diquark and triquark bound states. In particular we find that J/\psi and $ccc$ ground states survive up to T \sim 1.3 T_c, where T_c is the critical temperature, while the corresponding bottomonium states survive even up to higher temperature, T \sim 2.2 T_c.

Pedagogical applications of the one-dimensional Schrödinger’s equation to proximity effect systems: Comparison of Dirichlet and Neumann boundary conditions

Proximity effect systems in superconducting films can be modeled by a one-dimensional Schrödinger equation. Several systems are studied using Dirichlet and Neumann boundary conditions. It is observed that the two boundary conditions have a dramatic effect on the lowest eigenstate allowed in these systems, and this points to unusual behavior for solutions of Schrödinger’s equation in certain potential wells and proximity effect systems.

Cavity QED Based on Collective Magnetic Dipole Coupling: Spin Ensembles as Hybrid Two-Level Systems

We analyze the magnetic dipole coupling of an ensemble of spins to a superconducting microwave stripline structure, incorporating a Josephson junction based transmon qubit. We show that this system is described by an embedded Jaynes-Cummings model: in the strong coupling regime, collective spin-wave excitations of the ensemble of spins pick up the nonlinearity of the cavity mode, such that the two lowest eigenstates of the coupled spin wave-microwave cavity-Josephson junction system define a hybrid two-level system. The proposal described here enables new avenues for nonlinear optics using optical photons coupled to spin ensembles via Raman transitions. The possibility of strong coupling cavity QED with magnetic dipole transitions also opens up the possibility of extending quantum information processing protocols to spins in silicon or graphene, without the need for single-spin confinement.

Radial rescaling approach for the eigenvalue problem of a particle in an arbitrarily shaped box

In the present work we introduce a methodology for solving a quantum billiard with Dirichlet boundary conditions. The procedure starts from the exactly known solutions for the particle in a circular disk, which are subsequently radially rescaled in such a way that they obey the new boundary conditions. In this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision. Test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy.

An object-oriented C++ implementation of Davidson method for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix

A C++ class named Davidson is presented for determining a few eigenpairs with lowest or alternatively highest values of a large, real, symmetric matrix. The algorithm described by Stathopoulos and Fischer is used. The exception mechanism is involved to report the errors. The class is written in ANSI C++, so it is fully portable. In addition a console program as well as a program with graphical user interface for Microsoft Windows is attached, which allow one to calculate the lowest eigenstates of time-independent Schrödinger equation for a given binding potential in one, two or three spatial dimensions. The package contains the classes providing often used potential functions (model atom potential, Coulomb potential, square well potential and Kramers–Henneberger well potential) as well as a possibility to use any potential stored in a file (then any dimensionality of the problem is allowed). The described code is the subject of M.Sc. thesis of T.D. prepared under the supervision of J.M. Program summary Program title: Davidson Catalogue identifier: ADZM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZM_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3 037 055 No. of bytes in distributed program, including test data, etc.: 20 002 609 Distribution format: tar.gz Programming language: C++ Computer: All Operating system: Any RAM: User's parameters dependent Word size: 32 and 64 bits Supplementary material: Test results for the 2D and 3D cases is available Classification: 4, 4.8 Nature of problem: Finding a few extreme eigenpairs of a real, symmetric, sparse matrix. Examples in quantum optics (interaction of matter with a laser field). Solution method: Davidson algorithm Running time: The test example included in the distribution package (1D matrix) takes approximately 30 minutes to run. 2D matrix calculations can take hours and 3D, days, to run.

Pair condensation in a finite Fermi system

The lowest seniority-zero eigenstates of an exactly solvable multilevel pairing Hamiltonian for a finite Fermi system are examined at different pairing regimes. After briefly reviewing the form of the eigenstates in the Richardson formalism, we discuss a different representation of these states in terms of the collective pairs resulting from the diagonalization of the Hamiltonian in a space of two degenerate time-reversed fermions. We perform a two-fold analysis by working both in the fermionic space of these collective pairs and in a space of corresponding elementary bosons. On the fermionic side, we monitor the variations which occur, with increasing the pairing strength, in the structure of both these collective pairs and the lowest eigenstates. On the bosonic side, after reviewing a fermion-boson mapping procedure, we construct exact images of the fermion eigenstates and study their wave function. The analysis allows a close examination of the phenomenon of pair condensation in a finite Fermi system and gives new insights into the evolution of the lowest (seniority-zero) excited states of a pairing Hamiltonian from the unperturbed regime up to a strongly interacting one.

