Quantum information measures of the Aharonov-Bohm ring in uniform magnetic fields
O. Olendski




oolendski@sharjah.ac.ae
Department of Applied Physics and Astronomy, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates

Shannon quantum information entropies S_ρ,γ, Fisher informations I_ρ,γ, Onicescu energies O_ρ,γ and complexities e^SO are calculated both in the position (subscript ρ) and momentum (γ) spaces for the azimuthally symmetric two-dimensional nanoring that is placed into the combination of the transverse uniform magnetic field B and the Aharonov-Bohm (AB) flux ϕ_AB and whose  potential profile is modeled by the superposition of the quadratic and inverse quadratic dependencies on the radius r. The increasing intensity B flattens momentum waveforms Φ_nm( k) and in the limit of the infinitely large fields they turn to zero: Φ_nm( k)→0 at B→∞, what means that the position wave functions Ψ_nm( r), which are their Fourier counterparts, tend in this limit to the δ-functions. Position (momentum) Shannon entropy depends on the field B as a negative (positive) logarithm of ω_eff≡(ω_0^2+ω_c^2/4)^1/2, where  ω_0 determines the quadratic steepness of the confining potential and ω_c is a cyclotron frequency. This makes the sum S_ρ_nm+S_γ_nm a field-independent quantity that increases with the principal n and azimuthal m quantum numbers and does satisfy entropic uncertainty relation. Position Fisher information does not depend on m, linearly increases with n and varies as ω_eff whereas its n and m dependent Onicescu counterpart O_ρ_nm changes as ω_eff^-1. The products I_ρ_nmI_γ_nm and O_ρ_nmO_γ_nm are B-independent quantities. A dependence of the measures on the ring geometry is discussed. It is argued that a variation of the position Shannon entropy or Onicescu energy with the AB field uniquely determines an associated persistent current as a function of ϕ_AB at B=0. An inverse statement is correct too.


quantum ring Shannon information entropy Fisher informationOnicescu energymagnetic fieldAharonov-Bohm effect





§ INTRODUCTION

Magnetic field influence on the properties of the quantum rings (QRs) presents one of the most interesting problems in the physics of these artificial structures with nontrivial topology <cit.>. In addition to the traditional substances, such as superconductors <cit.>, normal metals <cit.> and semiconductors <cit.>, from which they are made, relatively new materials, such as graphene, are  being actively used in the fabrication of the QRs too <cit.>. An extreme configuration of the nonuniform field that is transverse to the ring plane is a geometry when the whole magnetic flux is concentrated in the centre of the QR. Despite the fact that this Aharonov-Bohm (AB) intensity <cit.> is not accessible for the charged quantum particles moving inside the ring, it strongly affects their properties leading, in particular, to the QR persistent currents <cit.> flowing along the azimuthal direction of the structure.

Among different characteristics of the nanosystems, a very special role is played by quantum-information measures, which are descriptors of miscellaneous facets of the distribution of position ρ( r) and wave vector γ( k) densities that are squared magnitudes of the corresponding waveforms Ψ_n( r) and Φ_n( k):



    ρ_n( r)   =|Ψ_n( r)|^2
    γ_n( k)   =|Φ_n( k)|^2,

 

where nonnegative integer index n counts all bound orbitals of the structure, and functions Ψ_n and Φ_n are related through the Fourier transformations:



    Φ_n( k)   =1/(2π)^l/2∫_ℝ^lΨ_n( r)e^-i krd r,
    Ψ_n( r)   =1/(2π)^l/2∫_ℝ^lΦ_n( k)e^i rkd k,
 


with the integration carried out over the whole available l-dimensional position, Eq. (<ref>), or momentum, Eq. (<ref>), region under the assumption that the waveforms are orthonormalized:


    ∫_ℝ^lΨ_n'^*( r)Ψ_n( r)d r=∫_ℝ^lΦ_n'^∗( k)Φ_n( k)d k=δ_nn',


where δ_nn'={[    1,  n=n';    0, n≠ n' ]. is a Kronecker delta, n,n'=0,1,2,….

Position S_ρ and momentum S_γ Shannon quantum-information entropies defined as



    S_ρ   =-∫_ℝ^lρ( r)lnρ( r)d r
    
    S_γ   =-∫_ℝ^lγ( k)lnγ( k)d k,



describe quantitatively the lack of our knowledge about location and motion of the particle: the smaller (larger) either of them is, the more (less) information is available about the corresponding property. These two functionals are not independent from each other but are related as <cit.>


    S_ρ+S_γ≥ l(1+lnπ).


A comparison of this inequality with the Heisenberg uncertainty relation reveals that Eq. (<ref>) presents a much more general base for defining 'uncertainty' <cit.> since the former always holds true whereas the latter is either violated or meaningless, among others, for the structures with non-Dirichlet boundary conditions <cit.>. The significance of the Shannon entropies stimulates their intensive detailed analysis for different geometries, such as one-dimensional (1D) harmonic oscillator <cit.> and its multidimensional generalizations <cit.>, 3D <cit.> and multidimensional hydrogen atom <cit.>, miscellaneous 1D potentials: one or two Dirac δ-functions <cit.>, Pöschl-Teller <cit.>, hyperbolic <cit.>, squared tangent <cit.>, symmetric <cit.> and asymmetric <cit.> trigonometric Rosen-Morse potentials, systems with position-dependent mass <cit.>, and many, many others. The literature of the Shannon entropies, as well as other measures mentioned below, is growing at a very impressive rate.

