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avoided crossing of energy levels is a universal phenomenon for the quantum non - integrable systems where the symmetry break leads to the splitting of degenerate energy levels forming a tiny energy gap . around the avoided crossing point of the two levels the landau - zener tunnelling ( lzt ) model provides an effective description for the tunnelling dynamics under assumption that the energy bias of two levels undergoes a linear change with time@xcite . it is a basic model in quantum mechanics and has versatile applications in quantum chemistry @xcite , collision theory @xcite , and more recently in the spin tunnelling of nanomagnets wensdorf1,wensdorf2 , bose - einstein condensates ( bec ) @xcite and quantum computing @xcite , to name only a few . lzt model has been extended to many versions taking diverse physical conditions into account : lzt problem with a time - varied sweeping rate@xcite , lzt model with a fast noise from the outer environment @xcite , lzt model with periodic modulation @xcite , and so on . among them , lzt in a nonlinear two - level system is one of most interesting models and attracts much attention recentlyraghavan , bwu , zobay , jliu1 . in this model , the level energies depend on the occupation of the levels , may arise in a meanfield treatment of a many - body system where the particles predominantly occupy two energy levels . the nonlinear lzt model not only demonstrate many novel behavior of great interest in theory but also has important applications in spin tunnelling of nanomagnets @xcite and a bose - einstein condensate in a double - well potential @xcite or in an optical lattice @xcite . however , since most of the problems of interests involve more than two energy levels , with transitions between several levels happening simultaneously@xcite , for example , becs trapped in multiple wells@xcite , spin tunnelling of nanomagnets with large spin , etc . it is naturally desirable to extend the above nonlinear tunnelling to the multi - level situation .    in present paper , we consider the simplest multi - level system  three - level system , to investigate its complicated tunnelling dynamics in the presence of nonlinearity . because quantum transitions may happen between several levels simultaneously , the lzt in the nonlinear three - level model show many striking properties distinguished from that of the two - level case . in the adiabatic limit we will show that , for a very small nonlinear parameter that the energy levels still keep the same topological structure as its linear counterpart , the adiabaticity breaks down manifesting the presence of a nonzero tunnelling probability . this is quite different from the two - level case , where the break down of the adiabaticity is certainly accompanied by a topological change on the energy levels . more interestingly , the tunnelling is irregular with showing an unresolved sensitivity on the sweeping rate , a phenomenon attributed to the existence of chaotic state . in the sudden limit , we derive an analytic expression for the tunnelling probability under stationary phase approximation and show that the nonlinearity can dramatically influence the tunnelling probability at the resonance between the nonlinear `` internal field '' and the external field . we also discuss the asymmetry of the tunnelling probability induced by the nonlinearity . the physical mechanism behind these phenomena is revealed and possible application of our model to triple - well trapped bose - einstein condensate is discussed . the paper is organized as follows . in sec.ii we introduce our nonlinear three - level lzt model and calculate its adiabatic levels . section iii discusses lzt among the levels . section iv gives a possible application of the model to the triple - well trapped bec . we consider following dimensionless schrdinger equation @xmath0 with the hamiltonian given by @xmath1where @xmath2 is the coupling constant between the neighboring levels ; @xmath3 is the nonlinear parameter ; the energy bias @xmath4 is supposed to be adjusted by a linearly external filed , i.e. , @xmath5 , @xmath6 is the sweeping rate ; @xmath7 , @xmath8 , @xmath9 is probability amplitude in each level and the total probability @xmath10 is conserved and set to be unit . when the nonlinear parameter vanishes , our model reduces to the linear case and the adiabatic energy levels @xmath11 ( fig.1(a ) ) derived by diagonalizing the hamiltonian ( 2 ) . tunnelling probability @xmath12 @xmath13 is defined as the occupation probability on the m - th level at @xmath14 for the state initially on the n - th level at @xmath15 . for the linear case , the above system is solvable analytically and the tunnelling probabilities can be explicitly expressed as @xcite @xmath16 ^{2 }   \label{euqation3}\ ] ] @xmath17   \label{euqation4}\ ] ] @xmath18 @xmath19 ^{2 }   \label{euqation6}\ ] ] the others are @xmath20 due to the symmetry of the levels .    