Dynamics of moving cavity solitons in two-level laser system from symmetric gaussian input: vectorial cubic-quintic complex Ginzburg–Landau equation

Abstract

This paper studies the interaction of an electromagnetic field with the matter in a laser cavity without assuming a fixed direction of the transverse electric field, described by the two-level Maxwell–Bloch equations. The derivation of the laser (3+1)-dimensional vectorial cubic-quintic complex Ginzburg–Landau equation is reported using a perturbative nonlinear analysis performed near the laser threshold. Considering the vector (2+1)D cubic-quintic complex Ginzburg–Landau equation, the stability of the moving dissipative solitons in the laser cavity is analyzed. Using the variational approximation, stability conditions and propagation trajectories of dissipative solitons are derived. Direct numerical simulations fully confirm analytical predictions of dissipative solitons trapped in an effective potential well. Potential applications of the obtained results related to spatial dissipative solitons, may be found in class B laser by considering solitons as individual addressable and shift registers of the all-optical data processing systems.

Appendix

1.1 A Multimodal method

The equations describing the interaction of the electromagnetic field with the matter are given by the Maxwell–Bloch Eqs. (1a)–(1c). The quantities E, P and D are taken as follows:

$$\begin{aligned} \mathbf{E}= &\, \sum \limits _{j = 1}^\infty {{ \in ^j}\sum \limits _{n = - j}^{ + j} {\mathbf{E }_j^n(r)\exp (inwt),} } \end{aligned}$$

(24)

$$\begin{aligned} \mathbf{P}= &\, \sum \limits _{j = 1}^\infty {{ \in ^j}\sum \limits _{n = - j}^{ + j} {\mathbf{P }_j^n(r)\exp (inwt),} } \end{aligned}$$

(25)

$$\begin{aligned} D= &\, \sum \limits _{j = 1}^\infty {{ \in ^j}\sum \limits _{n = - j}^{ + j} {D_j^n(r)\exp (inwt),} } \end{aligned}$$

(26)

under the conditions $\mathbf{E} _j^{ - n} = {(\mathbf{E} _j^n)^*}$, $\mathbf{P} _j^{ -n} = {(\mathbf{P} _j^n)^*}$, and $D_j^{ - n} = {(D_j^n)^*}$. We assume that the permanent electric field such that

$$\begin{aligned} \forall j > 0 \text { leads to} \,\mathbf{E} _j^0 =0.\end{aligned}$$

(27)

We focus our study to the case $E=E_1^{1}$, $D_1^{0}=D_0$. In the presence of the intense field in the system, we have $D_0<<\frac{2}{{\hbar {w_a}}}\left( {E\cdot \frac{{\partial P}}{{\partial t}}} \right)$. Inserting the relation of P and D given by Eqs. (25) and (26) into Eqs. (1b) and (1c), it comes, for any $e^{inw_at}$, the following relations:

$$\begin{aligned}&{w_a}\left[ {(1 - {n^2}){w_a} + i{\gamma _ \bot }} \right] (\epsilon P_1^n + {\epsilon ^2}P_2^n + {\epsilon ^3}P_3^n + \cdots )\nonumber \\&\quad = - g\sum \limits _{p + q = n} {(\epsilon D_1^q } + {\epsilon ^2}D_2^q)(\epsilon E_1^p + {\epsilon ^2}E_2^p) \end{aligned}$$

(28)

$$\begin{aligned}&({\gamma _\parallel } + in{w_a})(\epsilon D_1^n + {\epsilon ^2}D_2^n + \cdots ) \nonumber \\&\quad = \frac{{2i}}{\hbar }\sum \limits _{p + q = n} q (\epsilon E_1^q + {\epsilon ^2}E_2^q + \cdots )\nonumber \\&\qquad \times (\varepsilon P_1^q+ {\epsilon ^2}P_2^q + {\epsilon ^3}P_3^q + \cdots ) \end{aligned}$$

(29)

where p and q can take negative values, and $p+q=n$. For any power of $\epsilon$, solving these equations leads to:

$$\begin{aligned}&{\epsilon ^1},n = 0:\quad \mathbf{P} _1^0 = 0, \quad D_1^0 = {D_0}, \end{aligned}$$

(30)

$$\begin{aligned} {\epsilon ^1},&n = 1:\quad \mathbf{P} _1^1 = \frac{1}{{{\mu _0}{c^2}}}\left( { - 1 + \frac{{ik}}{{{w_a}}}} \right) \mathbf{E} _1^1,\quad D_1^1 = 0, \end{aligned}$$

