INTRODUCTION

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The interaction between a two-level atom and a quantized single-mode electromagnetic field is described by the Jaynes-Cummings model (JCM)\QCITE{cite}{}{Jaynes} which can be applied to investigating many quantum effects such as the quantum collapses and revivals of the atomic inversion, photon antibunching, squeezing of the radiation field, inversionless light amplification, electromagnetic induced transparency\QCITE{cite}{}{Eberly,Alexanian,Wodkiewicz,Imamolglu} and etc..We have investigated the exact solutions and the geometric phase factor of the two-level JCM whose Hamiltonian has time-dependent parameters, by making use of the generalized invariant theory. In addition, there exists a type of JCM (so-called two-level multiphoton Jaynes-Cummings model) who possesses supersymmetric structure\QCITE{cite}{}{Klein}. Some authors introduced a supersymmetric unitary transformation to diagonalize the Hamiltonian of this JCM and obtain the eigenfunctions of the stationary Schr\"{o}dinger equation$\QCITE{cite}{}{Lu1,Lu2}$. In the present paper, we generalize this method and obtain the exact solutions and the expression for the geometric phase factor of the time-dependent two-level multiphoton Jaynes-Cummings model (TLMJCM) through the invariant-related unitary transformation formulation.

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The invariant theory which is very appropriate to treat the time-dependent systems was first proposed by Lewis and Riesenfeld\QCITE{cite}{}{Lewis} in 1969. In 1991, one of the authors of this paper generalized the L-R invariant theory and proposed the invariant-related unitary transformation formulation\QCITE{cite}{}{Gao1,Gao3}.This formulation replaces the eigenstates of the time-dependent invariants by that of the time-independent invariants through the unitary transformation and obtain the exact solutions, which contain the dynamical and geometric phase factor, of the time-dependent Schr\"{o}dinger equation. One of the advantages of this unitary transformation method is that it can transform the hidden form, which is in connection with the chronological product, of the time-evolution operator $U(t)$ into the obvious expression. Many works have showed that the invariant-related unitary transformation approach is a powerful tool for treating the time-dependent systems and the geometric phase factor\QCITE{cite}{}{Gao2,Fu1,Fu2}.

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THE INVARIANT THEORY AND THE INVARIANT-RELATED UNITARY TRANSFORMATION FORMULATION

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For the sake of reviewing the L-R invariant theory\QCITE{cite}{}{Lewis}, we consider a one-dimensional system whose Hamiltonian $H(t)$ is time-dependent. According to L-R invariant theory, a Hermitian operator $I(t)$ is called invariant if it satisfies the following invariant equation

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\EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial I(t)}{\partial t}+\frac{1}{i}[I(t),H(t)]=0.}}{2}{(2.1)}{eq21}{}}The eigenvalue equation of the time-dependent invariant $\left| \lambda _{n},t\right\rangle $ is given \EQN{0}{1}{}{}{\RD{\CELL{I(t)\left| \lambda _{n},t\right\rangle =\lambda _{n}\left| \lambda _{n},t\right\rangle}}{2}{(2.2)}{eq22}{}}where \EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial \lambda _{n}}{\partial t}=0.}}{2}{(2.3)}{eq23}{}}The time-dependent Schr\"{o}dinger equation for the system is

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\EQN{0}{1}{}{}{\RD{\CELL{i\frac{\partial \left| \Psi (t)\right\rangle _{s}}{\partial t}=H(t)\left| \Psi (t)\right\rangle _{s}.}}{2}{(2.4)}{eq24}{}}In terms of the L-R invariant theory, the particular solution $\left| \lambda _{n},t\right\rangle _{s}$ of Eq.\QTSN{ref}{eq24} differs from the eigenfunction $\left| \lambda _{n},t\right\rangle $ of the invariant $I(t)$ only by a phase factor $\exp [i\phi _{n}(t)]$, then the general solution of the Schr\"{o}dinger equation \QTSN{ref}{eq24} can be written as

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\EQN{0}{1}{}{}{\RD{\CELL{\left| \Psi (t)\right\rangle _{s}=\tsum_{n}C_{n}\exp [i\phi _{n}(t)]\left| \lambda _{n},t\right\rangle ,}}{2}{(2.5)}{eq25}{}}where