with the presence of the nonlinear terms , we want to know how the tunnelling dynamics in the above system is affected . in our discussions , the coupling parameter is set to be unit , i.e. , @xmath21 . therefore , weak nonlinear case and strong nonlinear case mean that @xmath22 and @xmath23 , respectively . as to the external fields , we will consider three cases , namely , adiabatic limit , sudden limit , and moderate case , corresponding to @xmath24 , @xmath25 and @xmath26 , respectively . similar to the linear case , we need to analyze the adiabatic levels of the nonlinear model first . with @xmath27 , we introduce the relative phase @xmath28 . in terms of @xmath29 and @xmath30 , the nonlinear three - level system is casted into a classical hamiltonian system , @xmath31 @xmath32 @xmath29 and @xmath30 are two pairs of canonically conjugate variables of the classical hamiltonian system . the fixed points of the nonlinear classical hamiltonian correspond to the eigenstates of the nonlinear three - level system , and are given by the following equations :    @xmath33    @xmath34    @xmath35    @xmath36    by solving the equations ( [ euqation7])-([equation 10 ] ) the eigenstates of the system are obtained . accordingly , the eigenenergy is obtained by @xmath37 i.e. , the energy levels are gained as shown in fig.1 .    for weak nonlinearity , the levels structure is similar to its linear counterpart ( fig.1(b ) ) . for strong nonlinearity ( fig.1(c ) ) , in the mid - level a double - loop topological structure emerges and in the upper - level a butterfly structure appears . because of these topological distortions on the energy levels , we expect that the tunnelling dynamics will dramatically change . in this section we study lzt in the nonlinear three - level system both numerically and analytically . first , we consider two limit cases : adiabatic limit , sudden limit , respectively . then we will discuss the tunnelling probability in general case and investigate the symmetry of the tunnelling probability .      in adiabatic limit , the characters of the tunnelling probabilities should be entirely determined by the topology of the energy levels and the eigenstates properties ( corresponding to the stability of the fixed points in classical hamiltonian system ) , according to the adiabatic theorem@xcite . so , we expect that , for the weak nonlinearity case , an initial state started from any levels ( upper , mid or lower ) will follow the levels and evolves adiabatically , as a result , no quantum transition between levels occurs ; for the strong nonlinearity , an initial state from the lower level is expected to evolve adiabatically keeping stay on the ground state , leading to zero adiabatic tunnelling probability , whereas for the state initially from the mid or upper level , due to the topological change of the level , it can not move smoothly from left - side to the right - side . transition to other levels happens at the tip of the loop or butterfly . consequently , the adiabatic tunnelling probability is expected to be nonzero . however , the above picture is only partly corroborated by our directly solving the schrdinger equation using forth - fifth order runge - kutta adaptive - step algorithm , as shown in fig.2 . on the one hand , fig.2 clearly shows that , for the strong nonlinearity case , as we expect , no tunnelling for the state from the lower level , but a serious adiabatic tunnelling is observed for the states from upper two levels . in particular , we find that the tunnelling probability as a function of the sweeping rate shows an irregular oscillation . we associate this irregularity to the chaotic state . to demonstrate it , we plot in fig.3 the poincare section of the trajectories for @xmath39 before and after the tip of the butterfly structure of the upper level in fig.1c . it shows that , before the tip , the eigenstate corresponds to the fixed point surrounded by quasi - periodic orbit , therefore is stable . as the state evolves to the right tip of the butterfly , it contact with chaotic sea , after that the state become chaotic . the characteristics of the chaos is sensitive on the parameters , therefore the chaotic state is responsible for the irregular tunnelling probability exposed by fig.2h , i .    