Comparative analysis of information measures of the Dirichlet and Neumann two-dimensional quantum dots

Analytic representation of both position as well as momentum waveforms of the two-dimensional (2D) circular quantum dots with the Dirichlet and Neumann boundary conditions (BCs) allowed an efficient computation in either space of Shannon $S$, R\'{e}nyi $R(\alpha)$ and Tsallis $T(\alpha)$ entropies, Onicescu energies $O$ and Fisher informations $I$. It is shown that a transition to the 2D geometry lifts the 1D degeneracy of the position components $S_\rho$, $O_\rho$, $R_\rho(\alpha)$. Among many other findings, it is established that the lower limit $\alpha_{TH}$ of the semi-infinite range of the dimensionless R\'{e}nyi/Tsallis coefficient where one-parameter momentum entropies exist is equal to 2/5 for the Dirichlet requirement and 2/3 for the Neumann one. Since their 1D counterparts are $1/4$ and $1/2$, respectively, this simultaneously reveals that this critical value crucially depends not only on the position BC but the dimensionality of the structure too. As the 2D Neumann threshold $\alpha_{TH}^N$ is greater than one half, its R\'{e}nyi uncertainty relation for the sum of the position and wave vector components $R_\rho(\alpha)+R_\gamma\left(\frac{\alpha}{2\alpha-1}\right)$ is valid in the range $[1/2,2)$ only with its logarithmic divergence at the right edge whereas for all other systems it is defined at any coefficient $\alpha$ not smaller than one half. For both configurations, the lowest-energy level at $\alpha=1/2$ does saturate R\'{e}nyi and Tsallis entropic inequalities. Other properties are discussed and analyzed from the mathematical and physical points of view

R\'{e}nyi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells

A comparative analysis of the Dirichlet and Neumann boundary conditions (BCs) of the one-dimensional (1D) quantum well extracts similarities and differences of the R\'{e}nyi $R(\alpha)$ as well as Tsallis $T(\alpha)$ entropies between these two geometries. It is shown, in particular, that for either BC the dependencies of the R\'{e}nyi position components on the parameter $\alpha$ are the same for all orbitals but the lowest Neumann one for which the corresponding functional $R$ is not influenced by the variation of $\alpha$. Lower limit $\alpha_{TH}$ of the semi infinite range of the dimensionless R\'{e}nyi/Tsallis coefficient where {\em momentum} entropies exist crucially depends on the {\em position} BC and is equal to one quarter for the Dirichlet requirement and one half for the Neumann one. At $\alpha$ approaching this critical value, the corresponding momentum functionals do diverge. The gap between the thresholds $\alpha_{TH}$ of the two BCs causes different behavior of the R\'{e}nyi uncertainty relations as functions of $\alpha$. For both configurations, the lowest-energy level at $\alpha=1/2$ does saturate either type of the entropic inequality thus confirming an earlier surmise about it. It is also conjectured that the threshold $\alpha_{TH}$ of one half is characteristic of any 1D non-Dirichlet system. Other properties are discussed and analyzed from the mathematical and physical points of view.Comment: 7 figure

Comment on "On the realisation of quantum Fisher information"

It is shown that calculation of the momentum Fisher information of the quasione- dimensional hydrogen atom recently presented by Saha et al (2017 Eur. J. Phys. {\bf 38} 025103) is wrong. A correct derivation is provided and its didactical advantages and scientific significances are highlighted

Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

A theoretical analysis of the thermodynamic properties of the Robin wall characterized by the extrapolation length $\Lambda$ in the electric field $\mathscr{E}$ that pushes the particle to the surface is presented both in the canonical and two grand canonical representations and in the whole range of the Robin distance with the emphasis on its negative values which for the voltage-free configuration support negative-energy bound state. For the canonical ensemble, the heat capacity at $\Lambda<0$ exhibits a nonmonotonic behavior as a function of the temperature $T$ with its pronounced maximum unrestrictedly increasing for the decreasing fields as $\ln^2\mathscr{E}$ and its location being proportional to $(-\ln\mathscr{E})^{-1}$. For the Fermi-Dirac distribution, the specific heat per particle $c_N$ is a nonmonotonic function of the temperature too with the conspicuous extremum being preceded on the $T$ axis by the plateau whose magnitude at the vanishing $\mathscr{E}$ is defined as $3(N-1)/(2N)k_B$, with $N$ being a number of the particles. The maximum of $c_N$ is the largest for $N=1$ and, similar to the canonical ensemble, grows to infinity as the field goes to zero. For the Bose-Einstein ensemble, a formation of the sharp asymmetric feature on the $c_N$-$T$ dependence with the increase of $N$ is shown to be more prominent at the lower voltages. This cusp-like dependence of the heat capacity on the temperature, which for the infinite number of bosons transforms into the discontinuity of $c_N(T)$, is an indication of the phase transition to the condensate state. Qualitative and quantitative explanation of these physical phenomena is based on the variation of the energy spectrum by the electric field