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\EQN{6}{1}{}{}{\RD{\CELL{\phi _{n}(t)=\int_{0}^{t}\left\langle \lambda _{n},t^{^{\prime }}\right| i\frac{\partial }{\partial t^{^{\prime }}}-H(t^{^{\prime }})\left| \lambda _{n},t^{^{\prime }}\right\rangle dt^{^{\prime }},}}{1}{}{}{}}

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\EQN{0}{1}{}{}{\RD{\CELL{C_{n}=\langle \lambda _{n},t=0\left| \Psi (t=0)\right\rangle _{s}.}}{2}{(2.6)}{eq26}{}}$\left| \lambda _{n},t\right\rangle _{s}=\exp [i\phi _{n}(t)]\left| \lambda _{n},t\right\rangle $ $(n=1,2,\cdots )$ are said to form a complete set of the solutions of Eq.\QTSN{ref}{eq24}. The statement outlined above is the basic content of the L-R invariant theory.

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A time-dependent unitary transformation operator can be constructed to transform $I(t)$ into a time-independent invariant $I_{V}\equiv V^{\dagger }(t)V(t)V(t)$ with \EQN{1}{1}{}{}{\RD{\CELL{I_{V}\left| \lambda _{n}\right\rangle &=&\lambda _{n}\left| \lambda _{n}\right\rangle ,}}{2}{(2.7)}{eq27}{}\RD{\CELL{\left| \lambda _{n}\right\rangle &=&V^{\dagger }(t)\left| \lambda _{n},t\right\rangle .}}{2}{(2.8)}{eq28}{}}Under the unitary transformation $V(t),$ the Hamiltonian $H(t)$ is correspondingly changed into $H_{V}(t)$

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\EQN{0}{1}{}{}{\RD{\CELL{H_{V}(t)=V^{\dagger }(t)H(t)V(t)-V^{\dagger }(t)i\frac{\partial V(t)}{\partial t}.}}{2}{(2.9)}{eq29}{}}In accordance with this unitary transformation method\QCITE{cite}{}{Gao1}, it is very easy to verify that the particular solution $\left| \lambda _{n},t\right\rangle _{s0}$ of the time-dependent Schr\"{o}dinger equation associated with $H_{V}(t)$

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\EQN{0}{1}{}{}{\RD{\CELL{i\frac{\partial \left| \lambda _{n},t\right\rangle _{s0}}{\partial t}=H_{V}(t)\left| \lambda _{n},t\right\rangle _{s0}}}{2}{(2.10)}{eq210}{}}is different from the eigenfunction $\left| \lambda _{n}\right\rangle $ of $I_{V}$ only by the same phase factor $\exp [i\phi _{n}(t)]$ as that in Eq.\QTSN{ref}{eq25}, i.e.,

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\EQN{0}{1}{}{}{\RD{\CELL{\left| \lambda _{n},t\right\rangle _{s0}=\exp [i\phi _{n}(t)]\left| \lambda _{n}\right\rangle .}}{2}{(2.11)}{eq211}{}}Substitution of $\left| \lambda _{n},t\right\rangle _{s0}$ of Eq.\QTSN{ref}{eq210} into Eq.\QTSN{ref}{eq211} yields

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\EQN{0}{1}{}{}{\RD{\CELL{-\dot{\phi}(t)\left| \lambda _{n}\right\rangle =H_{V}(t)\left| \lambda _{n}\right\rangle ,}}{2}{(2.12)}{eq212}{}}which means that $H_{V}(t)$ differs from $I_{V}(t)$ only by a time-dependent multiplying c-number factor. It can be seen from Eq.\QTSN{ref}{eq212} that the particular solution of Eq.\QTSN{ref}{eq210} can be easily obtained by calculating the phase from Eq.\QTSN{ref}{eq212}. Thus, one is led to the conclusion that if the $V(t),$ $I_{V},$ $H_{V}(t)$ and the eigenfunction $\left| \lambda _{n}\right\rangle $ of $I_{V}$ have been found, the problem of solving the complicated time-dependent Schr\"{o}dinger equation \QTSN{ref}{eq24} reduces to that of solving the much simplified equation \QTSN{ref}{eq210}. This paper obtains the exact solutions of the time-dependent Schr\"{o}dinger equation describing TLMJCM and expression for its geometric phase factor by making use of this invariant-related unitary transformation method.