SU(2) Yang–Mills quantum mechanics of spatially constant fields

Abstract As a first step towards a strong coupling expansion of Yang–Mills theory, the SU ( 2 ) Yang–Mills quantum mechanics of spatially constant gauge fields is investigated in the symmetric gauge, with the six physical fields represented in terms of a positive definite symmetric 3 × 3 matrix S. Representing the eigenvalues of S in terms of elementary symmetric polynomials, the eigenstates of the corresponding harmonic oscillator problem can be calculated analytically and used as orthonormal basis of trial states for a variational calculation of the Yang–Mills quantum mechanics. In this way high precision results are obtained in a very effective way for the lowest eigenstates in the spin-0 sector as well as for higher spin. Furthermore I find, that practically all excitation energy of the eigenstates, independently of whether it is a vibrational or a rotational excitation, leads to an increase of the expectation value of the largest eigenvalue 〈 ϕ 3 〉 of S, whereas the expectation values of the other two eigenvalues, 〈 ϕ 1 〉 and 〈 ϕ 2 〉 , and also the component 〈 B 3 〉 = g 〈 ϕ 1 ϕ 2 〉 of the magnetic field, remain at their vacuum values.

Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases

We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.

FREE EXPANSION OF IMPENETRABLE BOSONS ON ONE-DIMENSIONAL OPTICAL LATTICES

We review recent exact results for the free expansion of impenetrable bosons on one-dimensional lattices, after switching off a confining potential. When the system is initially in a superfluid state, far from the regime in which the Mott-insulator appears in the middle of the trap, the momentum distribution of the expanding bosons rapidly approaches the momentum distribution of non-interacting fermions. Remarkably, no loss in coherence is observed in the system as reflected by a large occupation of the lowest eigenstate of the one-particle density matrix. In the opposite limit, when the initial system is a pure Mott insulator with one particle per lattice site, the expansion leads to the emergence of quasicondensates at finite momentum. In this case, one-particle correlations like the ones shown to be universal in the equilibrium case develop in the system. We show that the out-of-equilibrium behavior of the Shannon information entropy in momentum space, and its contrast with one of non-interacting fermions, allows to differentiate the two different regimes of interest. It also helps in understanding the crossover between them.

Heavy quark potential and quarkonia dissociation rates

Quenched lattice data for the quark-antiquark interaction (in terms of heavy quark free energies) in the color singlet channel at finite temperatures are fitted and used within the nonrelativistic Schroedinger equation formalism to obtain binding energies and scattering phase shifts for the lowest eigenstates in the charmonium and bottomonium systems in a hot gluon plasma. The partial dissociation rate due to the Bhanot-Peskin process is calculated using different assumptions for the gluon distribution function, including free massless gluons, massive gluons, and massive damped gluons. It is demonstrated that a temperature dependent gluon mass has an essential influence on the heavy quarkonia dissociation, but that this process alone is insufficient to describe the heavy quarkonia dissociation rates.

A coherent discrete variable representation method for multidimensional systems in physics

A coherent discrete variable representation (ZDVR) is proposed for constructing a multidimensional potential-optimized DVR basis. The multidimensional quadrature pivots are obtained by diagonalizing a complex coordinate operator matrix in a finite basis set, which is spanned by the lowest eigenstates of a two-dimensional reference Hamiltonian. Here a c-norm condition is used in the diagonalization procedure. The orthonormal eigenvectors define a collocation matrix connecting the localized ZDVR basis functions and the finite basis set. The method is applied to two vibrational models for computing the lowest bound states. Results show that the ZDVR method provides exponential convergence and accurate energies. Finally, a zeroth-order approximation method is also derived.

Effective computation of matrix elements between polynomial basis functions

Two methods of evaluating matrix elements of a function in a polynomial basis are considered: the expansion method, where the function is expanded in the basis and the integrals are evaluated analytically, and the numerical method, where the integration is performed directly using numerical quadrature. A reduced grid is proposed for the latter which makes use of the symmetry of the basis. Comparison of the two methods is presented in the context of evaluation of matrix elements in a non-direct product basis. If high accuracy of all matrix elements is required then the expansion method is the best choice. If however the accuracy of high order matrix elements is not important (as in variational ro-vibrational calculations where one is typically interested only in the lowest eigenstates), then the method based on the reduced grid offers sufficient accuracy and is much quicker than the expansion method.

