Finite Number Of Energy Levels Research Articles

Cooling with fermionic thermal reservoirs.

The quantum reservoirs commonly considered in open-quantum systems theory are those modeled by quantum harmonic oscillators, which are called bosonic reservoirs. Recently, quantum reservoirs modeled by two-level systems, the so-called fermionic reservoirs, have received attention due to their features. Given that the components of these reservoirs have a finite number of energy levels, unlike bosonic reservoirs, some studies are being carried out to explore the advantages of using this type of reservoir, especially in the operation of heat machines. In this paper, we carry out a case study of a quantum refrigerator operating in the presence of bosonic or fermionic thermal reservoirs, and we show that fermionic baths have advantages over bosonic ones.

Coherent control and creation of population gratings for a pair of attosecond pulses in a resonant medium based on one-dimensional rectangular quantum wells

Attosecond pulses can be used to create and control coherence in resonant media, since their duration is shorter than the population relaxation times T1 and medium polarization T2. Previously, the possibility of creating and ultrafast control of electromagnetically induced gratings (EMIG) of atomic populations in a resonant medium was shown using a sequence of extremely short light pulses, when the pulses coherently interact with the medium and do not simultaneously overlap in the medium. These studies were carried out in various approximations, when a finite number of energy levels of the medium is taken into account, or when the pulse amplitude is small. In this paper, based on a direct numerical solution of the time dependent Schredinger equation without the indicated approximations, we study the possibility of ultrafast coherent control of populations and the creation of an EMIG by a pair of attosecond pulses in a multilevel resonant medium with a low density of particles. The medium is modeled using a one-dimensional rectangular potential well with infinitely high walls. The studies performed show the possibility of ultrafast coherent control of the properties of resonant media based on quantum wells using attosecond pulses. Keywords: electromagnetically induced gratings, coherent interaction, extremely short pulses, unipolar pulses, attosecond pulses, medium coherence.

Когерентное управление и создание решеток населенностей парой аттосекундных импульсов в резонансной среде на основе одномерных прямоугольных квантовых ям

Attosecond pulses can be used to create and control coherence in resonant media, since their duration is shorter than the population relaxation times T_1 and medium polarization T_2. Previously, the possibility of creating and ultrafast control of electromagnetically induced gratings (EMIG) of atomic populations in a resonant medium was shown using a sequence of extremely short light pulses, when the pulses coherently interact with the medium and do not simultaneously overlap in the medium. These studies were carried out in various approximations, when a finite number of energy levels of the medium is taken into account, or when the pulse amplitude is small. In this paper, based on a direct numerical solution of the time dependent Schrödinger equation without the indicated approximations, we study the possibility of ultrafast coherent control of populations and the creation of an EMIG by a pair of attosecond pulses in a multilevel resonant medium with a low density of particles. The medium is modeled using a one-dimensional rectangular potential well with infinitely high walls. The studies performed show the possibility of ultrafast coherent control of the properties of resonant media based on quantum wells using attosecond pulses.

Systematic study of hadronic excitation energy using the Schottky anomaly

We extract the excitation energy scales of the hadron spectra in a less model-dependent method using Schottky anomaly. Schottky anomaly is a thermodynamical phenomenon that the specific heat of a system consisting of a finite number of energy levels has a peak at finite temperature due to the energy gaps. Using the masses of all hadrons that are experimentally established, we obtain the excitation energy scales of the hadron spectra and investigate their flavor dependence.

Effective mass of the discrete values as a hidden feature of the one-dimensional harmonic oscillator model: Exact solution of the Schrödinger equation with a mass varying by position

In this paper, exactly solvable model of the quantum harmonic oscillator is proposed. Wave functions of the stationary states and energy spectrum of the model are obtained through the solution of the corresponding Schrödinger equation with the assumption that the mass of the quantum oscillator system varies with position. We have shown that the solution of the Schrödinger equation in terms of the wave functions of the stationary states is expressed by the pseudo Jacobi polynomials and the mass varying with position depends from the positive integer [Formula: see text]. As a consequence of the positive integer [Formula: see text], energy spectrum is not only non-equidistant, but also there are only a finite number of energy levels. Under the limit, when [Formula: see text], the dependence of effective mass from the position disappears and the system recovers known non-relativistic quantum harmonic oscillator in the canonical approach where wave functions are expressed by the Hermite polynomials.

