Showing papers on "Ladder operator published in 2017"

Operator entanglement entropy of the time evolution operator in chaotic systems

Abstract: Entanglement entropy is a widely applicable indicator of the onset of quantum chaos. In particular, the entanglement entropy of the wave function after a quench from an initial state with low entanglement can be used to study the thermalization of the system. Here, the authors propose an initial-state independent quantity called the ``operator entanglement entropy'' (opEE) to extract the properties of the unitary evolution operator. They study the growth of the opEE in Floquet, chaotic, and many-body localized systems. They respectively have a linear, power-law, and logarithmic growth before reaching extensive saturation values. The most chaotic Floquet spin model has the maximal saturation value among the three classes and is identical to the value of a random unitary operator (the Page value). The authors interpret the opEE as the state EE of a quenched state living in a doubled Hilbert space, thus establishing its consistency with the existing state EE results. They conclude that the EE of the evolution operator should characterize the propagation of information in these systems.

Resonances in open quantum systems

Abstract: The Hamilton operator of an open quantum system is non-Hermitian. Its eigenvalues are generally complex and provide not only the energies but also the lifetimes of the states of the system. The states may couple via the common environment of scattering wave functions into which the system is embedded. This causes an external mixing (EM) of the states. Mathematically, EM is related to the existence of singular (the so-called exceptional) points. The eigenfunctions of a non-Hermitian operator are biorthogonal, in contrast to the orthogonal eigenfunctions of a Hermitian operator. A quantitative measure for the ratio between biorthogonality and orthogonality is the phase rigidity of the wave functions. At and near an exceptional point (EP), the phase rigidity takes its minimum value. The lifetimes of two nearby eigenstates of a quantum system bifurcate under the influence of an EP. At the parameter value of maximum width bifurcation, the phase rigidity approaches the value one, meaning that the two eigenfunctions become orthogonal. However, the eigenfunctions are externally mixed at this parameter value. The $S$ matrix and therewith the cross section do contain, in the one-channel case, almost no information on the EM of the states. The situation is completely different in the case with two (or more) channels where the resonance structure is strongly influenced by the EM of the states and interesting features of non-Hermitian quantum physics are revealed. We provide numerical results for two and three nearby eigenstates of a non-Hermitian Hamilton operator that are embedded in one common continuum and are influenced by two adjoining EPs. The results are discussed. They are of interest for an experimental test of the non-Hermitian quantum physics as well as for applications.

On Recovering the Dirac Operator with an Integral Delay from the Spectrum

Abstract: An inverse spectral problem for the Dirac operator with an integral delay is studied. We show, that the considered operator can be uniquely recovered from one spectrum, provide a constructive procedure for the solution of the inverse problem, and obtain necessary and sufficient conditions for its solvability.

Mass Ladder Operators from Spacetime Conformal Symmetry

Abstract: Ladder operators can be useful constructs, allowing for unique insight and intuition. In fact, they have played a special role in the development of quantum mechanics and field theory. Here, we introduce a novel type of ladder operators, which map a scalar field onto another massive scalar field. We construct such operators, in arbitrary dimensions, from closed conformal Killing vector fields, eigenvectors of the Ricci tensor. As an example, we explicitly construct these objects in anti–de Sitter (AdS) spacetime and show that they exist for masses above the Breitenlohner-Freedman bound. Starting from a regular seed solution of the massive Klein-Gordon equation, mass ladder operators in AdS allow one to build a variety of regular solutions with varying boundary condition at spatial infinity. We also discuss mass ladder operator in the context of spherical harmonics, and the relation between supersymmetric quantum mechanics and so-called Aretakis constants in an extremal black hole.

