States Of Harmonic Oscillator Research Articles

Clustering in structure and reactions using configuration interaction techniques

Background: Atomic nuclei are remarkable quantum many-body systems where clustering properties develop naturally from underlying interactions between the constituent nucleons. Clustering degrees of freedom manifest themselves in multiple structure and reaction observables.Purpose: Our goal is to study nuclear clustering and its emergence in many-nucleon dynamics from nucleon-nucleon interactions. Clustering is a phenomenon that is known to emerge on the boundary between structure and reactions, therefore developing appropriate techniques that bridge the structure-reaction divide and establishing connections to observables is among our principal objectives. Showing consistency and how the new techniques can be reduced to well established other methods is an important part of this work.Methods: The configuration-interaction technique based on second quantization is used to treat the quantum many-body problem assuring that fermionic antisymmetry is fully satisfied. The use of the harmonic oscillator single-particle basis allows for the center-of-mass coordinate to be separated and prepared in a desired oscillator state for each cluster. The relative motion reaction basis channels are constructed by coupling clusters in different harmonic oscillator states with respect to their relative motion. Finally, using a resonating group method strategy we solve the generalized eigenvalue problem to obtain scattering channels. Structural clustering characteristics are discussed and the modified harmonic oscillator representation for scattering equations method is used to extract scattering observables.Results: New methods for treating clustering problems have been put forward. We demonstrate broad applicability of the developed techniques. Examples highlight connections with algebraic techniques, and the role of approximations leading to algebraic limits is assessed using realistic examples. Various types of clustering characteristics are used to study alpha clustering in light nuclei that are relevant to currently ongoing experimental efforts. We demonstrate the emergence of strongly clustered bands of states in beryllium, triple alpha channels in $^{12}\mathrm{C}$, and molecular type clustering in $^{21}\mathrm{Ne}$. Starting from nucleon-nucleon interactions without any additional assumptions scattering phase shifts for alpha-alpha scattering are determined and shown to be consistent with those observed.Conclusions: In this work we put forward a new configuration-interaction-based method that targets the physics of clustering, and further unifies nuclear structure and reactions. We provide detailed discussions and many examples highlighting features and advantages of the approach.

Unitary group approach to describe interatomic potentials in 3D systems

ABSTRACTAn algebraic approach based on the unitary algebra is proposed to describe 3D systems for effective potentials. Our approach is based on the 3D vibron model where the addition of a scalar boson is introduced into the space of a 3D harmonic oscillator. Our approach is based on the mapping between the dynamical symmetry and the harmonic oscillator states. A minimisation procedure is used in order to determine the coefficients involved in the algebraic expansions. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations. As an application of this approach the eigensystem of the 3D-Morse potential is analysed in detail. The analyses of inertia moments as well as the dipole moment strengths are also included. To show the convenience of this approach, a comparison with one ab-initio calculation for vibrational and rotational levels for O is presented in the framework of an analytical potential. In addition, a notch test is done to show how the formalism allows to check the sensitivity of the ro-vibrational results as a function of the precise structure of the potential energy surface as a function of r.

Hermite–Gaussian model for quantum states

In order to characterize quantum states within the context of information geometry, we propose a generalization of the Gaussian model, which we called the Hermite–Gaussian model. We obtain the Fisher–Rao metric and the scalar curvature for this model, and we show its relation with the one-dimensional quantum harmonic oscillator. Using this model we characterize some families of states of the quantum harmonic oscillator. We find that for eigenstates of the Hamiltonian, mixtures of eigenstates and even or odd superpositions of eigenstates, the associated Fisher–Rao metrics – which are relevant in the context of quantum parameter estimation theory – are diagonal. Finally, we consider the action of the amplitude damping channel and we discuss the relationship between the quantum decay and the different geometric indicators.

Multiphoton supercoherent states

In this paper we are going to build the multiphoton supercoherent states for the supersymmetric harmonic oscillator as eigenstates of the $m$-th power of a special form (but still with a free parameter) of the Kornbluth-Zypman supersymmetric annihilation operator. They become expressed in terms of the multiphoton coherent states for the standard harmonic oscillator. The Heisenberg uncertainty relation and some statistical properties for these states will be studied. Since the multiphoton supercoherent states turn out to be periodic, then the associated geometric phases will be as well evaluated.

