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coherent conversion of an atomic to a molecular bose - einstein condensate ( bec ) is a first step towards ` superchemistry ' @xcite , which is the stimulated emission of molecules in a chemical reaction . a number of studies of this @xcite have shown that direct conversion via raman photoassociation @xcite appears feasible , based on stimulated free - bound and bound - bound transitions in the presence of two laser fields of different frequencies @xcite . here pairs of atoms from the two - atom continuum of the ground electronic potential are transferred  via an excited bound molecular state  to a bound molecular state of a lower energy in the ground potential . raman photoassociation allows coupling to a single molecular state , which can be selected by the raman laser frequencies . practical estimates using available lasers and transitions indicate that coherent transfer may be limited by spontaneous emission from the intermediate molecular excited electronic state . another mechanism that can result in coupled atomic - molecular bec systems @xcite is based on feshbach resonances @xcite . however , realistic analysis and experimental implementations @xcite indicate that the loss processes due to inelastic atom - molecule collisions occur at a significant rate .    a possible route towards minimizing losses and decoherence from spontaneous emission in photoassociation is stimulated raman adiabatic passage ( stirap ) @xcite , in which a counter - intuitive pulse sequence is used , where the first input pulse couples the molecular levels - even when there are no molecules present . in this situation , a dark superposition state is formed , due to interference effects between the atomic and molecular electronic ground states . this minimizes the probability of a real transition to the molecular excited state , and hence reduces spontaneous emission . previous analyses of this problem have not taken into account losses , collisions , or the full three - dimensional structure of the two bose condensates in a trap .    in this paper , we provide an analysis which is relevant in a physically appropriate model that does include the known physics of spontaneous emission losses , @xmath0-wave scattering processes and spatial diffusion of the condensates . the result is that the stirap process appears feasible at high laser pulse intensities , provided the rabi frequency is much greater than the two - photon detuning due to mean - field interactions . we give a detailed numerical calculation based on both homogeneous and inhomogeneous three - dimensional mean - field ( gross - pitaevskii type ) theories , including couplings to non - resonant intermediate levels , and show how the results scale with the two - photon detuning , pulse duration and raman pulse intensities . an optimal situation is found by considering an off - resonance operation and different effective rabi frequencies in the two raman channels . we show that these strategies can greatly enhance the conversion efficiency for given laser intensities , thus making the experimental requirements much more feasible . we start by considering the theory of coherently interacting atomic and molecular condensates needed to describe this process @xcite , and assume a specific coupling mechanism based on stimulated free - bound raman transitions @xcite , in which two atoms of energy @xmath1 collide to from a molecule of energy @xmath2 , with an excited molecular state forming as an intermediate step . the raman coupling is induced by two laser fields of frequencies @xmath3 and @xmath4 , and becomes resonant when the raman detuning @xmath5 goes to zero . this allows coupling to a single molecular state , which can be selected by the raman laser frequencies . we derive the atom - molecule coupling for a simplified model of the two - body interaction @xcite , in which the atoms interact in their electronic ground state through a single born - oppenheimer potential @xmath6 . molecules are formed in a single bound vibrational state of energy @xmath2 with radial wave function @xmath7 . two free atoms with zero relative kinetic energy have a total energy @xmath8 , and a relative radial wave function @xmath9 , normalized so that asymptotically @xmath10 . we assume that the laser field has two frequency components , with @xmath11   $ ] , @xmath12 . each couples the ground electronic state to a single electronically excited state described by a potential @xmath13 , with ` bare ' electronic rabi frequencies @xmath14 , where @xmath15 is the molecular electric dipole matrix element with a nuclear separation of @xmath16 . the excited state has vibrational levels @xmath17 with energies @xmath18 and radial wave functions @xmath19 . all bound levels are normalized so that @xmath20 . we consider the case of near resonant transitions @xmath21 and denote the resonant excited vibrational level @xmath22 via index @xmath23 ( see fig . [ diagram ] ) . the usual quantum field theory hamiltonian @xcite for noninteracting atomic ( @xmath24 ) and molecular ( @xmath25 ) species , in well - defined internal states described by annihilation operators @xmath26 , is given by : @xmath27 .