(31)

$$\begin{aligned}{\epsilon ^2}, &n = 0:\quad \mathbf{P} _2^0 = 0,\quad D_2^0 = \frac{{2i}}{{\hbar {\gamma _\parallel }}}\left( \mathbf{P _1^1\mathbf{E} _1^{ - 1} - \mathbf{P} _1^{ - 1}{} \mathbf{E} _1^1} \right) , \end{aligned}$$

(32)

$$\begin{aligned}&{\epsilon ^2},n = 1:\quad \mathbf{P} _2^1 = \frac{{ig}}{{{\gamma _ \bot }{w_a}}}\left( {D_1^0\mathbf{E} _1^1} \right) , \quad D_2^1 = 0, \end{aligned}$$

(33)

$$\begin{aligned}&{\epsilon ^2},n = 2:\quad \mathbf{P} _2^2 = 0, \quad D_2^2 = \frac{{2i}}{{\hbar \left( {{\gamma _\parallel } + 2i{w_a}} \right) }}\left( \mathbf{P _1^1\mathbf{E} _1^1} \right) , \end{aligned}$$

(34)

$$\begin{aligned}&{\epsilon ^3},n = 0:\quad \mathbf{P} _3^0 = 0,\quad D_3^0 = \frac{{2i}}{{\hbar {\gamma _\parallel }}}\left( \mathbf{P _2^1\mathbf{E} _1^{ - 1} - \mathbf{P} _2^{ - 1}{} \mathbf{E} _1^1} \right) , \end{aligned}$$

(35)

$$\begin{aligned}&{\epsilon ^3},n = 1:\quad \mathbf{P} _3^1 = \frac{{ig}}{{{\gamma _ \bot }{w_a}}}\left( {D_2^0\mathbf{E} _1^1 + D_2^2\mathbf{E} _1^{ - 1}} \right) , \quad \quad D_3^1 = 0, \end{aligned}$$

(36)

$$\begin{aligned}&{\epsilon ^3},n = 2:\quad \mathbf{P} _3^2 = 0,\quad D_3^2 = \frac{{2i}}{{\hbar \left( {{\gamma _\parallel } + 2i{w_a}} \right) }}\left( \mathbf{P _2^1\mathbf{E} _1^1} \right) , \end{aligned}$$

(37)

$$\begin{aligned}&{\epsilon ^3},n = 3:\quad \mathbf{P} _3^3 = \frac{{ig}}{{\left( {8{w_a} -3i{\gamma _ \bot }} \right) }}\left( {D_2^2\mathbf{E} _1^1} \right) ,\quad D_3^3 = 0, \end{aligned}$$

(38)

with $\mathbf{P} =\epsilon {\mathbf{P }_1}+{{\epsilon }^2}{\mathbf{P }_2}+{{\epsilon }^3}{\mathbf{P }_3}$

where $\mathbf{P} _1=\mathbf{P} _1^1,\quad \mathbf{P} _2=\mathbf{P} _2^1$    

and

$$\begin{aligned} \mathbf{P} _3=\mathbf{P} _3^1+\mathbf{P} _3^3. \end{aligned}$$

(39)

In the following, we perform the nonlinear perturbation analysis near the laser threshold by introducing a small parameter $\epsilon$ so that $D_0=D_{0C}+\epsilon ^2{\tilde{D}}_0$ ($\epsilon<<1$), $(\xi ,\zeta )=\epsilon (x,y)$, $(Z,\tau )=\epsilon ^2(z,t)$ [31]. Moreover,

$$\begin{aligned} \left( \begin{array}{l} \mathbf{E} \\ \partial _{t}{} \mathbf{E} \\ \mathbf{P} \\ \partial _{t}{} \mathbf{P} \\ D \end{array} \right) = \left( \begin{array}{l} 0\\ 0\\ 0\\ 0\\ D_0 \end{array} \right) + \epsilon \left( \begin{array}{l} \mathbf{E} _1\\ \partial _{t}{} \mathbf{E} _1\\ \mathbf{P} _1\\ \partial _{t}{} \mathbf{P} _1\\ D_1 \end{array} \right) + \epsilon ^2 \left( \begin{array}{l} \mathbf{E} _2\\ \partial _{t}{} \mathbf{E} _2\\ \mathbf{P} _2\\ \partial {_t}{} \mathbf{P} _2\\ D_2 \end{array} \right) + \end{aligned}$$