Geometric dynamics of a harmonic oscillator, arbitrary minimal uncertainty states and the smallest step 3 nilpotent Lie group**Dedicated to 70th birthday of Maurice de Gosson

The paper presents a new method of geometric solution of a Schrödinger equation by constructing an equivalent first-order partial differential equation with a bigger number of variables. The equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform. The method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three Lie group—the group (also known under many names, e.g. quartic group). We obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector. In contrast, it is shown that the well-known Fock–Segal–Bargmann transform and the Heisenberg group require a specific fiducial vector to produce a geometric solution. A technical aspect considered in this paper is that a certain modification of a coherent state transform is required: although the irreducible representation of the group is square-integrable modulo a subgroup , the obtained dynamic is transverse to the homogeneous space .

On the Role of Trap Anharmonicity in the Dynamics of a One-Dimensional Bose Gas Suddenly Released from a Power-Law Trap into a Box Potential

The effects of trapping anharmonicity on the expansion of a Bose–Einstein condensate (BEC) that is suddenly released from a one-dimensional power-law trap into a hard wall box potential are presented. It is chiefly found that trapping anharmonicity, from a general point of view, has no influence on the qualitative behavior of the dynamics apart from the fact that a higher power-law exponent leads to a faster expansion of the BEC. The evolution of the density distributions, both spatial and momentum, the self-similarity of the spatial density at later times, and its connection to energy components and their dynamics, such as the zero-point, interaction, and kinetic energy are examined. By a decomposition of the wavefunction via modal coefficients, the basis being the harmonic oscillator states, the dynamics of these coefficients are determined. The latter shows that high states become occupied as a result of the BEC expansion. Moreover, the information entropy is evaluated from the latter coefficients that particularly displays a time behavior in the form of small oscillations at long times arising from the interaction with the walls. When the BEC collides with the hard walls of the box potential, it is found to be in a very highly excited state. As such, it is demonstrated that a future need for the practical recapture of an expanding BEC requires methods that suppress the resulting excitations if a reexamination of the ground-state in the “recapturing confinement” is needed. In passing, the important role played by the width of the wavefunction is once again revealed as in previous work (Sakhel and Sakhel in J Phys B 50:105301, 2017). We propose experiments for the measurement of kinetic energy as a function of time for this type of investigation. For example, an in situ measurement of the momentum distribution reveals information about the latter. The current results on the density dynamics can be also verified experimentally.

Squeezing and resonance in a generalized Caldirola-Kanai type quantum parametric oscillator

The evolution operator of a Caldirola-Kanai type quantum parametric oscillator with a generalized quadratic Hamiltonian is obtained using the Wei-Norman Lie algebraic approach, and time evolution of the eigenstates of a harmonic oscillator and Glauber coherent states is found explicitly. Behavior of this oscillator is investigated under the influence of the external mixed term B(t)(q^p^+p^q^)/2, which affects the squeezing properties of the wave packets, and linear terms D0(t)q^, E0(t)p^ responsible for their displacement in time. According to this, we construct all exact quantum models with different parameters B(t), for which the structure of the Caldirola-Kanai oscillator in position space is preserved. Then, for each model, we obtain explicit solutions and analyze the squeezing and displacement properties of the wave packets according to the frequency modification by B(t) and periodic forces in the corresponding classical equation of motion.

Rényi α entropies of quantum states in closed form: Gaussian states and a class of non-Gaussian states.

In this work, we study the Rényi α entropies S_{α}(ρ[over ̂])=(1-α)^{-1}ln{Tr(ρ[over ̂]^{α})} of quantum states for N bosons in the phase-space representation. With the help of the Bopp rule, we derive the entropies of Gaussian states in closed form for positive integers α=2,3,4,⋯ and then, with the help of the analytic continuation, acquire the closed form also for real values of α>0. The quantity S_{2}(ρ[over ̂]), primarily studied in the literature, will then be a special case of our finding. Subsequently we acquire the Rényi α entropies, with positive integers α, in closed form also for a specific class of the non-Gaussian states (mixed states) for N bosons, which may be regarded as a generalization of the eigenstates |n〉 (pure states) of a single harmonic oscillator with n≥1, in which the Wigner functions have negative values indeed. Due to the fact that the dynamics of a system consisting of N oscillators is Gaussian, our result will contribute to a systematic study of the Rényi α entropy dynamics when the current form of a non-Gaussian state is initially prepared.

