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Bound magneto-polaron in triangular quantum dot qubit under an electric field
Superlattices and Microstructures 90 (2016) 20e29
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Bound magneto-polaron in triangular quantum dot qubit under an electric field A.J. Fotue a, *, N. Issofa a, M. Tiotsop a, S.C. Kenfack a, M.P. Tabue Djemmo a, c, A.V. Wirngo a, H. Fotsin b, L.C. Fai a a
Mesoscopic and Multilayers Structures Laboratory, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 479, Dschang, Cameroon b Laboratory of Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon c Laboratory of Mechanics and Modeling of Physical Systems, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 July 2015 Received in revised form 27 November 2015 Accepted 30 November 2015 Available online 8 December 2015
In this paper, we examine the time evolution of the quantum mechanical state of a magnetopolaron using the Pekar type variational method on the electric-LO-phonon strong coupling in a triangular quantum dot with Coulomb impurity. We obtain the Eigen energies and the Eigen functions of the ground state and the first excited state, respectively. This system in a quantum dot is treated as a two-level quantum system qubit and numerical calculations are done. The Shannon entropy and the expressions relating the period of oscillation and the electron-LO-phonon coupling strength, the Coulomb binding parameter and the polar angle are derived. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Polaron Quantum dot Electromagnetic field Pekar type variational method
1. Introduction With the exponential advancement of nanotechnology during these last years, the study of quantum computing and quantum information processing has generated widespread interest. The two-level system is usually employed as the elementary unit for storing information. Quantum computation will be based on the laws of quantum mechanics. Several schemes have been proposed for realizing quantum computers in recent years [1e8]. For quantum computers to have an edge over classical computers, they will need to carry thousands of qubits. Consequently, quantum computers with large numbers of qubits will be most feasible as solid-state systems. Self-assembled quantum dots (QDs) have attracted substantial attention due to their perfect crystal structures. Therefore, it is one of the most popular solid-state quantum information research fields that qubits can be realized by solid-state devices. Many schemes with widely varying content have been proposed for carrying out research on quantum dots [1,2,9,10]. Ezaki et al. [11] investigated the electronic structures in a triangular bound potential quantum dot. Jia-kui Sun et al. [12] and Zhi-Xin Li [13] respectively studied the decoherence and the effect of temperature on a polaron in a triangular quantum dot. However, self-assembled quantum dots (QDs) have also attracted substantial attention due to their perfect crystal structures [14e17]. One of the major concerns in QDs is the impurity states, which have attracted extensive attention in recent
* Corresponding author. E-mail address: [email protected] (A.J. Fotue). http://dx.doi.org/10.1016/j.spmi.2015.11.036 0749-6036/© 2015 Elsevier Ltd. All rights reserved.
A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
21
years [18e22]. More recently, the related problem of an optical polaron bound to a Coulomb impurity in a QD has also been considered in the presence of a magnetic field [23,24]. Imperfections being a rule rather than an exception, such an impuritybound polaron problem [25,26] is obviously more realistic and is therefore of much practical importance. After the pioneering work on information theory by Shannon [27] many studies have been carried out on the question of how information storage, processing and transmission tasks can be performed with macroscopic decohered resources [28,29]. However, the two level system and the influence of the Shannon entropy on the coherence of the polaron in a triangular quantum dot with a Coulomb impurity have not been taken into account in preceding works. In the present work, we obtain the Eigen energies of the ground state and the first-excited state, the Eigen functions of the ground state and the first-excited state by using variational method of Pekar type on the condition of electric-LO phonon strong coupling in a triangular quantum dot. This system in a quantum dot may be employed as a two-level quantum systemqubit [30e33]. We obtain the probability density of the polaron which oscillates with a given period when it is in a superposition of the ground and first excited states. The expressions relating the period of oscillation and the transition frequency of the polaron to the cyclotron frequency, electric field density parameter and Coulomb potential are derived. The phonon spontaneous emission causes the decoherence of the qubit. The Shannon entropy is derived and used to discuss the decoherence of the system. This paper has the following structure: In section 2, we describe the Hamiltonian of the system and use the Pekar variational method to derive the ground and first excited state energies. We also derive the probability density and the Shannon entropy. In section 3, we present and discuss the results and then, we end with the conclusion in section 4. 2. Theoretical model and calculation We consider a system in which the electrons is much more confined in the z direction than in the x and y directions. The electrons are assumed to be moving on the xy plane. The confining potential is taken as a triangular bound potential with the form [34]
1 2 mu20 r2 1 þ cos 3 w 2 7
Vrw ¼
(2.1)
The electrons are moving in this polar crystal and interacting with bulk LO phonons under the influence of an electric and a magnetic field. The electric field F is along the x direction while the magnetic field is along the z direction with vector potential A ¼ B(y/2, x/2, 0). The Hamiltonian of the electronephonon interaction system can be written as:
H¼
X 2 2 X 1 g2 1 g2 1 2 2 2 px y þ py þ x e* xF þ 1 þ cos 3 w þ ZuLO aþ Vq aq expðiqrÞ q aq þ mu0 r 2m 2m 2 7 4 4 q q b þ h:c: r (2.2)
whereg2 ¼ (2e/c)B and m is the band mass while u0 is the magnitude confinement strength of the potentials in the xy plane and w is the polar angle in the polar coordinate system. aþ q ðaq Þ denotes the creation (annihilation) operator of the bulk LO phonon with wave vector q. p ¼ (px, py, pz) and r ¼ (r,z) are the momentum and position vectors of the electron and r ¼ (x,y) is the position vector of the electron in the xy plane. Vq and a in (2.2) are respectively the amplitude of the electron€hlich coupling constant defined as phonon interaction and the Fro
Z 2muLO
1 1 4 4pa 2 V =
ZuLO q
=
Vq ¼ i
(2.3)
where
e2 2ZuLO
1 2muLO 2 1 1 ε∞ ε0 Z =
a¼
(2.4)
The last expression of the Hamiltonian (2.2) is the Coulomb impurity potential [35]between the electron and the hydrogen-like impurity. The Fourier transform of this expression is written as follows
X 4p b 1 b ¼ expðiq:rÞ r V q2 q here, b ¼ e2/ε∞ is the strength of the Coulombic impurity potential.