(40)

with

$$\begin{aligned} \begin{array}{l} \left( \begin{array}{l} {\mathbf {E}_1}\\ {\partial _t}{\mathbf {E}_1}\\ {P_1}\\ {\partial _t}{\mathbf {P}_1}\\ {D_1} \end{array} \right) = \left( \begin{array}{l} A\\ i{w_a}\mathbf {A}\\ \frac{1}{{{\mu _0}{c^2}}}( - 1 + \frac{{ik}}{{{w_a}}})\mathbf {A}\\ \frac{{i{w_a}}}{{{\mu _0}{c^2}}}( - 1 + \frac{{ik}}{{{w_a}}})\mathbf {A}\\ 0 \end{array} \right) {{\mathop {\mathrm {e}}\nolimits } ^{i\left( {{w_{}}t - {k_c}z} \right) }} + c.c.,\;\;A \bot {\hat{Z}}. \end{array} \end{aligned}$$

(41)

From the MB equations, some algebraic manipulations yield the following solvability condition

$$\begin{aligned} {\kappa }\frac{{\partial \mathbf{E _1}}}{{\partial \tau }}= &\, - 2i{w_a}\frac{{\partial \mathbf{E _1}}}{{\partial \tau }} - 2i{w_a}c\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2} \right) \mathbf{E _1} \nonumber \\&- {\mu _0}{c^2}\left( {2\frac{\partial }{{\partial \tau }}\frac{{\partial \mathbf{P _1}}}{{\partial t}}} \right) , \end{aligned}$$

(42)

$$\begin{aligned} 2\frac{\partial }{{\partial \tau }}\frac{{\partial \mathbf{P _1}}}{{\partial t}}= &\, - {\gamma _ \bot }2\frac{{\partial \mathbf{P _1}}}{{\partial \tau }} - g\left( {{{{{\tilde{D}}}}_0} + {D_2}} \right) \mathbf{E _1}, \end{aligned}$$

(43)

$$\begin{aligned} \frac{{\partial {D_2}}}{{\partial t}}= &\, - {\gamma _\parallel }{D_2} + \frac{2}{{\hbar {w_a}}}\left( {\mathbf{E _1}.\frac{{\partial \mathbf{P _1}}}{{\partial t}}} \right) . \end{aligned}$$

(44)

${D_2}$ is obtained by solving Eq. (44), and by combining Eqs. (42) and (43) just give

$$\begin{aligned} \frac{{\partial \mathbf{E _1}}}{{\partial \tau }}= &\, \frac{{2c({\gamma _ \bot } - i{w_a})}}{{k - {\gamma _ \bot } + 2i{w_a}}}\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2} \right) \mathbf{E _1}\nonumber \\&+\frac{{{\mu _0}{c^2}g}}{{k - {\gamma _ \bot } + 2i{w_a}}}\left( {{{{{\tilde{D}}}}_0} + {D_2}} \right) \mathbf{E} _1 \end{aligned}$$

(45)

The nonlinearities come from the interaction between the population inversion and the electric field. To analyze the higher order diffusive term in this system, the higher-order correction $\gamma _{_ \bot }^2\frac{{{\partial ^2}\mathbf{P _1}}}{{\partial {\tau ^2}}}$ is needed to the polarization Eq. (43)

$$\begin{aligned} \kappa \frac{{\partial \mathbf{E _1}}}{{\partial \tau }}= &\, - 2i{w_a}\frac{{\partial \mathbf{E _1}}}{{\partial \tau }}\nonumber \\&- 2i{w_a}c\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2} \right) \mathbf{E _1} - {\mu _0}{c^2}\left( {2\frac{\partial }{{\partial \tau }}\frac{{\partial \mathbf{P _1}}}{{\partial t}}} \right) . \end{aligned}$$

(46)

$$\begin{aligned} 2\frac{\partial }{{\partial \tau }}\frac{{\partial \mathbf{P _1}}}{{\partial t}}= &\, - {\gamma _ \bot }\frac{{\partial \mathbf{P _1}}}{{\partial t}} + \gamma _ \bot ^2\frac{{{\partial ^2}\mathbf{P _1}}}{{\partial {\tau ^2}}} - g\left( {{{{{\tilde{D}}}}_0} + {D_2}} \right) \mathbf{E _1}. \end{aligned}$$

(47)

$$\begin{aligned} \frac{{\partial {D_2}}}{{\partial t}}= &\, - {\gamma _\parallel }{D_2} + \frac{2}{{\hbar {w_a}}}\left( {\mathbf{E _1}.\frac{{\partial \mathbf{P _1}}}{{\partial t}}} \right) . \end{aligned}$$