\ ] ] here @xmath28 ( @xmath29 ) are the masses , @xmath30 , and @xmath31 is the trapping potential including the internal energy for the @xmath32-th species , where we define @xmath33 . including @xmath0-wave scattering processes and laser induced particle inter - conversion , together with the assumption of a momentum cut - off , results in the following terms in the effective interaction hamiltonian : @xmath34 , \\ \hat{h}_{int}^{(2 - 3 ) } & = & \int d^{3}{\mathbf{x}}\left [ \frac{-\hbar { \omega } _ { 2}}{2}e^{-i\omega _ { 2}t}\hat{\psi } _ { 2}^{{}}({\mathbf{x}})\hat{\psi } _ { 3}^{\dagger } ( { \mathbf{x}})+h.c.\right ] .\end{aligned}\ ] ] here @xmath35 ( @xmath12 )  are the molecular rabi frequencies . these can be treated using the franck - condon overlap integrals @xmath36 , if we take @xmath37 as the mean electronic rabi frequency . we note that @xmath38 , which connects the atomic and molecular condensates , has units of @xmath39 , and must be multiplied by the atomic condensate amplitude to obtain a true rabi frequency . the couplings @xmath40 in the diagonal case are given by @xmath41 , where @xmath42 is the @xmath32-th species @xmath0-wave scattering length , while the non - diagonal terms are given by @xmath43 , where @xmath44 is the inter - species scattering length and @xmath45 is the reduced mass . in addition , we account for losses from each state , at a rate @xmath46 . the resulting heisenberg equations for the field operators are treated within the mean - field approximation , in which the operators are replaced by their mean values , and a factorization is assumed . this approximation is expected to be valid at sufficiently high density . corrections due to quantum correlations @xcite have been treated in greater detail elsewhere @xcite . next , we introduce rotating frame detunings , defined so that : @xmath47 in the case of uniform condenstates , @xmath31 are equal to @xmath48 , and @xmath49 . this results in the following set of gross - pitaevskii type of equations for the mean - field amplitudes , in rotating frames such that @xmath50 $ ] : @xmath51 here @xmath52 is the @xmath32 -th gross - pitaevskii mean - field detuning in the rotating frame , defined so that : @xmath53 we also introduce the two - photon laser detuning at trap center : @xmath54    in addition to losses due to spontaneous emission from the electronic excited states , rotationally or vibrationally inelastic atom - molecule collisions may also give rise to losses . the magnitude of these rates is presently unknown , and we neglect them here . we note that these rates should decrease rapidly with increasing molecular binding energy and go to zero in the molecular ground state , so that it should be possible to obtain a very low rate by selecting the coupling to a deeply bound molecular level . this approximation means that we will set @xmath55 in the following treatment , where @xmath56 is the kronecker delta - function . the simplest stirap scheme employs exact tuning of the laser frequencies to the bare state resonances , i.e. , @xmath57 . this , however , is not necessarily required as the stirap can still occur with detuned intermediate levels . moreover , as we show below , in bec environments , the off - resonance operation turns out to be more efficient if the detunings compensate for the phase shifts due to mean - field energies . in effect this is equivalent to a renormalized two - photon `` on - resonant '' operation in which @xmath58 .    if , instead of considering near - resonance coupling , we consider a large intermediate level detuning so that the excited state can be adiabatically eliminated , we recover the basic terms in the set of equations analyzed in @xcite . simultaneously , this would give us the previously known result @xcite for the laser induced modification to the scattering length @xmath59 occurring in a simple single - laser photoassociation of pairs of atoms , thus justifying the above form of the interaction hamiltonian . in the present paper , however , we assume that the detunings @xmath60 are small compared to the characteristic separation between the vibrational levels so that all other vibrational levels can be neglected . the near - resonant excited level is