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Research Article
General Science
Material Science

# Modulation of off-diagonal components of static linear and nonlinear polarizabilities of doped quantum dots by coupled influence of noise and damped propagation of impurity

Jayanta Ganguly1, Surajit Saha2, Manas Ghosh3

# Abstract

We investigate the profiles of off-diagonal components of static linear (αxy, αyx), first nonlinear (βxyy, βyxx), and second nonlinear (γxxyy, γyyxx) polarizabilities of repulsive impurity doped quantum dot. We have considered propagation of the dopant in an ambience that damps the motion. Simultaneous presence of intrinsic noise of the system has also been considered. The dopant is characterized by a Gaussian potential and the noise considered is a Gaussian white noise. In order to compute the polarizability components the doped system is subjected to an external static electric field of given intensity. The internal noise directly couples with damping and the resulting noise-damping coupling strength appears to be a central parameter that happens to design the profiles of above polarizability components. The coupling strength modulates the dispersive character of the system and enforces relative dominance of noise effect and damping effect that depends heavily on order of polarizability. This results to diversities in the observed optical properties of static linear and nonlinear off-diagonal components. The present investigation is believed to reveal some useful features in the optical properties of doped quantum dots.

Keywords quantum dot, impurity, Gaussian white noise, damped propagation, polarizability

# Author and Article Information

Author info
1 Department of Chemistry, Brahmankhanda Basapara High School, West Bengal, India.
2 Department of Chemistry, Bishnupur Ramananda College, West Bengal, India.
3 Department of Chemistry, Physical Chemistry Section, Visva Bharati University, West Bengal, India.

ReceivedSep 2 2014　　AcceptedOct 18 2014　　PublishedNov 6 2014

CitationGanguly J, Saha S, Ghosh M (2014) Modulation of off-diagonal components of static linear and nonlinear polarizabilities of doped quantum dots by coupled influence of noise and damped propagation of impurity. Science Postprint 1(1): e00036. doi:10.14340/spp.2014.11A0001.

Copyright©2014 The Authors. Science Postprint is published by General Healthcare Inc. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs 2.1 Japan (CC BY-NC-ND 2.1 JP) License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

FundingNo significant funding.

Competing interestNo relevant competing interests were disclosed.

Corresponding authorManas Ghosh
AddressDepartment of Chemistry, Physical Chemistry Section, Visva Bharati University, Santiniketan, Birbhum 731 235, West Bengal, India.
E-mailpcmg77@rediffmail.com

# Introduction

Quantum dots (QDs) are the final destination so far as miniaturization of semiconductor devices is concerned. They exhibit more prolific nonlinear optical effects than the bulk materials, and possess widespread applications in high-speed electro-optical modulators, far infrared photodetectors, semiconductor optical amplifiers, and so on. The study of optical properties of these devices endows us with a lot of important information about their energy spectrum, the Fermi surface of electrons, and the value of electronic effective mass. Undoubtedly, QDs have justified their worth as a high performance semiconductor optoelectronic device. However, QDs are often contaminated with dopants during their manufacture which immensely alter their properties. This kind of huge alteration takes place because of subtle interplay between the impurity potential and the dot confinement potential. A large number of investigations on doped QD 1–3 therefore conform with increasing need of deciphering their properties. Within the domain of optoelectronic applications, impurity driven modulation of linear and nonlinear optical properties has been found to be extremely important in photodetectors and in varieties of high-speed electro-optical devices 4. A plethora of many important works on both linear and nonlinear optical properties of these structures was therefore an obvious outcome 5–12.

External electric field has often been found to significantly highlight an important aspect related with confined impurities 13. The electric field changes the energy spectrum of the carrier and regulates the intensity output of the optoelectronic devices. Moreover, the electric field often diminishes the symmetry of the system and leads to generation of nonlinear optical properties. Among the nonlinear optical properties the second-order quantity produces the lowest-order nonlinear effect with magnitude usually larger than higher-order ones. Thus, the applied electric field turns out to be highly relevant so far as optical properties of doped QDs are concerned 4, 7, 14–25.