(2.5)
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A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
To evaluate the polaron energy, we use the Pekar variational method. To achieve our goal, we choose the trial function of strong ecoupling that can be separated into two parts which individually describe the electron and the phonon. The trial function drawn from Ref. [36]is written as:
jj〉 ¼ jf〉U 0ph 〉
(2.6)
where jf〉 depends only on the electron coordinate, 0ph 〉 represents the phonon's vacuum state with aq 0ph 〉 ¼ 0, and U 0ph 〉 is the coherent state of the phonon,
" U ¼ exp
X
* aþ q fq aq fq
#
(2.7)
q
where fq ðfq* Þ is the variational function. We may choose the trial ground and the first-excited state wave functions of the electron to be
l l r2 jzðzÞ〉0ph 〉 jf0 〉 ¼ j0〉0ph 〉 ¼ p0ffiffiffi exp 0 2 p
(2.8)
l2 l r2 jzðzÞ〉 expð±i4Þ 0ph 〉 jf1 〉 ¼ j1〉0ph 〉 ¼ p1ffiffiffi r exp 1 2 p
(2.9)
wherel0 and l1 are the variational parameters. Equations (2.8) and (2.9) satisfy the following normalized relations:
〈0j0〉 ¼ 〈1j1〉 ¼ 1; 〈0j1〉 ¼ 0
(2.10)
Using the Pekar variational method, we have
H 0 ¼ U 1 HU
(2.11)
By minimizing the expectation value of the Hamiltonian, we obtain the ground state energy E0 ¼ 〈f0 jH0 jf0 〉 and the first excited state energy E1 ¼ 〈f1 jH 0 jf1 〉 . Performing the variations of E0 and E1 with respect to fq* , we obtain
q2 fq ðl0 Þ ¼ Vq* exp 4l0
(2.12)
q2 fq ðl1 Þ ¼ Vq* exp 4l1
(2.13)
Inserting (2.12) into E0 and (2.13) into E1, we then obtain the magnetopolaron ground and first excited state energies in the following forms: 1 pffiffiffi u2c 1 2 e* F al0 ð2puL0 Þ 2 b l0 p þ þ 2 4 1 þ cos 3 w cos w 7 2pl0 16l0 l0 l 2 2
=
E0 ¼
l20
(2.14)
0
and
1 b l31 u2c 2 2 e* F al þ 2 4 1 þ cos 3 w cos w 1 ð2puL0 Þ 2 pffiffiffi 7 4pl1 16l1 l1 l 32 2 p =
E1 ¼ 2l21 þ
(2.15)
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where l0 ¼ Z=mu0 is the effective confinement length of the QD. l0 and l1 are obtained by minimizing the ground and first excited state energies. The superposition state of electron can be expressed as:
1 jj01 〉 ¼ pffiffiffi ðj0〉 þ j1〉Þ 2
(2.16)
where
l2 r2 1 j0〉 ¼ pffiffiffi l0 exp 0 2 p and
! (2.17)
A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
Fig. 1. (a) ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron frequency uC for F ¼ 105.0; l0 ¼ 0.45; b ¼ 0.8;.
Fig. 2. a) ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron frequency uC for a ¼ 7.0; l0 ¼ 0.45; b ¼ 0.8.
Fig. 3. a) Ground state energy E0 and (b) First excited state energy E1 as a function of the cyclotron frequency uC for a ¼ 7.0; F ¼ 105.5; b ¼ 0.8.
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Fig. 4. a) Ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron frequency uC for a ¼ 7.0; F ¼ 105.5; l0 ¼ 0.45.
Fig. 5. Transition frequency u as a function of the cyclotron frequency uc for (a) F ¼ 105.0; l0 ¼ 0.45; b ¼ 0.8; w ¼ p/2; 4 ¼ 2p, (b) a ¼ 7.0; l0 ¼ 0.45; b ¼ 0.8; w ¼ p/ 2; 4 ¼ 2p, (c) a ¼ 7.0; F ¼ 105.5; b ¼ 0.8; w ¼ p/2; 4 ¼ 2p, (d) a ¼ 7.0; F ¼ 105.5; l0 ¼ 0.45; w ¼ p/2; 4 ¼ 2p.
A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
25
Fig. 6. Period of oscillation t as a function of the cyclotron frequency uC for (a) F ¼ 105.0; l0 ¼ 0.45; b ¼ 0.8; w ¼ p/2; 4 ¼ 2p, (b) a ¼ 7.0; l0 ¼ 0.45; b ¼ 0.8; w ¼ p/ 2; 4 ¼ 2p, (c) a ¼ 7.0; F ¼ 105.5; b ¼ 0.8; w ¼ p/2; 4 ¼ 2p, (d). a ¼ 7.0; F ¼ 105.5; l0 ¼ 0.45; w ¼ p/2; 4 ¼ 2p.
