Laser light and external magnetic field control of polaron in asymmetric quantum dot

Accepted Manuscript Laser light and external magnetic field control of Polaron in asymmetric quantum dot

M.F.C. Fobasso, A.J. Fotue, S.C. Kenfack, C.M. Ekengue, C.D.G. Ngoufack, D. Akay, L.C. Fai PII:

S0749-6036(18)32023-8

DOI:

10.1016/j.spmi.2018.12.023

Reference:

YSPMI 5985

To appear in:

Superlattices and Microstructures

Received Date:

05 October 2018

Accepted Date:

17 December 2018

Please cite this article as: M.F.C. Fobasso, A.J. Fotue, S.C. Kenfack, C.M. Ekengue, C.D.G. Ngoufack, D. Akay, L.C. Fai, Laser light and external magnetic field control of Polaron in asymmetric quantum dot, Superlattices and Microstructures (2018), doi: 10.1016/j.spmi. 2018.12.023

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ACCEPTED MANUSCRIPT Laser light and external magnetic field control of Polaron in asymmetric quantum dot M.F.C. FOBASSO1, A. J. FOTUE1*, S. C. KENFACK1,2 C.M. EKENGUE1, C.D.G. NGOUFACK1, D. AKAY3, L. C. FAI1 1 Mesoscopic

and Multilayers Structures Laboratory, Department of Physics, Faculty of Science,

University of Dschang, P.O. Box 479 Dschang, Cameroon 2 African

3

Institute for Mathematical Sciences, Biriwa, N1, Accra – Cape Coast Road, Ghana

Department of Physics, Faculty of Science, Ankara University, 06100 Tandogan, Ankara, Turkey

Abstract: The possibility of controlling a polaron in an asymmetric quantum dot with external magnetic field and laser light is examined in this paper. Analytical studies are done using a modified Lee-Low-Pines method. We calculate the fundamental and first state energies which form a single qubit and further determine the probability density and the Shannon’s entropy. These quantities as well as the mobility are influenced by the laser parameters, the size of the system and the magnetic field. The results performed show that the energy increases with the cyclotron frequency and the confinement strength of the potential. The analytical results highlight the fact that the application of the magnetic fields and laser enhances the possibility of finding discrete electronic states in quantum dots. The decoherence process is found to be greatly affected by the high-frequency laser field considered. Furthermore, it is reduced when particles are more confined in the nanostructure. It is established that a strong magnetic field and a low laser frequency enable the trapping and cooling of the polaron. Keywords: polaron; laser light; magnetic field; quantum dot.

1. Introduction Nanotechnology, optoelectronics and quantum computing are all rapidly evolving nowadays. In this framework, the fundamentals of polaron physics, which serve as a basis for analyzing the effects of polarons in polar crystals and ionic semiconductors are a major area of focus. Particular attention paid to large radius polarons which are characterized by the Fröhlich Hamiltonian. An induced polarization follows the charge carriers as they evolve in the medium. The carrier associated with the created polarization is considered to be a physical entity called the polaron, described by the physicist Landau, *Corresponding

Author: (A. J. FOTUE) Phone : +237 699456032. E-mail: [email protected]