ABC of ladder operators for rationally extended quantum harmonic oscillator systems

Abstract: The problem of construction of ladder operators for rationally extended quantum harmonic oscillator (REQHO) systems of a general form is investigated in the light of existence of different schemes of the Darboux-Crum-Krein-Adler transformations by which such systems can be generated from the quantum harmonic oscillator. Any REQHO system is characterized by the number of separated states in its spectrum, the number of `valence bands' in which the separated states are organized, and by the total number of the missing energy levels and their position. All these peculiarities of a REQHO system are shown to be detected and reflected by a trinity $(\mathcal{A}^\pm$, $\mathcal{B}^\pm$, $\mathcal{C}^\pm$) of the basic (primary) lowering and raising ladder operators related between themselves by certain algebraic identities with coefficients polynomially-dependent on the Hamiltonian. We show that all the secondary, higher-order ladder operators are obtainable by a composition of the basic ladder operators of the trinity which form the set of the spectrum-generating operators. Each trinity, in turn, can be constructed from the intertwining operators of the two complementary minimal schemes of the Darboux-Crum-Krein-Adler transformations.

Localization of a multi-dimensional quantum walk with one defect

Abstract: In this paper, we introduce a multi-dimensional generalization of Kitagawa's split-step discrete-time quantum walk, study the spectrum of its evolution operator for the case of one-defect coins, and prove localization of the walk. Using a spectral mapping theorem, we can reduce the spectral analysis of the evolution operator to that of a discrete Schrodinger operator with variable coefficients, which is analyzed using the Feshbach map.

Bi-Orthogonal Approach to Non-Hermitian Hamiltonians with the Oscillator Spectrum: Generalized Coherent States for Nonlinear Algebras

Abstract: A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media.

Gravitational dual of the Rényi displacement operator

Abstract: We give a recipe for computing correlation functions of the displacement operator localized on a spherical or planar higher dimensional twist defect using AdS/CFT. Such twist operators are typically used to construct the $n$'th Renyi entropies of spatial entanglement in CFTs and are holographically dual to black holes with hyperbolic horizons. The displacement operator then tells us how the Renyi entropies change under small shape deformations of the entangling surface. We explicitly construct the bulk to boundary propagator for the displacement operator insertion as a linearized metric fluctuation of the hyperbolic black hole and use this to extract the coefficient of the displacement operator two point function $C_D$ in any dimension. The $n \rightarrow 1$ limit of the twist displacement operator gives the same bulk response as the insertion of a null energy operator in vacuum, which is consistent with recent results on the shape dependence of entanglement entropy and modular energy.

Ab initio excited states from the in-medium similarity renormalization group

Abstract: We present two new methods for performing ab initio calculations of excited states for closed-shell systems within the in-medium similarity renormalization group (IMSRG) framework Both are based on combining the IMSRG with simple many-body methods commonly used to target excited states, such as the Tamm-Dancoff approximation (TDA) and equations-of-motion (EOM) techniques In the first approach, a two-step sequential IMSRG transformation is used to drive the Hamiltonian to a form where a simple TDA calculation (ie, diagonalization in the space of $1\mathrm{p}1\mathrm{h}$ excitations) becomes exact for a subset of eigenvalues In the second approach, EOM techniques are applied to the IMSRG ground-state-decoupled Hamiltonian to access excited states We perform proof-of-principle calculations for parabolic quantum dots in two dimensions and the closed-shell nuclei $^{16}\mathrm{O}$ and $^{22}\mathrm{O}$ We find that the TDA-IMSRG approach gives better accuracy than the EOM-IMSRG when calculations converge, but it is otherwise lacking the versatility and numerical stability of the latter Our calculated spectra are in reasonable agreement with analogous EOM-coupled-cluster calculations This work paves the way for more interesting applications of the EOM-IMSRG approach to calculations of consistently evolved observables such as electromagnetic strength functions and nuclear matrix elements, and extensions to nuclei within one or two nucleons of a closed shell by generalizing the EOM ladder operator to include particle-number nonconserving terms

The two rigid body interaction using angular momentum theory formulae

Abstract: This work presents an elegant formalism to model the evolution of the full two rigid body problem. The equations of motion, given in a Cartesian coordinate system, are expressed in terms of spherical harmonics and Wigner D-matrices. The algorithm benefits from the numerous recurrence relations satisfied by these functions allowing a fast evaluation of the mutual potential. Moreover, forces and torques are straightforwardly obtained by application of ladder operators taken from the angular momentum theory and commonly used in quantum mechanics. A numerical implementation of this algorithm is made. Tests show that the present code is significantly faster than those currently available in literature.

