You are an expert at summarizing long articles. Proceed to summarize the following text: 
the dynamics of a two - state quantum system resulting from the action of a resonant external pulse of precise temporal area is a classical example in quantum mechanics . this simple two - state quantum system has an interesting theoretical extension , from a nondegenerate two - level quantum system ( where only two two quantum states /ground and excited/ are available ) , to one with a degenerate ground level and a degenerate excited level . these physical systems can be found at isolated atoms and molecules , where the states of well defined angular momentum j , due to the rotational symmetry produces a degeneracy of 2j+1 magnetic sublevels . for correct description of laser - induced transitions between quantum states with defined angular momentum , one must also incorporate several magnetic sublevels ( labeled by m ) , that may serve as a possible initial states . hence each of these magnetic sublevels have a laser - driven excitation dynamic into to set of excited magnetic sublevels . other source of multilevel systems can be the systems , whose mathematical description uses the usual rotating - wave approximation ( rwa ) . within the rwa the degeneracies can also be found for more general multistate quantum systems .    to handle such multilevel systems , in 1983 morris and shore have introduced a coordinate transformation with the general property that could factorize the dynamic of any two degenerate sets of quantum states , to an equivalent description involving only independent uncoupled pairs of equations . generally , the morris - shore ( ms ) transformation is able to reduces the coherent quantum dynamics of a coupled degenerate two - level system to a set of independent nondegenerate two - state systems and a set of uncoupled states . the mathematical description of ms transformation only requires deriving of eigenvalues and the eigenstates of a hermitean matrix , which is a product of interaction matrices . the eigenstates of this hermitean matrix are the ms states , that represent the independent two - state systems and the dark states . the eigenvalues represent the ms interactions in the independent ms two - state systems . the ms transformation is quite general , but requires that all initial interactions are constant , or share the same time dependence , and also that all interactions are resonant , or equally detuned from the upper states . the condition of equally detuned set of states implies that the lower set of states is degenerate in rwa sense , and so is the upper set of states . having explain the constrains of ms transformation , it is natural to ask whether they can be relaxed . it turns out that within the framework of weak deviation of equally detuned set of statrs , perturbative solution can be derived . this paper is organized as follows . in sec . ii we review the two - level ms transformation and in details define the problem for its extension . in sec . iii we describe the perturbative solution to the problem of generalized ms transformation . section iv presents an examples for the generalized ms transformation . finally , sec . v presents a summary of the results . we consider a two - level quantum system in the general case when both levels are degenerate . the ground @xmath0 level consists of @xmath1 degenerate states , the excited @xmath2 level  of @xmath3 degenerate states and we assume , without any loss of generality , that @xmath4 every ground state @xmath5 @xmath6 is coupled via all excited states @xmath7 @xmath8 with pulsed interactions @xmath9 , each pair of which are on two - photon resonance , as depicted in [ fig1 ] . the excited level may be off single photon resonance by some detuning @xmath10 which should be the same for all couplings . we assume that the couplings share same time dependence @xmath11 @xmath12but we allow for different amplitudes @xmath13 and thereby pulse areas . the dynamics of the system is governed by the time - dependent schrdinger equation which in the usual rotating - wave approximation ( rwa ) has the form@xmath14where the elements of the @xmath15-dimensional state vector @xmath16 are the probability amplitudes of the states and the hamiltonian in a block - matrix form is given by@xmath17here @xmath18 is the @xmath1-dimensional square zero matrix , in which the zero off - diagonal elements indicate the absence of couplings between the @xmath0 states , while the zero diagonal elements show that the @xmath19 states have the same energy , which is taken as the zero of the energy scale . the matrix @xmath20 is an @xmath3-dimensional square diagonal matrix , with @xmath21 on the diagonal , @xmath20 @xmath22 @xmath23 in eq.