treated explicitly , rather than eliminated adiabatically as in @xcite . before carrying out simulations of the full @xmath61 equations in a trap , it is instructive to start with a simplified version of the theory - in which there are no kinetic energy terms . we expect this approximation to be valid in the thomas - fermi limit of large , relatively dense condensates , which is a regime of much current experimental interest . this is described by the following set of equations : @xmath62 where we have introduced as an effective thomas - fermi limit frequency shift , @xmath63 this corresponds to the treatment given in @xcite , except that we explicitly include the loss term @xmath64 due to spontaneous emission , the @xmath0-wave scattering processes due to @xmath40 , and the franck - condon integrals into the coupling coefficients for the free - bound and bound - bound transitions . in order to understand how this is related to the usual stirap technique in a three - level @xmath65 atomic system , we introduce a new wave - function @xmath66 , which corresponds to the coherent amplitude of a ( virtual ) molecular condensate with the same number of atoms as in the atomic bec . we then introduce a bose - stimulated rabi frequency , which includes a local coherent bec amplitude for the first free - bound transition : @xmath67 . this leads to the equations : @xmath68 these are precisely the usual stirap equations , except with additional detunings coming from the mean - field interactions , and a rabi frequency in the first transition which is proportional to the amplitude of the atomic bec wave - function . in practise , the rabi frequency may have an additional space - dependence due to the spatial variation of the laser phase and intensity . we therefore conclude that , provided we can satisfy the normal adiabatic stirap requirements of slow time - variation in the _ effective _ rabi frequencies in the above equations , the technique will also work for a bec . this is a simpler proof than previously @xcite . in particular , we can immediately deduce the expected solution for real rabi frequencies in the adiabatic limit . in order to have @xmath69 , one requires : @xmath70    here the space dependent mixing angle @xmath71 is obtained from the ratio of effective rabi frequencies : @xmath72 ^{1/2}\ , .\ ] ]    we can see that _ initially _ , while @xmath73 is still close to its initial value , the mixing angle is close to its expected value in normal stirap , since @xmath74 . however , at the final stages of the adiabatic passage , the nonlinear effects due to the atom - molecular coupling become important . as the atomic bec amplitude only varies on the time - scale of the input fields , the nonlinear atom - molecular coupling term by itself should not introduce new adiabatic restrictions . however , there will be time - dependent detunings introduced by the mean - field terms . finally , we can see that similar conclusions can also be reached in a non - uniform bec , by replacing the uniform detunings with appropriate gross - pitaevskii detunings , that include the spatial potentials . in a typical bec cooled in the thomas - fermi regime , we expect the kinetic energy terms to have relatively small effects . stirap can therefore be implemented as usually by using two laser pulses applied in counterintuitive order . we choose gaussian pulses of the form @xmath75,\ ; \ ; \ ; \ ; ( i=1,2),\ ] ] or @xmath76,\ ] ] where the peak values are related as follows : @xmath77 . the pulse at frequency @xmath4 is applied first , with the center at @xmath78 , while the second pulse at frequency @xmath3 is delayed by @xmath79 , i. e. , @xmath80 where @xmath81 is the delay coefficient , and @xmath82 is the pulse duration , which we assume is the same for both pulses .    in terms of the rabi frequencies , the adiabatic condition for stirap now reads as @xcite : @xmath83 where @xmath84 is the rms rabi frequency , @xmath85 is the duration during which the pulses overlap , and @xmath86 simply corresponds to the detuning of the single - photon transition . however , there is a second condition , which is often not stated explicitly . this is that stirap requires an effective two - photon resonance , to avoid dephasing between the initial and final states in the dark - state superposition . the two - photon resonance condition is different from the usual stirap case , since a detuning of @xmath87 causes a phase rotation _ both _ in _ @xmath88 _ and in _ @xmath89 _ as well , since this also includes a phase term from the condensate . as a result , the necessary condition for two - photon resonance is therefore : @xmath90    as is usually the case in stirap , these conditions can not be satisfied very early or late in the pulse sequence , when the rabi frequencies are small ; but they should be satisfied over most of the stirap interaction , and over most of the condensate volume . for simplicity , we will apply these conditions to the peak rabi frequency @xmath91 , and to the total pulse duration @xmath82 . further , since @xmath89 is itself a function of the stirap evolution , we introduce an effective first rabi frequency , defined in terms of the initial density @xmath92 . this is an upper bound to the stimulated rabi frequency ; thus @xmath93 ( and sometimes we write @xmath94 , for uniformity ) , where @xmath95,\ ; \ ; \ ; \ ; ( i=1,2)\ , , \ ] ] where @xmath96 and @xmath97 .    