(48)

Substituting Eq. (47) into Eq. (46), we obtain

$$\begin{aligned}&(\kappa - {\gamma _ \bot } + 2i{w_a})\left[ {1 + \frac{{2\gamma _ \bot ^2}}{{\kappa - {\gamma _ \bot } + 2i{w_a}}}\left( \frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2\right) } \right] \nonumber \\&\quad \frac{{\partial \mathbf{E _1}}}{{\partial T}} = 2c\left( {{\gamma _ \bot } - i{w_a}} \right) \left( \frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2\right) \mathbf{E _1}+ {\mu _0}{c^2}g\left( {{{{{\tilde{D}}}}_0} + {D_2}} \right) \mathbf{E _1}. \end{aligned}$$

(49)

Multiplying both sides of Eq. (49 ) by

$$\begin{aligned} \left( \left( {\kappa - {\gamma _ \bot } + 2i{w_a}} \right) \left[ {1 + \frac{{2\gamma _ \bot ^2}}{{\kappa - {\gamma _ \bot } + 2i{w_a}}}\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2} \right) } \right] \right) ^{-1} \end{aligned}$$

leads to the following amplitude equation derived by Gil [31]:

$$\begin{aligned} \frac{\partial }{{\partial \tau }}{} \mathbf{A}= &\, {C_1}A + {C_2}\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _{_ \bot }^2} \right) \mathbf{A} + {C_3}{\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _{_ \bot }^2} \right) ^2}{} \mathbf{A} \nonumber \\&+ {C_4}\left( \mathbf{A \cdot \mathbf{A ^*}} \right) \mathbf{A} + {C_5}\left( \mathbf{A \cdot \mathbf{A} } \right) \mathbf{A ^*}, \end{aligned}$$

(50)

with

$$\begin{aligned} {C_1}= &\, \frac{{{\mu _0}{c^2}g{{{{\tilde{D}}}}_0}\left( {{\kappa } - {\gamma _ \bot } + 2i{w_a}} \right) }}{\nonumber }\\&\times {{\left( {{{({\kappa } - {\gamma _ \bot })}^2} + 4w_a^2} \right) }}, \end{aligned}$$

(51)

$$\begin{aligned} {C_2}= &\, - \frac{{2c\left( {{\gamma _ \bot }\left( {{\gamma _ \bot } - {\kappa }} \right) + 2w_a^2 +i{w_a}\left( {\kappa - 3{\gamma _ \bot }} \right) } \right) }}{\nonumber }\\&\times {{\left( {{{({\kappa } - {\gamma _ \bot })}^2} + 4w_a^2} \right) }} , \end{aligned}$$

(52)

$$\begin{aligned} {C_3}= &\, - \frac{{4{c^2}{\gamma _ \bot }\left( {\gamma _ \bot ^2\left( {2\kappa - {\gamma _ \bot }} \right) + {\kappa }\left( {{\kappa }{\gamma _ \bot } - 4w_a^2} \right) - i{\gamma _ \bot }\left( {3\gamma _ \bot ^2 + 4w_a^2 - {\kappa }\left( {2\gamma _ \bot ^2 -{\kappa }} \right) } \right) } \right) }}{\nonumber }\\&\times {{{{\left( {{{({\kappa } - {\gamma _ \bot })}^2} + 4w_a^2} \right) }^2}}} , \end{aligned}$$

(53)

$$\begin{aligned} {C_4}= &\, \frac{{4kg\left( { - \left( {{\kappa } - {\gamma _ \bot }} \right) + 2i{w_a}} \right) }}{\nonumber }\\&\times {{\hbar {w_a}{\gamma _\parallel }\left( {{{({\kappa } - {\gamma _ \bot })}^2} + 4w_a^2} \right) }}, \end{aligned}$$

(54)

$$\begin{aligned} {C_5}= &\, \frac{{2g\left( {{\gamma _\parallel }\left( {2w_a^2 + {\kappa }\left( {{\kappa } - \gamma _\parallel ^2} \right) } \right) - 2w_a^2\left( {{\kappa } + {\gamma _ \bot }} \right) - i{w_a}\left( {{\gamma _\parallel }\left( {{\kappa } + {\gamma _ \bot }} \right) + 2{\kappa }\left( {{\kappa } - {\gamma _ \bot }} \right) + 4w_a^2} \right) } \right) }}{\nonumber }\\&\times {{\hbar {w_a}\left( {\gamma _\parallel ^2 + 4w_a^2} \right) \left( {{{({\kappa } - {\gamma _ \bot })}^2} + 4w_a^2} \right) }}. \end{aligned}$$