l2 r j1〉 ¼ p1ffiffiffi exp p
l21 r2 2
! (2.18)
The time evolution of the state of the electron can then be written as
1 E t 1 E t j01 ðt; rÞ ¼ pffiffiffi jf0 ðrÞ〉exp i 0 þ pffiffiffi jf1 ðrÞ〉exp i 1 Z Z 2 2
(2.19)
The probability density is in the following form:
Q ðt; r;Þ ¼ jj01 ðt; rÞj2 2 3 2 2 * 1 4 jf0 ðrÞj þ jf1 ðrÞj þ f0 ðrÞ41 ðrÞexpðiu01 tÞþ 5 ¼ 2 þ4 ðrÞ4* ðrÞexpðiu tÞ 01 0 1 where
(2.20)
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A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
Fig. 7. Spatiotemporal evolution of the electron probability density in a superposition state of j0〉 and j1〉 for l0 ¼ 0.25, uc ¼ 10.0, a ¼ 7.0, F ¼ 0.5, b ¼ 0.8, 4 ¼ 2p, w ¼ p/4, (a) t ¼ 0, (b) t ¼ T0/4, (c) t ¼ T0/2, (d) t ¼ 3T0/4 and (e) t ¼ T0.
u01 ¼
ðE1 E0 Þ Z
(2.21)
is the transition frequency from ground state to the first excited state. The Shannon entropy of the state given by the wave function j01 (t, r, z) leads to
Z SðtÞ ¼
dzdr jj01 ðt; r; zÞj2 lnjj01 ðt; r; zÞj2
(2.22)
3. Numerical results and discussions In this part, we show the plots of the ground and first excited state energies, the transition frequency, the probability density and the Shannon entropy, versus the confinement strength, the cyclotron frequency, the electric field strength parameter and the Coulomb bound potential. All parameters used in this part are dimensionless. In Figs. 1e4, we have plotted the ground and first excited state energies as functions of the cyclotron frequency for w ¼ p/2 and 4 ¼ 2p. The ground and first excited energies are increasing functions of the cyclotron frequency. Since the presence of a magnetic field is equivalent to introducing another confinement on electrons which leads to a greater overlap of the electron wave function, the electron-phonon interactions will be enhanced, and the ground state binding energy appears more obvious [37,38]. Fig. 1 also shows that the ground and first excited state energies are decreasing functions of the electronphonon coupling constant. This is because the larger the electron-phonon coupling strength, the stronger the electronphonon interaction. This leads to an increase of the electron's energy and makes the electron interact with more phonons. However, the contributions of the electron-phonon interaction are negative in (2.14) and (2.15). As a result, the states' energies will increase with decreasing coupling strength [30]. Fig. 2 shows that the ground and first excited state energies are decreasing functions of the electric field strength. This is because the electric field leads to an increase in the electron's energy and makes the electrons interact with more phonons. In this way, the states' energies and the transition frequency are increased [30]. Fig. 3 shows that, the ground and first excited state energies are decreasing functions of the confinement length. This is because the motion of the electrons is confined by the harmonic potential. From another point of view, since the presence of
A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
27
Fig. 8. A- Entropy as a function of time for l0 ¼ 0.75, uc ¼ 12.0, a ¼ 7.0, F ¼ 0.5, 4 ¼ 2p, w ¼ p/4(a) b ¼ 0.2, (b) b ¼ 0.9, B-Entropy as a function of time for l0 ¼ 0.75, uc ¼ 12.0, a ¼ 7.0, b ¼ 0.8, 4 ¼ 2p, w ¼ p/4(c) F ¼ 15.0, (d) F ¼ 30.0, C-Entropy as a function of time for l0 ¼ 0.75, F ¼ 5.0, a ¼ 7.0, b ¼ 0.8, 4 ¼ 2p, w ¼ p/4, (a) uc ¼ 1.0,(b). uc ¼ 10.0.
the triangular potential is equivalent to the introduction of another confinement on the electron, which leads to a greater of the electron's wavefunctions, the electron-phonon interactions will be enhanced. These results are in agreement with the results of Kervan et al. [39], Ren et al. [40], Kandemir et al. [41]and W. Xiao et al. [42]obtained respectively by using variational, Feynman-Haken path-integral, squeezed-state variational and linear combination operator methods. Fig. 4 shows that the ground and first excited state energies decrease with the Coulomb potential. This result is agreement with the results of Fotue et al. [38]. In Fig. 5, we have plotted the transition frequency as a function of the cyclotron frequency. The transition frequency is an increasing function of the cyclotron frequency, and this is because the magnetic field is considered as another confinement. The investigation indicates that the oscillating period decreases with an increasing electron-phonon coupling constant (Fig. 5a), electric field strength (Fig. 5b), effective confinement length (Fig. 5c) and Coulomb potential (Fig. 5d), meaning that
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A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