The performance of mesoscopic devices is largely affected by Noise specially when the size of the electronic system falls within nanometer scale 26. Quite often, noise perturbs the level of device performance 27 and hinders the manufacture of semiconductor heterostructure for numerous applications. Hastas et al. have demonstrated the relevance of noise in self-assembled InAs QDs embedded in GaAs 28. The noise can result externally, or it may be intrinsic. It is usually the change in the impurity configurations that gives rise to intrinsic　noise 29. There are also some important works that deal with low-frequency noise and its applications to defect related optoelectronic properties of materials and structures 30–33. The phenomenon of damping in QD attracted a lot of attention owing to its importance in fundamental physics as well as nanoelectronic applications. The applications include manufacture of high quality single electron transistors 34, logic elements (quantum bits) 35, memory cells 36, and lasers based on QD heterostructures 37. These nanoelectronic devices contain doped semiconductor fragments and the functioning of these impurities come under direct influence of damping 38–41. This happens as a result of sensitive interaction between environmental excitation in the vicinity of dopants and the QD carriers 41. Consequently, we find a good number of studies on the QD electronic damping dynamics very much linked with environmental excitation 35, 42.

Later we have made some investigations on optical properties of QD insisted by damped propagation of dopant 43, 44 and also by Gaussian white noise 45, 46. In the present manuscript we have explored the combined influence of damped drift of dopant and noise on some of the off-diagonal components of static linear (αxy, αyx), first nonlinear (second order) (βxyy, βyxx), and the second nonlinear (third order) (γxxyy, γyyxx) polarizabilities of doped QD. The said off-diagonal components have been computed by exposing QD to an external static electric field. The damped propagation of dopant eliminates the inversion symmetry of the dot leading to the emergence of non-zero nonlinear optical properties. For simplicity, here we have considered inherently linear drift of dopant from a fixed dot confinement center getting continually impeded owing to damping. The accompanying time- dependent modulation of spatial stretch of impurity (ξ-1) has also been taken into account 43, 44. In order to venture the aforesaid combined influence we consider Gaussian white noise generated internally that simply couples with the damping term. The present study focuses on investigating the role of coupling strength on the said polarizability components and reveals some important aspects.

# Methods

The model considers an electron subject to a harmonic confinement potential V (x, y) and a perpendicular magnetic field B. The confinement potential reads$V(x, y) =\frac{1}{2} m^* \omega_0^2(x^2 + y^2)$, where ω0 is the harmonic confinement frequency. Under the effective mass approximation, the Hamiltonian of the system becomes

\begin{eqnarray} H_0^\prime = \frac{1}{2m^*}\left[-i\hbar\nabla +\frac{e}{c}A \right]^2 + \frac{1}{2}m^* \omega_0^2(x^2+y^2). \end{eqnarray} (1)

m* is the effective electronic mass within the lattice of the material considered. We have taken m* = 0.067m0 and set $\hbar$ = e = m0 = a0 = 1. This value of m* closely resembles GaAs quantum dots. In Landau gauge [A = (By, 0, 0)] (A being the vector potential), the Hamiltonian reads

\begin{eqnarray} H_0^\prime = -\frac{\hbar^2}{2m^*}(\frac{\partial ^2} {\partial x^2}+ \frac{\partial^2}{\partial y^2})+ \frac{1}{2}m^* \omega_0^2x^2+\frac{1}{2}m^*(\omega_0^2+ \omega_c^2)y^2 - i\hbar\omega_cy\frac{\partial}{\partial x}, \end{eqnarray} (2)

ωc = eB/m*c being the cyclotron frequency (equivalent to magnetic confinement offered by B). The magnetic field in atomic unit corresponds to a field strength of miliTesla (mT) order. We may define Ω2 =ω02 + ωc2 as the effective frequency in the y-direction. The model Hamiltonian [cf. eq. (2)] represents a 2-d quantum dot with a single carrier electron 47 with lateral confinement (parabolic) of the electrons in the x-y plane. The parabolic confinement potential often serves as an appropriate representative of the potential in semiconductor structures 1, 3, 8, 18, 20 and has been actually invoked in the study of optical properties of doped QDs 11.
In the present problem we have considered that the QD is doped with a repulsive Gaussian impurity 48-52. Incorporating the impurity potential to the Hamiltonian [cf. eq. (2)] we obtain

