Quantum Dynamics in a 1D Dot/Antidot Lattice: Landau Minibands and Graphene Wave Packet Motion in a Magnetic Field

Abstract

This work is focused on the analysis of the quantum dynamics of a model one-dimensional lattice array of quantum dots on a two dimensional electron sheet/layer in a normal magnetic field. Our analysis is carried out with the derivation of the Green’s function for the quantum dot lattice subject to Landau quantization using the corresponding “no-lattice” Green’s function, for propagation along the axis of the 1D quantum dot lattice. The frequency/energy poles of this Green’s function provide the dispersion relation for the Landau quantized energy spectrum, which exhibits Landau minibands, rather than discrete Landau levels. In the case of nonrelativistic carriers, the dispersion relation is explicitly exhibited in a closed form in terms of the Jacobi Theta Function of the third kind, and an approximate solution displaying the Landau miniband formation is obtained in terms of Laguerre polynomials. We also examine the case of “relativistic” graphene carriers, employing the appropriate Landau quantized graphene Green’s function in a study of the associated wave packet dynamics in a graphene antidot lattice in a normal magnetic field. In this, we analyze the effects of pseudospin on the wave packet dynamics in Landau quantized graphene, including the role of Zitterbewegung in circular orbit motion.

Appendices

Appendix A

16.11 Zitterbewegung Phenomenon

Zitterbewegung is the jittery motion of the Dirac electron. It occurs when one tries to confine the Dirac electrons. According to the Heisenberg uncertainty principle, localization of the electron wave packet leads to uncertainty in momentum. For particles with zero rest mass (massless Dirac particles), uncertainty in momentum translates into uncertainty in energy of the particle (This should be contrasted with the nonrelativistic case, where the position-momentum uncertainty relation is independent of the energy-time uncertainty relation) [27].

16.1.1 16.11.1 Prediction and Interpretation of Zitterbewegung by Schrödinger

Schrödinger discovered a highly oscillatory motion of the electron with velocity c during his work on the time evolution of the position operator, which he named Zitterbewegung [43]. Schrödinger attempted to explain this phenomenon in terms of microscopic dynamical variables i.e coordinate and momentum. To determine the time evolution of the position operator, Schrödinger used Dirac’s Hamiltonian for the free electron-positron system, which is

$$\begin{aligned} H=c \mathbf {\alpha } . \textbf{q} + m c^2 \beta . \,\,\left[ \beta =\begin{pmatrix} I &{} 0\\ 0 &{} -I \end{pmatrix}\right] \end{aligned}$$

(16.77)

In the above equation $\mathbf {\alpha }$ and $\beta$ satisfy the following anti-commutation relations:

$$\begin{aligned} \{ \mathbf {\alpha }_i,\mathbf {\alpha }_j\}= 2 \delta _{ij} I; \qquad \{ \mathbf {\alpha }_i,\beta \}= 0; \qquad \beta ^2 =I, \end{aligned}$$

where $i,j=1,2,3$ and I is a unit matrix. Also the momentum operator $\textbf{q}$ and coordinate operator $\textbf{x}$ commute with $\mathbf {\alpha }$ and $\beta$, and satisfy the canonical anti-commutation relations for Fermions

$$\begin{aligned} \{ \textbf{x}_i,\textbf{x}_j\}=\{ \textbf{q}_i,\textbf{q}_j\} = 0 \qquad and \qquad \{ \textbf{x}_i,\textbf{q}_j\}= i\hbar \delta _{ij} I . \end{aligned}$$

In the Heisenberg picture, the time derivative of any one of these operators, say A, which does not have explicit time dependence is given by

$$\begin{aligned} \frac{dA}{dt} =\frac{ i}{\hbar } [H,A]. \end{aligned}$$

(16.78)

As a result

$$\begin{aligned} \frac{d\textbf{q}}{dt} = 0 , \qquad \frac{d H}{dt} = 0, \qquad \frac{d\textbf{x}}{dt} = c\mathbf {\alpha }, \end{aligned}$$