the qubit's lifetime is reduced, and then the decoherence process is quickened. One way of prolonging the qubit's lifetime and decreasing quantum decoherence is by adjusting the effective confinement length. This indicates a new way of controlling the triangular QD's state energies and the transition frequency via adjusting the effective confinement lengths. In Fig. 6, we have plotted the period of oscillation as a function of the cyclotron frequency. It is obvious from here that the period of oscillation is a decreasing function of the electric and magnetic fields. As a result of the presence of the electric and magnetic fields, the ground and the first-excited state energies increase, though the influence is greater on the first-excited state energy. The period of oscillation t0 decreases, that is to say, the life time of a qubit reduces, so the process of decoherence is quickened [43,44]. It can be very harmful to store information with the QD as an elementary unit. In Fig. 7, we have plotted the probability density when the electron is in the superposition state. From there, we can see that the electron probability density shows a periodic oscillation when the polar angle is varied, which arises from the influence of the triangular bound potential. This result is in accordance with that obtained by J. W. Yin et al. [45]and J. K. Sun et al. [12]. In Fig. 8, we have plotted the Shannon entropy as a function of time for different constant control parameter. This plot shows a decrease of the entropy with increasing time. The control of the coherence of the system can be done through the modulation of the electric and magnetic fields and the Coulomb binding potential [29,46]. 4. Conclusion In this paper, we have derived the ground and first excited state energies of the bound magnetopolaron under an electric field and the Coulomb potential in a triangular quantum dot and their relevant eigen-functions using the Pekar variational method. The single qubit can be envisaged as this kind of two-level quantum system in QDs. The electron probability density shows a periodic oscillation the polar angle is varied. The Shannon entropy is a decreasing function of time. The coherence of the system can be controlled by tunneling the electric and magnetic fields, the confinement length and the Coulomb binding potential. Because the electron phonon coupling constant and the phonon dispersion coefficient characterize the material, it is important to properly choose the material to use when making a quantum dot. Our results should be meaningful for the design and implementation of quantum computing both theoretically and experimentally and also for the control of decoherence in quantum systems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
W.P. Li, J.W. Yin, Y.F. Yu, Z.W. Wang, J.L. Xiao, J. Low Temp. Phys. 160 (2010) 112. Y.F. Yu, W.P. Li, J.W. Yin, J.L. Xiao, Int. J. Theor. Phys. 50 (2011) 3322. Y.J. Wen, J.L. Xiao, Y.F. Yu, Z.W. Wang, Chin. Phys. B18 (2009) 446. S. Jia-kui, L. Hong-Juan, J.L. Xiao, Physica B404 (2009) 1961. M.A. Nielsen, I.L. Chang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. I. D'Amico, Microelectron. J. 37 (2006) 1440. C.H. Bennett, D.P. DiVincenzo, Nature 404 (2000) 247. P. Hawrylak, M. Korkusinski, Solid State Commun. 136 (2005) 508. S.D. Sarma, R.D. Sousa, X.D. Hu, B. Koiller, Solid State Commun. 133 (2005) 737. Y.F. Yu, W.P. Li, J.W. Yin, J.L. Xiao, Int. J. Theor. Phys. 50 (2011) 3322. J.P. Barnes, W.S. Warren, Phys. Rev. A60 (1999) 4363. J.K. Sun, H.J. Li, J.L. Xiao, Mod. Phys. Lett. B23 (27) (2009) 3273. Z.X. Li, Mod. Phys. Lett. B26 (3) (2012) 1150015. M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849. M.W. Dellow, P.H. Beton, C.J.G.M. Langerak, T.J. Foster, P.C. Main, L. Eaves, M. Henini, S.P. Beaumont, C.D.W. Wilkinson, Phys. Rev. Lett. 68 (1992) 1754. S.S. Li, J.B. Xia, Phys. Rev. B 58 (1998) 3561. S.S. Li, J.B. Xia, Appl. Phys. Lett. 87 (2005) 043102. C. Sikorski, U. Merkt, Phys. Rev. Lett. 62 (1989) 2164. S.S. Li, J.B. Xia, Phys. Lett. A 366 (2007) 120. A. Lorke, J.P. Kotthaus, K. Ploog, Phys. Rev. Lett. 64 (1990) 2559. S.S. Li, X.J. Kong, J. Phys. Condens. Matter 4 (1992) 4815. S.S. Li, J.B. Xia, J. Appl. Phys. 100 (2006) 083714. K.D. Zhu, T. Kobayashi, Solid State Commun. 92 (1994) 353. T.C. Au-Yeung, S.L. Kho, S.W. Gu, L.H. Hong, E.M.C. Wong, J. Phys. Condens. Matter 6 (1994) 6761. S. Sil, A. Chatterjee, J. Phys. Condens. Matter 3 (1991) 9401. M. Bhattacharya, A. Chatterjee, T.K. Mitra, Phys. Rev. B 39 (1989) 8351. C.E. Shannon, Bell Syst. Tech. J. 27 (1) (1948), 379 and 623. ^, J. Mex. Chem. Soc. 52 (1) (2008) 60. R.P. Sagar, M. Ho L.C. Fai, M. Tchoffo, J.T. Diffo, G.C. Fouokeng, Phys. Rev. Res. Int. 4 (2) (2014) 267. J.L. Xiao, J. Low Temp. Phys. 172 (2013) 122. S.S. Li, J.B. Xia, F.H. Yang, Z.C. Niu, S.L. Feng, H.Z. Zheng, J. Appl. Phys. 90 (2001) 6151. S.S. Li, L. Gui- Lu, F.S. Bai, S.L. Feng, H.Z. Zheng, Proc. Natl. Acad. Sci. U. S. A. 98 (2001) 11847. Z.W. Wang, J.L. Xiao, Acta Phys. Sin. 56 (2007) 678. T. Ezaki, N. Mori, C. Hamaguchi, Phys. Rev. B 56 (1998) 6428. J.L. Xiao, Superlattices Microstruct. 70 (2014) 39. B.S. Kandemir, A. Cetin, J. Phys. Condens. Matter 17 (2005) 667. S.H. Chen, Int. J. Modern Phys. B 22 (16) (2008) 2611. A.J. Fotue, M. Tiotsop, N. Issofa, S.C. Kenfack, A.V. Wirngo, H. Fotsin, L.C. Fai, Am. J. Modern Phys. 4 (3) (2015) 109. N. Kervan, T. Altanhan, A. Chatterjee, Phys. Lett. A 315 (280) (2003) 4. Y.H. Ren, Q.H. Chen, Z.K. Jiao, Acta Phys. Soc. 7 (1998) 598.
A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29 [41] [42] [43] [44] [45] [46]
B.S. Kandemir, A. Cetin, J. Phys. Condens. Matter 17 (2003) 667. W. Xiao, J.L. Xiao, Int. J. Modern Phys. B 25 (2011) 3485. A. Grodecka, P. Machnikowski, Phys. Rev. B 73 (2006) 125306. M. Lovric, H.G. Krojanski, D. Suter, Phys. Rev. A 75 (2007) 042305. J.W. Yin, Y.F. Yu, H.J. Li, J. Low Temp. Phys. 177 (1e2) (2014) 72. Z.W. Wang, W.P. Li, J.W. Yin, J.L. Xiao, Commun. Theor. Phys. 49 (2008) 311.