Hidden symmetries of the Higgs oscillator and the conformal algebra

Abstract: We give a solution to the long-standing problem of constructing the generators of hidden symmetries of the quantum Higgs oscillator, a particle on a d-sphere moving in a central potential varying as the inverse cosine-squared of the polar angle. This superintegrable system is known to possess a rich algebraic structure, including a hidden SU(d) symmetry that can be deduced from classical conserved quantities and degeneracies of the quantum spectrum. The quantum generators of this SU(d) have not been constructed thus far, except at d = 2, and naive quantization of classical conserved quantities leads to deformed Lie algebras with quadratic terms in the commutation relations. The nonlocal generators we obtain here satisfy the standard su(d) Lie algebra, and their construction relies on a recently discovered realization of the conformal algebra, which contains a complete set of raising and lowering operators for the Higgs oscillator. This operator structure has emerged from a relation between the Higgs oscillator Schrodinger equation and the Klein–Gordon equation in Anti-de Sitter spacetime. From such a point-of-view, constructing the hidden symmetry generators reduces to manipulations within the abstract conformal algebra so(d, 2).

Symmetry, Geometry and Quantization with Hypercomplex Numbers

Abstract: These notes describe some links between the group SL2(R), the Heisenberg group and hypercomplex numbers - complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this framework. In particular, classical mechanics can be obtained as a theory with noncommutative observables and a non-zero Planck constant if we replace complex numbers in quantum mechanics by dual numbers. Our consideration is based on induced representations which are build from complex-/dual/-double-valued characters. Dynamic equations, rules of additions of probabilities, ladder operators and uncertainty relations are also discussed. Finally, we prove a Calderon-Vaillancourt-type norm estimation for relative convolutions.

Quantum entropy of non-Hermitian entangled systems

Abstract: Non-Hermitian Hamiltonians are an effective tool for describing the dynamics of open quantum systems. Previous research shows that the restrictions of conventional quantum mechanics may be violated in the non-Hermitian cases. We studied the entropy of a system of entangled qubits governed by a local non-Hermitian Hamiltonian operator. We find that local non-Hermitian operation influences the entropies of the two subsystems equally and simultaneously. This indicates that non-Hermitian operators possess the property of non-locality, which makes information exchange possible between subsystems. These information exchanges reduce the uncertainty of outcomes associated with two incompatible quantum measurements.

Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity

Abstract: To adopt a practical method to calculate the action of geometrical operators on quantum states is a crucial task in loop quantum gravity. In this paper, the graphical calculus based on the original Brink graphical method is applied to loop quantum gravity along the line of previous work. The graphical method provides a very powerful technique for simplifying complicated calculations. The closed formula of the volume operator and the actions of the Euclidean Hamiltonian constraint operator and the so-called inverse volume operator on spin-network states with trivalent vertices are derived via the graphical method. By employing suitable and non-ambiguous graphs to represent the action of operators as well as the spin-network states, we use the simple rules of transforming graphs to obtain the resulting formula. Comparing with the complicated algebraic derivation in some literature, our procedure is more concise, intuitive and visual. The resulting matrix elements of the volume operator is compact and uniform, fitting for both gauge-invariant and gauge-variant spin-network states. Our results indicate some corrections to the existing results for the Hamiltonian operator and inverse volume operator in the literature.

A New Integrable Equation Constructed via Combining the Recursion Operator of the Calogero-BogoyavlenskiiSchiff (CBS) Equation and its Inverse Operator

Abstract: In this work we construct a new integrable equation via combi ning the recursion operator of the Calogero-Bogoyavlenski iSchiff (CBS) equation and its inverse recursion operator. W e show that this equation nicely passes the Painlevé proper ty to emphasize its complete integrability. We formally derive multiple so lit n solutions by using the simplified Hirota’s direct meth od. We also use other techniques to obtain more solutions of distinct physi cal structures.