(16.79)

and (16.77) and (16.78) for the operator $\mathbf {\alpha }$ give

$$\begin{aligned} -i\hbar \frac{d\mathbf {\alpha }}{dt} = [H,\mathbf {\alpha }]= 2H\alpha -\{H,\mathbf {\alpha }\} = 2H \mathbf {\alpha }- 2c\textbf{q}. \end{aligned}$$

(16.80)

Above equation can be written as

$$\begin{aligned} -i \hbar \frac{d\mathbf {\alpha }}{dt} = 2H \mathbf {\eta }, \end{aligned}$$

where we have defined

$$\begin{aligned} \mathbf {\eta } =\mathbf {\alpha } - c H^{-1} \textbf{q}. \end{aligned}$$

(16.81)

Schrödinger noted that

$$\begin{aligned} -i \hbar \frac{d \mathbf {\eta }}{dt} = -i \hbar \frac{d \mathbf {\alpha }}{dt} = 2H \mathbf {\eta }, \end{aligned}$$

which yields

$$\begin{aligned} \mathbf {\eta } (t) = e^{\frac{2iHt}{\hbar }}\mathbf {\eta }_0 \qquad with \qquad \mathbf {\eta }_0(0)= \mathbf {\alpha }(0) - c H^{-1} \textbf{q}. \end{aligned}$$

(16.82)

Using $\{H,\mathbf {\eta }\}=0=\{H,\mathbf {\eta }_0\}$, (16.82) can also be written as

$$\begin{aligned} \mathbf {\eta }(t) = \mathbf {\eta }_0 e^{\frac{-2iHt}{\hbar }}. \end{aligned}$$

(16.83)

Combining (16.80), (16.81) and (16.83), Schrödinger obtained

$$\begin{aligned} \frac{d \textbf{x}}{dt} = c \mathbf {\alpha } = c^2 H^{-1} \textbf{q} + c \mathbf {\eta }_0 e^{\frac{-2iHt}{\hbar }}, \end{aligned}$$

which on integration yields

$$\begin{aligned} \textbf{x}(t) = \textbf{a} + c^2 H^{-1} \textbf{q} t + \frac{i}{\hbar } c \mathbf {\eta }_0 H^{-1} e^{\frac{-2iHt}{\hbar }}, \end{aligned}$$

(16.84)

where $\textbf{a}$ is an operator which is a constant of integration with the definition

$$\begin{aligned} \textbf{a} = \textbf{x}(0) - \frac{i}{2} \hbar c^2 H^{-2} \textbf{q}. \end{aligned}$$

(16.85)

Defining

$$\begin{aligned} \textbf{x}_A(t) = \textbf{a} + c^2 H^{-1} \textbf{q} t, \end{aligned}$$

equation (16.84) will become

$$\begin{aligned} \textbf{x}(t) = \textbf{x}_A(t) + \mathbf {\xi } (t), \end{aligned}$$

where we have defined

$$\begin{aligned} \mathbf {\xi } (t) = \frac{i}{2} \hbar c \mathbf {\eta }_0 H^{-1} e^{\frac{-2iHt}{\hbar }} = \frac{i}{2} \hbar c \mathbf {\eta } H^{-1}. \end{aligned}$$

(16.86)

Here, the term $\mathbf {\xi } (t)$ corresponds to a microscopic coordinate which oscillates at high frequency known as Zitterbewegung (ZB). This motion is superimposed on a macroscopic type of motion associated with the coordinate $\textbf{x}_A$. Characteristic amplitude associated with Zitterbewegung is $\frac{\hbar }{2mc}$ and this amplitude is equal to half of the Compton wavelength of an electron, while the characteristic angular frequency of the Zitterbewegung is $\frac{2mc^2}{\hbar }$.