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Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Bound magneto-polaron in triangular quantum dot qubit under an electric field A.J. Fotue a, *, N. Issofa a, M. Tiotsop a, S.C. Kenfack a, M.P. Tabue Djemmo a, c, A.V. Wirngo a, H. Fotsin b, L.C. Fai a a
Mesoscopic and Multilayers Structures Laboratory, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 479, Dschang, Cameroon b Laboratory of Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon c Laboratory of Mechanics and Modeling of Physical Systems, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 July 2015 Received in revised form 27 November 2015 Accepted 30 November 2015 Available online 8 December 2015
In this paper, we examine the time evolution of the quantum mechanical state of a magnetopolaron using the Pekar type variational method on the electric-LO-phonon strong coupling in a triangular quantum dot with Coulomb impurity. We obtain the Eigen energies and the Eigen functions of the ground state and the first excited state, respectively. This system in a quantum dot is treated as a two-level quantum system qubit and numerical calculations are done. The Shannon entropy and the expressions relating the period of oscillation and the electron-LO-phonon coupling strength, the Coulomb binding parameter and the polar angle are derived. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Polaron Quantum dot Electromagnetic field Pekar type variational method
1. Introduction With the exponential advancement of nanotechnology during these last years, the study of quantum computing and quantum information processing has generated widespread interest. The two-level system is usually employed as the elementary unit for storing information. Quantum computation will be based on the laws of quantum mechanics. Several schemes have been proposed for realizing quantum computers in recent years [1e8]. For quantum computers to have an edge over classical computers, they will need to carry thousands of qubits. Consequently, quantum computers with large numbers of qubits will be most feasible as solid-state systems. Self-assembled quantum dots (QDs) have attracted substantial attention due to their perfect crystal structures. Therefore, it is one of the most popular solid-state quantum information research fields that qubits can be realized by solid-state devices. Many schemes with widely varying content have been proposed for carrying out research on quantum dots [1,2,9,10]. Ezaki et al. [11] investigated the electronic structures in a triangular bound potential quantum dot. Jia-kui Sun et al. [12] and Zhi-Xin Li [13] respectively studied the decoherence and the effect of temperature on a polaron in a triangular quantum dot. However, self-assembled quantum dots (QDs) have also attracted substantial attention due to their perfect crystal structures [14e17]. One of the major concerns in QDs is the impurity states, which have attracted extensive attention in recent
* Corresponding author. E-mail address: [email protected] (A.J. Fotue). http://dx.doi.org/10.1016/j.spmi.2015.11.036 0749-6036/© 2015 Elsevier Ltd. All rights reserved.
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years [18e22]. More recently, the related problem of an optical polaron bound to a Coulomb impurity in a QD has also been considered in the presence of a magnetic field [23,24]. Imperfections being a rule rather than an exception, such an impuritybound polaron problem [25,26] is obviously more realistic and is therefore of much practical importance. After the pioneering work on information theory by Shannon [27] many studies have been carried out on the question of how information storage, processing and transmission tasks can be performed with macroscopic decohered resources [28,29]. However, the two level system and the influence of the Shannon entropy on the coherence of the polaron in a triangular quantum dot with a Coulomb impurity have not been taken into account in preceding works. In the present work, we obtain the Eigen energies of the ground state and the first-excited state, the Eigen functions of the ground state and the first-excited state by using variational method of Pekar type on the condition of electric-LO phonon strong coupling in a triangular quantum dot. This system in a quantum dot may be employed as a two-level quantum systemqubit [30e33]. We obtain the probability density of the polaron which oscillates with a given period when it is in a superposition of the ground and first excited states. The expressions relating the period of oscillation and the transition frequency of the polaron to the cyclotron frequency, electric field density parameter and Coulomb potential are derived. The phonon spontaneous emission causes the decoherence of the qubit. The Shannon entropy is derived and used to discuss the decoherence of the system. This paper has the following structure: In section 2, we describe the Hamiltonian of the system and use the Pekar variational method to derive the ground and first excited state energies. We also derive the probability density and the Shannon entropy. In section 3, we present and discuss the results and then, we end with the conclusion in section 4. 2. Theoretical model and calculation We consider a system in which the electrons is much more confined in the z direction than in the x and y directions. The electrons are assumed to be moving on the xy plane. The confining potential is taken as a triangular bound potential with the form [34]
1 2 mu20 r2 1 þ cos 3 w 2 7
Vrw ¼
(2.1)
The electrons are moving in this polar crystal and interacting with bulk LO phonons under the influence of an electric and a magnetic field. The electric field F is along the x direction while the magnetic field is along the z direction with vector potential A ¼ B(y/2, x/2, 0). The Hamiltonian of the electronephonon interaction system can be written as:
H¼
X 2 2 X 1 g2 1 g2 1 2 2 2 px y þ py þ x e* xF þ 1 þ cos 3 w þ ZuLO aþ Vq aq expðiqrÞ q aq þ mu0 r 2m 2m 2 7 4 4 q q b þ h:c: r (2.2)
whereg2 ¼ (2e/c)B and m is the band mass while u0 is the magnitude confinement strength of the potentials in the xy plane and w is the polar angle in the polar coordinate system. aþ q ðaq Þ denotes the creation (annihilation) operator of the bulk LO phonon with wave vector q. p ¼ (px, py, pz) and r ¼ (r,z) are the momentum and position vectors of the electron and r ¼ (x,y) is the position vector of the electron in the xy plane. Vq and a in (2.2) are respectively the amplitude of the electron€hlich coupling constant defined as phonon interaction and the Fro
Z 2muLO
1 1 4 4pa 2 V =
ZuLO q
=
Vq ¼ i
(2.3)
where
e2 2ZuLO
1 2muLO 2 1 1 ε∞ ε0 Z =
a¼
(2.4)
The last expression of the Hamiltonian (2.2) is the Coulomb impurity potential [35]between the electron and the hydrogen-like impurity. The Fourier transform of this expression is written as follows
X 4p b 1 b ¼ expðiq:rÞ r V q2 q here, b ¼ e2/ε∞ is the strength of the Coulombic impurity potential.