([h ] ) @xmath24 is an @xmath25-dimensional matrix , which elements are the couplings between the ground and the excited states , @xmath26 is its hermitian conjugate    @xmath27 ,   \label{v - vectors}\end{aligned}\ ] ]    where @xmath28 @xmath29 are @xmath3-dimensional vectors , composed of the interaction energies of the @xmath30th state of the @xmath2 set with all states of the @xmath0 set,@xmath31    if the hamiltonian describes the atom - laser interaction , in the presence of weak magnetic field the degeneracy of the states is lifted . due to the external field we have an extra terms , which can be assumed as a perturbation to the exact ms solution . in this case we have @xmath32where @xmath33 is diagonal matrix with dimension @xmath34 and generally @xmath33 @xmath35 @xmath36 . the aim of this paper is to analyze the ms transformation in the case of weak deviation of the @xmath20 @xmath37 @xmath23 condition , which in necessary for factorization towards set of two - state systems and set of decouple states . some elements concerning the general solution of the ms transformation will be also discussed . we will start our discussion by a brief description of the exact morris - shore ( ms ) transformation . morris and shore @xcite have shown that any degenerate two - level system , in which all couplings share the same time dependence and the same detuning , can be reduced with a constant unitary transformation @xmath38 to an equivalent system comprising only independent two - state systems and uncoupled ( dark ) states , as shown in fig . [ fig - system ] . this time - independent transformation is given by @xmath39where the tildas denote the ms basis hereafter . the constant transformation matrix @xmath38 can be represented in the block - matrix form @xmath40 ,   \label{w}\]]where @xmath41 is a unitary @xmath1-dimensional square matrix and @xmath42 is a unitary @xmath3-dimensional square matrix , @xmath43 and @xmath44 . the constant matrices @xmath41 and @xmath45 mix only sublevels of a given level : @xmath41 mixes the @xmath46 sublevels and @xmath45 mixes the @xmath47 sublevels . the transformed ms hamiltonian has the form@xmath48 ,   \label{h - ms}\]]where @xmath49the @xmath50 matrix @xmath51 may have @xmath52 null rows ( if @xmath53 ) , which correspond to decoupled states . the decomposition of @xmath54 into a set of independent two - state systems requires that , after removing the null rows , @xmath55 reduces to a @xmath3-dimensional diagonal matrix ; let us denote its diagonal elements by @xmath56 ( @xmath57 ) . it follows from eq . ( [ m ] ) that    [ mm ] @xmath58hence @xmath41 and @xmath45 are defined by the condition that they diagonalize @xmath59 and @xmath60 , respectively . it is important to note that the square matrices @xmath61 and @xmath60 have different dimensions , @xmath1 and @xmath3 , respectively . because all elements of @xmath62 are constant , @xmath41 and @xmath45 are also constant ; the elements @xmath56 are constant too . it is straightforward to show that the @xmath3 eigenvalues of @xmath60 are all non - negative ; according to eqs . ( [ m ] ) and ( [ mm ] ) they are @xmath63 ( @xmath57 ) . the matrix @xmath59 has the same eigenvalues and additional @xmath64 zero eigenvalues . the ms hamiltonian ( [ h - ms ] ) has the explicit form    @xmath65 . \label{h - ms explicit}\ ] ]    the structure of the ms hamiltonian ( [ h - ms explicit ] ) shows that in the ms basis the dynamics is decomposed into sets of @xmath66 decoupled single states , and @xmath3 independent two - state systems @xmath67 ( @xmath68 ) , each composed of an @xmath46 state @xmath69 and a @xmath47 state @xmath70 , and driven by the rwa hamiltonians , @xmath71 \quad ( n=1,2,\ldots , n_{b } ) . \label{hn}\]]each