\begin{eqnarray} H_0(x, y, \omega_c, \omega_0)= H_0^\prime(x,y,\omega_c, \omega_0)+ V_{imp}(x_0, y_0), \end{eqnarray} (3)

where $V_{imp}(x_0, y_0) = V_{imp} (0) = V_0 \; e^{-\xi_0 \left[(x-x_0)^2+(y-y_0)^2 \right]}$ with ξ0 > 0 and V0 > 0 for repulsive impurity, and (x0, y0) denotes the coordinate of the impurity center. V0 is a measure of the strength of impurity potential whereas ξ0-1 determines the spatial stretch of the impurity potential.
Thus, the problem boils down to modeling the energy eigenvalues and eigenvectors of the two dimensional Hamiltonian H0:

\begin{eqnarray} H_0\psi_n(x, y) = E_n\psi_n(x, y). \end{eqnarray} (4)

Equation (4) turns out to be the energy eigenvalue equation of a two dimensional harmonic　oscillator as ωc (i.e., B) → 0 and V0 → 0. Naturally, we seek diagonalization of H0 in the direct product basis of harmonic oscillator eigenfunctions. The time-independent Schrödinger equation has been solved using variational method expressing the trial wave function ψ(x, y)　in the product basis of harmonic oscillator eigenfunctions 43, 44 ϕn (αx) and ϕm (βy) respectively, as

\begin{eqnarray} \psi(x, y)=\sum_{n, m}C_{n, m}{\phi_n(\alpha x) \phi_m(\beta y)}, \end{eqnarray} (5)

where Cn, m are the variational parameters and $p = \sqrt{\frac{m^* \omega_0}{\hbar}}$ and $q = \sqrt{\frac{m^* \Omega}{\hbar}}$. The general expressions for the matrix elements of H0 and Vimp (0) in the chosen basis have been derived 43, 44, and in the linear variational calculation, an appreciably large number of basis functions have been exploited after making the required convergence test.
The dopant propagation can be realized in terms of time-dependent dopant coordinates x0x0(t) and y0y0(t). Such a dopant propagation, in turn, makes γ (the spatial spread of dopant) time-dependent 43, 44 as the latter depends on the instantaneous dopant location and therefore reads as $\gamma(t) = \gamma_0 \exp{\left \{- \eta \sqrt{x_0^2(t)+ y_0^2(t)} \right\}}$, where η is a very small parameter and γ0 is the initial value of γ. The time-dependence thus indicates a lazy extension of the domain over which the inuence of dopant is disseminated. Now the time-dependent Hamiltonian reads

\begin{eqnarray} H(t)= \left[H_0 - V_{imp}(0)\right] + V_1(t), \end{eqnarray} (6)

where

\begin{eqnarray} V_1(t) = V_0 \; e^{-\gamma(t) \left[\left\{x-x_0(t) \right\}^2+ \left\{y-y_0(t)\right\}^2 \right]}. \end{eqnarray} (7)

In the present work the intrinsic time-dependence of dopant propagation has been considered to be linear for convenient handling of the problem so that x0(t) = x0 + at, y0(t) = y0 + bt 43-44. The hindrance offered by damping to the dopant propagation has been modeled by introducing an exponentially decaying term phenomenologically through the parameter ζ (a measure of damping strength) giving rise to x0(t) = (x0 + at) e-ζt and y0(t) = (y0 + bt) e-ζt. Such representation causes a faster fall of dopant propagation with increase in the damping strength in comparison with the undamped motion. The spatial stretch of impurity (γ-1) also changes accordingly.
The noise comprises of random term [σ(t)] quite often assumed to be following a Gaussian distribution and characterized by the equations 45, 46:

\begin{eqnarray} \langle \sigma(t) \rangle = 0, \end{eqnarray} (8)

the zero average condition, and

\begin{eqnarray} \langle \sigma(t) \sigma(t^\prime) \rangle = 2 \mu \delta(t-t^\prime), \end{eqnarray} (9)