(55)

To analyze higher order nonlinearities in the system, the nonlinear polarization term $\mathbf{P} _3$ is needed. Therefore, taking into account the nonlinear polarization into the population inversion Eq. (48) yields

$$\begin{aligned} k\frac{{\partial \mathbf{E _1}}}{{\partial \tau }}= &\, - 2i{w_a}\frac{{\partial \mathbf{E _1}}}{{\partial \tau }} - 2i{w_a}c\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2} \right) \mathbf{E _1} \nonumber \\&- {\mu _0}{c^2}\left( {2\frac{\partial }{{\partial \tau }}\frac{{\partial \mathbf{P _1}}}{{\partial t}}} \right) , \end{aligned}$$

(56)

$$\begin{aligned} 2\frac{\partial }{{\partial \tau }}\frac{{\partial \mathbf{P _1}}}{{\partial t}}= &\, - {\gamma _ \bot }\frac{{\partial \mathbf{P _1}}}{{\partial \tau }} + \gamma _{_ \bot }^2\frac{{{\partial ^2}\mathbf{P _1}}}{{\partial {T^2}}} - g\left( {{{{{\tilde{D}}}}_0} + {D_2}} \right) \mathbf{E _1}, \end{aligned}$$

(57)

$$\begin{aligned} \frac{{\partial {D_2}}}{{\partial t}}= &\, - {\gamma _\parallel }{D_2} + \frac{2}{{\hbar {w_a}}}\left( {\mathbf{E _1}.\frac{{\partial {(\mathbf{P} _1+\mathbf{P} _3)}}}{{\partial t}}} \right) ,\end{aligned}$$

(58)

Where $D_2$ is again obtained by solving Eq. (58), i.e.,

$$\begin{aligned} {D_2}= &\, {D_{20}} + {D_{22}}{e^{2i\left( {{w_a}t - {k_c}z} \right) }} + D_{22}^*{e^{ - 2i\left( {{w_a}t - {k_c}z} \right) }}\nonumber \\&+ {D_{24}}{e^{4i\left( {{w_a}t - {k_c}z} \right) }} + D_{24}^*{e^{ - 4i\left( {{w_a}t - {k_c}z} \right) }}, \end{aligned}$$

(59)

with

$$\begin{aligned} {D_{20}}= &\, \frac{4}{{\hbar {\mu _0}{c^2}{w_a}{\gamma _\parallel }}}\Bigg ( - k\mathbf{A} \mathbf{A ^*} + \frac{{kg\mathbf{A ^2}\mathbf{A ^{*2}}}}{{\hbar {w_a}{\gamma _ \bot }}}\Bigg (\frac{4}{{{\gamma _\parallel }}}+ \frac{1}{{\left( {{\gamma _\parallel } - 2i{w_a}} \right) }}+ \frac{1}{{\left( {{\gamma _\parallel } + 2i{w_a}} \right) }}\Bigg )\nonumber \\&+ \frac{{ig\mathbf{A ^2}\mathbf{A ^{*2}}}}{{\hbar {w_a}{\gamma _ \bot }}}\Bigg (\frac{1}{{\left( {{\gamma _\parallel } + 2i{w_a}} \right) }} - \frac{1}{{\left( {{\gamma _\parallel } - 2i{w_a}} \right) }}\Bigg )\Bigg ), \end{aligned}$$

(60)

$$\begin{aligned} {D_{22}}= &\, \frac{2}{{\hbar {\mu _0}{c^2}{w_a}\left( {{\gamma _\parallel } + 2i{w_a}} \right) }}( - \mathbf{A ^2}(k + i{w_a}) \nonumber \\&+ \frac{{2g\mathbf{A ^3}\mathbf{A ^*}}}{\hbar }\Bigg (\frac{k}{{{\gamma _ \bot }{w_a}}}\Bigg (\frac{1}{{\left( {{\gamma _\parallel } 2i{w_a}} \right) }} + \frac{2}{{{\gamma _\parallel }}}\Bigg )\nonumber \\&+ \frac{3}{{\left( {{\gamma _\parallel } + 2i{w_a}} \right) \left( {8{w_a} - 3i{\gamma _ \bot }} \right) }}\nonumber \\&+ \frac{i}{{\left( {{\gamma _\parallel } + 2i{w_a}} \right) }}\Bigg (\frac{1}{{{\gamma _ \bot }}} - \frac{{3k}}{{{w_a}\left( {8{w_a} - 3i{\gamma _ \bot }} \right) }}\Bigg )\Bigg )\Bigg ), \end{aligned}$$