of these two - state hamiltonians has the same detuning @xmath10 , but they differ in the couplings @xmath56 . each of the new @xmath46 states @xmath72 is the eigenstate of @xmath61 corresponding to the eigenvalue @xmath73 , whereas each of the new @xmath47 states @xmath70 is the eigenstate of @xmath60 , corresponding to the same eigenvalue @xmath73 . the square root of this common eigenvalue , @xmath74 , represents the coupling in the respective independent ms two - state system @xmath75 ( @xmath57 ) . the @xmath66 zero eigenvalues of @xmath59 correspond to decoupled ( dark ) states in the @xmath46 set ( we assume thoughout that @xmath76 ; therefore , dark states , if any , are in the @xmath46 set ) . the dark states are decoupled from the dynamical evolution because they are driven by one - dimensional null hamiltonians . the ms decomposition described above allows us to reduce the degenerate two - level problem to a set of @xmath66 nondegenerate two - state problems , wherein the detuning is unchanged while the couplings @xmath56 are combinations of the initial couplings between the @xmath46 and @xmath47 states and defined as the square roots of the eigenvalues of @xmath77 .    from the vector form ( [ v - vectors ] ) of @xmath78 we obtain    @xmath79 . \label{v+v}\end{aligned}\ ] ]    note that @xmath60 is the gramm matrix for the set of vectors @xmath80 . thus if all these vectors are linearly independent then @xmath81 and all eigenvalues of @xmath82 are nonzero @xcite ; however , this assumption is unnecessary . we assume that we can find the eigenvalues @xmath73 @xmath83 of the matrices ( [ vv+ ] ) and ( [ v+v ] ) , and the corresponding orthonormalized eigenvectors : the @xmath3 coupled eigenstates @xmath84 of @xmath59 and @xmath85 of @xmath60 , and the @xmath66 decoupled eigenstates @xmath86 of @xmath61 . we use these eigenstates to construct the transformation matrices as    @xmath87 , \quad \mathbf{b}=\left [ \begin{array}{c } \langle \widetilde{\psi } _ { 1}^{b}| \\ \vdots \\ \langle \widetilde{\psi } _ { n_{b}}^{b}|% \end{array}% \right ] . \label{a b}\ ] ]      the propagator the independent ms  two - state systems @xmath67 ( @xmath68 ) , defined by    @xmath88 = \widetilde{\mathbf{u}}_{n}(t_{f},t_{i})\left [ \begin{array}{c } \widetilde{c}_{n}^{a}(t_{i } ) \\ \widetilde{c}_{n}^{b}(t_{i})% \end{array}% \right ] , \ ] ]    is unitary and can be parameterized in terms of the cayley - klein parameters@xmath89here , @xmath90 and @xmath91 are the cayley - klein parameters and they obey the relation@xmath92      by taking into account the ms propagators ( [ un ] ) for the two - state ms systems , the ordering of the states , and the ms hamiltonian ( [ h - ms explicit ] ) , the propagator of the entire system in the ms basis is written as@xmath93 . \label{u - ms}\]]by taking into account the completeness relations    [ completeness ] @xmath94it is straightforward to show that the propagator in the original basis @xmath95 reads    @xmath96 . \label{u}\ ] ]    note that the propagator does not depend on the decoupled states @xmath97 @xmath98 , which are excluded by using eq . ( [ completeness a ] ) . this has to be expected because , owing to their degeneracy , the choice of the decoupled states is not unique , since any superposition of them is also a zero - eigenvalue eigenstate of @xmath59 . because the dynamics in the original basis must not depend on such arbitrariness , the propagator @xmath99 must not depend on the decoupled states at all . we    @xmath100    where @xmath101 is diagonal matrix , given by    @xmath102    @xmath103 ,   \label{d } \\ \mathbf{d}_{2}(t ) & = & \text{diag}\left [ d_{1}^{2}(t),d_{2}^{2}(t), .. ,d_{n_{b}}^{2}(t)\right ] , \end{aligned}\ ] ]    the main difficulty for finding exact solution is that in the ms basis we have@xmath104 \\ & = & \mathbf{s}(\mathbf{h}_{0}(t)+\mathbf{d}(t))\mathbf{s}^{\dagger } = \mathbf{h% } _ { 0}^{ms}(t)+\mathbf{sd}(t)\mathbf{s}^{\dagger } \end{aligned}\ ] ]    the matrices @xmath105 @xmath106 and @xmath107 @xmath108 are not diagonal this simply means in the ms basis the dynamics can not be decomposed into sets of @xmath66 decoupled single states , and @xmath3 independent two - state systems . hereafter for convenience we will assume @xmath109 and the notation @xmath110 will be used .    