Visual–Motor Coordination Using a Quantum Clustering Based Neural Control Scheme

Visual–Motor Coordination is a problem considered analogous to the hand-eye coordination in biological systems. In this work we propose a novel approach to this problem using Quantum Clustering and an extended Kohonen's Self-Organizing Feature Map (K-SOFM). This facilities the use of the method in varying workspaces by considering the joint angles of the robot arm. Unlike previous work, where a fixed topology for the input space is considered, the proposed approach determines a topology as the workspace varies. Quantum Clustering is a method which constructs a scale-space probability function and uses the Schroedinger equation and its lowest eigenstate to obtain a potential whose minimum gives the cluster centers. It transforms the input space into a Hilbert space, where it searches for its minimum. The motivation of this work is to identify the implicit relationship existing between the end-effector positions and the joint angles through Quantum Clustering and Neural Network methods to fine-tune the system to correctly identify the mapping.

Duality of two types ofSU(1, 1) coherent states and an intermediate type

The two types of SU(1, 1) coherent states of Barut and Girardello and of Perelomov are dual in a sense that the operators in the eigenvalue equation and in the exponentials which create these types of coherent states from the lowest eigenstate (vacuum) form an asymmetric Heisenberg–Weyl algebra. A new type of SU(1, 1) coherent states which takes on an intermediate position between the two dual types of SU(1, 1) coherent states already mentioned is established and investigated. In this new type, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg–Weyl algebra. Properties of the different SU(1, 1) coherent states are investigated for the realization of SU(1, 1) by one-dimensional quantum-mechanical potential problems leading to a quadratic law of energy-level spacing. Coherent SU(1, 1) phase states are discussed for this realization of SU(1, 1). It is shown that no wavepackets corresponding to any of the SU(1, 1) coherent states can exactly preserve their shape during time evolution.

Ten-Electron Central Field Problem: An Inhomogeneous Electron Liquid

Recent work has been carried out on the exchange energy density epsive;x(r) of a ten-electron atomic ion in the (bare Coulomb) limit of large atomic number Z [Howard, I. A. et al (2000). Phys. Rev. A, 62, 062512]. This analytical study of epsive; x(r) was made possible by the existence of a closed form of the first-order (idempotent) density matrix (IDM). Here, some generalizations are effected to a central potential energy V(r) which (a) localizes the ten electrons and (b) yields closed K and L shells for these ten electrons occupying the lowest eigenstates with spin compensation. In particular, it is shown that p-shell properties alone determine the IDM in this example of a confined inhomogeneous electron liquid.

Algorithm for data clustering in pattern recognition problems based on quantum mechanics.

We propose a novel clustering method that is based on physical intuition derived from quantum mechanics. Starting with given data points, we construct a scale-space probability function. Viewing the latter as the lowest eigenstate of a Schrödinger equation, we use simple analytic operations to derive a potential function whose minima determine cluster centers. The method has one parameter, determining the scale over which cluster structures are searched. We demonstrate it on data analyzed in two dimensions (chosen from the eigenvectors of the correlation matrix). The method is applicable in higher dimensions by limiting the evaluation of the Schrödinger potential to the locations of data points.

Wigner localization and dynamics in two‐electron semiconductor rings

AbstractWe present a theoretical analysis of the electronic structure and the linear‐response dynamics for two‐dimensional rings in a perpendicular magnetic field containing two electrons. The exact lower eigenstates are found by numerically solving the Schrödinger equation and are used to build the pertubative dipole response. We show the equivalence of this technique with the integration in real time used in a previous work. Comparison with the predictions of local spin density approximation theory is also done. In narrow rings the solution is shown to correspond to a rigid rotor and thus to a localized structure in the intrinsic frame, i.e., a rotating Wigner molecule. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

The Existence and Stability of Noncommutative Scalar Solitons

We establish existence and stabilty results for solitons in noncommutative scalar field theories in even space dimension $2d$. In particular, for any finite rank spectral projection $P$ of the number operator ${\mathcal N}$ of the $d$-dimensional harmonic oscillator and sufficiently large noncommutativity parameter $\theta$ we prove the existence of a rotationally invariant soliton which depends smoothly on $\theta$ and converges to a multiple of $P$ as $\theta\to\infty$. In the two-dimensional case we prove that these solitons are stable at large $\theta$, if $P=P_N$, where $P_N$ projects onto the space spanned by the $N+1$ lowest eigenstates of ${\mathcal N}$, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to $P=P_0$ for all $\theta$ in its domain of existence. Finally, for arbitrary $d$ and small values of $\theta$, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on $\theta$.