(2.5)
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To evaluate the polaron energy, we use the Pekar variational method. To achieve our goal, we choose the trial function of strong ecoupling that can be separated into two parts which individually describe the electron and the phonon. The trial function drawn from Ref. [36]is written as:
jj〉 ¼ jf〉U 0ph 〉
(2.6)
where jf〉 depends only on the electron coordinate, 0ph 〉 represents the phonon's vacuum state with aq 0ph 〉 ¼ 0, and U 0ph 〉 is the coherent state of the phonon,
" U ¼ exp
X
* aþ q fq aq fq
#
(2.7)
q
where fq ðfq* Þ is the variational function. We may choose the trial ground and the first-excited state wave functions of the electron to be
l l r2 jzðzÞ〉0ph 〉 jf0 〉 ¼ j0〉0ph 〉 ¼ p0ffiffiffi exp 0 2 p
(2.8)
l2 l r2 jzðzÞ〉 expð±i4Þ 0ph 〉 jf1 〉 ¼ j1〉0ph 〉 ¼ p1ffiffiffi r exp 1 2 p
(2.9)
wherel0 and l1 are the variational parameters. Equations (2.8) and (2.9) satisfy the following normalized relations:
〈0j0〉 ¼ 〈1j1〉 ¼ 1; 〈0j1〉 ¼ 0
(2.10)
Using the Pekar variational method, we have
H 0 ¼ U 1 HU
(2.11)
By minimizing the expectation value of the Hamiltonian, we obtain the ground state energy E0 ¼ 〈f0 jH0 jf0 〉 and the first excited state energy E1 ¼ 〈f1 jH 0 jf1 〉 . Performing the variations of E0 and E1 with respect to fq* , we obtain
q2 fq ðl0 Þ ¼ Vq* exp 4l0
(2.12)
q2 fq ðl1 Þ ¼ Vq* exp 4l1
(2.13)
Inserting (2.12) into E0 and (2.13) into E1, we then obtain the magnetopolaron ground and first excited state energies in the following forms: 1 pffiffiffi u2c 1 2 e* F al0 ð2puL0 Þ 2 b l0 p þ þ 2 4 1 þ cos 3 w cos w 7 2pl0 16l0 l0 l 2 2
=
E0 ¼
l20
(2.14)
0
and
1 b l31 u2c 2 2 e* F al þ 2 4 1 þ cos 3 w cos w 1 ð2puL0 Þ 2 pffiffiffi 7 4pl1 16l1 l1 l 32 2 p =
E1 ¼ 2l21 þ
(2.15)
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where l0 ¼ Z=mu0 is the effective confinement length of the QD. l0 and l1 are obtained by minimizing the ground and first excited state energies. The superposition state of electron can be expressed as:
1 jj01 〉 ¼ pffiffiffi ðj0〉 þ j1〉Þ 2
(2.16)
where
l2 r2 1 j0〉 ¼ pffiffiffi l0 exp 0 2 p and
! (2.17)
A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
Fig. 1. (a) ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron frequency uC for F ¼ 105.0; l0 ¼ 0.45; b ¼ 0.8;.
Fig. 2. a) ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron frequency uC for a ¼ 7.0; l0 ¼ 0.45; b ¼ 0.8.
Fig. 3. a) Ground state energy E0 and (b) First excited state energy E1 as a function of the cyclotron frequency uC for a ¼ 7.0; F ¼ 105.5; b ¼ 0.8.
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Fig. 4. a) Ground state energy E0 and (b) first excited state energy E1 as a function of the cyclotron frequency uC for a ¼ 7.0; F ¼ 105.5; l0 ¼ 0.45.
Fig. 5. Transition frequency u as a function of the cyclotron frequency uc for (a) F ¼ 105.0; l0 ¼ 0.45; b ¼ 0.8; w ¼ p/2; 4 ¼ 2p, (b) a ¼ 7.0; l0 ¼ 0.45; b ¼ 0.8; w ¼ p/ 2; 4 ¼ 2p, (c) a ¼ 7.0; F ¼ 105.5; b ¼ 0.8; w ¼ p/2; 4 ¼ 2p, (d) a ¼ 7.0; F ¼ 105.5; l0 ¼ 0.45; w ¼ p/2; 4 ¼ 2p.
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25
Fig. 6. Period of oscillation t as a function of the cyclotron frequency uC for (a) F ¼ 105.0; l0 ¼ 0.45; b ¼ 0.8; w ¼ p/2; 4 ¼ 2p, (b) a ¼ 7.0; l0 ¼ 0.45; b ¼ 0.8; w ¼ p/ 2; 4 ¼ 2p, (c) a ¼ 7.0; F ¼ 105.5; b ¼ 0.8; w ¼ p/2; 4 ¼ 2p, (d). a ¼ 7.0; F ¼ 105.5; l0 ¼ 0.45; w ¼ p/2; 4 ¼ 2p.
l2 r j1〉 ¼ p1ffiffiffi exp p
l21 r2 2
! (2.18)
The time evolution of the state of the electron can then be written as
1 E t 1 E t j01 ðt; rÞ ¼ pffiffiffi jf0 ðrÞ〉exp i 0 þ pffiffiffi jf1 ðrÞ〉exp i 1 Z Z 2 2
(2.19)
The probability density is in the following form:
Q ðt; r;Þ ¼ jj01 ðt; rÞj2 2 3 2 2 * 1 4 jf0 ðrÞj þ jf1 ðrÞj þ f0 ðrÞ41 ðrÞexpðiu01 tÞþ 5 ¼ 2 þ4 ðrÞ4* ðrÞexpðiu tÞ 01 0 1 where
(2.20)
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Fig. 7. Spatiotemporal evolution of the electron probability density in a superposition state of j0〉 and j1〉 for l0 ¼ 0.25, uc ¼ 10.0, a ¼ 7.0, F ¼ 0.5, b ¼ 0.8, 4 ¼ 2p, w ¼ p/4, (a) t ¼ 0, (b) t ¼ T0/4, (c) t ¼ T0/2, (d) t ¼ 3T0/4 and (e) t ¼ T0.