a typical pulse sequence is shown in fig . [ stirap - pulses ] . from the definition of @xmath98 , the franck - condon overlap integrals are an important issue from the point of view of employing a realistic set of parameters . since the overlap integrals enter into the definition of the effective rabi frequencies , their values will affect the adiabatic condition rewritten in terms of the `` bare '' electronic rabi frequencies @xmath37 . we will analyze this in more detail in the next section . we start by considering a uniform condensate , described by a similar equation to the thomas - fermi case , except with a uniform trap potential for simplicity :    @xmath99    here the uniform detuning term @xmath100 is defined as the thomas - fermi detuning , evaluated at the trap center . we start by considering a uniform condensate in which the @xmath0-wave scattering interactions are negligible ( i.e. , @xmath101 ) , and assume exact resonances with respect to bare state transitions , @xmath102 . we first simulate the above simplified equations ( [ j-3 ] ) with an initial condition of a pure atomic condensate . this model is more realistic than that of m. mackie et al . @xcite , as it includes spontaneous emission . we find that including the loss term @xmath64 imposes restrictions on the effective rabi frequencies @xmath103 and the duration of the pulses @xmath82 . the results are best understood with reference to table [ stirap_param ] , which gives the values of typical stirap parameters characteristic of a condensate of @xmath104rb atoms @xcite ) , and corresponding to the pulse sequence in fig . [ stirap - pulses ] . .typical parameter values for efficient stirap . [ cols="<,<",options="header " , ]     the most favorable case that we find corresponds to tuning the raman transitions to @xmath105 in the excited potential bound by @xmath106 @xmath107 or @xmath108 s@xmath109 from the dissociation limit , and @xmath110 in the ground potential bound by @xmath111 ghz or @xmath112 s@xmath109 . the binding energy of the level @xmath110 sets up the frequency difference @xmath113 @xmath114 s@xmath109 , and the values of all relevant detunings . the nearby levels around @xmath105 in the excited potential are separated by about @xmath115 s@xmath109 , which is much larger than @xmath116 . for this arrangement of the target levels , the resonant franck - condon overlap integrals are equal to @xmath117 m@xmath118 and @xmath119 , so that the effective peak rabi frequencies are equal to : @xmath120 s@xmath109 , for @xmath121 m@xmath122 , and @xmath123 s@xmath109 . the resulting values of calculated loss and light shift coefficients are given in table [ losses and light shifts ] . an important factor in minimizing the most significant loss coefficient @xmath124 and its `` cousin '' term in eq . ( [ condition - gamma-2(1 ) ] ) is the relatively small value of @xmath125 and of the franck - condon overlaps @xmath126 of the closest nearby levels . while this is favorable for the undesired loss terms , the small value of @xmath125 also affects the strength of the bound - bound coupling of the primary raman transition , @xmath97 , which must be kept large . however , the small @xmath125-value is compensated here by a strong bare electronic rabi frequency @xmath127 s@xmath109 so that @xmath128 is still large and the adiabaticity condition is maintained .    