Exactly solvable new classes of potentials with finite discrete energies

In this work, we propose more realistic models with discrete and finite number of energy levels that could fit well to molecules with potentials that were modeled previously as harmonic oscillator. The considered potentials could be also used as good models in quantum physics, statistical and condensed matter physics, atomic physics, nuclear physics, particle physics, high energy physics, mathematical physics, as well as in chemistry of complex molecules. More precisely, we derive the solutions of two families of Schrödinger equations using supersymmetric quantum mechanics technique for superpotentials having shape invariance properties, and where their eigenvalues and eigenfunctions are exactly determined. The range of their finite number of bound states is given explicitly. Furthermore, this result will contribute in extending the already small list of exactly solvable Schrödinger equations, where we have summarized in a table all well-known potentials having exact solutions and their superpotentials, their partner potentials, and their energies, as well as, the newly discovered potentials proposed here.

Approximations of the Sum of States by Laplace\u2019s Method for a System of Particles with a Finite Number of Energy Levels and Application to Limit Theorems

We consider a generic system composed of a fixed number of particles distributed over a finite number of energy levels. We make only general assumptions about system’s properties and the entropy. System’s constraints other than fixed number of particles can be included by appropriate reduction of system’s state space. For the entropy we consider three generic cases. It can have a maximum in the interior of system’s state space or on the boundary. On the boundary we can have another two cases. There the entropy can increase linearly with increase of the number of particles and in the another case grows slower than linearly. The main results are approximations of system’s sum of states using Laplace’s method. Estimates of the error terms are also included. As an application, we prove the law of large numbers which yields the most probable state of the system. This state is the one with the maximal entropy. We also find limiting laws for the fluctuations. These laws are different for the considered cases of the entropy. They can be mixtures of Normal, Exponential and Discrete distributions. Explicit rates of convergence are provided for all the theorems.

Multipartite Quantum Correlations: Symplectic and Algebraic Geometry Approach

We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as identification of states with `the same correlations properties' i.e. ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one to another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question `what can be transformed into what \textit{via} available means?'. Exactly such an interpretations, i.e, in terms of mutual transformability can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods.

A collisional-radiative model for lithium impurity in plasma boundary region of Experimental Advanced Superconducting Tokamak

A green emission layer caused by lithium impurity is universally observed in plasma boundary region of Experimental Advanced Superconducting Tokamak (EAST) via a visible-light camera, where lithium coating is normally adopted as a routine technique of wall conditioning. In this article, in order to estimate the spatial distribution of green light intensity of this emission layer according to the given real parameter distributions of edge plasmas, a practicable method is proposed based on a collisional-radiative model. In this model, a finite number of energy levels of lithium are taken into account, and proper simplifications of convection-diffusion equations are made according to the order-of-magnitude analysis. We process the atomic data collected from the OPEN-ADAS database, and develop a corresponding program in Mathematica 10.4.1 to solve the simplified one-dimensional problem numerically. Estimation results are obtained respectively for the two sets of edge plasma profiles of EAST in L-mode and H-mode regimes, and both clearly show a good unimodal structure of the spatial distribution of green light intensity of this emission layer. These analyses actually provide the spatial distributions of lithium impurities at different energy levels, not only indicating the spatial distribution of the intensity of this emission layer induced by lithium impurity but also revealing the physical processes that lithium experiences in edge plasma. There are some different and common characteristics in the spatial distribution of the intensity of this emission layer in these two important cases. This emission layer is kept outside the last closed magnetic surface in both cases while it becomes thinner with a higher intensity peak in H-mode case. Besides, the sensitivity of this algorithm to the measurement error of edge plasma profile is also explored in this work. It is found that the relative errors of the numerical results obtained by our proposed method are comparable to those of edge plasma profiles. This work provides important theoretical references for developing a new practical technique of fast reconstructing edge plasma configurations in EAST based on the emission of lithium impurity, and may further contribute a lot to the studies of edge plasma behaviors when three-dimensional perturbation fields are adopted.