the two-time correlation condition with a negligible correlation time, μ being the noise strength. The highly uctuating term σ(t) is called white noise because of a monotonous spectrum in frequency space, like that of white light. The noise strength μ becomes simply a measure of fluctuation intensity 53. We have invoked Box-Muller algorithm to generate σ(t). Consideration of noise [σ(t)] which is generated internally allows us envisage a coupling between itself and damping through the coupling parameter λ. The coupling modifies the drift of the dopant as x0(t) = (x0 + at) e -ζt ・ {1 +λ σ(t)} and y0(t) = (y0 + bt) e -ζt ・ {1 +λ σ(t)}.
A variation of λ from a very low fractional value to unity provides us a means to tune the extent of coupling between damping and noise. Such coupling ensures retrieval of initial condition (t = 0) if we assume that σ(t) → 0 as t → 0 and also splits the drift of the dopant into two well-separated components; the first of them being the purely damping controlled term (no noise effect) [(x0 + at) e-ζt] whereas the second term represents the coupling between noise and damping [(x0 + at) e -ζtλσ(t)]. Similar arguments also hold good for y0(t).
The matrix element involving any two arbitrary eigenstates r and s of H0 due to V1(t) now reads 43, 44

\begin{eqnarray} V_{r, s}^{imp}(t) &=& \langle \psi_r(x, y) \vert V_1(t) \vert \psi_s(x, y) \rangle \nonumber \\ &=& \sum_{nm} \sum_{n^\prime m^\prime} C_{nm, r}^* C_{n^\prime m^\prime, s} \langle \phi_n(p x) \phi_m(q y) \vert V_1(t) \vert \phi_{n^\prime}(p x) \phi_{m^\prime}(q y) \rangle \nonumber \\ &=& V_0(t) \sum_{j=1}^{16} V_{r, s}^j(t) \end{eqnarray} (10)

However, the expressions of various related terms 43, 44 are modified and given in Appendix A. Using above relations definite expressions of $V_{r, s}^{imp}(t)$ can be obtained [cf. eq. (10)]. The external static electric field V2 of strength ϵ is now switched on where

\begin{eqnarray} V_2 = -\{\epsilon_x|e|x+\epsilon_y|e|y\}, \end{eqnarray} (11)

ϵx and ϵy are the field intensities along x and y directions. Now the time-dependent Hamiltonian reads

\begin{eqnarray} H(t)= \left[H_0 - V_{imp}(0)\right] + V_1(t)+ V_2. \end{eqnarray} (12)

The matrix elements due to V2 reads

\begin{eqnarray*} \left(V_2\right)_{n, m; n^\prime, m^\prime} &=& \langle \psi_p(x, y) \vert V_2 \vert \psi_q(x, y) \rangle \nonumber \\ &=& \sum_{nm} \sum_{n^\prime m^\prime} C_{nm, p}^* C_{n^\prime m^\prime, q} \langle \phi_n(\alpha x) \phi_m(\beta y) \vert V_2\vert \phi_{n^\prime}(\alpha x) \phi_{m^\prime}(\beta y) \rangle \nonumber \\ &=& \epsilon_x \sum_{nm} \sum_{n^\prime m^\prime} C_{nm, p}^* C_{n^\prime m^\prime, q} \left[\frac{1}{\alpha} \left\{\sqrt{\frac{n^\prime+1}{2}}\delta_{n^\prime+1,n}+ \sqrt{\frac{n^\prime}{2}}\delta_{n^\prime-1,n}\right\} \delta_{m,m^\prime} \right] \nonumber \\ & + & \epsilon_y \sum_{nm} \sum_{n^\prime m^\prime} C_{nm, p}^* C_{n^\prime m^\prime, q} \left[\frac{1} {\beta}\left\{\sqrt{\frac{m^\prime+1}{2}}\delta_{m^\prime+1, m}+ \sqrt{\frac{m^\prime}{2}}\delta_{m^\prime-1,m}\right\} \delta_{n,n^\prime} \right] \end{eqnarray*}

The time-dependent wave function can now be described by a superposition of the eigenstates of H0, i.e.