Expressions for the position I_ρ and momentum I_γ Fisher informations <cit.> read:



    I_ρ   =∫_ℝ^l|∇_ rρ( r)|^2/ρ( r)d r
    
    I_γ   =∫_ℝ^l|∇_ kγ( k)|^2/γ( k)d k.



A presence of the gradients makes these functionals local measures of uncertainty that are sensitive to the speed of change of the corresponding density. Contrary to the sum of the Shannon entropies, the product of two Fisher informations does not have a rigorous lower bound: widely discussed inequality 


    I_ρ I_γ≥4


is violated, e.g., for the non-Dirichlet systems <cit.>.

Physical meaning of the Onicescu energies <cit.>, or disequilibria,



    O_ρ   =∫_ℝ^lρ^2( r)d r
    
    O_γ   =∫_ℝ^lγ^2( k)d k



lies in the fact that they describe deviations from the uniform distributions.

Based on these functionals, some combined measures could be defined; for example, a complexity


    CGL=e^SO


introduced by R. G. Catalán, J. Garay and R. López-Ruiz <cit.> tries to simultaneously quantify an uncertainty represented by the first factor as well as disequilibrium due to the presence of the Onicescu energy. Remarkably, for any l-dimensional space either position or momentum component of this dimensionless quantity can not be smaller than unity <cit.>


    CGL≥1.



In this Letter, an exact analysis of the above mentioned quantum-information measures is provided for the QR in the superposition of the transverse uniform magnetic field B and the Aharonov-Bohm flux ϕ_AB. For the azimuthally symmetric 2D potential profile U( r) of the ring [with r=(r,φ_r) being position polar coordinates with the origin of this frame of reference coinciding with the geometrical center of the structure] we use a combination of the quadratic and inverse quadratic depedendencies on the radius r <cit.>:


    U(r)=1/2m^*ω_0^2r^2+ħ^2/2m^*r^2 a-ħω_0a^1/2,


where m^* is an effective mass of a charge carrier, frequency ω_0 defines a steepness of the confining in-plane outer surface of the QR with its characteristic radius r_0=[ħ/(2m^*ω_0)]^1/2, and positive dimensionless constant a describes a strength of the repulsive potential near the origin. The last negative item in Eq. (<ref>) guarantees that the sole minimum of the potential that is achieved at 


    r_min=2^1/2a^1/4r_0


is zero <cit.>. It is shown that for each quantum orbital the sum of the Shannon components and products of the Fisher (Onicescu) position and momentum informations (energies) do not depend on the uniform intensity since the change of one item is exactly compensated by the opposite variation with B of the conjugate counterpart. In vein of our discussion above it means, in particular, that the total amount of information that one has about the position and momentum of the nanoparticle can not be altered by the uniform magnetic field. In the framework of this model, some analytic results can be derived and construed from physical point of view; among others, a simple expression for the position Fisher information, Eq. (<ref>), manifests that it does not depend on the azimuthal quantum number and its product with the momentum component is a B-independent quantity, as is the multiplication of the two parts of the Onicescu energy too. It is deducted that the position Shannon entropy and the energy spectrum depend on the AB flux in a very similar way what can be used for deducing one of them from the knowledge about the second item. Possible generalization of the obtained results is also discussed.



§ FORMULATION AND DISCUSSION



 §.§ Position and momentum waveforms






We start our discussion by noting that the variation of the parameters of the potential from Eq. (<ref>) allows one to describe different structures; first, as pointed out in the Introduction, for aω_0≠0 it represents an isolated ring of the finite width and average radius from Eq. (<ref>). Next, a model with a=0 corresponds to the quantum dot (QD) whereas an absence of the external boundary, ω_0=0, portrays a quantum antidot <cit.> with its repulsive ability being proportional to the dimensionless parameter a. Moreover, the limit ω_0→∞ with the requirement of the constant radius r_min reduces the problem to the 1D ring whereas a suitable form for the 2D straight wire is given by the condition of the constant ω_0 and unrestrictedly enlarging r_min. It is known that the QR with the small antidot strength describes a "thick" ring whereas the structure with the large a is deemed as a "thin" annulus <cit.>. A huge advantage of the theoretical consideration of Eq. (<ref>) is due to the fact that it allows an exact (and quite simple and tractable, as is re-shown below) analytical solutions for the energy spectrum and position functions. Despite its simplicity, such representation of the potential correctly explains experimental data; in particular, it was successful in construing the beating effect in the oscillation pattern <cit.> and a magnitude of the persistent current in a single GaAs/Al_xGa_1-xAs loop <cit.>. Fig. <ref>(a) schematically depicts the azimuthally symmetric profile U(r) in terms of the Cartesian components x=rcosφ_r and y=rsinφ_r for the QR and a comparison with its QD counterpart, window (b), explicitly shows that any non-zero antidot strength converts the geometry from the singly-connected configuration at a=0 to the doubly-connected one. Panel (c) demonstrates a transformation of the QD, a=0, through the "thick" ring (finite small and moderate a) to the "thin" one at a→∞.