Polarization operator in the 2+1 dimensional quantum electrodynamics with a nonzero fermion density in a constant uniform magnetic field

The polarization operator (tensor) for planar charged fermions in a constant uniform magnetic field is calculated in the one-loop approximation of $$2+1$$ -dimensional quantum electrodynamics (QED $$_{2+1}$$ ) with a nonzero fermion density. We construct the Green function of the Dirac equation with a constant uniform external magnetic field in QED $$_{2+1}$$ at a finite chemical potential, find the imaginary part of this Green function, and then obtain the polarization tensor related to the combined contribution from real particles occupying the finite number of energy levels and magnetic field. We expect that some physical effects under consideration seem likely to be revealed in a monolayer graphene sample in the presence of an external constant uniform magnetic field $$B$$ perpendicular to it.

Quantum dynamics in the partial Wigner picture

Recently we have shown how the partial Wigner representation of quantum mechanics can be used to study hybrid quantum models where a system with a finite number of energy levels is coupled to linear or nonlinear oscillators (Beck and Sergi 2013 Phys. Lett. A 377 1047). The purpose of this work is to provide a detailed derivation of the partially Wigner-transformed quantum equations of motion for nonlinear oscillator subsystems under the action of general polynomial potentials. Such equations can be written in terms of a propagator, which can then be expanded in a power series. The linear terms of the series describe quantum-classical dynamics while the nonlinear terms provide the corrections needed to restore the fully quantum character of the evolution. In the case of polynomial potentials and position dependent couplings, the number of nonlinear terms is finite and the corrections can be calculated explicitly. In this work we show how to implement numerically the above scheme where, in principle, no assumption about the strength of the coupling must be taken. We illustrate the formalism by studying a two-level system interacting with an asymmetric quartic oscillator. We integrate the quantum dynamics of the total system and provide a comparison with the case of the quantum-classical dynamics of the quartic oscillator. The approach presented here is expected to be effective for studying hybrid quantum circuits in quantum information theory and for witnessing the quantum-to-classical transition in nano-oscillators coupled to pseudo-spins.

Quantum Integrability in Systems with Finite Number of Levels

We consider the problem of defining quantum integrability in systems with finite number of energy levels starting from commuting matrices and construct new general classes of such matrix models with a given number of commuting partners. We argue that if the matrices depend on a (real) parameter, one can define quantum integrability from this feature alone, leading to specific results such as exact solvability, Poissonian energy level statistics and to level crossings.

Neutron spectroscopy of molecular nanomagnets

This short overview gives an account on the use of neutron spectroscopy for the examination of molecular nanomagnets, systems constructed by crystalline arrangement of finite size clusters (usually with regular form) of interacting moment carrying atoms – magnetic molecules. Opposed to extended magnetic systems with bands of collective excitations such as spin-waves the molecular nanomagnets are entities with local properties, each magnetic molecule possessing a finite number of energy levels that can be solved exactly for small enough systems. In essence, the number of states remains finite despite growing rapidly with increasing number of magnetic centers and the value of the spin quantum number. Increasingly large numbers of states and more complex exchange networks lead to the need for approximative treatments, the validity of which can be checked with neutron spectroscopy. Molecular nanomagnets provide interesting examples of physics and magnetochemistry, illustrated here with a few examples that highlight the power of neutron spectroscopy for precise investigation of the energy level structure and spatial configuration of the magnetic exchange parameters.

Modification of Crum's Theorem for 'Discrete' Quantum Mechanics

Crum's theorem in one-dimensional quantum mechanics asserts the existence of an associated Hamiltonian system for any given Hamiltonian with the complete set of eigenvalues and eigenfunctions. The associated system is iso-spectral to the original one except for the lowest energy state, which is deleted. A modification due to Krein-Adler provides algebraic construction of a new complete Hamiltonian system by deleting a finite number of energy levels. Here we present a discrete version of the modification based on the Crum's theorem for the `discrete' quantum mechanics developed by two of the present authors.