Influence of the interplay between de Gennes boundary conditions and cubicity of Ginzburg-Landau equation on the properties of superconducting films

Exact solutions of the Ginzburg-Landau (GL) equation for the straight film subjected at its edges to the Robin-type boundary conditions characterized by the extrapolation length $\Lambda$ are analyzed with the primary emphasis on the interaction between the coefficient $\beta$ of the cubic GL term and the de Gennes distance $\Lambda$ and its influence on the temperature $T$ of the strip. Very substantial role is played also by the carrier density $n_s$ that naturally emerges as an integration constant of the GL equation. Physical interpretation of the obtained results is based on the $n_s$-dependent effective potential $V_{eff}({\bf r})$ created by the nonlinear term and its influence on the lowest eigenvalue of the corresponding Schr\"{o}dinger equation. In particular, for the large cubicities, the temperature $T$ becomes $\Lambda$ independent linearly decreasing function of the growing $\beta$ since in this limit the boundary conditions can not alter very strong $V_{eff}$. It is shown that the temperature increase, which is produced in the linear GL regime by the negative de Gennes distance, is wiped out by the growing cubicity. In this case, the decreasing $T$ passes through its bulk value $T_c$ at the unique density $n_s^{(0)}$ only, and the corresponding extrapolation length $\Lambda_{T=T_c}$ is an analytical function of $\beta$ whose properties are discussed in detail. For the densities smaller than $n_s^{(0)}$, the temperature stays above $T_c$ saturating for the large cubicities to the value determined by $n_s$ and negative $\Lambda$ while for $n_s>n_s^{(0)}$ the superconductivity is destroyed by the growing GL nonlinearity at some temperature $T>T_c$, which depends on $\Lambda$, $n_s$ and $\beta$. It is proved that the concentration $n_s^{(0)}$ transforms for the large cubicities into the density of the bulk sample.Comment: 26 pages, 8 figure

Comparative analysis of electric field influence on the quantum wells with different boundary conditions. II. Thermodynamic properties

Thermodynamic properties of the one-dimensional (1D) quantum well (QW) with miscellaneous permutations of the Dirichlet (D) and Neumann (N) boundary conditions (BCs) at its edges in the perpendicular to the surfaces electric field $\mathscr{E}$ are calculated. For the canonical ensemble, analytical expressions involving theta functions are found for the mean energy and heat capacity $c_V$ for the box with no applied voltage. Pronounced maximum accompanied by the adjacent minimum of the specific heat dependence on the temperature $T$ for the pure Neumann QW and their absence for other BCs are predicted and explained by the structure of the corresponding energy spectrum. Applied field leads to the increase of the heat capacity and formation of the new or modification of the existing extrema what is qualitatively described by the influence of the associated electric potential. A remarkable feature of the Fermi grand canonical ensemble is, at any BC combination in zero fields, a salient maximum of $c_V$ observed on the $T$ axis for one particle and its absence for any other number $N$ of corpuscles. Qualitative and quantitative explanation of this phenomenon employs the analysis of the chemical potential and its temperature dependence for different $N$. It is proved that critical temperature $T_{cr}$ of the Bose-Einstein (BE) condensation increases with the applied voltage for any number of particles and for any BC permutation except the ND case at small intensities $\mathscr{E}$ what is explained again by the modification by the field of the interrelated energies. It is shown that even for the temperatures smaller than $T_{cr}$ the total dipole moment $\langle P\rangle$ may become negative for the quite moderate $\mathscr{E}$. For either Fermi or BE system, the influence of the electric field on the heat capacity is shown to be suppressed with $N$ growing

Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic-field-induced restoration

Solutions of the scalar Helmholtz wave equation are derived for the analysis of the transport and thermodynamic properties of the two-dimensional disk and three-dimensional infinitely long straight wire in the external uniform longitudinal magnetic field $\bf B$ under the assumption that the Robin boundary condition contains extrapolation length $\Lambda$ with nonzero imaginary part $\Lambda_i$. As a result of this complexity, the self-adjointness of the Hamiltonian is lost, its eigenvalues $E$ become complex too and the discrete bound states of the disk characteristic for the real $\Lambda$ turn into the corresponding quasibound states with their lifetime defined by the eigenenergies imaginary parts $E_i$. Accordingly, the longitudinal flux undergoes an alteration as it flows along the wire with its attenuation/amplification being $E_i$-dependent too. It is shown that, for zero magnetic field, the component $E_i$ as a function of the Robin imaginary part exhibits a pronounced sharp extremum with its magnitude being the largest for the zero real part $\Lambda_r$ of the extrapolation length. Increasing magnitude of $\Lambda_r$ quenches the $E_i-\Lambda_i$ resonance and at very large $\Lambda_r$ the eigenenergies $E$ approach the asymptotic real values independent of $\Lambda_i$. The extremum is also wiped out by the magnetic field when, for the large $B$, the energies tend to the Landau levels. Mathematical and physical interpretations of the obtained results are provided; in particular, it is shown that the finite lifetime of the disk quasibound states stems from the $\Lambda_i$-induced currents flowing through the sample boundary. Possible experimental tests of the calculated effect are discussed; namely, it is argued that it can be observed in superconductors by applying to them the external electric field $\cal E$ normal to the surface

Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum-information measures

Information-theoretical concepts are employed for the analysis of the interplay between a transverse electric field $\mathscr{E}$ applied to a one-dimensional surface and Robin boundary condition (BC), which with the help of the extrapolation length $\Lambda$ zeroes at the interface a linear combination of the quantum mechanical wave function and its spatial derivative, and its influence on the properties of the structure. For doing this, exact analytical solutions of the corresponding Schr\"{o}dinger equation are derived and used for calculating energies, dipole moments, position $S_x$ and momentum $S_k$ quantum information entropies and their Fisher information $I_x$ and $I_k$ and Onicescu information energies $O_x$ and $O_k$ counterparts. It is shown that the weak (strong) electric field changes the Robin wall into the Dirichlet, $\Lambda=0$ (Neumann, $\Lambda=\infty$), surface. This transformation of the energy spectrum and associated waveforms in the growing field defines an evolution of the quantum-information measures; for example, it is proved that for the Dirichlet and Neumann BCs the position (momentum) quantum information entropy varies as a positive (negative) natural logarithm of the electric intensity what results in their field-independent sum $S_x+S_k$. Analogously, at $\Lambda=0$ and $\Lambda=\infty$ the position and momentum Fisher informations (Onicescu energies) depend on the applied voltage as $\mathscr{E}^{2/3}$ ($\mathscr{E}^{1/3}$) and its inverse, respectively, leading to the field-independent product $I_xI_k$ ($O_xO_k$). Peculiarities of their transformations at the finite nonzero $\Lambda$ are discussed and similarities and differences between the three quantum-information measures in the electric field are highlighted with the special attention being paid to the configuration with the negative extrapolation length.Comment: 10 Figures, 1 Tabl

Electric-Field Control of Bound States and Optical Spectrum in Window-Coupled Quantum Waveguides

Properties of the bound states of two quantum waveguides coupled via the window of the width $s$ in their common boundary are calculated under the assumption that the transverse electric field $\pmb{\mathscr{E}}$ is applied to the structure. It is shown that the increase of the electric intensity brings closer to each other fundamental propagation thresholds of the opening and the arms. As a result, the ground state, which in the absence of the field exists at any nonzero $s$, exhibits the energy $E_0$ decrease for the growing $\mathscr{E}$ and in the high-field regime $E_0$ stays practically the same regardless of the size of the connecting region. It is predicted that the critical window widths $s_{cr_n}$, $n=1,2,\ldots$, at which new excited localized orbitals emerge, strongly depend on the transverse voltage; in particular, the field leads to the increase of $s_{cr_n}$, and, for quite strong electric intensities, the critical width unrestrictedly diverges. This remarkable feature of the electric-field-induced switching of the bound states can be checked, for example, by the change of the optical properties of the structure when the gate voltage is applied; namely, both the oscillator strength and absorption spectrum exhibit a conspicuous maximum on their $\mathscr{E}$ dependence and turn to zero when the electric intensity reaches its critical value. Comparative analysis of the two-dimensional (2D) and 3D geometries reveals their qualitative similarity and quantitative differences.Comment: 21 pages, 12 figure

Magnetic field control of the intraband optical absorption in two-dimensional quantum rings

Linear and nonlinear optical absorption coefficients of the two-dimensional semiconductor ring in the perpendicular magnetic field $\bf B$ are calculated within independent electron approximation. Characteristic feature of the energy spectrum are crossings of the levels with adjacent nonpositive magnetic quantum numbers as the intensity $B$ changes. It is shown that the absorption coefficient of the associated optical transition is drastically decreased at the fields corresponding to the crossing. Proposed model of the Volcano disc allows to get simple mathematical analytical results, which provide clear physical interpretation. An interplay between positive linear and intensity-dependent negative cubic absorption coefficients is discussed; in particular, critical light intensity at which additional resonances appear in the total absorption dependence on the light frequency, is calculated as a function of the magnetic field and levels' broadening.Comment: 17 pages, 5 figure