In addition to the electrostatic potential, the 2D charged particle motion  is influenced by the magnetic field B_tot that is directed perpendicular to the (x,y)-plane, B_tot=B_tot k, and presents a superposition of the uniform component B and extremely localized AB part <cit.>:


    B_tot=B+ϕ_ABδ(x)δ(y)=B+ϕ_ABδ(r)/π r,


where δ(ξ) is a 1D δ-function and ϕ_AB is the AB flux concentrated at the QR origin only.

For finding energies E and position waveforms Ψ( r), one needs to address the 2D Schrödinger equation


    [1/2m^*(-iħ∇_ r+e A_tot)^2+U(r)]Ψ( r)=EΨ( r)


(here, e is an absolute value of the electronic charge), that is most conveniently solved in the polar coordinates introduced above; in particular, in the symmetric gauge, the vector potential A_tot( r) that describes the influence of the magnetic fields on a charged particle via the relation B_tot=∇_ r× A_tot has a nonzero azimuthal component only which depends solely on the distance from the center:


    A_tot_r=0,  A_tot_φ_r(r)=1/2Br+ϕ_AB/2π r.


Here, the first item in A_tot_φ_r is due to the uniform intensity B of the total field from Eq. (<ref>) and the AB flux yields the inverse-distance-like dependence in Eq. (<ref>). Note that this zero-divergence potential satisfies the Stokes theorem:


    ∮_C A_tot· d l=F_C,


where d l is an elementary displacement along the closed arbitrary contour C and F_C is a net magnetic flux piercing it.

Then, as a result of representation from Eq. (<ref>), the normalized, Eq. (<ref>),


    ∫_0^2πdφ_r∫_0^∞ dr rΨ_n'm'^*(r,φ_r)Ψ_nm(r,φ_r)=δ_nn'δ_mm',


position wave function is separated as a product of the angular and radial dependencies:



    Ψ_nm(r,φ_r)=1/(2π)^1/2e^imφ_rR_nm(r)
    with the latter one being expressed as:   R_nm(r)=1/r_eff[n!/Γ(n+λ+1)]^1/2exp(-1/4r^2/r_eff^2)(1/2r^2/r_eff^2)^λ/2L_n^λ(1/2r^2/r_eff^2),



where Γ(x) is Γ-function <cit.>, L_n^α(x) is a generalized Laguerre polynomial  <cit.>,



    r_eff   =(ħ/2m^*ω_eff)^1/2
    ω_eff   =(ω_0^2+1/4ω_c^2)^1/2andω_c   =eB/m^*is the cyclotron frequency,λ   =(m_ϕ^2+a)^1/2
    
    m_ϕ   =m+νwith ν being a dimensionless AB flux, i.e., the latter one is expressed in units of the elementary flux quantum ϕ_0=h/e:ν   =ϕ_AB/ϕ_0,



and n=0,1,…, and m=0,±1,…, are the principal and magnetic quantum numbers, respectively. Corresponding energy spectrum reads:


    E_nm(a,ν;ω_c)=ħω_eff(2n+λ+1)+1/2m_ϕħω_c-ħω_0a^1/2.
 

For completeness, expressions for the persistent current J_nm and magnetization M_nm are given below too:



    J_nm   ≡-e/h∂ E/∂ m≡-∂ E/∂ϕ_AB=-eω_0/2π[m_ϕ/λ√(1+1/4(ω_c/ω_0)^2)+1/2ω_c/ω_0]
    
    M_nm   ≡-∂ E/∂ B=-eħ/2m^*[1/2ω_c/ω_eff(2n+λ+1)+m_ϕ].



Note that Eqs. (<ref>), (<ref>) and (<ref>) stay invariant under the transformation


    m→ m-1,  ν→ν+1;


accordingly, for analyzing the properties of these physical quantities, it is sufficient to study the AB flux inside the range


    -1/2≤ν≤1/2.


Observe also that at the zero uniform field, B=0, edges of this interval are characterized by the two infinite sets of the degeneracy of the levels:



    E_nm(a,-1/2;0)   =E_n,-m+1(a,-1/2;0)
    
    E_nm(a,1/2;0)   =E_n,-m-1(a,1/2;0).




For finding momentum quantum-information measures, one needs to evaluate first the corresponding waveform, which, according to Eq. (<ref>), is:


    Φ_nm( k)≡Φ_nm(k,φ_k)=1/2π∫_0^2πdφ_r∫_0^∞drrR_nm(r)e^i[mφ_r-krcos(φ_r-φ_k)].