ACCEPTED MANUSCRIPT (Landau L. D., 1933). The effective properties of the quasiparticle polaron are different from those of band carriers in that a polaron has as intrinsic characteristics: an effective mass, a binding energy and a characteristic response to magnetic an external electric fields (these induce properties like optical absorption coefficient and mobility) (Devreese J. T., 2005). From this, it is logical to postulate that polarons can develop other new characteristics in the presence of laser and magnetic fields. Many investigations in theoretical physics for the study of fields in laser-matter interaction has been strongly motivated by the fast progress of laser technology over the years, and there is a good reason to believe that this progress will evolve continuously. The rapid progress of laser technology in recent years has motivated several investigations on fields in laser-matter interactions, and it is reasonable to affirm that this progress will continue. Laser semiconductors are mostly important in image communications, computer networks and interconnection of cable television signals, optical stacked circuits, laser printers, telecommunications and signal processing, and also have a wide range of applications in medicine and in military domains. Considering nanotechnology in building nano devices, there exist zero dimensional lasers commonly called quantum dot lasers. Quantum dot lasers with discrete density of states possess high temperature dependence and low threshold current. Also, quantum efficiency, high modulation rate and optical gain are superior to those of certain lasers. The effects of factors like temperature (Chen SH and Xiao JL, (2007), Kumar D. et al., 2015, Narayanan M. and Peter AJ, 2012, Rossetti M. et al., 2009), the size of quantum dot nanostructures (Baskoutas S. and Terzis AF, 2006, Pryor C., 1998), the stoichiometric percentage of the elements constituting the active area of the laser (Shi Z. et al., 2011), the quantum dot distribution and density on the energies of different levels and on the performance of quantum dot lasers are both important and interesting. Consequently, exploring the effects of given fields will be helpful to optimize the efficiency and power of lasers built with quantum dots. Special attention must be paid to the impact of temperature because any perturbation which results from changing the laser operating conditions must be anticipated (Mahdi A B et al., 2017). Recently, the electronic characteristics of quantum dot (QD) semiconductors have drawn a lot of attention as objects with zero-dimension have special physical properties which are rarely found in electron atomic systems. The discrete nature of quantum dot energy levels gives them certain properties of atoms, even though they can easily be introduced into miniaturized nano devices (Destefani C.F. and Ulloa S.E., 2006). QDs are also considered most special and interesting candidates for the quantum bits needed for quantum information processing and future quantum device applications (Rastelli G. et al., 2012). Many other important applications such as lasers are used to improve the optoelectronic properties of these zero dimensional systems or QDs (Schillak P. and Czajkowski G. 2009). Lasers made from quantum dot nanostructures are more robust to temperature fluctuations. The dot is generally considered as a sphere or point laterally confined by a parabolic potential whose direction

ACCEPTED MANUSCRIPT becomes normal to the direction of growth in the quantum well. Theoretically, the spherical model can easily be solved due to its rather prominent symmetry. Despite this advantage, this model could be difficult to design and implement. The lateral confinement described by the parabolic form of the potential is then more realistic. A number of studies have shown the particular influence on optical and transport properties of micro-structures like quantum well (2D) by the electron-phonon interaction (Rodriguez Suàrez RL and Matos-Abiague A, 2003, al, 1997, Fliyou M. et al., 1998; Xie HJ et al, 2000), quantum wires (1D) (Fai LC et al, 2005, Pokatilov EP et al, 1998, PhaniMurali KR, Ashok C, 2005, Buonocore F. et al, 2002) and quantum dots (0D) (Fai LC et al 2005, Devreese JT et al, 2001). Quantum dot nanostructures have become a major subject of research interest at present due to the optical characteristics resulting from the confinement of electronic particles and holes (Zíková M., 2012, Ma YJ et al., 2013, Danesh K. and Rajaei E, 2010, Nedzinskas R et al., 2012). Currently, quantum dot materials have promising applications in optical amplifiers and semiconductor lasers (Bimberg D et al., 2000, Gioannini M, 2006, Danesh K and Rajaei E, 2011, Asryan LV and Luryi S, 2001). They are very important in solar cells and in new laser Nano devices. Thus, having the relevant information on the energy of the different states deformation and physical characteristics that could be changed by modifying certain parameters like temperature should have an important bearing on the laser processes of a quantum dot. Based on this, research groups are trying to optimize and develop quantum dots for the production of optoelectronic structures with better performances (Mahdi A B and Esfandiar R). It is both relevant and important to find ways of ameliorating the efficiency of QDs having a fixed and known size. However, lasers can be used at very low (Le-Van Q et al., 2015, Rossetti M. et al., 2009, Tong CZ et al., 2007) or very high (Ohse RW, 1988, Rouillard Y. et al., 2000) temperature conditions. At the same time, it is proven that the temperature variation modifies the laser emission procedure both by the tuning of the output photoluminescence and by the laser characteristics (Rossetti M et al., 2009) due to the behavior of the carriers depending on the temperature. It is very important that quantum systems should be adequately separated from any external perturbations, because this could deform or even annihilate the superposed states. In fact, quantum systems are mainly affected by the interaction between the quantum memory and the environment. This exchange sometimes perturbs and destroys the stored information in quantum memory as it induces decoherence. Indeed, decoherence plays a key role in the area of quantum computing. , Consequently, significant experimental and theoretical efforts have been made (see Barnes JP and Warren WS, 1999, Tolkunov D. and Privman V., 2004, Grodecka A. and Machnikowski P., 2006, Lovric M. et al, 2007), to

investigate quantum coherence and the possibility of extending the

decoherence time. Fotue et al studied the decoherence resulting from different types of potentials within which polarons are confined (Fotue A.J. al., 2016). They concluded that quantum nanostructures are very sensitive to the confinement structure and proposed an adequate structure in which decoherence could be reduced or even suppressed. Several researchers have studied decoherence using Shannon's entropy (see, for instance, Fotue A.J. et al, 2015, Chen S et al, 2009) and