using the above parameter values and simulating the stirap equations with the additional terms given by eqs . ( [ losses1])-([losses3 ] ) , results in about @xmath129 conversion efficiency ( @xmath130 ) , which is rather high and is an encouraging result . in this simulation , the estimated optimum pulse duration @xmath82 , the pulse delay coefficient @xmath81 , and the two - photon detuning @xmath131 , were taken as follows : @xmath132 @xmath133 s@xmath109 , @xmath134 , and @xmath135 s@xmath109 . the effective rabi frequencies and the particle number densities for this calculation are given in figs . [ multi - level ] ( a ) and ( b ) . we note that these results are essentially identical whether the exact complex form or the far - off - resonant approximations to the additional coefficients are employed . in other words , any coherent cancellation process via stirap in the extra vibrational levels , is negligible compared to the incoherent losses via non - stirap excitation of these levels . the final step in our analysis is to include the trap potential and the kinetic energy terms and simulate the full set of coupled inhomogeneous mean - field equations ( [ psi_3 ] ) in three space dimensions . the initial state in these simulations is a pure atomic bec , with no molecules present , as given by the standard steady - state gross - pitaevskii equation in a trap . we consider spherically symmetric trap potentials @xmath136 and choose the trap oscillation frequencies @xmath137 equal to each other : @xmath138 hz ( @xmath29 ) . including these terms , we simulate eqs . ( [ psi_3 ] ) assuming that the initial peak density of the atomic bec is @xmath139 m@xmath122 at the trap center . this corresponds to the total initial number of atoms @xmath140 equal to @xmath141 . we note that the characteristic time scale associated with the trap potential can be estimated as @xmath142 s. the time scale associated with the kinetic energy term is estimated from the ` healing ' length @xmath143 which corresponds to a healing time scale of @xmath144 . under adiabatic conversion of an equilibrium ( thomas - fermi like ) atomic bec , this healing time scale coincides with the dephasing time @xmath145 associated with the mean field energy potential and discussed before . both these time scales ( @xmath146 and @xmath144 ) are longer than the duration of pulses in stirap we employed earlier . therefore , addition of these terms can not dramatically change the results and conclusions obtained above for uniform condensates .    to show this we first consider the idealized three - level model with parameter values given in tables [ stirap_param ] and [ table_u ] , i.e. the case of equal and relatively strong effective rabi frequencies . the results of simulations of eqs . ( [ psi_3 ] ) are shown in figs . [ density ] and [ 3d - numbers ] , where we see a rather high conversion efficiency @xmath147 . this should be compared with the uniform condensates result of fig . [ conversion93 ] . next , we consider the case of more realistic parameter values ( lower rabi frequencies ) , and include the effects of incoherent radiative losses and dephasings as described in the previous sub - section . using the calculated values of all relevant parameters for the raman transitions tuned to @xmath105 and @xmath110 , and adding the trap potential and kinetic energy terms to the earlier eqs . ( [ losses1])-([losses3 ] ) , we simulate the process of stirap governed by the resulting full set of mean - field equations in three dimensions : @xmath148 @xmath149 @xmath150    we switch on the sequence of two raman pulses , as shown in fig . 9 , corresponding to : @xmath120 s@xmath109 , @xmath123 s@xmath109 , @xmath132 @xmath133 s @xmath109 , and @xmath134 . the values of @xmath82 and @xmath81 , together with the choice of the two - photon detuning @xmath135 s@xmath109 , correspond to the optimized set of parameters as in table [ table_main ] and are obtained for @xmath105 and @xmath110 levels using @xmath151 s@xmath109 and @xmath127 s@xmath109 . the corresponding values of all other relevant coefficients are given in table [ losses and light shifts ] and [ table_u ] , while the spontaneous decay rate is @xmath152 s@xmath109 , as before . the optimum value of @xmath131 accounts for nonzero values of the light shift coefficients @xmath153 , so that the actual detuning to be optimized using the table [ table_main ] is the overall effective detuning @xmath154 . the results of simulations with this set of parameter values are given in fig . [ final - numbers ] where we plot the total occupation numbers in the atomic and molecular becs as a function of time . the obtained conversion efficiency is @xmath155 . this is @xmath156 lower than the efficiency in the corresponding homogeneous case of fig . [ multi - level ] , but still is a rather encouraging result given the fact that @xmath157 conversion of about @xmath158 atoms would give a molecular bec with the total of @xmath159 molecules , with a peak density of about @xmath160 m@xmath122 . to summarize , stirap is a potential route towards the coherent conversion of an atomic to a molecular bec . this process involves stimulated emission of molecules , and is very different to normal chemical kinetics . as such , it is a form of ` superchemistry ' @xcite . stirap in an atomic bec can be treated in a very similar way to normal