Schrödinger went on to note, that if the orbital-angular momentum $\textbf{L}$ and spin vector $\textbf{S}$ are introduced as

$$\begin{aligned} \textbf{S} = -\frac{i}{4}\hbar \mathbf {\alpha } \times \mathbf {\alpha } \qquad and \qquad \textbf{L} =\textbf{x} \times \textbf{q} \end{aligned}$$

then both $\textbf{L}$ and $\textbf{S}$ alone are not constant but their sum $\textbf{L} + \textbf{S}$ is a constant of the motion. Schrödinger found

$$\begin{aligned} \textbf{S}(t) = \textbf{S}_A - \mathbf {\eta } (t) \times \textbf{q} \end{aligned}$$

and

$$\begin{aligned} \textbf{L}(t) = \textbf{L}_A + \mathbf {\eta } (t) \times \textbf{q}, \end{aligned}$$

where $\textbf{S}_A$ and $\textbf{L}_A$ are constants and the term $\mathbf {\eta } (t) \times \textbf{q}$ is oscillatory. Finally he found that

$$\begin{aligned} \textbf{L} + \textbf{S} = \textbf{L}_A + \textbf{S}_A . \end{aligned}$$

(16.87)

This constant of the motion allowed Schrödinger to explain Zitterbewegung in terms of microscopic dynamical variables spin and orbital angular momentum [43].

16.1.2 16.11.2 Zitterbewegung: Interpretation in Terms of Interference Between Positive and Negative Energy States

It can be easily verified that (16.84) along with (16.82) and (16.85) can be written as

$$\begin{aligned} \textbf{x} (t) = \textbf{x} (0)+ \frac{c^2 \textbf{q}}{H}t + \frac{\hbar c}{2 i H} \left( e^{\frac{2iHt}{\hbar }} -1 \right) \left( \mathbf {\alpha }(0) - \frac{c \textbf{q}}{H} \right) . \end{aligned}$$

(16.88)

Also (16.80) can be brought to the form

$$\begin{aligned} H\mathbf {\alpha } +\mathbf {\alpha } H = 2 c \textbf{q}, \end{aligned}$$

and hence

$$\begin{aligned} H \left( \mathbf {\alpha } - \frac{c \textbf{q}}{H} \right) - \left( \mathbf {\alpha } - \frac{c \textbf{q}}{H} \right) H = 0. \end{aligned}$$

(16.89)

Along with the initial term $\textbf{x}(0)$ at $t = 0$, (16.88) also carries two more terms, one of them is linear in time t and corresponds to group velocity, while the second term is oscillatory in nature and is known as Zitterbewegung. $\mathbf {\alpha } - \frac{c \textbf{q}}{H}$ is an operator whose matrix elements are required to evaluate the average value $\int \psi ^\dagger (0,\textbf{x}) \textbf{x}(t) \psi (0,\textbf{x}) d^3x$. Nonvanishing matrix elements of this operator only lie between states of the same momentum. Also from (16.89), the anticommutator only vanishes when the energies are of opposite sign. Hence it can be concluded that ZB is a consequence of interference between positive and negative energy states as predicted by relativistic quantum mechanics [44].

Appendix B

16.12 Contour Integration

To solve the second integral in (16.69), let us denote the integral by I [21]

$$\begin{aligned} I = \frac{\alpha d \omega _g}{2\pi } \sum _{n^\prime =-\infty }^\infty \int _{-\infty }^\infty d\eta e^{- i \omega _{g}\eta t}&\mathcal {G}^0(x_1;n^\prime d;\eta ) \int _{-\frac{\pi }{d}}^{\frac{\pi }{d}} dp e^{- ipn^\prime d} \nonumber \\&\times \left[ I_2-\alpha \dot{\tilde{\mathcal {G}}}^0(p;0,0;\eta )\right] ^{-1} \tilde{\mathcal {G}}^0(p;x_2;\eta ). \end{aligned}$$

(16.90)