ACCEPTED MANUSCRIPT have proposed ideal quantum models to avoid decoherence and design nano devices. In this work, we intend to control the decoherence process with the help of decoherence time and Shannon’s entropy; the system under study will be subjected to many environmental effects. Besides, we will propose a means of improving the efficiency of the QD nanostructure. Another improvement of the efficiency of a QD can be obtained with linearly polarized laser fields. In this way, Bandrauk et al, (Bandrauk A.D, 1997, Dejan B. M. and Anthony F. S., 1999) proposed a way of checking the high harmonic generation by a motionless magnetic field. It stands out however, that decoherence in the presence of linear polarized laser and magnetic fields has not so far been studied for the purpose of optimizing polaronic control. Thus, we have carried out the present investigation on how to control the QD polaron with both a laser field and a magnetic field and study its dynamics and coherence in the presence of such an environmental effect. The rest of this paper is organized as follows: Section 2 consists of the characteristic Hamiltonian of the system and employs the modified Lee Low Pines method to derive the properties of the polaron. In Section 3, detailed analysis of the structural properties are given. The conclusion and some open questions are discussed in section 4. 2. Theoretical model and calculations The quantum system under consideration consists of a free electronic particle of a Bloch sphere, moving in an asymmetric quantum dot. The confinement is taken to be in the x - y plane and along the

z -direction, with 0 and  z being the respective frequencies in the x - y plane and the z  direction under the laser field. This electron interacts with the longitudinal optical phonon having  ph as frequency in the absence of dispersion. The Fröhlich total Hamiltonian in the presence of an electromagnetic field (at Feynman units   m   ph  1 ) reads: 2

1 e 1  1 H      A  r    0 2  2    z 2 z 2     ph aq  aq   2 c 2  2 q     q aq  exp(iq.r )  h.c   exE cos  t    cos  3t     

(1)

q





where A is a potential vector chosen in two directions. Taking B the time independent magnetic field applied

in

the

z  direction

2 1 e   2   A  r    c2  c lz  2 c 2 

where C 

e B0 B with   , c B0





as B  Bez ,

the

first

term

of

expression

(1)

becomes

ACCEPTED MANUSCRIPT C stands for the cyclotron frequency. With this transformation of the Hamiltonian in (1), we obtain:

H

2 e2 1 1  2  2  2  02  2   z2 z 2     ph aq† aq    q aq† eiqr   q* aq e  iqr    2 2c 2 2 q (2) q

 exE cos  t    cos  3t     The electric field as chosen by Bandrauk et al (Bandrauk A.D, 1997) has the following form:

 E  t   E cos  t    cos  3t    

(3)

Here, we have selected a combination of fields. The term E expresses the maximum strength of laser and  represents the frequency of the laser light. Here, we consider a superposition of lasers that have as frequencies  and 3 varying with the phase

 . This choice is motivated by the fact that first of all we need intense and coherent laser field in the system (Bandrauk A.D, 1997). The Hamiltonian in (2) exhibits some useful effects: cloistering or

e2 2 2 imprisonment by a diamagnetic potential 2   ; resonance when the cyclotron frequency is equal 2c to the laser frequency, a given amplitude of the laser driven in x  direction by a linear electric field and finally the term which represents the interaction between electrons and phonons (Bandrauk A.D, 1997).



aq  aq  stands for the annihilation (creation) operator with q being the wave vector of the volume longitudinal optical phonon. Pˆ represents the momentum and 𝑟 = (𝑥,𝑦,𝑧) denotes the electron position vector. The amplitude of the electron-phonon interaction is  q can be seen in (1) and have been defined by:

1 1 4  2  q  i  2   q  V 

1

2

(4)

where

 e2  2  ph

 

1   1 1 2   (5)   2 ph     0  

In (5),   is the electronic dielectric constant of the ionic crystal and  0 is the static dielectric constant of the ionic crystal or polar semiconductor, while  ph stands for the frequency of the longitudinal optical phonon.