on the other hand , fig.2 also shows that for the weak nonlinearity , even though the adiabatic levels keeps the same topological structure as the linear case , there is still nonzero tunnelling probability for the state started from the mid - level . the tunnelling also shows some kind of irregularity . this phenomenon counter to our naive conjecture from observing the topological structure of the adiabatic levels .    to explain this unusual phenomenon , we need make detailed analysis on the property of the fixed points of the classical system hamiltonian(7 ) , corresponding to the eigenstates of the mid - level . we plot quantity @xmath40 as the function of @xmath41 in fig.4 ( a , b ) , we see the adiabatic evolution of the eigenstate breaks down around @xmath42 due to the nonlinearity ( fig.4 ( b ) ) . this adiabaticity breakage is caused by the change on the property of the fixed point corresponding to the eigenstate of the mid - level . this is revealed by investigating the hamiltonian - jaccobi matrix obatined by linearizing the nonlinear equations ( [ euqation7])-([equation 10 ] ) at fixed points , @xmath43we solve the eigenvalues of @xmath44 for different @xmath41 and plot our results in fig.4c . these eigenvalues can be real , complex or pure imaginary . only pure imaginary eigenvalues corresponds to the stable fixed point , others indicate the unstable ones . in fig.4 ( c),we can see the eigenvalues are complex number ( i.e. , their real parts are not zero ) around @xmath45 . the corresponding fixed points are unstable . for other regions , the eigenvalues of @xmath44 are pure imaginary . therefore , even though no topological structure changes on the level structures , the instability of the fixed point corresponding to the mid - level leads to the breakdown of the adiabaticity manifesting the irregular nonzero tunnelling probability exposed by fig.2e in the adiabatic limit . the above instability mechanism occurs for any smaller nonlinear perturbation . let us make some analytic deduction as follows . note that the fixed points of equations ( [ euqation7])-([equation 10 ] ) can be accurately calculated if @xmath46 @xmath47 @xmath48 @xmath49 for @xmath50 and @xmath51 @xmath52 for @xmath53 . by employing the perturbation theory using @xmath3 as small parameter , we can get the fixed points for small @xmath54 @xmath55 @xmath56 @xmath48 @xmath57 for nonlinear case . substituting them into equation ( [ euqation9 ] ) , we can obtain the eigenvalues of @xmath44 by solving the following quartic equation : @xmath58the useful quadratic discriminant is @xmath59 in linear case , @xmath60 , @xmath61 is always larger than zero , which means that the solutions for @xmath62 are pure imaginary , thus the fixed points are stable . for small @xmath3 , @xmath63 the real part of the solutions @xmath64 while the imaginary part @xmath65 as a result , the fixed point corresponding to mid - level becomes unstable around @xmath66 for any small nonlinearity , implying the break down of the adiabatic evolution of states on the mid - level . the sudden limit corresponds to nonadiabatic lzt . the tunnelling probability does not relate much to the structure of the levels . in this limit a weak nonlinearity does not affect the tunnelling probability , however , a strong nonlinearity can dramatically influence the tunnelling dynamics .    in this limit , we can derive the analytical expression of the tunnelling probabilities using the stationary phase approximation ( spa ) . as a demonstration , we concentrate on the mid - level , i.e. to calculate @xmath68 which equals to @xmath69 . because of the large sweeping rate @xmath70 , a quantum state would stay on the initial level most of the time . thus the amplitudes @xmath7 and@xmath71 in the schrdinger equation ( 1 ) remain small and @xmath72 all the time . a perturbation treatment of the problem becomes adequate . we begin with the variable transformation , @xmath73,\]]@xmath74,\]]@xmath75.