Quantum Ring: A Unique Playground for the Quantum-Mechanical Paradigm

Fomin, V. M.

doi:10.1007/978-3-319-95159-1_1

Cited by 17 publications

(26 citation statements)

References 224 publications

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“…In an attempt to expand quantum-information theory to the study of the quantum rings (QRs) [1], a recent analysis [2] addressed an influence of the combination of the transverse uniform magnetic field B and the AB flux φ AB [3] on the position and momentum components of, among others, Shannon entropy [4] of the one-particle orbitals of the flat 2D annulus whose rotationally symmetric potential profile U(r) is modeled in the position of polar coordinates (r, ϕ r ) by the superposition of the quadratic and inverse quadratic dependencies on the radius r [5][6][7][8][9][10][11][12][13][14][15]:…”

Section: Introductionmentioning

confidence: 99%

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

Olendski

2019

Entropy

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One-parameter functionals of the Rényi R ρ,γ (α) and Tsallis T ρ,γ (α) types are calculated both 1 in the position (subscript ρ) and momentum (γ) spaces for the azimuthally symmetric 2D nanoring that 2 is placed into the combination of the transverse uniform magnetic field B and the Aharonov-Bohm (AB) 3 flux φ AB and whose potential profile is modelled by the superposition of the quadratic and inverse 4 quadratic dependencies on the radius r. Position (momentum) Rényi entropy depends on the field B as 5 a negative (positive) logarithm of ω e f f ≡ ω 2 0 + ω 2 c /4 1/2 , where ω 0 determines the quadratic steepness 6 of the confining potential and ω c is a cyclotron frequency. This makes the sum R ρ nm (α)field-independent quantity that increases with the principal n and azimuthal m quantum numbers and 8 does satisfy corresponding uncertainty relation. In the limit α → 1 both entropies in either space tend to 9 their Shannon counterparts along, however, different paths. Analytic expression for the lower boundary 10 of the semi-infinite range of the dimensionless coefficient α where the momentum entropies exist reveals 11 that it depends on the ring geometry, AB intensity and quantum number m. It is proved that there is the 12 only orbital for which both Rényi and Tsallis uncertainty relations turn into the identity at α = 1/2 and 13 which is not necessarily the lowest-energy level. At any coefficient α, the dependence of the position 14 Rényi entropy on the AB flux mimics the energy variation with φ AB what, under appropriate scaling, can 15 be used for the unique determination of the associated persistent current. Similarities and differences 16 between the two entropies and their uncertainty relations are discussed too. 17 Note that, as it follows from Eqs. (49a) and (49c), the sum of the entropies of the generalized Gaussian 111 approaches its edge values (which are equal to each other due to the fact that each item in it has the 112 same dependence on the Rényi parameter and, as a result, due to the condition from Eq. (18), at the rims 113 α and β simply interchange their places) from above and since the expression in the square brackets in 114 Eq. (49b) is always positive, left-hand side of Eq. (21) reaches its maximum at the Shannon entropy. Also, 115 the leading terms in all three cases are increasing functions of the magnetic index what means that at 116 the greater |m| the corresponding curve lies higher satisfying, of course, the uncertainty relation. As our 117 numerical results show, the same statement holds true at the fixed quantum number m and the increasing 118 principal index n. 119For the QR, a > 0, a comparison of Eqs. (25), (39) and (48) proves that the n = m = 0 orbital does 120 convert at α = 1/2 the Rényi uncertainty into the identity, as it did for the Tsallis inequality too. This is 121 also exemplified in Fig. 2(a), which shows that its sum R ρ (α) + R γ (β) at any parameter α is the smallest 122 one as compared to other levels. Dependence of the left-hand side of Eq. (21) on n and |m| is...

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Section: Introductionmentioning

confidence: 99%

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

Olendski

2019

Entropy

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“…Despite the advances achieved in growth techniques such as molecular beam epitaxy and metal chemical vapor deposition, that have allowed to fabricate QRs with geometries close to the circular one -which have been studied by many groups in recent years 8–16 , the actual rings do have a more complex structure due to the presence of impurities and, more importantly, geometrical imperfections 10,17,18 . For instance, the deviations from the ideal toric-shape can be observed from AFM images of QRs obtained via droplet epitaxy 19,20 .…”

Section: Introductionmentioning

confidence: 99%

Effects of Geometry on the Electronic Properties of Semiconductor Elliptical Quantum Rings

Vinasco

Radu

Kasapoğlu

et al. 2018

Sci Rep

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The electronic states in GaAs-AlxGa1−xAs elliptically-shaped quantum rings are theoretically investigated through the numerical solution of the effective mass band equation via the finite element method. The results are obtained for different sizes and geometries, including the possibility of a number of hill-shaped deformations that play the role of either connected or isolated quantum dots (hills), depending on the configuration chosen. The quantum ring transversal section is assumed to exhibit three different geometrical symmetries - squared, triangular and parabolic. The behavior of the allowed confined states as functions of the cross-section shape, the ring dimensions, and the number of hills-like structures are discussed in detail. The effective energy bandgap (photoluminescence peak with electron-hole correlation) is reported as well, as a function of the Al molar fraction.

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“…Quantum rings are slender, annular structures that can be fabricated with predetermined radius and thickness [1]. Fullerene or, more properly, C 60 , is formed by 60 carbon atoms located at the corner of hexagons and pentagons and closely resembles a soccer ball.…”

Section: Introductionmentioning

confidence: 99%

On the dynamics of confined particles: a laser test

Fiordilino¹

2017

Laser Phys. Lett.

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Quantum Ring: A Unique Playground for the Quantum-Mechanical Paradigm

Cited by 17 publications

References 224 publications

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

Effects of Geometry on the Electronic Properties of Semiconductor Elliptical Quantum Rings

On the dynamics of confined particles: a laser test

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