($I_2$ is unit matrix of order 2) where

$$\begin{aligned} \eta = \frac{\omega }{\omega _g} \qquad and \qquad d\omega = \omega _g d\eta . \end{aligned}$$

Note that in the above integral, each “no lattice” Green’s function has real poles at $\eta$=$\pm \sqrt{n}$. These “no lattice” poles of the term

$$\begin{aligned} T_1 (\eta ) = \left[ I_2 -\alpha \dot{\tilde{\mathcal {G}}}^0(p;0,0;\eta )\right] ^{-1} \tilde{\mathcal {G}}^0(p;x_2;\eta ) \end{aligned}$$

(16.91)

can be seen to cancel by re-expressing the “no lattice” Green’s functions $\dot{\tilde{\mathcal {G}}}^0(p;0,0;\eta )$ and $\tilde{\mathcal {G}}^0(p;x_2;\eta )$. For this purpose, by using (16.58) and (16.59), one can easily write $\alpha \dot{\tilde{\mathcal {G}}}^0(p;0,0;\eta )$ matrix as

$$\begin{aligned} \alpha \dot{\tilde{\mathcal {G}}}^0(p;0,0;\eta ) = \frac{1}{\prod _{n=0}^\infty (\eta ^2-n) } \begin{pmatrix} a_{11} &{} a_{12} \\ a_{12} &{} a_{11} \end{pmatrix} \end{aligned}$$

(16.92)

where we have defined (j corresponds to the index r in (16.58, 16.59))

$$\begin{aligned} a_{11}(\eta )= \alpha _1 \eta \sum _{m=0}^\infty L_m \left[ \frac{\omega ^2_g l^2}{4\gamma ^2}\left( j\frac{d}{l} \right) ^2\right] \prod _{n=0 \atop n\ne m}^\infty (\eta ^2-n), \end{aligned}$$

and

$$\begin{aligned} a_{12}(\eta )= \alpha _2\sum _{m=1}^\infty L_{m-1}^1 \left[ \frac{\omega ^2_g l^2}{4\gamma ^2}\left( j\frac{d}{l} \right) ^2\right] \prod _{n=0 \atop n\ne m+1}^\infty (\eta ^2-n), \end{aligned}$$

with

$$\begin{aligned} \alpha _1= \frac{\alpha \omega _g}{4\pi \hbar \gamma ^2} \sum _{j=-\infty }^\infty e^{i pjd} e^{-\frac{\omega ^2_g l^2}{8\gamma ^2}\left( j\frac{d}{l} \right) ^2}, \end{aligned}$$

and

$$\begin{aligned} \alpha _2= \frac{i \alpha \omega _ g^2 l}{8\pi \hbar \gamma ^3} \sum _{j=-\infty }^\infty \left( j \frac{d}{l} \right) e^{i pjd} e^{-\frac{\omega ^2_g l^2}{8\gamma ^2}\left( j\frac{d}{l}\right) ^2}. \end{aligned}$$

Similarly, the $\tilde{\mathcal {G}}^0(p;x_2;\eta )$ matrix can be written as

$$\begin{aligned} \tilde{\mathcal {G}}^0(p;x_2;\eta )= \frac{1}{\prod _{n=0}^\infty (\eta ^2-n) } \begin{pmatrix} b_{11} &{} b_{12} \\ b_{12} &{} b_{11} \end{pmatrix}, \end{aligned}$$

(16.93)

where we have defined

$$\begin{aligned} b_{11}(\eta )=\beta _1\eta \sum _{m=0}^\infty L_m \left[ \frac{\omega ^2_g l^2}{4\gamma ^2}\left( j\frac{d}{l} \right) ^2\right] \prod _{n=0\atop n\ne m}^\infty (\eta ^2-n), \end{aligned}$$

and

$$\begin{aligned} b_{12}(\eta )= \beta _2 \sum _{m=1}^\infty L_{m-1}^1 \left[ \frac{\omega ^2_g}{4\gamma ^2 l^2}\left( j\frac{d}{l} \right) ^2\right] \prod _{n=0\atop n\ne m+1}^\infty (\eta ^2-n) , \end{aligned}$$