On perturbation theory for an asymmetric anharmonic oscillator

A modified perturbation theory is formulated for an asymmetric anharmonic oscillator based on a choice of the main approximation constructed by analogy with the self-consistent field model. This perturbation theory has a much wider range of application in comparison with the standard approach and considers a finite number of energy levels in the potential well already in the main approximation. This approach is used for the construction of a two-atomic model of a quantum crystal. A quantum analog of the Lindeman criterion is obtained.

Positive and Negative Temperatures in a Two-Level System: Thermodynamic and Statistical-Mechanical Perspectives

Transient negative temperature states have been reported for a range of systems having a finite number of energy levels. While such systems are rare and seem to contradict the common notion that temperature is always positive, they provide an effective platform for illustrating the relationship between the thermodynamic and statistical-mechanical formulations of temperature. In this article we present a set of calculations for a two-level system containing N particles (1 ≤ N ≤ ∞) that graphically illustrates the statistical nature of temperature as well as the fundamental equivalence of its thermodynamic and statistical-mechanical formulations. These calculations, which we have applied in our undergraduate- and graduate-level physical chemistry courses, provide pedagogically useful insights into the meaning of a variety of thermodynamic and statistical mechanical concepts that students frequently have difficulty grasping.

Solvable scalar and spin models with near-neighbors interactions

We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass frame, while the rational models are shown to possess an equally spaced infinite algebraic spectrum. For the trigonometric and one of the rational models, the corresponding eigenfunctions are explicitly computed. We also study the scalar reductions of the models, some of which had already appeared in the literature, and compute their algebraic eigenfunctions in closed form. In the rational cases, for which only partial results were available, we give concise expressions of the eigenfunctions in terms of generalized Laguerre and Jacobi polynomials.

Quantum control in infinite dimensions

Accurate control of quantum evolution is an essential requirement for quantum state engineering, laser chemistry, quantum information and quantum computing. Conditions of controllability for systems with a finite number of energy levels have been extensively studied. By contrast, results for controllability in infinite dimensions have been mostly negative, stating that full control cannot be achieved with a finite-dimensional control Lie algebra. Here we show that by adding a discrete operation to a Lie algebra it is possible to obtain full control in infinite dimensions with a small number of control operators.

Superconductivity in Mesoscopic Metal Particles

Recently, it has been possible to construct single-electron transistors to study electronic properties, including superconductivity, in metallic grains of nanometer size. Among several theoretical results are suppression of superconductivity with decreasing grain size and parity effect, that is, dependence on the parity of the number of electrons on the grain. We study how these results are affected by degeneracy of energy levels. In addition to the time-reversal symmetry, for certain energy spectra and more generally for lattice symmetries, energy levels are strongly degenerate near the Fermi energy. For a parabolic dispersion, degeneracy d is of the order of kFL, whereas the typical distance between the levels is of the order of ∈F/(kFL)2 where kF and ∈F are the Fermi wave vector and energy, respectively, and L is the particle size. First, using an exact solution method for BCS Hamiltonian with finite number of energy levels, we find a new feature for the well studied nondegenerate case. In that case, parity effect exhibits a minimum instead of a monotonic behavior. For d-fold degenerate states we find that the ratio of two successive parity-effect parameters Δp is nearly 1 + 1/d. Our numerical solutions for the exact ground state energy of negative-U Hubbard model on a cubic cluster also give very similar results. Hence we conclude that parity effect is a general property of small Fermi systems with attractive interaction, and it is closely related to degeneracy of energy levels.

Negative Absolute Temperatures, a Novelty

The concept of absolute temperature is developed on the basis of the second law. It is shown that although, in general, the absolute temperature can take only positive values, it can achieve negative values for particular cases where only a finite number of energy levels are available.