u01 ¼
ðE1 E0 Þ Z
(2.21)
is the transition frequency from ground state to the first excited state. The Shannon entropy of the state given by the wave function j01 (t, r, z) leads to
Z SðtÞ ¼
dzdr jj01 ðt; r; zÞj2 lnjj01 ðt; r; zÞj2
(2.22)
3. Numerical results and discussions In this part, we show the plots of the ground and first excited state energies, the transition frequency, the probability density and the Shannon entropy, versus the confinement strength, the cyclotron frequency, the electric field strength parameter and the Coulomb bound potential. All parameters used in this part are dimensionless. In Figs. 1e4, we have plotted the ground and first excited state energies as functions of the cyclotron frequency for w ¼ p/2 and 4 ¼ 2p. The ground and first excited energies are increasing functions of the cyclotron frequency. Since the presence of a magnetic field is equivalent to introducing another confinement on electrons which leads to a greater overlap of the electron wave function, the electron-phonon interactions will be enhanced, and the ground state binding energy appears more obvious [37,38]. Fig. 1 also shows that the ground and first excited state energies are decreasing functions of the electronphonon coupling constant. This is because the larger the electron-phonon coupling strength, the stronger the electronphonon interaction. This leads to an increase of the electron's energy and makes the electron interact with more phonons. However, the contributions of the electron-phonon interaction are negative in (2.14) and (2.15). As a result, the states' energies will increase with decreasing coupling strength [30]. Fig. 2 shows that the ground and first excited state energies are decreasing functions of the electric field strength. This is because the electric field leads to an increase in the electron's energy and makes the electrons interact with more phonons. In this way, the states' energies and the transition frequency are increased [30]. Fig. 3 shows that, the ground and first excited state energies are decreasing functions of the confinement length. This is because the motion of the electrons is confined by the harmonic potential. From another point of view, since the presence of
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27
Fig. 8. A- Entropy as a function of time for l0 ¼ 0.75, uc ¼ 12.0, a ¼ 7.0, F ¼ 0.5, 4 ¼ 2p, w ¼ p/4(a) b ¼ 0.2, (b) b ¼ 0.9, B-Entropy as a function of time for l0 ¼ 0.75, uc ¼ 12.0, a ¼ 7.0, b ¼ 0.8, 4 ¼ 2p, w ¼ p/4(c) F ¼ 15.0, (d) F ¼ 30.0, C-Entropy as a function of time for l0 ¼ 0.75, F ¼ 5.0, a ¼ 7.0, b ¼ 0.8, 4 ¼ 2p, w ¼ p/4, (a) uc ¼ 1.0,(b). uc ¼ 10.0.
the triangular potential is equivalent to the introduction of another confinement on the electron, which leads to a greater of the electron's wavefunctions, the electron-phonon interactions will be enhanced. These results are in agreement with the results of Kervan et al. [39], Ren et al. [40], Kandemir et al. [41]and W. Xiao et al. [42]obtained respectively by using variational, Feynman-Haken path-integral, squeezed-state variational and linear combination operator methods. Fig. 4 shows that the ground and first excited state energies decrease with the Coulomb potential. This result is agreement with the results of Fotue et al. [38]. In Fig. 5, we have plotted the transition frequency as a function of the cyclotron frequency. The transition frequency is an increasing function of the cyclotron frequency, and this is because the magnetic field is considered as another confinement. The investigation indicates that the oscillating period decreases with an increasing electron-phonon coupling constant (Fig. 5a), electric field strength (Fig. 5b), effective confinement length (Fig. 5c) and Coulomb potential (Fig. 5d), meaning that
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A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29
the qubit's lifetime is reduced, and then the decoherence process is quickened. One way of prolonging the qubit's lifetime and decreasing quantum decoherence is by adjusting the effective confinement length. This indicates a new way of controlling the triangular QD's state energies and the transition frequency via adjusting the effective confinement lengths. In Fig. 6, we have plotted the period of oscillation as a function of the cyclotron frequency. It is obvious from here that the period of oscillation is a decreasing function of the electric and magnetic fields. As a result of the presence of the electric and magnetic fields, the ground and the first-excited state energies increase, though the influence is greater on the first-excited state energy. The period of oscillation t0 decreases, that is to say, the life time of a qubit reduces, so the process of decoherence is quickened [43,44]. It can be very harmful to store information with the QD as an elementary unit. In Fig. 7, we have plotted the probability density when the electron is in the superposition state. From there, we can see that the electron probability density shows a periodic oscillation when the polar angle is varied, which arises from the influence of the triangular bound potential. This result is in accordance with that obtained by J. W. Yin et al. [45]and J. K. Sun et al. [12]. In Fig. 8, we have plotted the Shannon entropy as a function of time for different constant control parameter. This plot shows a decrease of the entropy with increasing time. The control of the coherence of the system can be done through the modulation of the electric and magnetic fields and the Coulomb binding potential [29,46]. 