Azimuthal integration is carried out analytically what allows to represent the function Φ as a product of the angular and radial dependencies too:



    Φ_nm(k,φ_k)=(-i)^m/(2π)^1/2e^imφ_kK_nm(k),
    where the latter factor is:   K_nm(k)=r_eff[n!/Γ(n+λ+1)]^1/2∫_0^∞e^-z/2z^λ/2L_n^λ(z)J_|m|(2^1/2r_effkz^1/2)dz,



where J_m(ξ) is m-th order Bessel function <cit.>. Analytic expressions for the integrals of this type are missing in known literature <cit.>; accordingly, below their direct numerical quadrature was employed. The only exception is the ground band, n=0, of the QD, a=0, without the AB flux, ϕ_AB=0 (when, accordingly, λ=|m|):



    .K_0m(k)|_a=ν=0   =r_eff2^|m|/2+1/(|m|!)^1/2(r_effk)^|m|exp(-r_eff^2k^2);
    in particular:.K_00(k)|_a=ν=0   =2r_effexp(-r_eff^2k^2).



Comparison of the last formula with Eq. (<ref>) shows that these are position and momentum waveforms of the lowest-energy orbital of the 2D harmonic oscillator what leads, as will be shown below, to the saturation of the entropic uncertainty relation, Eq. (<ref>).





Figs. <ref> and <ref> depict position R_nm(r) and momentum K_nm(k) waveforms, respectively, as functions of the normalized cyclotron frequency ω_c/ω_0 with zero AB flux, ϕ_AB=0, for several n and m. As it follows from the general laws of quantum mechanics <cit.>, the principal index n determines a number of nodes of the radial dependence R_nm (excluding the points at the origin, r=0, and infinity, r=∞) whereas the higher absolute values of the azimuthal index correspond to the location of the electric charge further away from the origin. Increasing field pushes the particle closer to the geometrical center of the ring with the simultaneous increase of the function R_nm(r) extrema. In the wave vector domain, each dependence K_nm(k) with |m|≥1 is characterized by n+1 extrema whereas its K_n0(k) counterpart has at k=0 an additional minimum or maximum with the largest amplitude. As it follows from Eq. (<ref>), in the limit of the infinitely strong magnetic intensities, the waveform Φ_nm( k) turns to zero:



    Φ_nm( k)   0.
    This flattening of the radial parts of the momentum functions is clearly seen in Fig. <ref>. This also means that its Fourier transform Ψ_nm( r) tends in the same limit to the 2D δ-function:Ψ_nm( r)   δ( r),



what physically means an extremely strong field-induced squeezing of the charge near the origin.











 §.§ Measures in uniform magnetic field B

Knowledge of the waveforms Ψ_nm( r) and Φ_nm( k) and, accordingly, of the corresponding densities ρ_nm( r) and γ_nm( k) paves the way to calculating quantum-information measures; for example, position and momentum Shannon entropies are:



    S_ρ_nm=2ln r_eff+ln2π-lnn!/Γ(n+λ+1)
       -n!/Γ(n+λ+1)∫_0^∞e^-zz^λ L_n^λ(z)^2ln(e^-zz^λ L_n^λ(z)^2)dz
       S_γ_nm=-2ln r_eff+ln2π-lnn!/Γ(n+λ+1)
       -n!/Γ(n+λ+1)∫_0^∞dξξ[∫_0^∞e^-z/2z^λ/2L_n^λ(z)J_|m|(2^1/2ξ z^1/2)dz]^2
       ×ln([∫_0^∞e^-z/2z^λ/2L_n^λ(z)J_|m|(2^1/2ξ z^1/2)dz]^2).
    These expressions manifest that the whole dependence of the position (momentum) entropy on the magnetic intensity B is reflected in the negative (positive) term ln(1+1/4ω_c^2/ω_0^2) only what means that their sum is a field-independent quantity:   S_ρ_nm+S_γ_nm=2ln2π-2lnn!/Γ(n+λ+1)
       -n!/Γ(n+λ+1)∫_0^∞e^-zz^λ L_n^λ(z)^2ln(e^-zz^λ L_n^λ(z)^2)dz
       -n!/Γ(n+λ+1)∫_0^∞ dξξ[∫_0^∞e^-z/2z^λ/2L_n^λ(z)J_|m|(2^1/2ξ z^1/2)dz]^2
       ×ln([∫_0^∞e^-z/2z^λ/2L_n^λ(z)J_|m|(2^1/2ξ z^1/2)dz]^2).



Thus, an amount of the total information that is available simultaneously about position and motion of the nanoparticle can not be altered by the uniform magnetic fields. For the strong B, position (momentum) Shannon entropy decreases (increases) as ∓lnω_c, what is consistent with the interpretation of the entropies as measures of uncertainty: in this regime, we know more (less) about particle location (momentum), as it follows from Figs. <ref> and <ref>; accordingly, the corresponding measure gets smaller (larger) tending to the negative (positive) infinity. However, the loss at the growing fields of the information about the electron momentum is exactly compensated by the extra knowledge we acquire about its position. As a result, the sum of the two entropies does not depend on B. For the ground band, n=0, the position component can be calculated analytically:



    S_ρ_0m   =2ln r_eff+1+ln2π+lnΓ(λ+1)+λ[1-ψ(λ+1)],
    where ψ(x)=d[lnΓ(x)]/dx=Γ'(x)/Γ(x) is psi, or digamma function, which is a derivative of the logarithm of the Γ-function <cit.>. Asymptotic form of this expressionS_ρ_0m   →2ln r_eff+3/2ln2π+1/2+1/2lnλ,  λ→∞,
    shows that at the very large magnitudes of the magnetic indices the entropy S_ρ_0m increases as 1/2ln|m|. In the opposite limit of the azimuthally symmetric state, m=0, the QD (a=0) entropy without the AB flux, ϕ_AB=0, is:.S_ρ_00|_a=ν=0   =2ln r_eff+1+lnπ+ln2.