ACCEPTED MANUSCRIPT Following the framework of the LLPH method, the trial wave function has been selected as:













  exp  ia  qraq† aq  exp    f q aq†  f q* aq   0 i  r  



q

 q





(6)

The parameter a in Eq.(6) is variational and helps to distinguish different types of coupling, and can be obtained when the energy is minimized; 0 appearing in the trial wave function stands for a



vacuum state, and i 0,1  r  stands for the electronic wave function. To calculate the total energy of the system we have used the energy eigenvalues relation E   H  . The term in (6) with variational function f q permits to displace the phonon variables. The trial electronic wave functions of the fundamental and first-excited states are chosen of the same form as those used by Kenfack et al (2017):

 x  y z 0   32  

  x2 x 2   y2 y 2   z2 z 2   ip z   e 0 (7)  exp   2   

 2  x  y  z3 1    32 

   x2 x 2   y2 y 2   z2 z 2   ip z  exp    ze 1 (8) 2   

where  x ,  y ,  z , p0

and  x ,  y ,  z , p1 are variational constants and can be obtained by

minimizing the energy of the system. Here, we work with p0  p1 to satisfy the following relations:

0 0  1 1  1 and 0 1  0 . After some straightforward calculation and minimization, we obtained the explicit expression for ground state energy as follows:

  x2  y2  2  02   2   2 e2 e2 eE E0     z   2  2 2  2    02  2 2  2   z 2  cos  t   cos 3  t             0  4  4 16 1 2    4 4 4 4  4  c 4  c x x y y z x       (9) with





2 2 2 exp   1  ax  qx2 2  x2  1  a y  q y2 2  y2  1  az  qz2 2  z2     0  2  dqx dq y dqz 2 2 2 2 2 2   ax qx  a y q y  az qz    ph  2  

ACCEPTED MANUSCRIPT Finally, the virtual number of phonons around the electron in the ground state can be calculated as:





2 2 2 exp   1  ax  qx2 2  x2  1  a y  q y2 2  y2  1  az  qz2 2  z2    (10) N 0  2  dqx dq y dqz  ax2 qx2  a y2 q y2  az2 qz2     ph   2  

The mobility is expressed as the inverse of the number of phonons. It is important to state that the behavior of the system’s mobility is given a simple development which highlights some of its major features; especially the temperature dependence of that mobility. In fact, the existence of the lattice vibration shows that there is a temperature variation, which has an impact on the mobility. Its expression is given by: 1

    2 2 2 2 2 2 2 2 2   exp  1  ax  qx 2 x  1  a y  q y 2 y  1  az  qz 2 z      (11) 0   2  dqx dq y dqz    ax2 qx2  a y2 q y2  az2 qz2     ph     2      





At finite temperature, however, all electrons cannot be located in the ground state, which is an indication of the fact that the lattice vibrations perturb not only real phonons but also electrons in an asymmetric parabolic potential. Based on the theory of quantum statistics (Sun Y. et al, 2014, Cai Ch.Y. et al, 2016), the mean number of quantum vibrations is

N0 

1  exp( LO )  1 kBT

,

(12)

where T stands for temperature and k B is the Boltzmann constant. Physically, there is a relations between (10) and (12) showing the consistence of both equations and proving the temperature dependence of phonons. The effective mass of the polaron is calculated using an arbitrary integral functional J 0 taken as:

J 0  0 H 0  U 0 0 Pˆz 0

(13)

where U 0 is the Lagrange multiplier, which as before can be identified as the polaron velocity and the momentum in the longitudinal direction is given by:

Pz  i

   qz aq† aq z q

ACCEPTED MANUSCRIPT Upon calculation, we obtain 2  x2  y  z2 J0     P02 4 4 4

 2  2   2 e2 e2 eE   02  2 2  2    02  2 2  2   z 2  cos t   cos  3t       4 16 1 2     z x  4 x 4 x c   4 y 4 y c 

 a2 q2  2  iqr 1 a  2    hc   U 0 P0   U 0 qz f q q  ph 2  fq  q q fq* e  q  

2

  2 a2 q2  iqr 1 a  J 0  C0  P02     ph   U 0 qz  f q    q f q* e  hc   U 0 P0 (14)   2 q  q  where

 y2

 02  2  02 e2 e2 eE 2  C0      2  2 2     2  2 2  2   z2  cos t   cos  3t       4  16 1 2     4 4 4  4 x 4 x c z x   4 y 4 y c 

 x2

 z2

(14) Minimizing the function J 0 with respect to f q and P0 give

  q k U q U 02 U 02  1 2 P0 J 0  C0   U 0 P0     2  U 2 2 3 4 2 q   0 a q  2   ph    2   2

2

2 0

2 z

  q k q    C0 (15)  2 2 3    q  a q    ph    2    2

2

2 z

Replacing all the terms by their explicit forms, we obtain the effective mass as:

M *  1  2 q



2 q





2 2 2 qz2 exp   1  ax  2  x2 qx2  1  a y  2  y2 q y2  1  az  2  z2 qz2    (16) 2 2 2 2 2 2 3  ax qx  a y q y  az qz   ph     2  

The first excited state energy is given as follows: 2   2  eE   2  x2  y 3 z2  02 e2 e2 E1      2  2 2  2    0 2  2 2  2   z 2  z 0 cos   t    cos  3t     1 4 4 4  4 x 4 x c   4 y 4 y c  4 z 16 x

(17) with

ACCEPTED MANUSCRIPT  1  az 2 qz2 1  2  dqx dq y dqz 1   2  z2 





2 2 2 2 2 2 2 2 2    exp   1  ax  / 4  x qx  1  a y  / 4  y q y  1  az  / 4  z qz    ax2 qx2  a y2 q y2  az2 qz2   ph  2

We can also calculate the mean number of phonons in this state and the mobility as it has been done for the ground state using equation (12). Between the ground state energy and the first excited state energy, we can have the frequency of the transition by relation:

01  E1  E0

(18)

Therefore, this transition frequency is an indicator of the fact that the ground and first excited state energies are theoretically possible candidates for the formation of the single qubit required by the twolevel system in quantum computation. The wave function for the electron existing in both states is given by:

 01 

1 2



0

 1



(19)

where 1/ 2

 x  y z  0  0    32     2  x  y  z3 1  1    32 

  x2 x 2   y2 y 2   z2 z 2   ip z exp   e 0 2  

1/ 2

  

  x2 x 2   y2 y 2   z2 z 2   ip z exp    ze 1 2  

(19-a)

(19-b)

Then, we can calculate the system’s wave function in time evolution by:

 01  t , x, y, z  

1 2



0 exp  i 

E0 

 1  E  t 1 exp  i 1 t  2    

(20)

The time evolution of the wave function does not give information on the nature of the particle despite the fact that it establishes a QD qubit. From this, it is important to deal with physical parameters indicating the presence of quasi-particles in the given region of space, namely the probability density given by:

Q  x, y, z , t    01  t , x, y, z 

2

(21)

In order to study thedecoherence brought about by external parameters, we will calculate the decoherence time and Shannon’s entropy in order to open up possible applications in information

ACCEPTED MANUSCRIPT theory. Decoherence is examined based on Fermi’s Golden rule in the dipole approximation through the evaluation of spontaneous emission rate as:

e 2  E  0 r 1 3



2

(22)

2 c 0

In (20), c is the speed of light in vacuum,  0 represents the vacuum dielectric constant,  refers to the coefficient of dispersion, E is the difference between energies for the functions

0 and 1 , and

 1 is the decoherence time. The Shannon entropy of the system is evaluated as: S  t    dzdydx  01  t , x, y, z  ln  01  t , x, y, z  2

2

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ACCEPTED MANUSCRIPT 3. Numerical results and discussion In order to clarify the results that have been obtained, in this section, we numerically investigate the energy as a function of the other parameters, namely the laser and magnetic fields. The probability density and mobility are also plotted as a function of laser parameters  , E and the size of the asymmetric parabolic quantum dot. To interpret the great efficiency of decoherence on the system, we have displayed the decoherence time and Shannon entropy. Our goal is to highlight the effects of the laser field, the effects of the magnetic field and the magnitude of the confinement in this system. In this part we have taken dimensionless values of    10 . 350  =1  =10  =20

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Figure 1: Ground state energy versus confinement strength in the x-direction at constant cyclotron frequency. Figure 1 shows the variation of the ground state energy with confinement strength for several values of the cyclotron frequency. We observe that the energy increases with the frequency and magnetic field. This result is interesting because it brings to light the fact the energy increases as the dimension of the structure is reduced. The behavior observed here agrees with that obtained by Fotue et al (2016) and Phani Murali K. (2006). We equally observe that the energy increases in the presence of the magnetic field. Indeed, the cyclotron frequency in this system is an efficient parameter for controlling the energy of the system. In other to increase the system’s energy, especially that of the the ground state, it is possible to modulate either the size of the system or the cyclotron frequency which characterizes the

ACCEPTED MANUSCRIPT strength of the magnetic field. Another parameter which equally plays an interesting role and whose presence affects the system is the laser frequency. 61.6  z =2