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THE EXACT SOLUTIONS OF THE TIME-DEPENDENT TLMJCM

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The Hamiltonian of the TLMJCM under the rotating wave approximation is given by

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\EQN{0}{1}{}{}{\RD{\CELL{H(t)=\omega (t)a^{\dagger }a+\frac{\omega _{0}(t)}{2}\sigma _{z}+g(t)(a^{\dagger })^{k}\sigma _{-}+g^{\ast }(t)a^{k}\sigma _{+}}}{2}{(3.1)}{eq31}{}}where $a^{\dagger }$ and $a$ are the creation and annihilation operators for the electromagnetic field, and obey the commutation relation $\left[ a,a^{\dagger }\right] =1$; $\sigma _{\pm }$ and $\sigma _{z}$ denote the two-level atom operators which satisfy the commutation relation $\left[ \sigma _{z},\sigma _{\pm }\right] =\pm 2\sigma _{\pm }$ ; $g(t)$ and $g^{\ast }(t)$ are the coupling coefficients and $k$ is the photon number in each atom transition process; $\omega _{0}(t)$ and $\omega (t)$ are respectively the transition frequency and the mode frequency. All the parameters in \QTSN{ref}{eq31} are time-dependent.

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The supersymmetric structure can be found in the TLMJCM by defining the following supersymmetric transformation generators\QCITE{cite}{}{Lu1,Lu2}:

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\EQN{1}{1}{}{}{\RD{\CELL{N &=&a^{\dagger }a+\frac{k-1}{2}\sigma _{z}+\frac{1}{2}=\left( \MATRIX{2,2}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,,}\CELL{a^{\dagger }a+\frac{k}{2}}\CELL{0}\CELL{0}\CELL{a^{\dagger }a-\frac{k}{2}}\right) ,N^{^{\prime }}=\left( \MATRIX{2,2}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,,}\CELL{a^{k}(a^{\dagger })^{k}}\CELL{0}\CELL{0}\CELL{(a^{\dagger })^{k}a^{k}}\right) ,}}{0}{}{}{}\RD{\CELL{Q &=&(a^{\dagger })^{k}\sigma _{-}=\left( \MATRIX{2,2}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,,}\CELL{0}\CELL{0}\CELL{(a^{\dagger })^{k}}\CELL{0}\right) ,Q^{\dagger }=a^{k}\sigma _{+}=\left( \MATRIX{2,2}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,,}\CELL{0}\CELL{a^{k}}\CELL{0}\CELL{0}\right) .}}{2}{(3.2)}{eq32}{}}It is easily verified that $(N,N^{^{\prime }},Q,Q^{\dagger })$ form supersymmetric generators and have supersymmetric Lie algebra properties, i.e.,

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\EQN{1}{1}{}{}{\RD{\CELL{Q^{2} &=&(Q^{\dagger })^{2}=0,\left[ Q^{\dagger },Q\right] =N^{^{\prime }}\sigma _{z},\left[ N,N^{^{\prime }}\right] =0,\left[ N,Q\right] =Q,}}{0}{}{}{}\RD{\CELL{\left[ N,Q^{\dagger }\right] &=&-Q^{\dagger },\left\{ Q^{\dagger },Q\right\} =N^{^{\prime }},\left\{ Q,\sigma _{z}\right\} =\left\{ Q^{\dagger },\sigma _{z}\right\} =0,}}{0}{}{}{}\RD{\CELL{\left[ Q,\sigma _{z}\right] &=&2Q,\left[ Q^{\dagger },\sigma _{z}\right] =-2Q^{\dagger },\left( Q^{\dagger }-Q\right) ^{2}=-N^{^{\prime }}}}{2}{(3.3)}{eq33}{}}where $\left\{ {}\right\} $ denotes the anticommuting bracket. By the aid of \QTSN{ref}{eq32} and \QTSN{ref}{eq33}, the Hamiltonian \QTSN{ref}{eq31} of the TLMJCM can be rewritten as

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\EQN{0}{1}{}{}{\RD{\CELL{H(t)=\omega (t)N+\frac{\omega (t)-\delta (t)}{2}\sigma _{z}+g(t)Q+g^{\ast }(t)Q^{\dagger }-\frac{\omega (t)}{2}}}{2}{(3.4)}{eq34}{}}with $\delta (t)=k\omega (t)-\omega _{0}(t).$

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The equation which describes the time-evolution for the TLMJCM is the time-dependent Schr\"{o}dinger equation \QTSN{ref}{eq24}. We will show the solvability of Eq.\QTSN{ref}{eq24} by using the generalized invariant formulation in what follows.