Hypernuclear no-core shell model

We extend the No-Core Shell Model (NCSM) methodology to incorporate strangeness degrees of freedom and apply it to single-$\Lambda$ hypernuclei. After discussing the transformation of the hyperon-nucleon (YN) interaction into Harmonic-Oscillator (HO) basis and the Similarity Renormalization Group transformation applied to it to improve model-space convergence, we present two complementary formulations of the NCSM, one that uses relative Jacobi coordinates and symmetry-adapted basis states to fully exploit the symmetries of the hypernuclear Hamiltonian, and one working in a Slater determinant basis of HO states where antisymmetrization and computation of matrix elements is simple and to which an importance-truncation scheme can be applied. For the Jacobi-coordinate formulation, we give an iterative procedure for the construction of the antisymmetric basis for arbitrary particle number and present the formulae used to embed two- and three-baryon interactions into the many-body space. For the Slater-determinant formulation, we discuss the conversion of the YN interaction matrix elements from relative to single-particle coordinates, the importance-truncation scheme that tailors the model space to the description of the low-lying spectrum, and the role of the redundant center-of-mass degrees of freedom. We conclude with a validation of both formulations in the four-body system, giving converged ground-state energies for a chiral Hamiltonian, and present a short survey of the $A\le7$ hyper-helium isotopes.

Zombie states for description of structure and dynamics of multi-electron systems.

Canonical Coherent States (CSs) of Harmonic Oscillator have been extensively used as a basis in a number of computational methods of quantum dynamics. However, generalising such techniques for fermionic systems is difficult because Fermionic Coherent States (FCSs) require complicated algebra of Grassmann numbers not well suited for numerical calculations. This paper introduces a coherent antisymmetrised superposition of "dead" and "alive" electronic states called here Zombie State (ZS), which can be used in a manner of FCSs but without Grassmann algebra. Instead, for Zombie States, a very simple sign-changing rule is used in the definition of creation and annihilation operators. Then, calculation of electronic structure Hamiltonian matrix elements between two ZSs becomes very simple and a straightforward technique for time propagation of fermionic wave functions can be developed. By analogy with the existing methods based on Canonical Coherent States of Harmonic Oscillator, fermionic wave functions can be propagated using a set of randomly selected Zombie States as a basis. As a proof of principles, the proposed Coupled Zombie States approach is tested on a simple example showing that the technique is exact.

The coherent states of the two-dimensional isotropic harmonic oscillator and the classical limit of the Landau theory of a charged particle in a uniform magnetic field

In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z-axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a+ and a−. We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a+ in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a−. The phase of the circular motion is the same as the phase of the complex eigen value of a+. So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z-axis. We also show that as the magnitude of the eigen value of a+ becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.

Coherent light squeezing states within a modified microring system

Abstract We have proposed the simple method of the squeezed light generation in the modified microring resonator, which is known as the microring conjugate mirror (MCM). When the monochromatic light is input into the MCM, the general form of the squeezed coherent states for a quantum harmonic oscillator can be generated by controlling the additional two side rings, which are the phase modulators. By using the graphical method called the Optiwave program, the coherent squeezed states of coherent light within an MCM can be obtained and interpreted as the amplitude, phase, quadrature and photon number-squeezed states. This method has shown potentials for microring related device design, which can be used before practical applications.

Tighter Einstein-Podolsky-Rosen steering inequality based on the sum-uncertainty relation

We consider the uncertainty bound on the sum of variances of two incompatible observables in order to derive a corresponding steering inequality. Our steering criterion when applied to discrete variables yields the optimum steering range for two qubit Werner states in the two measurement and two outcome scenario. We further employ the derived steering relation for several classes of continuous variable systems. We show that non-Gaussian entangled states such as the photon subtracted squeezed vacuum state and the two-dimensional harmonic oscillator state furnish greater violation of the sum steering relation compared to the Reid criterion as well as the entropic steering criterion. The sum steering inequality provides a tighter steering condition to reveal the steerability of continuous variable states.

Two-mode generalization of the Jaynes–Cummings and Anti-Jaynes–Cummings models

We introduce two generalizations of the Jaynes–Cummings (JC) model for two modes of oscillation. The first model is formed by two Jaynes–Cummings interactions, while the second model is written as a simultaneous Jaynes–Cummings and Anti-Jaynes–Cummings (AJC) interactions. We study some of its properties and obtain the energy spectrum and eigenfunctions of these models by using the tilting transformation and the Perelomov number coherent states of the two-dimensional harmonic oscillator. Moreover, as physical applications, we connect these new models with two important and novel problems: The relativistic non-degenerate parametric amplifier and the relativistic problem of two coupled oscillators.