stirap , by introducing an effective rabi frequency for the first photo - association transition . a potential difficulty is the relatively low values of the effective rabi frequency in the first or photo - association transition . this can be regarded as physically due to the low densities of atoms in a typical weakly interacting bec , compared with atoms in a molecule . this means that the corresponding franck - condon coefficient , after multiplying by the relevant bec amplitude , has a very small value . thus , the laser intensities required may be quite high , in order to obtain rabi frequencies comparable to those used in atomic transitions . this by itself is not critical , since deep in the adiabatic limit it is normally permissible to use low rabi frequencies , as long as the associated time - scales are long enough . from this point of view , the use of stirap , and consequent reduction of spontaneous emission , is a physically sensible idea . however , including @xmath0-wave scattering or mean - field processes into the model for stirap sets up a characteristic two - photon dephasing time scale , so that the pulse durations in stirap have to be _ shorter _ than a certain critical value . short pulse durations necessarily involve high values for the effective rabi frequencies in the usual symmetric case of @xmath161 , thus requiring a very high laser power for the first transition - if the adiabaticity condition is to be maintained .    in order to ease this demanding requirement and be able to achieve highest possible conversion efficiency at smaller total laser power , we propose to use an off - resonance operation ( thus cancelling part of the mean - field detuning effect ) and effective rabi frequencies @xmath162 and @xmath128 of different magnitudes . this has the effect of dramatically increasing coherent molecule production , in a physically accessible regime of moderate laser intensity . further improvements may be possible by tailoring the input pulse frequencies to the time - dependent two - photon detuning , caused by inter - atomic and intermolecular scattering . we have carried out mean - field calculations in three dimenisons to verify that trap inhomogeneity should not have adverse effects on the stirap process . finally , we stress the importance of radiative losses and dephasing due to incoherent processes that occur during stirap . these processes are usually assumed to be negligible in ordinary stirap between purely atomic or molecular states . in the present case of coupled atomic / molecular becs , this assumption can not be easily justified since the free - bound transition typically involves a relatively low effective rabi frequency , which necessitates long pulse durations . instead , we find that the incoherent couplings can be rather destructive unless special care is taken to minimize their effect . this involves detailed knowledge of the structure of the free - bound and bound - bound transitions and subsequent identification of optimum target levels in stirap , so that the overall conversion efficiency remains comparable to the predictions of the simplified three - level model . d. and k. k. gratefully acknowledge the arc for the support of this work . d. h. and r. w. acknowledge the support by the u. s. national science foundation , the r. a. welch foundation , and the nasa microgravity research program . 10 d. j. heinzen , r. wynar , p. d. drummond , and k. v. kheruntsyan , phys . lett . * * 8**4 , 5029 ( 2000 ) . k. v. kheruntsyan and p. d. drummond , phys . a * * 5**8 , r2676 ( 1998 ) . p. d. drummond , k. v. kheruntsyan , and h. he , phys . rev . lett . * * 8**1 , 3055 ( 1998 ) . j. javanainen and m. mackie , phys . a * * 5**9 , r3186 ( 1999 ) . r. h. wynar , r. s. freeland , d. j. han , c. ryu , and d. j. heinzen , science * * 2**87 , 1016 ( 2000 ) . y. b. band and p. s. julienne , phys . a * * 5**1 , r4317 ( 1995 ) ; 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the process of stimulated raman adiabatic passage ( stirap ) provides a possible route for the generation of a coherent molecular bose - einstein condensate ( bec ) from an atomic bec . 
 we analyze this process in a three - dimensional mean - field theory , including atom - atom interactions and non - resonant intermediate levels . 
 we find that the process is feasible , but at larger rabi frequencies than anticipated from a crude single - mode lossless analysis , due to two - photon dephasing caused by the atomic interactions . 
 we then identify optimal strategies in stirap allowing one to maintain high conversion efficiencies with smaller rabi frequencies and under experimentally less demanding conditions . 
 pacs numbers : 03.75.fi , 05.30.jp , 03.65.ge .