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Figure 2: Ground state energy versus laser frequency at constant confinement frequency (in the zdirection). Figure 2 shows a plot of the ground state energy against laser frequency at constant confinement strength in the z-direction. Here, we observe that the energy oscillates with the laser frequency and increases as the confinement strength of the system increases. It is seen that as the dimension of the structure is reduced in a quantum dot, the energy of the system becomes more and more discrete which appreciably reveals the quantum nature of the system. The form of the energy obtained has a wave-like behavior because of the harmonic form of the laser field. The plot particularly reveals the fact that the energy of the polaron fluctuates like the laser, showing that the latter has an impact on the properties of the polaron. The curve gives information on the fact that energy is then affected by the presence of the laser field especially when we combine different forms of laser. The energy of the system is affected by the laser field, the magnetic field and the size of the system. Therefore, modulating those parameters can serve to control a polaron confined in an asymmetric quantum dot. Such a system can be applied to control the efficiency of the semiconductor laser. This work shows compatibility with the work of Kenfack S.C et al, 2017.

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Figure 3: Probability density versus cyclotron frequency and laser frequency when the polaron is in superposed states 0 and 1 . Figure 3 shows the probability density as a function of the cyclotron frequency and laser frequency. The polaron is confined and localized in space and the probability density oscillates with the laser frequency. Clearly, we see that while this density oscillates and decreases with  , it increases with  . This means that the location of the polaron is fixed in space, which also highlights the effects of the magnetic and laser fields. These effects are also influenced by the choice of the dot and the form of the laser. In fact, with the reduction of the dimensions of the system, the nature of the quantum state is much more discrete and the localization of the polaron is rendered more possible by the control of the laser and magnetic fields.

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Figure 5: Electron probability density versus cyclotron frequency and time. In figure 5, we have plotted the probability density in three dimensional space as a function of time and cyclotron frequency when the formed states are superposed. The presence of several peaks in the form of the probability density is clear proof of its oscillatory behavior. From figure 5, it is easily noticed that a variation of the probability density with time and the spatial coordinates

 and z .

Besides, with the presence of the asymmetric potential in the direction of the quantum dot, the ensuing probability density exhibits a maximum configuration in the presence of the magnetic field which is characterized by the cyclotron frequency. The location of the polaron in space is appreciably influenced by laser light and magnetic field. We have suppose that the polaron inside the structure can be controlled, this enable to enhance quantum effects for nanotechnology purposes.

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Figure 6: Mobility versus confinement strength in the x-direction for ground state (on the left) and excited state (on the right). Figure 6 is a plot of the polaron mobility as a function of the confinement frequency of the polaron. The subfigure on the left shows the polaron mobility in the ground state while the figure on the right shows the polaron mobility in the first excited state. It is seen from both curves that the polaron mobility reduces with the frequency of the asymmetric quantum dot potential. The polaron mobility is found to have the same behavior in both cases. It is also seen that at higher frequency, the smaller the radius and the slower the motion of the polaron. It is also possible to reduce the motion of a polaron confined in a quantum dot and is very useful in the study of transport properties of the polaron. We thus propose a model in which the laser field enables a restriction of the motion of the particles in a dot.

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Figure 7: Ground state mobility versus coupling constant at fixed laser frequency. 1 0.95 0.9

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Figure 8: Ground state mobility versus coupling constant at fixed cyclotron frequency.

ACCEPTED MANUSCRIPT Figures 7 and 8 show the plots of the mobility in the ground state as a function of the coupling constant  . Figure 7 shows the mobility at fixed laser frequency while figure 8 shows the mobility at fixed cyclotron frequency. Here, is it seen that the ground state mobility increases with the coupling constant. Mobility increases when the cyclotron frequency decreases and when the laser light frequency increases. It can be deduced from that reciprocal impact of the total electromagnetic field in the system gives an idea on the polaron’s mobility and it is concluded from the evidence that the stronger the coupling constant, the higher the mobility. Following the results obtained for the mobility, it is observed that the latter increases for large values of the coupling constant at a rate which is faster than that predicted by Shultz (Shultz, 1959), in which the augmentation starts even for small coupling constant. This behavior described is expected since the strength of coupling constant appears strongly, the phonon’s wavelength reduces steeply than localization of the polaron. Since the wavelength contributes effectively in polaron scattering, the polaron becomes insensitive to phonon scattering, as shown by Santhi et al (Santhi et al 2009). With the presence of laser light and magnetic field, new effects on the polaron such as sensitivity to phonon scattering are observed. Indeed, we observe that a low laser field and a high magnetic field both reduce the motion of quasi particles. The presence of both effects can serve as a means to cool the polaron. 6  =5  =100