\begin{eqnarray} \psi(x, y, t)=\sum_q a_q(t)\psi_q, \end{eqnarray} (13)

and the time-dependent Schrödinger equation (TDSE) containing the evolving wave function [cf. eq. (13)] has now been solved numerically by 6-th order Runge-Kutta-Fehlberg method with a time step size △t = 0.01 a.u. and numerical stability of the integrator has been verified (see Appendix B). The numerical solution of TDSE gives the time-dependent superposition coefficients with the initial conditions ap(0) = 1, aq(0) = 0, for all qp, where p may be the ground or any other excited states of H0. The quantity Pk(t) = │ak(t)│2 can be used as a measure of population of kth state of H0 at time t.

The energy of the system E displays typical dynamics when represented as

\begin{eqnarray} E(t)=\sum_k E _k (0)\;P_k(t), \end{eqnarray} (14)

where Ek(0) is the energy of kth eigenstate of H0 at t = 0. The time-average energy of the dot <E> is computed by numerically integrating the following expression:

\begin{eqnarray} \langle E \rangle = \frac{1}{T} \int_0 ^T E(t)dt, \end{eqnarray} (15)

where T is the final time up to which the dynamic evolution of the system is monitored. We have determined the energy eigenvalues for various combinations of ϵx and ϵy and used them to compute some of the off-diagonal components of linear and nonlinear polarizabilities of the dot by the following relations obtained by numerical differentiation. For linear polarizability:

\begin{eqnarray} \alpha_{xy}\epsilon_x\epsilon_y & = & \frac{1}{48} \left[E(2\epsilon_x, 2\epsilon_y)-E(2\epsilon_x, -2\epsilon_y)- E(-2\epsilon_x, 2\epsilon_y)+E(-2\epsilon_x, -2\epsilon_y)\right] \nonumber \\ && - \frac{1}{3}\left[E(\epsilon_x, \epsilon_y)- E(\epsilon_x, -\epsilon_y)-E(-\epsilon_x, \epsilon_y)+ E(-\epsilon_x, -\epsilon_y)\right] \end{eqnarray} (16)

and a similar expression is used for computing αyx component.
The off-diagonal components of first nonlinear polarizability (second order/quadratic hy-perpolarizability) are calculated from following expressions.

\begin{eqnarray} \beta_{xyy}\; \varepsilon_x \varepsilon_y^2 & = & \frac{1}{2}\left[E(-\varepsilon_x, -\varepsilon_y)- E(\varepsilon_x,\varepsilon_y)+ E(-\varepsilon_x, \varepsilon_y)-E(\varepsilon_x, -\varepsilon_y)\right] \nonumber \\ && + \left[E(\varepsilon_x, 0)-E(-\varepsilon_x, 0) \right] \end{eqnarray} (17)

and a similar expression is used for computing βyxx component.
The off-diagonal components of second nonlinear polarizability (third order/cubic hyper-polarizability) are given by

\begin{eqnarray} \gamma_{xxyy}\; \epsilon_x^2\epsilon_y^2 &=& 2 \left[E(\epsilon_x) +E(-\epsilon_x)\right]+2 \left[E(\epsilon_y)+ E(-\epsilon_y)\right] \nonumber \\ && -\left[E(\epsilon_x, \epsilon_y)+E(-\epsilon_x, -\epsilon_y)+E(\epsilon_x, -\epsilon_y)+E(-\epsilon_x, \epsilon_y)\right]-4E(0) \end{eqnarray}
(18)

and a similar expression is used for computing γyyxx component.

# Results and Discussion

In this section we would like to discuss the variations of off-diagonal components of static linear (αxy, αyx), second order nonlinear (βxyy, βyxx), and third order nonlinear (γxxyy, γyyxx) polarizabilities with coupling constant (λ). λ appears to be the central parameter in the present study since it is the switch with which we can modulate the noise-damping interaction. Moreover, a variation of λ in turn implies a camouflaged variation of noise strength (μ).
Figure 1 displays the variation of αxy component with λ. The component exhibits a steadily decreasing trend with increase in λ and displays minimization at λ ~ 0.32. After minimization, the component enhances with further increase in λ. Damping restricts the motion of the dopant away from the dot confinement center. Consequently, the dopant suffers some enhanced confinement which reduces the dispersive nature of the system forcing the α component to diminish. On the other hand, noise, owing to its random nature, enhances the dispersive character of system and thereby the polarizability components. The observed behavior of said α component indicates that a gradual increase in λ up to ~ 30% (moderate coupling regime) causes an overwhelming dominance of damping over noise. Thus, the said α component falls sharply. The scenario changes completely beyond λ ~ 30% (the strong coupling regime) whence noise begins to surmount the hindrance offered by damping and αxy increases. αyx component, on the other hand, displays nearly similar behavior and therefore the corresponding profile is not presented.