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According to the invariant theory, we should first construct an invariant $I(t).$ It can be seen from the invariant equation \QTSN{ref}{eq21} that $I(t)$ is the linear combination of $N,\sigma _{z},Q$ and $Q^{\dagger }.$ However, it should be emphasized that the generalized invariant theory can only be applied to the study of the system for which there exists the quasialgebra defined in Ref.\QCITE{cite}{}{Mizrahi}. It is easily seen from \QTSN{ref}{eq33} that there is no such quasialgebra for the TLMJCM. For the sake of treating the time-dependent TLMJCM we extend, in the present paper, the method used for finding the dynamical algebra $O(4)$ of the hydrogen atom. In the case of hydrogen, the dynamical algebra $O(4)$ was found by working in the sub-Hilbert-space corresponding to a particular eigenvalue of the Hamiltonian \QCITE{cite}{}{Schiff}. In this paper, we will show that in the case of the TLMJCM, a generalized quasialgebra, which enables one to obtain the complete set of the exact solutions for the TLMJCM, can be found by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the supersymmetric generator $N^{^{\prime }}$.

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Use is made of $a^{k}(a^{\dagger })^{k}\left| m\right\rangle =\frac{(m+k)!}{m!}\left| m\right\rangle $ and $(a^{\dagger })^{k}a^{k}\left| m\right\rangle =\frac{(m+k)!}{m!}\left| m+k\right\rangle ,$ then one can arrive at

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\EQN{0}{1}{}{}{\RD{\CELL{N^{^{\prime }}\binom{\left| m\right\rangle }{\left| m+k\right\rangle }=\lambda _{m}\binom{\left| m\right\rangle }{\left| m+k\right\rangle }}}{2}{(3.5)}{eq35}{}}with $\lambda _{m}=\frac{(m+k)!}{m!}.$ Thus we obtain the supersymmetric quasialgebra $(N,Q,Q^{\dagger },\sigma _{z})$ in the sub-Hilbert-space corresponding to the particular eigenvalue $\lambda _{m}$ of $N^{^{\prime }}, $ by replacing the generator $N^{^{\prime }}$ with $\lambda _{m}$ in the commutation relations in \QTSN{ref}{eq33}, namely,

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\EQN{0}{1}{}{}{\RD{\CELL{\left[ Q^{\dagger },Q\right] =\lambda _{m}\sigma _{z},\left\{ Q^{\dagger },Q\right\} =\lambda _{m},\left( Q^{\dagger }-Q\right) ^{2}=-\lambda _{m}.}}{2}{(3.6)}{eq36}{}}

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In accordance with the invariant theory, the invariant $I(t)$ is of the form

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\EQN{0}{1}{}{}{\RD{\CELL{I(t)=c(t)Q^{\dagger }+c^{\ast }(t)Q+b(t)\sigma _{z}}}{2}{(3.7)}{eq37}{}}where $c^{\ast }(t)$ is the complex conjugation of $c(t),$ and $b(t)$ is real. The substitution of the expressions \QTSN{ref}{eq34} and \QTSN{ref}{eq37} for $I(t) $ and $H(t)$ into Eq.\QTSN{ref}{eq21} leads to the following set of auxiliary equations

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\EQN{1}{1}{}{}{\RD{\CELL{\overset{.}{c}-\frac{1}{i}[c(\omega -\omega _{0})-2bg] &=&0,\overset{.}{c^{\ast }}+\frac{1}{i}[c^{\ast }(\omega -\omega _{0})-2bg^{\ast }]=0,}}{0}{}{}{}\RD{\CELL{\overset{.}{b}+\frac{1}{i}(c^{\ast }g-cg^{\ast })\lambda _{m} &=&0}}{2}{(3.8)}{}{}}where the dot denotes the time derivative. Hence, the three time-parameters $c,c^{\ast }$ and $b$ in $I(t)$ are determined by these three auxiliary equations.