\]]as a result , the diagonal terms in hamiltonian are transformed away , and the evolution equations of @xmath76 , @xmath77 , @xmath78become : @xmath79\]]@xmath80 \\ & & -\frac{v}{2i}a_{3}^{^{\prime } } \exp [ i\int_{0}^{t}\left ( \frac{\gamma } { 2}+% \frac{c}{4}(\left| a_{2}\right| ^{2}-\left| a_{3}\right| ^{2})\right ) dt]\end{aligned}\]]@xmath81\ ] ]    we need to calculate the above integrals self - consistently . due to the large @xmath70 , the nonlinear term in the exponent generally gives a rapid phase oscillation , which makes the integral small . the dominant contribution comes from the stationary point @xmath82 of the phase around which we have @xmath83.\]]@xmath84with @xmath85 _ { t_{0}}.\]]since @xmath86 , then we have @xmath87 \right| ^{2}\]]this expression can be differentiated and evaluated its result at time @xmath88 . a standard fresnel integral with the result @xmath89 _ { t_{0}}=\left ( \frac{v}{2}\right ) ^{2}\sqrt{% \frac{\pi } { \alpha _ { 1}}}$ ] is obtained . combining this with the relation ( 15 ) , we come to a closed equation for @xmath90 , @xmath91for given @xmath70 , @xmath3 , @xmath2 , @xmath90 in the above equation can be obtained . the tunnelling probability @xmath92 the alliance of eq.(20)and ( 21 ) gives the analytic expression on the tunnelling probability @xmath93 in the sudden limit . compared with our numerical simulation it shows good agreement at @xmath94 , @xmath95 a clear deviation is observable ( fig.5a ) . it is due to the resonance between the `` internal field '' and the external field leads to the invalidity of our assumption @xmath96 , as we show latter . similarly , to calculate @xmath97 , we consider following equation , @xmath98\]]@xmath99@xmath100    differently , in this case we may have three stationary phase points that are solutions of equation ( 24 ) when @xmath101 , but only one solution otherwise , as demonstrated in fig.6 . we denote them as @xmath102 , @xmath103 , @xmath104 from smallest to largest . for small c , @xmath102 is around @xmath105 , and the other two solutions locates at the two sides of the origin . in this case , we simply take @xmath106 .    then @xmath107.\ ] ]    the tunnelling probability @xmath108 the alliance of the eq.(24,26 ) will give the approximate solution of the @xmath97 . compared with our numerical simulation it shows a good agreement at @xmath109 , whereas for @xmath110 a clear deviation is observed ( fig.5b ) .    what happens around @xmath111 that leads to the break down of our stationary phase approximation ? the reason is the resonance between the `` internal field '' and the external field . let us recall the exponent in the integrand of eq.(22 ) , we find the effective sweeping rate should be the difference between the change rate of the `` internal field '' ( i.e. , @xmath112 ) and the sweeping rate of the external field . at @xmath111 , we find the two frequencies become almost identical , leading to the invalidity of spa assumption of rapid phase oscillation . this resonance accompanied by the bifurcation of the stationary phase points . crossing @xmath111 we observe the number stationary phase points changes from three to one , as shown in fig.6 . the resonance breaks the spa leading to serious transition from level 2 to level 1 , consequently , at @xmath95 , our assumption @xmath96 become invalid , and our approximation on the @xmath93 from spa is no longer good as shown in fig.5a . the nonlinear tunnelling probability as the function of the two scaled quantities @xmath113 and @xmath114 , show many unusual properties . taking mid - level tunnelling @xmath68 for example , we make a large numerical exploration for a wide range of parameters , to demonstrate the general property of the nonlinear tunnelling probability in fig.7 . in general , increasing the sweeping rate will reduce the probability of tunnelling to upper or lower level and the positive nonlinearity usually suppresses the probability of the state s staying in the mid - level , because that the nonlinearity with positive @xmath3 can be regarded as a kind of repulsive potential . this repulsive self - interaction make particle tend to transition to lower level more easily , and this transition becomes more serious at the occurrence of the resonance