with

$$\begin{aligned} \beta _1= \frac{ \omega _g}{4\pi \hbar \gamma ^2} \sum _{j=-\infty }^\infty e^{i pjd} e^{-\frac{\omega ^2_g l^2}{8\gamma ^2}\left( \frac{jd-x_2}{l} \right) ^2}, \end{aligned}$$

and

$$\begin{aligned} \beta _2=\frac{ i \omega _g^2 l}{8\pi \hbar \gamma ^3} \sum _{j=-\infty }^\infty \left( \frac{jd-x_2}{l} \right) e^{i pjd} e^{-\frac{\omega ^2_g l^2}{8\gamma ^2}\left( \frac{jd-x_2}{l}\right) ^2}. \end{aligned}$$

Note that in the above equations, n is a Landau index and m is a dummy index for the Landau levels; the maximum value of n and m is the same.

Finally substituting (16.92) and (16.93) in (16.91), the matrix $T_1$ becomes

$$\begin{aligned} T_1 (\eta ) = \Bigg [ \begin{pmatrix} 1 &{} 0 \\ 0 &{} 1 \end{pmatrix} - \frac{1}{\prod _{n=0}^\infty (\eta ^2-n)} \begin{pmatrix} a_{11}(\eta ) &{} a_{12}(\eta ) \\ a_{12}(\eta ) &{} a_{11}(\eta ) \end{pmatrix} \Bigg ]^{-1} \frac{1}{\prod _{n=0}^\infty (\eta ^2-n)} \begin{pmatrix} b_{11}(\eta ) &{} b_{12}(\eta ) \\ b_{12}(\eta ) &{} b_{11}(\eta ) \end{pmatrix} \end{aligned}$$

$$\begin{aligned} = \left( \frac{1}{\prod _{n=0}^\infty (\eta ^2-n)}\right) ^{-1} \Bigg [ \begin{pmatrix} \prod _{n=0}^\infty (\eta ^2-n) &{} 0 \\ 0 &{} \prod _{n=0}^\infty (\eta ^2-n) \end{pmatrix} - \begin{pmatrix} a_{11}(\eta ) &{} a_{12}(\eta ) \\ a_{12}(\eta ) &{} a_{11}(\eta ) \end{pmatrix} \Bigg ]^{-1} \\ \times \frac{1}{\prod _{n=0}^\infty (\eta ^2-n)} \begin{pmatrix} b_{11}(\eta ) &{} b_{12}(\eta ) \\ b_{12}(\eta ) &{} b_{11}(\eta ) \end{pmatrix} \end{aligned}$$

$$\begin{aligned} = \Bigg [ \begin{pmatrix} \prod _{n=0}^\infty (\eta ^2-n) &{} 0 \\ 0 &{} \prod _{n=0}^\infty (\eta ^2-n) \end{pmatrix} - \begin{pmatrix} a_{11}(\eta ) &{} a_{12}(\eta ) \\ a_{12}(\eta ) &{} a_{11}(\eta ) \end{pmatrix} \Bigg ]^{-1} \begin{pmatrix} b_{11}(\eta ) &{} b_{12}(\eta ) \\ b_{12}(\eta ) &{} b_{11}(\eta ) \end{pmatrix}. \end{aligned}$$

In the above expression, the real “no lattice” poles ($\eta$=$\pm \sqrt{n}$) cancel. The above expression can be solved numerically. In this computation, it is also necessary to address the poles of the actual lattice Green’s function, which are defined by the zeros of the denominator, $det\left[ I_2-\alpha \dot{\tilde{\mathcal {G}}}^0(p;0,0;\eta )\right] =0$, and these roots depend on p (and are not spaced by integer multiples of $\omega _g$): These poles are treated using the line broadening discussion of Sect. 16.5.2. The resultant $2 \times 2$ matrix with $c_{ij}(\eta )$ (where $i,j=1,2$) as matrix elements can be written as (substituting $q=pd$)

$$\begin{aligned} Q (\eta ) = \int _{-\pi }^{\pi } dq e^{{- iqn^\prime }}T_1 (\eta ) = \begin{pmatrix} c_{11}(\eta ) &{} c_{12}(\eta ) \\ c_{21}(\eta ) &{} c_{22}(\eta ) \end{pmatrix}. \end{aligned}$$