Figure 2 depicts the similar profile for the βxyy and βyxx components. The two components are found to exhibit contrasting features in their profiles. The βyxx component evinces a maximization at λ ~ 0.4 whereas βxyy undergoes minimization at λ ~ 0.8. It appears that the environmental influence on β components is not as streamline as in case of α and γ components. Since the two β components are mutually nonequivalent [cf. eq. (18)] the environmental inuences on them are different under the guise of λ. The plots thus suggest that at λ ~ 0.4 (low-medium coupling regime) the noise effect prominently excels over damping for βyxx component leading to its distinct maximization. The maxima, however, has a transient existence since a small departure of the coupling strength from the zone of occurrence of maxima results in dominance of damping effect and βyxx subsides. It needs mention that within the same coupling regime we observe kind of compromise between noises and damping effects leading to steady behavior of βxyy component. Within strong coupling regime (λ ~ 0.8) the damping effect completely outweighs noise giving rise to minimization of βxyy component. However, as before, the said dominance of damping over noise appears evanescent as shortly after minimization the βxyy component begins to increase with λ. The ambience consisting of noise and damping behaves equivalently to both the β components over a somewhat stretched domain of 0.45≤ λ ≤0.75 where they exhibit persistent decrease with increase in λ. We can therefore infer that inside this domain the coupling occurs in such a way that damping prevails over noise for both the β components.

Figure 3 displays the variation of γxxyy with λ. The component displays a more or less steady behavior with λ up to λ ~ 0.45 beyond which it falls sharply with further increase in coupling strength and reaches a minima at λ ~ 0.84. After minimization the component begins to rise noticeably in the strong coupling regime as λ exceeds the value 0.84. The γyyxx component exhibits nearly similar behavior and we therefore abstain from showing the figures. Thus, although both the off-diagonal α and γ components exhibit minimization, it is the noise-damping coupling that discriminates the location of minima. The minimization occurs at a coupling strength of λ ~ 0.32 for αxy component and at λ ~ 0.84 for γxxyy component. Hence, the relative dominance of damping and noise in designing the polarizability profiles appears to be very much linked with the order of polarizability.

# Conclusions

The static off-diagonal linear and nonlinear polarizability components of impurity doped quantum dots under the combined influence of damped propagation of dopant and noise reveal some noteworthy aspects. In practice, the components are found to be strongly dependent on the noise-damping coupling strength. The noise-damping coupling parameter (λ) emerges as the chief tool with which the dispersive nature of the system can be tuned. With static electric field, the overall influence of noise-damping interaction is manifested through the emergence of sharp rise, sharp fall, and kind of steady behavior of off-diagonal polarizability components. We can therefore infer that it is the extent of coupling between damping and noise that ultimately tailors the profiles of off-diagonal polarizability components. However, the tailoring manifestly depends on the order of polarizability and can design the polarizability profile in diverse manners. The results appear interesting and expected to be important in the related field of research. It needs to be mentioned that there are no experimental data available so far to verify our model considerations. It will be a challenge in future to await verification.

# Author contributions

Ganguly S, Saha S and Ghosh M have equivalently contributed the work.