(61)

$$\begin{aligned} {D_{24}}= &\, \frac{{12g\mathbf{A ^4}}}{{{\hbar ^2}{\mu _0}{c^2}{w_a}\left( {{\gamma _\parallel } + 4i{w_a}} \right) \left( {8{w_a} - 3i{\gamma _ \bot }} \right) }}\left( {1 - \frac{{ik}}{{{w_a}}}} \right) . \end{aligned}$$

(62)

Substituting Eq. (59) into Eq. (57), and after some algebra, we obtain the following (3+1)D vectorial cubic-quintic CGL equation

$$\begin{aligned} \begin{aligned} \frac{{\partial \mathbf{A} }}{{\partial \tau }}&= {z_1}{} \mathbf{A} + {z_2}\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2} \right) \mathbf{A} + {z_3}{\left( {\frac{\partial }{{\partial Z}} + \frac{i}{{2{k_c}}}\nabla _ \bot ^2} \right) ^2}{} \mathbf{A} + {z_4}\left( \mathbf{A \cdot \mathbf{A ^*}} \right) \mathbf{A} \\&\quad + {z_5}\left( \mathbf{A \cdot \mathbf{A} } \right) \mathbf{A ^*}+ {z_6}\left( {\mathbf{A ^2}\cdot \mathbf{A ^{*2}}} \right) \mathbf{A} + {z_7}\left( {\mathbf{A ^3}\cdot \mathbf{A ^*}} \right) \mathbf{A ^*}, \end{aligned} \end{aligned}$$

(63)

with

$$\begin{aligned} {z_1}= &\, \frac{{{\mu _0}{c^2}g{{{{\tilde{D}}}}_0}\left( {k - {\gamma _\bot } +2i{w_a}} \right) }}{{{{\left( {k - {\gamma _ \bot }} \right) }^2} + 4w_a^2}}, \end{aligned}$$

(64)

$$\begin{aligned} {z_2}= &\, \frac{{2c\left( {2w_a^2 + {\gamma _ \bot }\left( {{\gamma _ \bot } - k} \right) + i{w_a}\left( {k - 3{\gamma _ \bot }} \right) } \right) }}{{{{\left( {k - {\gamma _ \bot }} \right) }^2} + 4w_a^2}}, \end{aligned}$$

(65)

$$\begin{aligned} {z_3}= &\, \frac{{4{c^2}{\gamma _ \bot }\left( \begin{array}{l} \gamma _ \bot ^2(2\kappa - {\gamma _ \bot }) + \kappa \left( {\kappa {\gamma _ \bot } - 4w_a^2} \right) \\ - i{\gamma _ \bot }\left( {3\gamma _ \bot ^2 + 4w_a^2 - \kappa (2{\gamma _ \bot } - \kappa )} \right) \end{array} \right) }}{{{{(\kappa - {\gamma _ \bot })}^2} + w_a^2}}, \end{aligned}$$

(66)

$$\begin{aligned} {z_4}= &\, \frac{{4kg\left( { \left( {k - {\gamma _ \bot }} \right) - 2i{w_a}} \right) }}{{\hbar {w_a}{\gamma _\parallel }\left( {{{\left( {k - {\gamma _ \bot }} \right) }^2} + 4w_a^2} \right) }}, \end{aligned}$$

(67)

$$\begin{aligned} {z_5}= &\, \frac{{2g\left( \begin{array}{l} {\gamma _\parallel }(\kappa (\kappa - \gamma _ \bot ^2) + 2w_a^2) - 2w_a^2(\kappa + {\gamma _ \bot })\\ - i{w_a}\left( {{\gamma _\parallel }(\kappa + {\gamma _ \bot }) + \kappa (\kappa - {\gamma _ \bot }) + 42w_a^2} \right) \end{array} \right) }}{{\hbar {w_a}\left( {{{(\kappa - {\gamma _ \bot })}^2} + w_a^2} \right) \left( {\gamma _\parallel ^2 + 4w_a^2} \right) }}, \end{aligned}$$