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According to the invariant-related unitary transformation method, we define the unitary transformation operator as follows

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\EQN{0}{1}{}{}{\RD{\CELL{V(t)=\exp [\beta (t)Q-\beta ^{\ast }(t)Q^{\dagger }]}}{2}{(3.9)}{eq39}{}}with $\beta ^{\ast }(t)$ being the complex conjugation of $\beta (t).$ With the help of the commutation relations \QTSN{ref}{eq33}, it can be found that, by the complicated and lengthy computations, if $\beta (t)$ and $\beta ^{\ast }(t)$ satisfy the following equations

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\EQN{0}{1}{}{}{\RD{\CELL{\sin (4\beta \beta ^{\ast }\lambda _{m})^{\frac{1}{2}}=\frac{(c\beta ^{\ast }+c^{\ast }\beta )}{(4\beta \beta ^{\ast }\lambda _{m})^{\frac{1}{2}}},\cos (4\beta \beta ^{\ast }\lambda _{m})^{\frac{1}{2}}=b,}}{2}{(3.10)}{eq310}{}}a time-independent invariant can be obtained as follows

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\EQN{0}{1}{}{}{\RD{\CELL{I_{V}\equiv V^{\dagger }(t)I(t)V(t)=\sigma _{z}.}}{2}{(3.11)}{eq311}{}}From Eq.\QTSN{ref}{eq310}, we substitute the time-dependent parameters $\theta $ and $\phi $ for $c,c^{\ast }$ and $b$ in $I(t)$ for simplicity, and the results are

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\EQN{1}{1}{}{}{\RD{\CELL{\beta &=&-\frac{\frac{\theta }{2}\exp (-i\phi )}{\lambda _{m}^{\frac{1}{2}}},\beta ^{\ast }=-\frac{\frac{\theta }{2}\exp (i\phi )}{\lambda _{m}^{\frac{1}{2}}},}}{0}{}{}{}\RD{\CELL{c &=&-\frac{\sin \theta \exp (-i\phi )}{\lambda _{m}^{\frac{1}{2}}},c^{\ast }=-\frac{\sin \theta \exp (i\phi )}{\lambda _{m}^{\frac{1}{2}}}.}}{2}{(3.12)}{eq312}{}}Thus, the invariant $I(t)$ in \QTSN{ref}{eq37} can be rewritten

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\EQN{0}{1}{}{}{\RD{\CELL{I(t)=-\frac{\sin \theta }{\lambda _{m}^{\frac{1}{2}}}[\exp (-i\phi )Q+\exp (i\phi )Q^{\dagger }]+\cos \theta \sigma _{z}.}}{2}{(3.13)}{eq313}{}}In the meanwhile, under the unitary transformation \QTSN{ref}{eq39}, the Hamiltonian \QTSN{ref}{eq34} can be transformed into

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\EQN{1}{1}{}{}{\RD{\CELL{H_{V}(t) &\equiv &V^{\dagger }(t)H(t)V(t)-V^{\dagger }(t)i\frac{\partial }{\partial t}V(t)}}{0}{}{}{}\RD{\CELL{&=&\omega N+\frac{\omega }{2}(\sigma _{z}-1)+\{-\frac{1}{2}\lambda _{m}^{\frac{1}{2}}[g\exp (i\phi )+g^{\ast }\exp (-i\phi )]\sin \theta +}}{0}{}{}{}\RD{\CELL{&&+\frac{\omega _{0}-\omega }{2}\cos \theta -\frac{\overset{.}{\phi }}{2}(1-\cos \theta )\}\sigma _{z}.}}{2}{(3.14)}{eq314}{}}The eigenstates of $\sigma _{z}$ corresponding to the eigenvalue $\sigma =+1$ and $\sigma =-1$ are $\binom{1}{0}$ and $\binom{0}{1},$ and the eigenstate of $N^{^{\prime }}$ is $\binom{\left| m\right\rangle }{\left| m+k\right\rangle }$ in terms of \QTSN{ref}{eq35}. From Eq.\QTSN{ref}{eq26}, \QTSN{ref}{eq211}, \QTSN{ref}{eq212}, we obtain two particular solutions of the time-dependent Schr\"{o}dinger equation of the TLMJCM which are written in the form

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\EQN{0}{1}{}{}{\RD{\CELL{\Psi _{m,\sigma =+1}(t)=\exp \{\frac{1}{i}\int_{0}^{t}[\overset{.}{\varphi }_{d,\sigma =+1}(t^{^{\prime }})+\overset{.}{\varphi }_{g,\sigma =+1}(t^{^{\prime }})]dt^{^{\prime }}\}V(t^{^{\prime }})\binom{\left| m\right\rangle }{0}}}{2}{(3.15)}{eq315}{}}with $\overset{.}{\varphi }_{d,\sigma =+1}(t^{^{\prime }})=(m+\frac{k}{2})\omega (t^{^{\prime }})-\frac{1}{2}\lambda _{m}^{\frac{1}{2}}\{g(t^{^{\prime }})\exp [i\phi (t^{^{\prime }})]+g^{\ast }(t^{^{\prime }})\exp [-i\phi (t^{^{\prime }})]\}\sin \theta (t^{^{\prime }})$