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# Appendix A

$\xi(t) = \xi_0. \; \exp(-\eta \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}). \; \exp(-\zeta t). \left\{1+ \lambda \sigma(t)\right\}$, where $\Lambda_1 = x_0^2+y_0^2$, $\Lambda_2 = 2(a x_0+b y_0)$, and $\Lambda_3 = a^2+b^2$. The relevant $w(t)$ and $v(t)$ functions 43, 44 grossly look like $w_1(t)=\frac{\xi(t)} {p^2+\xi(t)}$, $w_2(t)= \frac{\xi(t)} {q^2+\xi(t)}$, $v_1(t) = p x_0(t) \sqrt{\frac{\xi(t)}{p^2+\xi(t)}}$, and $v_2(t) = q y_0(t) \sqrt{\frac{\xi(t)}{q^2+\xi(t)}}$.

The modified expressions of corresponding derivatives read as follows 43, 44:

\begin{eqnarray*} w_1^{\prime}(t) &=& \frac{\eta p^2 \xi(t) e^{-\zeta t}}{\left[p^2+ \xi(t)\right]^2}. \left[\zeta \left\{1+ \lambda \sigma(t) \right\} \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \right. \nonumber \\ &-& \lambda \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \sigma^{\prime}(t) - \left. \frac{1}{2} \frac{\left\{1+ \lambda \sigma(t)\right\} (\Lambda_2 + 2\Lambda_3 t)} {\sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}} \right], \end{eqnarray*} \begin{eqnarray*} w_2^{\prime}(t) &=& \frac{\eta q^2 \xi(t) e^{-\zeta t}}{\left[q^2+ \xi(t)\right]^2}. \left[\zeta \left\{1+ \lambda \xi(t) \right\} \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \right. \nonumber \\ &-& \lambda \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \sigma^{\prime}(t) - \left. \frac{1}{2} \frac{\left\{1+ \lambda \sigma(t)\right\} (\Lambda_2 + 2\Lambda_3 t)} {\sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}} \right], \end{eqnarray*} \begin{eqnarray*} v_1^{\prime}(t) &=& \frac{p \xi^{\frac{1}{2}}(t) e^{-\zeta t}}{\sqrt{p^2+ \xi(t)}} \left[(x_0 + at) \lambda \sigma^{\prime}(t) + a\left\{1+ \lambda \sigma(t) \right\} - \zeta(x_0+ at)\left\{1+ \lambda \sigma(t) \right\} \right] \nonumber \\ &-& \frac{1}{2}\frac{p^3 \eta x_0(t) \xi^{\frac{1}{2}}(t) e^{-\zeta t}}{\left\{p^2+ \xi(t)\right\}^{\frac{3}{2}}} \left[\zeta \left\{1+ \lambda \sigma(t) \right\} \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \right. \nonumber \\ &-& \lambda \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \sigma^{\prime}(t) - \left. \frac{1}{2} \frac{\left\{1+ \lambda \sigma(t)\right\} (\Lambda_2 + 2\Lambda_3 t)} {\sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}} \right], \end{eqnarray*}

and

\begin{eqnarray*} v_2^{\prime}(t) &=& \frac{q \xi^{\frac{1}{2}}(t) e^{-\zeta t}}{\sqrt{q^2+ \xi(t)}} \left[(y_0 + at) \lambda \sigma^{\prime}(t) + a\left\{1+ \lambda \sigma(t) \right\} - \zeta(y_0+ at)\left\{1+ \lambda \sigma(t) \right\} \right] \nonumber \\ &-& \frac{1}{2}\frac{q^3 \eta y_0(t) \xi^{\frac{1}{2}}(t) e^{-\zeta t}}{\left\{q^2+ \xi(t)\right\}^{\frac{3}{2}}} \left[\zeta \left\{1+ \lambda \sigma(t) \right\} \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \right. \nonumber \\ &-& \lambda \sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2} \sigma^{\prime}(t) - \left. \frac{1}{2} \frac{\left\{1+ \lambda \sigma(t)\right\} (\Lambda_2 + 2\Lambda_3 t)} {\sqrt{\Lambda_1 + \Lambda_2 t + \Lambda_3 t^2}} \right], \end{eqnarray*}

# Appendix B

\begin{eqnarray*} i\hbar\frac{\partial \psi}{\partial t} &=& H\psi \;\;\;{\rm or \;\;\;equivalently} \nonumber \\ i\hbar \dot{a_q}(t) & = & H a_q(t), \end{eqnarray*}
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