4. Conclusion In this paper, we have derived the ground and first excited state energies of the bound magnetopolaron under an electric field and the Coulomb potential in a triangular quantum dot and their relevant eigen-functions using the Pekar variational method. The single qubit can be envisaged as this kind of two-level quantum system in QDs. The electron probability density shows a periodic oscillation the polar angle is varied. The Shannon entropy is a decreasing function of time. The coherence of the system can be controlled by tunneling the electric and magnetic fields, the confinement length and the Coulomb binding potential. Because the electron phonon coupling constant and the phonon dispersion coefficient characterize the material, it is important to properly choose the material to use when making a quantum dot. Our results should be meaningful for the design and implementation of quantum computing both theoretically and experimentally and also for the control of decoherence in quantum systems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
W.P. Li, J.W. Yin, Y.F. Yu, Z.W. Wang, J.L. Xiao, J. Low Temp. Phys. 160 (2010) 112. Y.F. Yu, W.P. Li, J.W. Yin, J.L. Xiao, Int. J. Theor. Phys. 50 (2011) 3322. Y.J. Wen, J.L. Xiao, Y.F. Yu, Z.W. Wang, Chin. Phys. B18 (2009) 446. S. Jia-kui, L. Hong-Juan, J.L. Xiao, Physica B404 (2009) 1961. M.A. Nielsen, I.L. Chang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. I. D'Amico, Microelectron. J. 37 (2006) 1440. C.H. Bennett, D.P. DiVincenzo, Nature 404 (2000) 247. P. Hawrylak, M. Korkusinski, Solid State Commun. 136 (2005) 508. S.D. Sarma, R.D. Sousa, X.D. Hu, B. Koiller, Solid State Commun. 133 (2005) 737. Y.F. Yu, W.P. Li, J.W. Yin, J.L. Xiao, Int. J. Theor. Phys. 50 (2011) 3322. J.P. Barnes, W.S. Warren, Phys. Rev. A60 (1999) 4363. J.K. Sun, H.J. Li, J.L. Xiao, Mod. Phys. Lett. B23 (27) (2009) 3273. Z.X. Li, Mod. Phys. Lett. B26 (3) (2012) 1150015. M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849. M.W. Dellow, P.H. Beton, C.J.G.M. Langerak, T.J. Foster, P.C. Main, L. Eaves, M. Henini, S.P. Beaumont, C.D.W. Wilkinson, Phys. Rev. Lett. 68 (1992) 1754. S.S. Li, J.B. Xia, Phys. Rev. B 58 (1998) 3561. S.S. Li, J.B. Xia, Appl. Phys. Lett. 87 (2005) 043102. C. Sikorski, U. Merkt, Phys. Rev. Lett. 62 (1989) 2164. S.S. Li, J.B. Xia, Phys. Lett. A 366 (2007) 120. A. Lorke, J.P. Kotthaus, K. Ploog, Phys. Rev. Lett. 64 (1990) 2559. S.S. Li, X.J. Kong, J. Phys. Condens. Matter 4 (1992) 4815. S.S. Li, J.B. Xia, J. Appl. Phys. 100 (2006) 083714. K.D. Zhu, T. Kobayashi, Solid State Commun. 92 (1994) 353. T.C. Au-Yeung, S.L. Kho, S.W. Gu, L.H. Hong, E.M.C. Wong, J. Phys. Condens. Matter 6 (1994) 6761. S. Sil, A. Chatterjee, J. Phys. Condens. Matter 3 (1991) 9401. M. Bhattacharya, A. Chatterjee, T.K. Mitra, Phys. Rev. B 39 (1989) 8351. C.E. Shannon, Bell Syst. Tech. J. 27 (1) (1948), 379 and 623. ^, J. Mex. Chem. Soc. 52 (1) (2008) 60. R.P. Sagar, M. Ho L.C. Fai, M. Tchoffo, J.T. Diffo, G.C. Fouokeng, Phys. Rev. Res. Int. 4 (2) (2014) 267. J.L. Xiao, J. Low Temp. Phys. 172 (2013) 122. S.S. Li, J.B. Xia, F.H. Yang, Z.C. Niu, S.L. Feng, H.Z. Zheng, J. Appl. Phys. 90 (2001) 6151. S.S. Li, L. Gui- Lu, F.S. Bai, S.L. Feng, H.Z. Zheng, Proc. Natl. Acad. Sci. U. S. A. 98 (2001) 11847. Z.W. Wang, J.L. Xiao, Acta Phys. Sin. 56 (2007) 678. T. Ezaki, N. Mori, C. Hamaguchi, Phys. Rev. B 56 (1998) 6428. J.L. Xiao, Superlattices Microstruct. 70 (2014) 39. B.S. Kandemir, A. Cetin, J. Phys. Condens. Matter 17 (2005) 667. S.H. Chen, Int. J. Modern Phys. B 22 (16) (2008) 2611. A.J. Fotue, M. Tiotsop, N. Issofa, S.C. Kenfack, A.V. Wirngo, H. Fotsin, L.C. Fai, Am. J. Modern Phys. 4 (3) (2015) 109. N. Kervan, T. Altanhan, A. Chatterjee, Phys. Lett. A 315 (280) (2003) 4. Y.H. Ren, Q.H. Chen, Z.K. Jiao, Acta Phys. Soc. 7 (1998) 598.
A.J. Fotue et al. / Superlattices and Microstructures 90 (2016) 20e29 [41] [42] [43] [44] [45] [46]
B.S. Kandemir, A. Cetin, J. Phys. Condens. Matter 17 (2003) 667. W. Xiao, J.L. Xiao, Int. J. Modern Phys. B 25 (2011) 3485. A. Grodecka, P. Machnikowski, Phys. Rev. B 73 (2006) 125306. M. Lovric, H.G. Krojanski, D. Suter, Phys. Rev. A 75 (2007) 042305. J.W. Yin, Y.F. Yu, H.J. Li, J. Low Temp. Phys. 177 (1e2) (2014) 72. Z.W. Wang, W.P. Li, J.W. Yin, J.L. Xiao, Commun. Theor. Phys. 49 (2008) 311.
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