(16.94)

Note that the above integral can be numerically calculated by applying the trapezoidal rule in the limits $-\pi$ to $\pi$, while kee** the trapezoidal step equal to $\frac{\pi }{10}$; this gives an accuracy up to five decimal points for each value of $n^\prime$.

Putting (16.94) in (16.90), the integral I becomes

$$\begin{aligned} I = \frac{\alpha \omega _g}{2\pi } \sum _{n^\prime =-\infty }^\infty \int _{-\infty }^\infty d\eta e^{- i \omega _g\eta t} \mathcal {G}^0(x_1,n^\prime d;\eta ) . Q(\eta ). \end{aligned}$$

(16.95)

It can be seen that the “no lattice” poles of the second integral (p integral) in (16.90) have been removed; and the poles of the actual lattice Green’s function have been dealt with using the line broadening of Sect. 16.5.2, so the only remaining poles that we have to deal with now are the poles of the matrix $\mathcal {G}^0(x_1,n^\prime d;\eta )$.

Using (16.67) and (16.68), the matrix $\alpha \mathcal {G}^0(x_1,n^\prime d;\eta )$ can be written as

$$\begin{aligned} \alpha \mathcal {G}^0(x_1,n^\prime d;\eta )= \begin{pmatrix} \gamma _{1}\frac{1}{\eta } &{} 0 \\ 0 &{} \gamma _{1} \frac{1}{\eta } \end{pmatrix} + \sum _{n=1}^\infty \begin{pmatrix} \gamma _{1} \eta \frac{ L_n \left[ \Upsilon \right] }{\eta ^2 -n} &{} \gamma _{2} \frac{ L_{n-1}^1 \left[ \Upsilon \right] }{\eta ^2 -n} \\ \gamma _{2} \frac{ L_{n-1}^1 \left[ \Upsilon \right] }{\eta ^2-n} &{} \gamma _{1} \eta \frac{ L_n \left[ \Upsilon \right] }{\eta ^2 -n} \end{pmatrix}, \end{aligned}$$

(16.96)

where $\Upsilon$, $\gamma _1$ and $\gamma _2$ are defined in (16.74) and (16.75). Using the matrices $Q(\eta )$ and $\alpha \mathcal {G}^0(x_1,n^\prime d;\eta )$ in (16.95), and breaking the matrix into two matrices, we get

$$\begin{aligned} I = \frac{ \omega _g}{2\pi } \sum _{n^\prime =-\infty }^\infty \int _{-\infty }^\infty d\eta e^{- i \omega _g\eta t} \begin{pmatrix} \gamma _{1}\frac{1}{\eta } c_{11}(\eta ) &{} \gamma _{1}\frac{1}{\eta } c_{12}(\eta ) \\ \gamma _{1}\frac{1}{\eta } c_{21}(\eta ) &{} \gamma _{1}\frac{1}{\eta } c_{22}(\eta ) \nonumber \end{pmatrix} \\ + \frac{ \omega _g}{2\pi } \sum _{n^\prime =-\infty }^\infty \int _{-\infty }^\infty d\eta e^{- i \omega _g\eta t} \begin{pmatrix} M_{11}(\eta ) &{} M_{12}(\eta ) \\ M_{21}(\eta ) &{} M_{22}(\eta ) \end{pmatrix}. \end{aligned}$$

(16.97)