Quasinormal modes of generalized Pöschl–Teller potentials

Abstract: Using algebraic techniques we obtain quasinormal modes and frequencies associated to generalized forms of the scattering Poschl–Teller potential. This approach is based on the association of the corresponding equations of motion with Casimir invariants of differential representations of the Lie algebra . In the presented development, highest weight representations are constructed and fundamental states are calculated. An infinite tower of quasinormal mode solutions is obtained by the action of a lowering operator. The algebraic results are used in the analysis of the Cauchy initial value problem associated to the generalized Poschl–Teller potentials. For the scattering potentials considered, there are no late-time tails and the dynamics is always stable.

Arik–Coon q-oscillator cat states on the noncommutative complex plane ℂq−1 and their nonclassical properties

Abstract: The normalized even and odd q-cat states corresponding to Arik–Coon q-oscillator on the noncommutative complex plane ℂq−1 are constructed as the eigenstates of the lowering operator of a q-deformed su(1, 1) algebra with the left eigenvalues. We present the appropriate noncommutative measures in order to realize the resolution of the identity condition by the even and odd q-cat states. Then, we obtain the q-Bargmann–Fock realizations of the Fock representation of the q-deformed su(1, 1) algebra as well as the inner products of standard states in the q-Bargmann representations of the even and odd subspaces. Also, the Euler’s formula of the q-factorial and the Gaussian integrals based on the noncommutative q-integration are obtained. Violation of the uncertainty relation, photon antibunching effect and sub-Poissonian photon statistics by the even and odd q-cat states are considered in the cases 0 1.

Linear quantum optical bare raising operator

Abstract: We propose a simple implementation of the bare raising operator on coherent states via conditional measurement, which succeeds with high probability and fidelity. This operation works well not only on states with a Poissonian photon number distribution but also for a much wider class of states. As a part of this scheme, we highlight an experimentally testable effect in which a single photon is induced through a highly reflecting beamsplitter by a large amplitude coherent state, with probability in the limit of large coherent state amplitude.

Weights, raising and lowering operators, and K-types for automorphic forms on SL(3,R)

Abstract: We give a fully explicit description of Lie algebra derivatives (generalizing raising and lowering operators) for representations of SL(3,R) in terms of a basis of Wigner functions. This basis is natural from the point of view of principal series representations, as well as computations in the analytic theory of automorphic forms (e.g., with Whittaker functions). The method is based on the Clebsch-Gordan multiplication rule for Wigner functions, and applies to other Lie groups whose maximal compact subgroup is isogenous to a product of SU(2) and U(1) factors. As an application, we give a complete and explicit description of the K-type structure of certain cohomological representations.

Coupled Supersymmetric Quantum Mechanics and ladder structures beyond the harmonic oscillator

Abstract: The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general---except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalize the Heisenberg uncertainty principle. Coupled SUSY is explicitly realized by an infinite family of differential operators.

Operator form of the three-nucleon scattering amplitude

Abstract: To extend the applications of the so-called ``three-dimensional'' formalism to the description of three-nucleon scattering within the Faddeev formalism, we develop a general form of the three-nucleon scattering amplitude. This form significantly decreases the numerical complexity of the ``three-dimensional'' calculations by reducing the scattering amplitude to a linear combination of momentum-dependent spin operators and scalar functions of momenta. The number and structure of the spin operators is fixed and the scalar functions can be represented numerically using standard methods such as multidimensional arrays. In this paper, we show that all orders of the iterated Faddeev equation can be written in this general form. We argue that calculations utilizing the three-nucleon force will also conform to the same general form. Additionally, we show how the general form of the scattering amplitude can be used to transform the Faddeev equation to make it suitable for numerical calculations using iterative methods.

Inverse Weyl transform/operator

Abstract: The Weyl procedure associates a function of two ordinary variables, called the c-function or symbol, with an operator, called the Weyl operator of the symbol. One generally formulates this association by defining the operator corresponding to a given symbol. In this paper we consider the reverse problem: Given the Weyl operator, what is the matching symbol? We give a number of explicit formulas for obtaining the symbol that would generate an arbitrary Weyl operator, and we illustrate each form with an example.