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Figure 9: Decoherence time versus coupling constant at fixed laser frequency. Figure 9 presents the plot of the decoherence time with the coupling constant for two different values of laser frequency. Two zones are observed in both cases: one with a reduction of the decoherence

ACCEPTED MANUSCRIPT time and the other with an increase in decoherence time. Decoherence time decreases in the weak coupling regime. However, when the coupling constant increases, decoherence time slowly increases and then tends to a constant value. Decoherence time decreases rather quickly with higher values of laser frequency. This means that the presence of the laser affects the coupling between electrons and phonons. The system is thus sensitive to environmental effects, especially the presence of the laser. It is worthwhile to note that the present result, which shows the increase of the coupling constant when the studied polaron is in an asymmetric potential is quite increasing. Furthermore, increasing the laser frequency could also be one way of controlling decoherence. This figure indicates that there are two critical points where the system is completely stable or coherent. In fact, when the frequency of the laser is high enough, decoherence reduces faster and although there is a point where the system is coherent it is quite short-lived because at high coupling strength decoherence increases. This should be the case of semiconductors like GaAs and ZnS which are characterized by small values of the coupling constant (J.-S Pan 1985). When the frequency decreases, we observe also a critical point which shows a null value of decoherence; this is observed in the case of AgBr semiconductor, but at high coupling this decoherence time increases faster than at high frequency (C. Lee et al, 1998). Materials with high coupling constant like RbCl, KCl can be useful to build artificial quantum dot (J.-S Pan 1985; M. Tiotsop 2016). We observe clearly that the decoherence process is controlled in the presence of laser. For practical purposes, a period where the system keeps its quantum state over a certain period of time can be identified. Decoherence can be reduced if the frequency of the laser is maximal, which could help to modify the properties of semiconductors which are characterized by fixed coupling constant. 2.5  =10  =100

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ACCEPTED MANUSCRIPT Figure 10: Decoherence time versus time at constant value of magnetic field   10 and   100 . Figure 10 shows a plot of the decoherence time versus time at constant cyclotron frequency. Here, it is seen that decoherence evolves periodically in time. This figure also shows that as the cyclotron frequency is increased, the peak of the decoherence time increases and decreases in the same period of time, showing that with the laser we can control the system in order to reduce decoherence. The magnetic field chosen in the z-direction enhances the electron-phonon exchange and thus plays the role of a new confinement for that polaron. The increasing magnetic field creates an interesting behavior, namely the moving away of the electron from the center of the dot so that it becomes closer to the surface area along the coordinate axes. Then, reducing the ground state is important experimentally for controlling and modulating the characteristics of optoelectronic devices. This result agrees with that obtained by Fotue et al(Fotue et al 2016). 2.5  =10  =100

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t Figure 11: Decoherence time versus time at constant laser frequency   10 and   100 . Figure 11 is a plot of the decoherence time as a function of time and at constant laser frequency. Worthy of note is the fact that a coherent evolution of the decoherence time with the laser time is observed as the laser frequency is increased. This means that all the laser parameters affect the studied system, especially the laser frequency. It appears that laser control helps to modify the properties of a quantum system and restore coherence in this particular system. The form of the laser here is taken as the usual form of electric field seen in the literature, which is known to cause the self-induced

ACCEPTED MANUSCRIPT emission of polarons and the high emission rate at resonance. This result is similar to that of Fotue et al, (Fotue et al, 2016) whose work confirmed that coherence is evidently controlled by tunneling the magnetic and electric fields. Decoherence of electrons encircled by phonons faced the problem of loss of information because of the electron-phonon exchange. Therefore, important protection of the stored information permits to avoid unpleasant effects like loss of the two level system formed. 0.073  =2

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t Figure 12: Time evolution of entropy at constant laser frequency. Figure 12 shows a curve of Shannon entropy S versus time for two values of the laser frequency. The behavior observed is that entropy or information is transferred with time by maintaining a fixed amplitude. At the higher value of the laser frequency, oscillations show a continuity in function of time and a shift when the value of frequency changes; this shift is more pronounced at t=[6,8]. The reduction of this parameter thus helps to obtain a coherent evolution of the entropy. The process of decoherence can be regarded as a good consequence of information evolution in an environment where electrons and phonons exist. The presence of the laser principally affects the coherence of this system.