instead of exact factorization we look at solution that can be expressed in perturbative form . the hamiltonian in a block - matrix form is given by @xmath111 we will look for perturbative solution for the ms problem and for this reason we will have the following definition for the transformation matrix @xmath112    @xmath113    where the additional condition @xmath114 is imposed . this simply means that all terms from the series expansion in eq.([s - series ] ) are constant matrices . also in reason to simplify the calculations and without loss of generality the @xmath115 matrices have the block - diagonal form , similar to ( [ w])@xmath116 . \label{s - matrix - form}\ ] ]    according to the eq.([transformation ] ) and using the series expansion for the transformation matrix @xmath38 given in eq.([s - series ] ) we will obtain @xmath117    from the expression above is clear the meaning of the @xmath118 state vectors .    using the series expansion for the transformation matrix @xmath38 we will obtain the following expression for the hamiltonian in the new basis@xmath119 \left ( \sum_{i=0}\varepsilon ^{i}% \mathbf{s}_{i}^{\dagger } \right ) \\ & = & \mathbf{s}_{0}\mathbf{h}_{0}\mathbf{s}_{0}^{\dagger } + \varepsilon \left [ \mathbf{s}_{0}\mathbf{ds}_{0}^{\dagger } + \mathbf{s}_{1}\mathbf{h}_{0}\mathbf{% s}_{0}^{\dagger } + \mathbf{s}_{0}\mathbf{h}_{0}\mathbf{s}_{1}^{\dagger } % \right ]   \notag \\ & & + \varepsilon ^{2}\left [ \mathbf{s}_{0}\mathbf{ds}_{1}^{\dagger } + \mathbf{s}% _ { 1}\mathbf{ds}_{0}^{\dagger } + \right . \notag \\ & & \left . + \mathbf{s}_{1}% \mathbf{h}_{0}\mathbf{s}_{1}^{\dagger } + \mathbf{s}_{2}\mathbf{h}_{0}\mathbf{s% } _ { 2}^{\dagger } \right ] + o(\varepsilon ^{3 } )   \notag\end{aligned}\ ] ]    in the above expression for the hamiltonian only the first two terms in the power expansion of @xmath120 are given , but the reader can easily take more terms for this perturbative expansion .    let the matrix @xmath121 be chosen such that @xmath122and @xmath123 has the form    @xmath124 , \ ] ]    where @xmath51 is given by    @xmath125 .\ ] ]    the square matrix @xmath126 is diagonal , @xmath127\ ] ]    this choice for the matrix @xmath121 corresponds to neglecting the perturbative term @xmath128    having in mind that the matrix @xmath121 is computed , using eq.(h - msbasis - eps ) we can continue and take the corrections up to first order of @xmath120 by choosing matrix @xmath129 from the equation    @xmath130    using the block - matrix forms for @xmath131 @xmath132 @xmath133 and @xmath129 matrices and writing the third term to the rhs of the equation the above expression reads    @xmath134 + \left [ \begin{array}{cc } \mathbf{0 } & \mathbf{a}_{0}\mathbf{\mathbf{v}b}_{1}^{\dagger } \\ \mathbf{b}_{0}\mathbf{\mathbf{v}}^{\dagger } \mathbf{a}_{1}^{\dagger } & \mathbf{b}_{0}\mathbf{\delta b}_{1}^{\dagger } % \end{array}% \right ] = -\left [ \begin{array}{cc } \mathbf{0 } & \mathbf{0 } \\ \mathbf{0 } & \mathbf{b}_{0}\mathbf{db}_{1}^{\dagger } % \end{array}% \right ]   \label{s1-matrix - form}\ ] ]    the above block - matrix equation is equivalent to the system of matrix equations@xmath135    the first and the second equation from the above system are equivalent under the hermitian conjugate operation . we want to note that from eq.([s0-def ] ) the @xmath136 and @xmath137 matrices are known . hence the system of equation given by eq.([s1-b ] ) and eq.([s1-c ] ) , can be solved for @xmath138 and @xmath139 matrices , which defines the transformation matrix @xmath140    before we continue with @xmath141 matrix computation we would like to simplify the matrix equation eq.([s1-c ] ) . remember that the matrix @xmath142 @xmath22 @xmath23 , one can use the commutation relations @xmath143 and @xmath144 , and the unitary properties @xmath145 and @xmath139  to transform eq.([s1-c ] ) . @xmath146    using the unitary properties for the matrices @xmath137 and @xmath147  for eq.