On the other hand, with the help of the function from Eq. (<ref>) one evaluates the momentum entropy as



    .S_γ_0m|_a=ν=0   =-2ln r_eff+1+ln(|m|!/2^|m|+1π)-2^|m|+1|m|ψ(|m|+1);
    in particular:.S_γ_00|_a=ν=0   =-2ln r_eff+1+lnπ-ln2.



Comparing Eqs. (<ref>) and (<ref>), one sees that the QD ground state does saturate the entropic uncertainty relation, Eq. (<ref>), regardless of the magnetic intensity B. As mentioned above, this takes place since the corrresponding position and momentum dependencies are those of the 2D harmonic oscillator. 





Solid line in Fig. <ref> depicts a variation of the field-indepdendent quantity S_ρ_00+S_γ_00 with the strength of the antidot a for the zero AB flux, ϕ_AB=0. At the small a, corresponding to the "thick" ring <cit.>, the sum increases quite rapidly from its threshold QD value 2(1+lnπ)=4.2894… whereas as the annulus  gets thinner <cit.>, the growth of the total entropy slows down. Different behaviour is  observed for the sum of the two entropies of the azimuthally asymmetric states, m≠0; namely, as Fig. <ref> demonstrates, it has a pronounced minimum whose location on the a axis for the greater quantum indices |m| is shifted to the right with the simultaneous broadening of this extremum. Entropy dependencies depicted in Figs. <ref> and <ref> for the ground band, n=0, are qualitatively repeated for the orbitals with the nonzero principal numbers n.





Table <ref> lists the sums S_ρ_nm+S_γ_nm for n ranging from zero to five and |m|=0..5 and 10 for the QR with a=20. It is seen that the total entropy is an increasing function of both the principal as well as magnetic quantum numbers.

A remarkable property of the position Fisher information


    I_ρ_nm=8/r_eff^2n!/Γ(n+λ+1)∫_0^∞e^-zz^λ+1[1/2(λ/z-1)L_n^λ(z)-L_n-1^λ+1(z)]^2dz


is the fact that it does not depend on the azimuthal quantum number and the AB flux:



    I_ρ_nm=4n+2/r_eff^2,
    as a direct evaluation <cit.> of Eq. (<ref>) reveals. Momentum Fisher information reads:   I_γ_nm=r_eff^28n!/Γ(n+λ+1)∫_0^∞dξξ[∫_0^∞e^-z/2z^(λ+1)/2L_n^λ(z)J_|m|'(2^1/2ξ z^1/2)dz]^2,
    what makes the product of the two, similar to the sum of the Shannon entropies, a field-independent quantity:   I_ρ_nmI_γ_nm=8(4n+2)n!/Γ(n+λ+1)∫_0^∞dξξ[∫_0^∞e^-z/2z^(λ+1)/2L_n^λ(z)J_|m|'(2^1/2ξ z^1/2)dz]^2.



One can interpret the m-independence of the position Fisher information by the fact that this quantum number determines a spatial localization of the particle but not a number of oscillations that are described by the principal index n, as stated above. Increasing magnetic intensity sharpens waveforms R_nm(r), as Fig. <ref> vividly demonstrates, what leads to the linear growth of I_ρ with ω_eff that at the high B reduces to the linear dependence of the position component on the field. Simultaneously, the slope of the variation of the momentum functions K_nm(k) becomes more gentle approaching in the limit B→∞ the (zero) uniform k-independent value, Eq. (<ref>), what means a vanishing of the momentum Fisher information. However, the product of the two stays the same since the increasing intensity  of the oscillations of the position waveform Ψ_nm( r) is exactly compensated by the flattening of its momentum counterpart. Using Eq. (<ref>), one derives the momentum Fisher information of the ground band of the QD as


    .I_γ_0m|_a=ν=0=8r_eff^2,


which is again a m-independent measure. Dashed line in Fig. <ref> shows that the quantity I_ρ_00I_γ_00 increases with the transformation of the QR from the 'thick' to the 'thin' geometry. Our results also reveal that for the greater |m| a slope of the I_ρ_0mI_γ_0m dependence on a does decrease. In addition, an analysis of the products I_ρ_nmI_γ_nm that are provided in Table <ref> uncovers that it increases with the quantum number n and gets smaller for the growing |m|.