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$+\frac{\omega _{0}(t^{^{\prime }})-\omega (t^{^{\prime }})}{2}\cos \theta (t^{^{\prime }})$ and $\overset{.}{\varphi }_{g,\sigma =+1}(t^{^{\prime }})=-\frac{\overset{.}{\phi (t^{^{\prime }}})}{2}[1-\cos \theta (t^{^{\prime }})]; $ and

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\EQN{0}{1}{}{}{\RD{\CELL{\Psi _{m,\sigma =-1}(t)=\exp \{\frac{1}{i}\int_{0}^{t}[\overset{.}{\varphi }_{d,\sigma =-1}(t^{^{\prime }})+\overset{.}{\varphi }_{g,\sigma =-1}(t^{^{\prime }})]dt^{^{\prime }}\}V(t^{^{\prime }})\binom{0}{\left| m+k\right\rangle }}}{2}{(3.16)}{eq316}{}}with $\overset{.}{\varphi }_{d,\sigma =-1}(t^{^{\prime }})=(m+\frac{k}{2})\omega (t^{^{\prime }})+\frac{1}{2}\lambda _{m}^{\frac{1}{2}}\{g(t^{^{\prime }})\exp [i\phi (t^{^{\prime }})]+g^{\ast }(t^{^{\prime }})\exp [-i\phi (t^{^{\prime }})]\}\sin \theta (t^{^{\prime }})$

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$-\frac{\omega _{0}(t^{^{\prime }})-\omega (t^{^{\prime }})}{2}\cos \theta (t^{^{\prime }})$ and $\overset{.}{\varphi }_{g,\sigma =-1}(t^{^{\prime }})=\frac{\overset{.}{\phi (t^{^{\prime }}})}{2}[1-\cos \theta (t^{^{\prime }})]. $

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These two particular solutions of the Schr\"{o}dinger equation \QTSN{ref}{eq24} contain corresponding dynamical phase factor $\exp [\frac{1}{i}\int_{0}^{t}\overset{.}{\varphi }_{d,\sigma }(t^{^{\prime }})dt^{^{\prime }}]$ and the geometric phase factor $\exp [\frac{1}{i}\int_{0}^{t}\overset{.}{\varphi }_{g,\sigma }(t^{^{\prime }})]dt^{^{\prime }}$ with $\sigma =\pm 1.$ Apparently, it can be seen that the former is dependent on the transition frequency $\omega _{0}(t)$ and the mode frequency $\omega (t)$, and the coupling coefficients $g(t)$ and $g^{\ast }(t)$ as well, whereas the latter is independent of these frequency parameters and the coupling coefficients.

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CONCLUDING REMARKS

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We construct an invariant in the sub-Hilbert-space corresponding to a particular eigenvalue of $N^{^{\prime }}$ and get the exact solutions of the time-dependent TLMJCM by making use of the invariant-related unitary transformation formulation. This formulation replaces eigenstates of the time-dependent invariants by that of the time-independent invariants through the unitary transformation. In view of the above calculation, we can see that this unitary transformation formulation has some useful applications, for instance, it can solve the time-dependent systems and treat the geometric phase factor, and obtain the obvious expressions, instead of the hidden form, for the evolution operator of the wave functions. This method can also be generalized to study the time-evolutions of the quantum Klein-Gordon field and Dirac field in time-dependent backgrounds such as spatially homogeneous electric field and Friedmann-Robertson-Walker flat spacetimes\QCITE{cite}{}{Gao4,Gao5,Gao6}.

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Since the three-level two-mode Jaynes-Cummings model plays an important role in Quantum Optics, the supersymmetric structure and the exact solutions of the time-dependent three-level two-mode multiphoton JCM deserves further investigations by the formalism suggested in the present paper.

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Achnowledgment This project was supported by the National Natural Science Foundation of China under the project No.$19775040$. The project was also supported by the Zhejiang Provincial Natural Science Foundation of China.

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