([s1-c ] ) the system of matrix equation ,  which defines the transformation matrix @xmath129 can be written in the form    @xmath148    the algorithm for deriving expression for the @xmath141 matrix is similar to this for @xmath129 matrix . having in mind that the matrices @xmath121 and @xmath129 are computed , we can continue and take the corrections up to @xmath149 by choosing matrix @xmath150 from the equation    @xmath151    by similar methods to the presented for the matrix @xmath141 , one can derive the higher order correction i.e. , find matrices @xmath152 @xmath153 etc . above , we described the method for deriving solution to the generalized ms problem , where the evolution of a two - level quantum system is subjected to pulsed external field and in the general case the ground and excited levels have arbitrary level of degeneracy , @xmath1 and @xmath3 respectively . in this section , we will illustrate these results with a specific example : when the @xmath2 set consists of two degenerate states , i.e. @xmath155 this case is interesting , because of the possible realizations of our model in different real physical systems . moreover , this special case has an exact analytical solution , which is very rare for multidimensional systems . we label the ground states @xmath5 @xmath156 where @xmath157 is arbitrary and we denote the two excited states @xmath158 and @xmath159 . in this case the interaction operator @xmath78 ( [ v ] ) has @xmath157 columns and two rows , and its explicit form is given by      where @xmath161 and @xmath162 are @xmath157-dimensional vectors comprising the couplings between the ground states and the corresponding excited state@xmath163 , ~~~\left\vert \omega ^{\prime \prime } \right\rangle = \left [ \begin{array}{c } \omega _ { 1}^{\prime \prime } \\ \omega _ { 2}^{\prime \prime } \\ \vdots \\ \omega _ { n}^{\prime \prime } % \end{array}% \right ] .\]]we write the product of @xmath164@xmath165 , \]]with eigenvalues , @xmath166 ,   \label{eigenvalues}\]]where @xmath167 denotes            \begin{array}{c } \left\langle \omega ^{\prime } |\omega ^{\prime \prime } \right\rangle \\ \lambda _ { -}-\left\vert \omega ^{\prime } \right\vert ^{2}% \end{array}% \right ] = \left [ \begin{array}{c } e^{i\sigma } \cos \theta \\ -\sin \theta% \end{array}% \right ] ,   \label{b-}\end{aligned}\]]with @xmath175 and @xmath176 normalization factors . the next step is to find the householder vectors @xmath177 and @xmath178 which are the ground states in the ms  two - state systems and define the propagator @xmath179 they are the eigenstates of the @xmath157-dimensional matrix ,      corresponding to the same non - zero eigenvalues ( [ eigenvalues ] ) . we construct them as superpositions of the interaction vectors with coefficients @xmath181 @xmath182 and @xmath183 @xmath184we determine these coefficients from the eigenvalue equations , @xmath185and from the normalization conditions , @xmath186as a result we obtain@xmath187    the results given in equations eq.([b+ ] ) , eq.([b- ] ) , eq.([a+ ] ) and eq.([a- ] ) define the block - structure of the @xmath121 matrix via eq.([s - matrix - form ] ) . using eq.([b+ ] ) , eq.([b- ] ) the @xmath145 matrix reads                    c.p . williams and s.h . clearwater , _ explorations in quantum computing _ , ( springer - verlag , berlin , 1997 ) ; a. steane , rep . . phys . * 61 * , 117 ( 1998 ) ; m.a . nielsen and i.l . chuang , _ quantum computation and quantum information _ ( cambridge university press , cambridge , 2000 ) . k. bergmann , h. theuer , and b.w . shore , rev . phys . * 70 * , 1003 ( 1998 ) ; n.v . vitanov , t. halfmann , b.w . shore , and k. bergmann , ann . phys . chem . * 52 * , 763 ( 2001 ) ; n.v . vitanov , m. fleischhauer , b.w . shore and k. bergmann , adv . at . . phys . * 46 * , 55 ( 2001 ) . s. stenholm , in _ frontiers of laser spectroscopy _ , les houches summer school session xxvii , edited by r. bailian , s. haroche and s. liberman ( amsterdam , north holland , 1975 ) , p. 399 ; r. lefebvre and j. savolainen , j. chem . phys . * 60 * , 2509 ( 1974 ) ; m. bixon and j. jortner , j. chem . phys . * 48 * , 715 ( 1968 ) .
the aim of this paper is to analyze the morris - shore ( ms ) transformation in the case of weak deviation of the condition for equal detunings . for this generalization of the ms transformation a perturbative solution 
 is derived . 
 some elements concerning the general solution of the ms transformation are also discussed .