Expressions for the position



    O_ρ_nm   =1/r_eff^21/2π[n!/Γ(n+λ+1)]^2∫_0^∞ e^-2zz^2λ[L_n^λ(z)]^4dz
    and momentumO_γ_nm   =r_eff^2/2π[n!/Γ(n+λ+1)]^2∫_0^∞dξξ[∫_0^∞e^-z/2z^λ/2L_n^λ(z)J_|m|(2^1/2ξ z^1/2)dz]^4



QR Onicescu energies demonstrate that their dependencies on the uniform field are the same as for the Fisher informations what makes the product O_ρ_nmO_γ_nm a B-independent quantity again. A growth (lessening) with the field of the position (momentum) component is explained by the increasing (decreasing) nonuniformity of the function R_nm(r) [K_nm(k)] in this regime. Similar to the previous functionals, the variation of one measure is exactly counterbalanced by the change in the opposite direction of the second fellow in such a way that their product stays intact by the uniform field. Data in Table <ref> show that O_ρ_nmO_γ_nm is a decreasing function of either of the quantum indices. For n=0, Eq. (<ref>) yields:



    O_ρ_0m   =1/r_eff^21/4π^3/2Γ(λ+1/2)/Γ(λ+1),
    and its asymptoteO_ρ_0m   →1/r_eff^21/4π^3/21/λ^1/2,  λ→∞,



shows that at the large |m| the position Onicescu energy fades as |m|^-1/2. For the QD without the AB flux, its momentum counterpart reads as



    .O_γ_0m|_a=ν=0   =r_eff^2/π(2|m|)!/2^2|m|(|m|!)^2with its limit at the huge |m| being.O_γ_0m|_a=ν=0   →r_eff^2/π^3/21/|m|^1/2,  |m|→∞.



Solid line in Fig. <ref> exemplifies that the field-independent product O_ρ_00O_γ_00 smoothly decreases as the QR becomes thinner; however, its m≠0 companions exhibit the maximum in their a dependence, which at the larger |m| is shifted further to the right with its simultaneous broadening (not shown here). These features are qualitatively reproduced for the n≠0 orbitals.

Comparing Eqs. (<ref>) and (<ref>), one sees that both position and momentum parts of the complexity CGL do not depend on the uniform field either. Their values are provided in the corresponding columns of Table <ref>. As expected, each of them is greater than unity. Dependence of the n=0, m=0 complexities on the QAD strength is shown in Fig. <ref> by the dashed, dotted and dash-dotted lines for CGL_ρ, CGL_γ and their product, respectively, under the assumption that the AB flux is zero. For the QD, both position and momentum complexities have the value of e/2=1.3591… with their product e^2/4=1.8472…. The former one quite precipitously decreases as a gets stronger approaching at a→∞ the limit of (e/2)^1/2=1.1658…; however, the growth of the momentum part guarantees that their product smoothly increases as the QR becomes thinner.



 §.§ AB measures





Turning to the discussion of the measures' dependence on the AB flux, first let us write Taylor expansions of the position Shannon entropy, Eq. (<ref>), and Onicescu energy, Eq. (<ref>), with respect to the AB parameter ν for the azimuthally symmetric orbital, m=0:


    S_ρ_00(ν)   =2ln r_eff+ln2π+1+lnΓ(a^1/2+1)+a^1/2[1-ψ(a^1/2)]
       +1/2[1/a^1/2+1/a-ψ^(1)(a)]ν^2
    O_ρ_00(ν)   =1/r_eff^21/4π^3/2Γ(a^1/2+1/2)/Γ(a^1/2+1)(1-1/2a^1/2[ψ(a^1/2+1)-ψ(a^1/2+1/2)]ν^2),


where the series have been truncated at the first nonvanishing power of the flux and ψ^(n)(x)=d^nψ(x)/dx^n, n=1,2,… is a polygamma function <cit.>. It can be shown that the expression in the second square brackets of the right-hand side of Eq. (<ref>) is always positive what makes the entropy of the corresponding level a convex function of the AB flux. On the other hand, since ψ(x) is an increasing function of its positive argument <cit.>, Onicescu energy O_ρ_00 is a concave function of ν. At the quite large a, utilizing properties of the polygamma function <cit.>, one has:


    S_ρ_00(ν)   =2ln r_eff+3/2ln2π+1/2+1/4ln a+1/4a(1-1/3a^1/2)ν^2, a≫1,
    O_ρ_00(ν)   =1/r_eff^21/4π^3/2a^1/4[1-1/4a(1-1/4a^1/2)ν^2],   a≫1.


These equations manifest that the opposite directions of the S_ρ_00-ν and O_ρ_00-ν characteristics have almost the same magnitude of their inclines. Comparing them with the Taylor expansion of the energy, Eq. (<ref>), at the zero uniform magnetic field:


    E_n0(a,ν;0)=ħω_0(2n+1+1/2a^1/2ν^2),


one sees that both the entropy and (normalized in units of ħω_0) energy are the AB convex functions with, however, different slopes.