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 Figure 13:Evolution of entropy versus cyclotron frequency at constant confinement strength in the xdirection. The figure 13 displays the curve of information entropy versus cyclotron frequency at constant values of confinement strength in x-direction. This entropy evolves with cyclotron frequency at different oscillation amplitudes. The observed oscillatory behavior increases and decreases in amplitude over successive periods. This translates the fact that sometimes the quantum states are spontaneously lost after their apparition. From a quantum information theoretic point, the studied system can be said to be alternately loosing and gaining energy. The presence of the magnetic field thus becomes a source of destructive interference producing decoherence because, despite its new role of a confinement for the system, the magnetic field is first of all an environmental effect and then decoherence control factor in this system. The continuously evolving amplitude of the entropy with strong coupling between the particles is an indicator of a bid to restore coherence of the initially formed states. Thus, a modulation of the magnetic field and quantum dimension could help to efficiently control the coherence of the system, what ties with the predictions of Fotue and collaborators (Fotue et al, 2015). The complete coherence of the studied system is controlled by varying the laser field, the quantum size of the asymmetric potential and the magnetic field. Because the electron phonon coupling constant is a property of the particular material, it is important to adequately select a material for designing a QD. The results obtained here should be significant for the control of decoherence in quantum systems in both theoretical and experimental work in this area.

ACCEPTED MANUSCRIPT Summary The influence of both the laser and magnetic fields on a system of confined polaron particles in an asymmetric quantum dot has been examined with the aim of proposing an efficient control scheme for polarons in nanostructures. After calculating the energies of the concerned states by means of a trial eigenfunction for the system, the former were transformed using a modified Lee Low Pines method. This enabled the formation of a single qubit with the characteristics of a two-level quantum dot. The Shannon entropy has been determined and shown to be strongly influenced by the laser frequency, the cyclotron frequency and the size of the structure. The numerical results have revealed that the probability density, which is itself influenced by the presence of electrons, oscillates with a given periodicity when the particles are localized in the considered two-level system. The mobility of the polaron decreases with the frequency of the asymmetric quantum dot potential. The influence of the laser field on the polaronic state and hence, on QD nanostructures has been shown to be considerable. The frequency of the laser has also been seen to have an impact on the energy of the system. Given that the mobility of the particles is quite reduced at low laser field and high magnetic field, a cooling of the polaron can be achieved. Because the polaron coupling strength is a particular property of any given semiconductor material, a judicious way of selecting the latter for the purpose of fabricating QDs has been proposed. The calculations have revealed that the strong localization of particles which guarantees their presence in QDs is enhanced with laser confinement. These results have important implications on the possible applications of infrared detectors and derived modules, and other solid state systems based on quantum computers which are known to rely on a good manipulation of the electronic states. Therefore, some improvements can be obtained the applications of nano devices which use QD structures when the fields are well selected. Suitable experimental work is however required in order to support the theoretical results here obtained. References [1] L.V. Asryan and S. Luryi, Tunneling-injection quantum-dot laser: ultrahigh temperature stability, IEEE Journal of Physics, 37 (2001) 905-910. [2] A. D. Bandrauk et al, John Wiley & Sons, Inc, 1997. [3] J.P. Barnes and W.S. Warren, Decoherence and Programmable Quantum Computation, Phys. Rev. A, 60 (1999) 4363-4374 [4] S. Baskoutas and A. F. Terzis, Size-dependent band gap of colloidal quantum dots, J. Appl. Phys, 99 (2006) 013708. [5] W. Becker et al, Light at the end of the tunnel: two- and three-step models in intense-field laseratom physics, Quantum Semiclassic.Opt 7 (1995) 423. [6] D. Bimberg et al, Quantum dot lasers: breakthrough in optoelectronics,Thin Solid Films, 367 (2000) 235-249.

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We investigate in this work the possibility of controlling polaron in asymmetric quantum dot with laser light and external magnetic field. Analytical studies have been done via the application of modified Lee-Low-Pines method. We calculate the ground and first excited state energy that form a single qubit; the probability density and also the entropy. Our results show that energy is an increase function of cyclotron frequency and confinement strength. All the properties of the system calculated here are more sensitive to the laser field frequency and external magnetic field. Our calculations reveal that the strong localization of the electronic states in the quantum dot is enhanced due to application of laser and magnetic field. Decoherence process is greatly affect by the presence of high frequency of laser field and is reduced when the particles are more confined in our system. This work also reveals that strong magnetic field and low laser frequency permit to trap and cool polaron. We propose materials with weak decoherence.