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arXiv:1703.09158

Compared study of Shannon, Tsallis and Gaussian entropy of bound magnetopolaron in nanostructures

Maurice Tiotsop, A. J. Fotué, Hilaire Bertrand Fotsin, Lukong Cornelius Fai

Published Mar 27, 2017• 0 citations
arXivHTMLPDF

Abstract

Many methods have been experimented to study decoherence in nanostructures. Tsallis, Shannon and Gaussian entropy have been used to study decoherence separately; in this paper, we compared the results of the sus-mentioned entropies in nanostructures. The linear combination operator and the unitary transformation was used to derive the magnetopolaron spectrum that strongly interact with the LO phonons in the presence of electric field in the pseudo harmonic and delta quantum dot. Numerical results revealed for the quantum pseudo dot that: (i) The amplitude of Gauss entropy is greater than the amplitude of Tsallis entropy which inturn is greater than the amplitude of Shannon entropy. The Tsallis entropy is not more significant in nanostructure compared to Shannon and Gauss entropies, (ii) With an increase of the zero point, the dominance of the Gauss entropy on the Shannon entropy was observed on one hand and the dominance of the Shannon entropy on the Tsallis entropy on the other hand ; this suggested that in nanostructures, Gauss entropy is more suitable in the evaluation of the average of information in the system, for the delta quantum dot it was observed that (iii) when the Gauss entropy is considered, a lot of information about the system is missed. The collapse revival phenomenon in Shannon entropy was observed in RbCl and GaAs delta quantum dot with the enhancement of delta parameter; with an increase in this parameter, the system in the case of CsI evolved coherently; with Shannon and Tsallis entropies , information in the system is faster and coherently exchanged; (iv) The Shannon entropy is more significant because its amplitude outweighs on the others when the delta dimension length enhances. The Tsallis entropy involves as wave bundle; which oscillate periodically with an increase of the oscillation period when delta dimension length is improved.