Panels (a) and (b) of Fig. <ref> provide a comparative analysis of the energy spectrum and position Shannon entropy dependencies on the AB field at the zero uniform component, B=0. A very strong similarity between the two is clearly seen; for example, for the ground level, m=0, if both E_00 and S_ρ_00 are appropriately scaled, the two lines practically coincide with each other. Thus, for evaluating the persistent current, Eq. (<ref>), one can use, in addition to (or instead of) the speed of change of the energy levels, the corresponding rate of variation of the associated position Shannon entropies multiplied by the normalizing coefficient -eω_0a^1/2/π. Of course, for getting the information on the persistent current, one can also use the Onicescu energy too but the related factor will have the opposite sign (and different value) as that for the entropy. For the inverse problem, measuring the current J_nm(ν), one can deduce what are the Shannon entropy and Onicescu energy of the corresponding orbital. Observe that S_ρ_nm, similar to the energy spectrum, persistent current and magnetization, stays invariant under the transformation from Eq. (<ref>). The same is true for the position Onicescu energy. In addition, the degeneracy of the energies expressed by Eqs. (<ref>), is inherited by S_ρ_nm too. However, as Eqs. (<ref>), (<ref>) and (<ref>) show, it is not the case for the momentum components of all three measures. This remarkable difference is traced back to the corresponding waveforms, Eqs. (<ref>) and (<ref>), respectively. Momentum Shannon entropies are depicted in Fig. <ref>(c). It is seen that S_γ_00 is a concave function of the AB flux and that at ν=-1/2 (ν=1/2) it does not turn into S_γ_0,+1 (S_γ_0,-1), as just described above. For all other orbitals, the direction of change of the momentum Shannon entropy is opposite to that of its position counterpart too but since the relative magnitude of the former variation is smaller, the sum of the two entropies repeats as a function of ν the behaviour of the position component, as panel (d) of Fig. (<ref>) demonstrates. Obviously, the invariance of S_ρ_nm+S_γ_nm under the transformation from Eq. (<ref>) is not conserved either.





Panel (a) of Fig. <ref> that exhibits position Onicescu energy confirms our analytic result about the concavity of the function O_ρ_00(ν), Eq. (<ref>). Similar to this orbital, for any other level the direction of change of O_ρ_nm is opposite to that of the position entropy too, cf. Figs. <ref>(b) and <ref>(a). As was the Shannon case, the deviation with ν of the  momentum Onicescu component is inverse to that of the position one; in particular, as window (b) of Fig. <ref> shows, O_γ_00(ν) is a convex function of the flux; however, the product O_ρ_00(ν)O_γ_00(ν) demonstrates a concave behaviour again (not shown here).





Dependence of the complexities CGL_ρ_0m and CGL_γ_0m on the AB flux is illustrated in panels (a) and (b) of Fig. <ref>, respectively. It is seen that qualitatively they reproduce the Onicescu energies behaviour. As it follows from Eqs. (<ref>) and (<ref>), the ground-state position part for the thin rings is a concave function of ϕ_AB:


    CGL_ρ_00(ν)=(e/2)^1/2(1-ν^2/48a^3/2),   a≫1.


For the larger ring radius, the variation of the complexities with the flux ϕ_AB, similar to all other measures, is being suppressed; for example, at a→∞ the lowest level position component approaches, as it directly follows from Eq. (<ref>), the value of (e/2)^1/2, irrespectively of the AB intensity.





Next, as Fig. <ref> demonstrates, momentum Fisher information I_γ_00 has its minimum at the zero AB flux and, similar to all other m=0 measures is symmetric with respect to ν=0 whereas for the orbitals with the positive (negative) magnetic index m it increases (decreases) with ν. In addition, as mentioned above, the degeneracy expressed by equations (<ref>) for the energies, does not hold for the momentum component I_γ_nm. Since the position components depend neither on B nor ϕ_AB, all these properties are also characteristic of the product I_ρ_nmI_γ_nm.





As a last remark, let us point out that in the presence of the uniform magnetic intensity, B≠0, the quantum-information measures preserve the shape of their B=0 dependence on the AB flux, as is seen, for example, from Eqs. (<ref>),  (<ref>) and (<ref>) for the Shannon entropy, Onicescu energy and complexity. However, the structure of the energy spectrum is drastically modified by the field B; namely, since it introduces the preferred azimuthal direction, the energies E_n0 are not anymore symmetric functions of ϕ_AB demonstrating instead at the quite strong B a continuous increase with the dimensionless flux ν. The same is correct for the orbitals with the nonzero magnetic index m too. A representative example of the field such that ω_c/ω_0=20 is shown in Fig. <ref>. The invariance under the transformation from Eq. (<ref>) is not destroyed by the uniform field but the degeneracies from Eqs. (<ref>) are lifted by it.



§ CONCLUSIONS

Analysis of the quantum-information measures of the nanostructure requires a correct evaluation of the corresponding one-particle waveforms. Their position components Ψ_nm( r) for the 2D electron in the QR subject to the superposition of the transverse uniform magnetic field and the AB flux have been known very well <cit.>. Above, momentum wave functions Φ_nm( k) have been calculated straightforwardly, Eq. (<ref>), as Fourier transforms of their position counterparts, Eq. (<ref>). It was shown, in particular, that the increasing intensity B subdues oscillations of the radial part of Φ_nm( k) transforming them in the limit of the infinitely strong fields into the flat zero-amplitude surface, Eq. (<ref>). Importantly, in the case of the AB-free QD the expression for the ground-band momentum waveform is nothing else but the modified Gaussian, Eq. (<ref>), what means, that its lowest orbital, m=0, does saturate Shannon entropy uncertainty relation, Eq. (<ref>). Obviously, all other states satisfy this relation too as an inequality in which the sum of the two Shannon entropies S_ρ_nm+S_γ_nm is a B-independent quantity since the first and second items in it contain ±ln(1+1/4ω_c^2/ω_0^2), respectively. Other main findings include independencies of the position Fisher information on the azimuthal quantum number m and of the products I_ρ_nmI_γ_nm and O_ρ_nmO_γ_nm and both position CGL_ρ_nm and momentum CGL_γ_nm parts of the complexity on the uniform field B. A comparison of the analytic expressions at B=0 of the dependencies of the ground-state position Shannon entropy, Onicescu energy and  the energy spectrum on the AB flux reveals a strong similarity of the former and latter variations what can be used in deducing the magnitude of either of them if the other quantity is known; in particular, from the measurement of the persistent current one can infer what Shannon entropy and Onicescu energies are.