(68)

$$\begin{aligned} {z_6}= &\, \frac{{8k{g^2}\left( {{\gamma _\parallel }\left( {k +2{\gamma _\parallel }} \right) + 10w_a^2} \right) \left( {k - {\gamma _ \bot } - 2i{w_a}} \right) }}{{{\hbar ^2}w_a^2{\gamma _ \bot }\gamma _\parallel ^2\left( {{{\left( {k - {\gamma _ \bot }} \right) }^2} + 4w_a^2} \right) \left( {\gamma _\parallel ^2 + 4w_a^2} \right) }}, \end{aligned}$$

(69)

$$\begin{aligned}&\begin{aligned} {z_{7r}}&= \frac{{4{g^2}\left( \begin{aligned}&({\gamma _\parallel }(k(3\gamma _\parallel ^2 + 4w_a^2) + 4w_a^2{\gamma _\parallel })(9\gamma _ \bot ^2 + 64w_a^2) + 3{\gamma _ \bot }{\gamma _\parallel }((8w_a^2 \\&+ 3k{\gamma _ \bot })(\gamma _{_\parallel }^2 - 4w_a^2) + 4{\gamma _\parallel }w_a^2( - 8k + 3{\gamma _ \bot })))(k - {\gamma _ \bot })\\&+ 2w_a^2(((\gamma _\parallel ^2 - 4w_a^2 - 4k{\gamma _\parallel }){\gamma _\parallel } - 4k(\gamma _\parallel ^2 + 4w_a^2))(9\gamma _ \bot ^2 + 64w_a^2) \\&+ 3{\gamma _\parallel }{\gamma _ \bot }(( - 8k + 3{\gamma _ \bot })(\gamma _\parallel ^2 - 4w_a^2) - 4(8w_a^2 + 3k{\gamma _ \bot }){\gamma _\parallel })) \end{aligned} \right) }}{{{\hbar ^2}w_a^2{\gamma _ \bot }{\gamma _\parallel }{{(\gamma _\parallel ^2 + 4w_a^2)}^2}(9\gamma _ \bot ^2 + 64w_a^2)({{(k - {\gamma _ \bot })}^2} + 4w_a^2)}}\end{aligned} \end{aligned}$$

(70)

$$\begin{aligned} {z_{7i}}= &\, \frac{{4{g^2}\left( \begin{aligned}&({w_a}(((\gamma _{_\parallel }^2 - 4(w_a^2 + k{\gamma _\parallel })) - 4k(\gamma _\parallel ^2 + 4w_a^2))(9\gamma _ \bot ^2 + 64w_a^2)\\&+ 3{\gamma _\parallel }{\gamma _ \bot }(( - 8k + 3{\gamma _ \bot })(\gamma _{_\parallel }^2 - 4w_a^2)- 4{\gamma _\parallel }(8w_a^2 + 3k{\gamma _ \bot })))(\mathrm{{k - }}{\gamma _ \bot } \\&\mathrm{{) - 2(}}{\gamma _\parallel }\mathrm{{(k(3}}\gamma _\parallel ^2 + 4w_a^2) + 4w_a^2{\gamma _\parallel })(9\gamma _{\bot }^2 + 64w_a^2) + 3{w_a}{\gamma _\parallel }{\gamma _ \bot }((8w_a^2 \\&+3k{\gamma _ \bot })(\gamma _\parallel ^2 - 4w_a^2) + 4w_a^2{\gamma _\parallel }(3{\gamma _ \bot } - 8k))) \end{aligned} \right) }}{{{\hbar ^2}w_a^2{\gamma _ \bot }{\gamma _\parallel }{{(\gamma _\parallel ^2 + 4w_a^2)}^2}(9\gamma _ \bot ^2 + 64w_a^2)({{(k - {\gamma _ \bot })}^2} + 4w_a^2)}} \end{aligned}$$

(71)

1.2 B Effective potential

The expression of the effective potential derived in Sec. 4 is as follows:

$$\begin{aligned}\begin{array}{l} \begin{aligned} &{}U = \Bigg (\frac{{32X_0^4}}{{81{X^2}}} + \frac{1}{{{X^2}}} - \frac{{28X_0^2}}{{81}} + \frac{{4X_0^2}}{{9{X^2}}} - \frac{{13}}{{36}} + \frac{3}{{128X_0^2}}\Bigg )\\ &{}\quad {A^8}\delta _i^2{e^{ - \frac{{16X_0^2}}{{{X^2}}}}} + \Bigg (\frac{{8X_0^4}}{{{X^2}}}\\ &{}\quad + \frac{1}{{4{X^2}}} + \Bigg (\frac{{17}}{{18{X^2}}} - \frac{{79}}{{126}}\Bigg )X_0^2 - \frac{{5{X^2}}}{4} - \frac{{53}}{{84}} - \frac{9}{{56X_0^2}}\Bigg )\\ &{} {A^6}{\delta _i}{\gamma _i}{e^{ - \frac{{14X_0^2}}{{3{X^2}}}}} \\ &{}\quad + (( - \frac{1}{8}(7{X^2} + \frac{{15}}{{8X_0^2}})\delta _i^2- \frac{1}{9}(1 + 8{X^2} + \frac{{16}}{9}X_0^2){\delta _i}\mu )\\ &{} {A^8} - (1 + 2X_0^2 \\ &{}\quad + 7{X^2}){A^6}\varepsilon {\delta _i} + (\frac{8}{9}(4C+ \frac{\beta }{{{X^2}}} - 4\beta {\delta _i}{C^2}{X^2})X_0^2{\delta _i} - \frac{{16}}{9}(1 \\ &{}\quad + 5\beta C{X^2}){\delta _i}NX_0^2 - \frac{{8\beta {\delta _r}C{X^2}}}{3} \\ &{}\quad +2\beta {\delta _i}{C^2}{X^2}({X^2} - 1) + 2{\delta _i}C \\ &{} \quad - \frac{4}{3}{\delta _r} + 2{\delta _i}C{X^2} + {\delta _i}\beta {N^2}{X^2} - \delta {\delta _i}{X^2} \\ &{}\quad + \frac{{\beta {\delta _i}}}{{2{X^2}}} - \frac{{11}}{6}\beta {\delta _i} - \frac{{{\delta _i}N}}{{{X_0}}} \\ &{}\quad - \frac{{3\beta {\delta _i}}}{{4X_0^2}}){A^4}){e^{ - \frac{{8X_0^2}}{{{X^2}}}}} \\ &{}\quad + ( - (1 + 2X_0^2 - \frac{{11}}{2}{X^2})\mu {\gamma _i}{A^6} - (1 +4{X^2} \\ &{}\quad +2X_0^2){A^4} + ( - {\gamma _r} - \frac{3}{2}{\gamma _i}\beta + 2{\gamma _i}C \\ &{}\quad + 4{\gamma _i}CX_0^2- 2{\gamma _i}N{X_0} + 2\beta {\gamma _i}{C^2}{X^4} \\ &{}\quad - \frac{{{\gamma _i}N}}{{{X_p}}} - \frac{{{\gamma _i}\beta }}{{2X_0^2}} + \frac{{\beta {\gamma _i}}}{{2{X^2}}} + \frac{{{\gamma _i}\beta X_0^2}}{{{X^2}}}){A^2}\\ &{}\quad - (\frac{{{\gamma _i}\delta }}{2} + 8\beta {\gamma _i}C{X_0}N+ 2\beta {\gamma _i}{C^2}\\ &{}\quad +\beta {\gamma _r}C + 4\beta {\gamma _i}{C^2}X_0^2 - \frac{{\beta {\gamma _i}N}}{2})){e^{ - \frac{{2X_0^2}}{{{X^2}}}}} + (\frac{{\gamma _i^2X_0^2}}{{2{X^2}}} \\ &{}\quad + \frac{{\gamma _i^2}}{{8{X^2}}} - \frac{{\gamma _i^2}}{4} \\ &{}\quad -\frac{3}{{32}}\gamma _i^2{X^2}- \frac{{\gamma _i^2}}{{16X_0^2}} - \frac{{\gamma _i^2X_0^2}}{4} \\ &{}\quad + \frac{{\gamma _i^2X_0^2}}{{2{X^2}}}){e^{ - \frac{{4X_0^2}}{{{X^2}}}}}- \frac{{2{\mu ^2}{X^2}{A^8}}}{9} - \frac{{5\mu \varepsilon {X^2}{A^6}}}{{12}}\\ &{}\quad +4{\beta ^2}{C^2}{X^2}(2{C^2}{X^4} + 3) - \frac{2}{{{X^2}}} + \frac{{{\beta ^2}}}{{2{X^2}}}\\ &{} \quad - 8\beta {C^3}{X^4} - 24\beta C\ln (X) \\ &{}\quad + (\frac{{{X^2}\varepsilon }}{4}(\beta {N^2} - \delta )\\ &{}\quad + \beta C{X^2}( - 1 + \varepsilon C{X^2}) + (1 + \beta \varepsilon )\ln (X)){A^2} \end{aligned} \end{array}\end{aligned}$$