(The first and second matrices correspond to n=0 and $n>0$ Landau minibands, respectively.) In the above expression

$$\begin{aligned} M_{11 \atop 22}(\eta ) = \sum _{n=1}^\infty \Bigg ( \gamma _{1} \eta \frac{ L_n \left[ \Upsilon \right] }{\eta ^2 -n} c_{11 \atop 22 }(\eta ) + \gamma _{2} \frac{ L_{n-1}^1 \left( \Upsilon \right) }{\eta ^2 -n} c_{21 \atop 12}(\eta )\Bigg ) , \end{aligned}$$

and

$$\begin{aligned} M_{12 \atop 21}(\eta ) = \sum _{n=1}^\infty \Bigg ( \gamma _{1} \eta \frac{ L_n \left[ \Upsilon \right] }{\eta ^2 -n} c_{12 \atop 21}(\eta ) + \gamma _{2} \frac{ L_{n-1}^1 \left[ \Upsilon \right] }{\eta ^2 -n} c_{22 \atop 11}(\eta ) \Bigg ). \end{aligned}$$

In (16.97), the first matrix has a pole at $\eta$=0, while the second matrix has poles at $\eta$=$\pm \sqrt{n}$. We now use contour integration with the Jordan lemma (closing the contour in the lower half plane for $t>0$) to evaluate the integrals. Results for the two terms in (16.97) are

$$\begin{aligned} \int _{-\infty }^\infty d\eta e^{- i \omega _g\eta t} \frac{c_{ij}(\eta )}{\eta } =- i\pi \eta _+(t) \left[ c_{ij}(\eta )\right] _{\eta =0} , \end{aligned}$$

(16.98)

$$\begin{aligned} \int _{-\infty }^\infty d\eta e^{- i \omega _g\eta t} M_{11 \atop 22 }(\eta )&= -i \pi \eta _+(t) \sum _{n=1}^\infty \cos \left( \frac{\omega _g l}{\gamma } \frac{t}{\tau _o} \sqrt{n} \right) \nonumber \\&\times \left[ \gamma _1 L_n \left[ \Upsilon \right] c_{11 \atop 22}(\eta ) + \frac{\gamma _2}{\sqrt{n}} L_{n-1}^1 \left[ \Upsilon \right] c_{21 \atop 12} (\eta ) \right] _{\eta =\sqrt{n}} \end{aligned}$$

(16.99)

and

$$\begin{aligned} \int _{-\infty }^\infty d\eta e^{- i \omega _g\eta t} M_{12 \atop 21}(\eta )&=- \pi \eta _+(t) \sum _{n=1}^\infty \sin \left( \frac{\omega _g l}{\gamma } \frac{t}{\tau _o} \sqrt{n} \right) \nonumber \\&\times \left[ \gamma _1 L_n \left[ \Upsilon \right] c_{12 \atop 21}(\eta ) + \frac{\gamma _2}{\sqrt{n}} L_{n-1}^1 \left[ \Upsilon \right] c_{22 \atop 11} (\eta ) \right] _{\eta =\sqrt{n}}. \end{aligned}$$

(16.100)

($\eta _+(t)$ is the Heaviside unit step function and i, j=1, 2). In the calculation of the expressions given by (16.98), (16.99) and (16.100), we have also used

$$\begin{aligned} \left[ c_{11 \atop 22}(\eta )\right] _{\eta =-\sqrt{n}} = \left[ c_{11 \atop 22}(\eta )\right] _{\eta =\sqrt{n}}, \end{aligned}$$

and

$$\begin{aligned} \left[ c_{12 \atop 21}(\eta )\right] _{\eta =-\sqrt{n}} = - \left[ c_{12 \atop 21}(\eta )\right] _{\eta =\sqrt{n}}, \end{aligned}$$

which we found during the calculations [21].

Hence, (16.98) along with (16.99) and (16.100) provide the complete solution for the integral I, which is the time representation of the second term of the full Green’s function $\mathcal {G}(x_1,x_2; t)_{K}$ given in (16.69).