In addition to the measures discussed above, in a similar way the quantities



    ⟨ r^2⟩   =∫_ℝ^lr^2ρ( r)d r
    ⟨ k^2⟩   =∫_ℝ^lk^2γ( k)d k



that enter the uncertainty relation <cit.>


    √(⟨ r^2⟩)√(⟨ k^2⟩)≥|m|+1


can be considered too; in particular, explicit evaluation yields for the position component:


    ⟨ r^2⟩=2(2n+λ+1)r_eff^2.


Its momentum counterpart



    ⟨ k^2⟩=1/r_eff^2n!/Γ(n+λ+1)∫_0^∞dξξ^3[∫_0^∞e^-z/2z^λ/2L_n^λ(z)J_|m|(2^1/2ξ z^1/2)dz]^2
    can be derived analytically for the lowest band, n=0, of the QD (a=0) without the AB flux:   .⟨ k^2⟩_0m|_a=ν=0=1/r_eff^2|m|+1/2.



From Eqs. (<ref>) and (<ref>) it is seen that the product ⟨ r^2⟩⟨ k^2⟩ is a B-independent quantity again. Let us also note that the lowest band of the AB-free QD does saturate the uncetainty relation, Eq. (<ref>), since the position and momentum waveforms represent in this case modified Gaussian functions <cit.>, see, e.g., Eq. (<ref>).

Finally, let us point out that the wave functions introduced here can be used in the analysis of the Rényi <cit.> R_ρ,γ and Tsallis <cit.> T_ρ,γ entropies whose expressions in terms of the densities ρ( r) and γ( k) are:




    R_ρ(α)   =1/1-αln(∫_ℝ^lρ^α( r)d r)
    
    R_γ(α)   =1/1-αln(∫_ℝ^lγ^α( k)d k)
    
    T_ρ(α)   =1/α-1(1-∫_ℝ^lρ^α( r)d r)
    
    T_γ(α)   =1/α-1(1-∫_ℝ^lγ^α( k)d k).

 

For each bound quantum orbital, the position and momentum components of these one-parameter functionals are not independent from each other but obey the following uncertainties <cit.>:



    R_ρ(α)+R_γ(β)≥-l/2(1/1-αlnα/π+1/1-βlnβ/π)
       (α/π)^l/(4α)[1+(1-α)T_ρ(α)]^1/(2α)≥(β/π)^l/(4β)[1+(1-β)T_γ(β)]^1/(2β),



where the non-negative coefficients α and β obey the constraint:


    1/α+1/β=2,


which for the case of the Tsallis relation, Eq. (<ref>), has to be supplemented by the additional requirement:


    1/2≤α≤1.


In the limit α→1 (and, accordingly, β→1) Eqs. (<ref>) and (<ref>) degenerate into their Shannon counterparts, Eqs. (<ref>) and (<ref>), respectively, whereas the Tsallis inequality turns into the identity with each of its sides equal to π^-l/4. Recently, it was conjectured <cit.> that the lowest orbital of the l-dimensional quantum structure in the limit α=1/2 transforms Rényi, Eq. (<ref>), and Tsallis, Eq. (<ref>),  uncertainty relations into the equalities. One can check this conjecture for the AB-free (ν=0) QD (a=0) lowest band (n=0) for which, for example, Rényi entropies read:



    .R_ρ_0m(α)|_a=ν=0   =2ln r_eff+ln2π+1/1-αlnΓ(|m|α+1)/(|m|!)^αα^|m|α+1
    .R_γ_0m(α)|_a=ν=0   =-2ln r_eff+lnπ/2+1/1-αlnΓ(|m|α+1)/(|m|!)^αα^|m|α+1.



Asymptote α→1/2 degenerates right-hand side of Eq. (<ref>) into 2ln2π whereas its left-hand side for the functionals from Eqs. (<ref>) yields in the same limit:


    .[R_ρ_0m(α)+R_γ_0m(β)]|_a=ν=0=2ln2π+|m|(1+ln2)+lnΓ^2(|m|/2+1)/|m|^|m|.


As a function of the magnetic quantum number, the expression from Eq. (<ref>) monotonically increases with |m| and for the azumthally symmetric state, m=0, which is the lowest-energy orbital, it converts Rényi inequality, Eq. (<ref>), into the identity confirming in this way the above mentioned conjecture. For the QR, a≠0, an analysis of the Rényi and Tsallis entropies requires separate careful consideration.



§ ACKNOWLEDGMENTS

Research was supported by SEED Project No. 1702143045-P from the Research Funding Department, Vice Chancellor for Research and Graduate Studies, University of Sharjah.

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