Quantum Harmonic Oscillator State Control in a Squeezed Fock Basis.

We demonstrate control of a trapped-ion quantum harmonic oscillator in a squeezed Fock state basis, using engineered Hamiltonians analogous to the Jaynes-Cummings and anti-Jaynes-Cummings forms. We demonstrate that for squeezed Fock states with low n the engineered Hamiltonians reproduce the sqrt[n] scaling of the matrix elements which is typical of Jaynes-Cummings physics, and also examine deviations due to the finite wavelength of our control fields. Starting from a squeezed vacuum state, we apply sequences of alternating transfer pulses which allow us to climb the squeezed Fock state ladder, creating states up to excitations of n=6 with up to 8.7dB of squeezing, as well as demonstrating superpositions of these states. These techniques offer access to new sets of states of the harmonic oscillator which may be applicable for precision metrology or quantum information science.

Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems

The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large D. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given, and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical ( D → ∞ ) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.

Lindblad Equation for Harmonic Oscillator: Uncertainty Relation Depending on Temperature

Specific nonequilibrium states of the quantum harmonic oscillator described by the Lindblad equation have been hereby suggested. This equation makes it possible to determine time-varying effects produced by statistical operator or statistical matrix. Thus, respective representation-varied equilibrium statistical matrixes have been found. Specific mean value equations have been found and their equilibrium solutions have been obtained.

Resonant detectors of gravitational wave as a possible probe of the noncommutative structure of space

We report the plausibility of using quantum mechanical transitions, induced by the combined effect of gravitational waves (GWs) and noncommutative (NC) structure of space, among the states of a 2-dimensional harmonic oscillator, to probe the spatial NC geometry. The phonon modes excited by the passing GW within the resonant bar-detector or spherical detectors are formally identical to forced harmonic oscillator and they represent a length variation of roughly the same order of magnitude as the characteristic length-scale of spatial noncommutativity estimated from the phenomenological upper bound of the NC parameter. This motivates our present work. We employ various GW wave-forms that are typically expected from possible astronomical sources. We find that the transition probablities are quite sensitive to the nature of polarization of the GW. We also elaborate on the particular type of sources of GW, radiation from which one can induce such transitions. We speculate that this can be used as an effective probe of the spatial noncommutative structure when the quantum limit of sensitivity is achieved/surpassed in resonant bar/spherical detectors of GWs in the near future.

Scaling Transform and Stretched States in Quantum Mechanics

We consider the Husimi Q(q, p)-functions which are quantum quasiprobability distributions on the phase space. It is known that, under a scaling transform (q; p) → (⋋q; ⋋p), the Husimi function of any physical state is converted into a function which is also the Husimi function of some physical state. More precisely, it has been proved that, if Q(q, p) is the Husimi function, the function ⋋2Q(⋋q; ⋋p) is also the Husimi function. We call a state with the Husimi function ⋋2Q(⋋q; ⋋p) the stretched state and investigate the properties of the stretched Fock states. These states can be obtained as a result of applying the scaling transform to the Fock states of the harmonic oscillator. The harmonic-oscillator Fock states are pure states, but the stretched Fock states are mixed states. We find the density matrices of stretched Fock states in an explicit form. Their structure can be described with the help of negative binomial distributions. We present the graphs of distributions of negative binomial coefficients for different stretched Fock states and show the von Neumann entropy of the simplest stretched Fock state.

Deterministic nonclassicality for quantum-mechanical oscillators in thermal states

Quantum nonclassicality is the basic building stone for the vast majority of quantum information applications and methods of its generation are at the forefront of research. One of the obstacles any method needs to clear is the looming presence of decohorence and noise which act against the nonclassicality and often erase it completely. In this paper we show that nonclassical states of a quantum harmonic oscillators initially in thermal equilibrium states can be deterministically created by coupling it to a single two level system. This can be achieved even in the absorption regime in which the two level system is initially in the ground state. The method is resilient to noise and it may actually benefit from it, as witnessed by the systems with higher thermal energy producing more nonclassical states.