between the `` internal field '' and external field . the occurrence of the resonance is clearly exposed by the boundary between the white regime and dark regime in fig.7 . in the white regime , due to the resonance , the nonlinearity dramatically changes the tunnelling probability . the other issue we want to address is the symmetry . the nonlinearity makes levels deform and therefore break the symmetry between upper level and lower level , consequently , the relations @xmath115 hold in linear case break in presence of the nonlinearity . for our three - level system , the symmetry breaking is clearly exposed by fig.8 of showing the tunnelling probability @xmath12 as the functions of @xmath116 for @xmath39 , @xmath117 . in the linear case , we have @xmath118 , however , with the presence of the nonlinear , @xmath119 increases whereas the @xmath120 decreases . the similar things happen for the @xmath121 and @xmath122 . the above symmetry breaking may be observed experimentally@xcite . in conclusion , we have made a comprehensive analysis of the landau - zener tunnelling in a nonlinear three - level system , both analytically and numerically . many novel tunnelling properties are demonstrated and behind dynamical mechanism is revealed .    our model can be directly applied to the triple - well trapped bec and to explain the tunnelling dynamics between the traps @xcite . in a triple - trap @xmath123 , a bec is described by gross - pitaevskii equation ( gpe ) @xmath124 \psi ( r , t ) $ ] under the mean - field approximation , where @xmath125 , @xmath126 the atomic mass and @xmath127 the scattering length of the atom - atom interaction . the wave function @xmath128 of gpe is the superposition of three wave functions describing the condensate in each trapraghavan , i.e. , @xmath129 . when we study the tunnelling of three weakly coupled bec in traps @xmath130 , @xmath131 and @xmath132 , the dynamics of the system is described by the nonlinear schdinger equation with the hamiltonian , @xmath133where @xmath134 @xmath135 is the ground state energy for each trap . @xmath136 @xmath135 stands for atom - atom interaction , i.e. , nonlinear parameter . @xmath137 is the coupling matrix element between trap @xmath130 and @xmath131 . @xmath138 is the coupling matrix element between trap @xmath131 and @xmath132 . for simplicity , we only consider the case that these two coupling matrix elements are the same and there is no coupling between trap @xmath130 and @xmath132 , i.e. , @xmath139 , @xmath140 . the energy bias can be adjusted by tilting the trapping well and the nonlinearity can be adjusted by the feshbach resonance technique . we hope our theory will stimulate the experiment in the direction . this work was supported by national natural science foundation of china ( no.10474008,10445005 ) , science and technology fund of caep , the national fundamental research programme of china under grant no . 2005cb3724503 , the national high technology research and development program of china ( 863 program ) international cooperation program under grant no.2004aa1z1220 . w. wernsdorfer , s. bhaduri , c. boskovic , g. christou , and d. n. hendrickson , phys . b * 65 * , 180403 ( 2002 ) ; w. wernsdorfer , r. sessoli , a. caneschi , d. gatteschi , and a. cornia , europhys . lett . * 50 * , 552 ( 2000 ) .
we present a comprehensive analysis of the landau - zener tunnelling of a nonlinear three - level system in a linearly sweeping external field . 
 we find the presence of nonzero tunnelling probability in the adiabatic limit ( i.e. , very slowly sweeping field ) even for the situation that the nonlinear term is very small and the energy levels keep the same topological structure as that of linear case . 
 in particular , the tunnelling is irregular with showing an unresolved sensitivity on the sweeping rate . for the case of fast - sweeping fields 
 , we derive an analytic expression for the tunnelling probability with stationary phase approximation and show that the nonlinearity can dramatically influence the tunnelling probability when the nonlinear `` internal field '' resonate with the external field . 
 we also discuss the asymmetry of the tunnelling probability induced by the nonlinearity . 
 physics behind the above phenomena is revealed and possible application of our model to triple - well trapped bose - einstein condensate is discussed .