DOI
10.48550/arxiv.1703.09158

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Many methods have been experimented to study decoherence in nanostructures. Tsallis, Shannon and Gaussian entropy have been used to study decoherence separately; in this paper, we compared the results of the sus-mentioned entropies in nanostructures. The linear combination operator and the unitary transformation was used to derive the magnetopolaron spectrum that strongly interact with the LO phonons in the presence of electric field in the pseudo harmonic and delta quantum dot. Numerical results revealed for the quantum pseudo dot that: (i) The amplitude of Gauss entropy is greater than the amplitude of Tsallis entropy which inturn is greater than the amplitude of Shannon entropy. The Tsallis entropy is not more significant in nanostructure compared to Shannon and Gauss entropies, (ii) With an increase of the zero point, the dominance of the Gauss entropy on the Shannon entropy was observed on one hand and the dominance of the Shannon entropy on the Tsallis entropy on the other hand ; this suggested that in nanostructures, Gauss entropy is more suitable in the evaluation of the average of information in the system, for the delta quantum dot it was observed that (iii) when the Gauss entropy is considered, a lot of information about the system is missed. The collapse revival phenomenon in Shannon entropy was observed in RbCl and GaAs delta quantum dot with the enhancement of delta parameter; with an increase in this parameter, the system in the case of CsI evolved coherently; with Shannon and Tsallis entropies , information in the system is faster and coherently exchanged; (iv) The Shannon entropy is more significant because its amplitude outweighs on the others when the delta dimension length enhances. The Tsallis entropy involves as wave bundle; which oscillate periodically with an increase of the oscillation period when delta dimension length is improved.
In order to protect coherence of quantum states and reduce the impact of environment on quantum information, we investigate decoherence and relaxation time of magnetopolaron in the presence of three dimensional impurity under strong parabolic potential. The first states energies have been evaluated using the Lee Low Pine transformation and Pekar-type variational method. Parameters such as: decoherence time, transition frequency, spontaneous emission, Shannon entropy, relaxation time and probability density, have been evaluated. It has been seen that the impurity and electron-phonon coupling constant have a considerable effect on formation, protection of quantum qubit and quantum transport. The information exchange measured by the rate of Shannon entropy, has a great dependence on impurity and with its interaction with electrons. The relaxation time τr exhibits increasing behavior as a function of, α, β, and ωc. The electron-phonon coupling constant, impurity and cyclotron frequency are useful parameters to prevent decoherence phenomena. This study paves the way to prolong quantum effect in nanostructure and favor the realization of the future quantum computer.
Polaron in CsI quantum pseudo dot under an electromagnetic field was considered, and the ground and first excited state energies were derived by employing the combining Pekar variational and unitary transformation methods. With the two-level system obtained, single qubit was envisioned and the decoherence was studied using non-extensive entropy (Tsallis entropy). Numerical results showed: (i) the increase (decrease) of the energy levels (period of oscillation) with the increase of chemical potential, the zero point of pseudo dot, cyclotron frequency, and transverse and longitudinal confinements; (ii) the Tsallis entropy evolved as a wave envelop that increase with the increase of non-extenxive parameter and with the increase of electric field strength, zero point of pseudo dot and cyclotron frequency the wave envelop evolve periodically with reduction of period; (iii) The transition probability increases from the boundary to the centre of the dot where it has its maximum value. It was also noted that the probability density oscillate with period T0=ℏ/ΔΕ with the tunnelling of the chemical potential and zero point of the pseudo dot. These results are helpful in the control of decoherence in quantum systems and may also be useful for the design of quantum computers.
Abstract A comparative study of 1D quantum structures which allows analytic expressions for the position and momentum Rényi R ( α ) and Tsallis T ( α ) entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms the special status of the Gaussian distribution: at any parameter α it converts into the equality both Rényi and Tsallis uncertainty relations removing for the latter an additional requirement 1/2 ≤ α ≤ 1 that is a necessary condition for all other geometries. It is shown that the lowest limit of the semi-infinite range of the dimensionless parameter α where momentum components exist strongly depends on the position potential and/or boundary condition for the position wave function. Asymptotic limits reveal that in either space the entropies R ( α ) and T ( α ) approach their Shannon counterpart, α = 1, along different paths. Similarities and differences between the two entropies and their uncertainty relations are exemplified. Some unsolved problems are also indicated.
With the development of quantum technology and information science, significant attention is focused on studying the quantum information properties in confined systems. In this work, we present a systematic investigation of quantum information measures – including position-momentum uncertainty (Δx,Δp), Shannon entropy (Sx, Sp), and Onicescu energy (Ox, Op) in a confined Robin harmonic oscillator (CRHO) with symmetric boundary conditions λL = λR = λ. It is interesting to find that the Heisenberg uncertainty relation ΔxΔp≥ℏ/2, which holds under the Dirichlet boundary condition (λ = 0), is violated for the ground state of the CRHO. In contrast, the entropic uncertainty relation (Sx + Sp ≥ 1 + lnπ) remains satisfied at any Robin length λ and quantum state, demonstrating its fundamental advantage and broader applicability in quantum systems. For a fixed Robin length, we observe that as the quantum well width increases within a finite range, the position-space Shannon entropy Sₓ increases, while the Onicescu energy Oₓ decreases. Despite their opposing tendencies, both Shannon entropy and Onicescu energy reflect the same physical insight regarding the localisation-delocalisation property for the CRHO system. In addition, the dependence of these information measures on the Robin length is discussed, and their physical implications are highlighted. This study demonstrates that boundary conditions play a critical role in quantum information metrics, thereby informing the design and optimisation of quantum devices such as quantum dots, nanoscale resonators and atom-based sensors.
We calculate information measures using the Tsallis, Rényi, and Shannon entropies for two classes (Lane–Emden and Caldirola–Kanai) of time-dependent mesoscopic RLC circuits. To determine the expressions for the entropies, we used the dynamical invariant method to obtain the exact Schrödinger wave function |(ψn(x,t))\big (\psi _n (x,t)\big )(ψn​(x,t))|⁠. For the state |n=0n=0n=0|⁠, all the expressions found are given in terms of |$\rho $|⁠, a c-number quantity satisfying a nonlinear differential equation. For the Caldirola–Kanai system, the entropies do not vary with time, but decrease with increasing damping factor. For the Lane–Emden system, the entropies decrease with increasing time. We also analyze the behavior of the Shannon and Rényi lengths in the charge and magnetic flux spaces and compare them with the respective standard deviations.
We calculate Shannon information entropy of trapped interacting bosons in both the position and momentum spaces, Sr and Sk, respectively. The total entropy maintains the functional form S = a + b ln N for repulsive bosons. At the noninteracting limit the lower bound of entropic uncertainty relation is also satisfied whereas the diverging behavior of Sr and Sk at the critical point of collapse for attractive condensate accurately calculates the stability factor. Next we study the dynamics of Shannon information entropy with varying interparticle potential. We numerically solve the time-dependent Gross–Pitaevskii equation and study the influence of increasing nonlinearity in the dynamics of entropy uncertainty relation (EUR). We observe that for small nonlinearity the dynamics is regular. With increase in nonlinearity although Shannon entropy shows large variation in amplitude of the oscillation, the EUR is maintained throughout time for all cases and it confirms its generality. We also study the dynamics in a very tight trap when the condensate becomes highly correlated and strongly inhomogeneous. Time evolution of total entropy exhibits aperiodic and fluctuating nature in very tight trap. We also calculate Landsberg's order parameter for various interaction strengths which supports earlier observation that entropy and order are decoupled.