Brazilian Journal of Physics vol no 4 Deceber 87 An Exactly Soluble Multiato-Multiphoton Coupling Model A N F Aleixo Instituto de Física Universidade Federal do Rio de Janeiro Rio de Janeiro RJ Brazil A B Balantekin Departent of Physics University of Wisconsin Madison Wisconsin 576 USA M A Cândido Ribeiro Departaento de Física Instituto de Biociências Letras e Ciências Exatas UNESP São José do Rio Preto SP Brazil Received on 7 May A class of bound-state probles which represent the interaction between a syste of N two-level atos an electroagnetic radiation via an -photon process is studied with = We also evaluate soe nonlinear effects usually related with the polarisability of the ediu with the dependence on the intensity of the radiation field on the atter-radiation interaction We obtain exact values for the eigenstates eigenvalues for all values of total angular oentu of the syste all possible nuber of photons involved in the interaction We give explicit analytic expressions for sall nubers of atos discuss soe aspects of the coposition of the eigenstates of the spectra obtained in these cases Introduction A stard odel ost used in quantu optics idealizes the interaction of the atter with the radiation by a siple Hailtonian of a two-level ato coupled to a single bosonic ode [] This Hailtonian is a central ingredient in the quantized description of any optical syste involving the interaction between light atos For this Hailtonian ostly the single-particle situation has been studied [-] The purpose of this paper is to give the exact coplete solution for this odel when we consider the presence of N two-level systes which interact with a single-ode radiation via a single or ulti-photon process that take place when the energy separation between the atoic levels is close to the energy of radiation quanta In this case the effects of the spatial distribution of the particles it is not taken into account the spin angular oentu Ŝi of each particle contributes to for a total angular oentu Ĵ of the syste We consider in this study the effects due to the nonlinear radiation polarisability with the inclusion of a nonlinear ter in the Hailtonian [8] This kind of nonlinear effect is usually considered in the study of icroasers Another nonlinear extension of our study takes the coupling between atter the radiation to depend on the intensity of the radiation field [4-69] It is interesting since this kind of interaction eans that effectively the coupling is proportional to the aplitude of the field corresponding to a ore realistic physical situation We obtain exact values for the eigenstates eigenvalues for all values of j for any nuber of photon involved in the ato-radiation interaction We give explicit analytic expressions for sall nubers of atos discuss soe aspects of the spectra obtained in these cases This paper is organized in the following way in Section II we introduce the ultiato Hailtonian with ultiphoton interaction its solution; in Section III we do the sae for an intensity-dependent radiation interaction; application are presented in Section IV soe nuerical results with its discussion are presented in section V Finally conclusion brief rearks close the paper in Section VI Multi-ato odel with ultiphoton interaction The Hailtonian describing the interaction between a single-ode radiation N two-level systes via an - photon process when we include nonlinear effects can be written as Ĥ = Ĥ( +Ĥ( where with = { Ĥ( Ĥ int = Ω (â Ĵ + + â Ĵ = ωâ â + ωĵ Ĥ ( = Ĥint + ˆk Ĵ ( ˆk = Θ â â ( Here Ω is the atter-radiation interaction paraeter Θ is the nonlinear ediu paraeter related to the third-order nonlinear susceptibility = ω ω is a constant related to the detuning of the syste ω (ω is the field (atoic transition frequency In the above expressions
874 ANF Aleixo AB Balantekin MA Cândido Ribeiro Ĵ ± = Ĵ ± iĵ ( where Ĵi (with i = are the coponents of the total angular oentu of the syste obtained by the su of the spin angular oentu of the N two-level atos To obtain the eigenstates eigenvalues for the Hailtonian ( we first reeber that the bosonic operators when applied to the nuber states satisfy the following properties â n = n n â n = n + n + (4 where the excited bosonic states are written as n = Now if we introduce the operators (â n n! (5  = â  = â ˆN = (â â + ââ (6 that satisfy the following coutation relations [ Â] ˆN =  [ ] ˆN  =  (7 ] [  = { } (ˆn +! ˆn! ˆn! (ˆn! where ˆn = â â (8 then we can rewrite the coponents of the total Hailtonian ( as ( Ĥ ( = ω ˆN + Ĵ Ĥ int = Ω ( Ĵ + +  Ĵ ( ( ˆk = Θ ˆN ˆN (9 Note that for = the new operators reproduce the stard bosonic coutation relations [ˆn â] = â [ˆn â ] = â [â â ] = characteristic of the Heisenberg-Weyl algebra for = these coutation relations which can be written as [ ˆN Â] =  [ ˆN  ] =  [  ] = ˆN represent the bosonic realization of the SU( algebra With the new for (9 of the total Hailtonian eleents it is easy to show that the two ters in the Hailtonian ( coute with each other Ĥ(] = [Ĥ( therefore it is possible to find a coon set of eigenstates for the The siplest way to deterine the eigenstates eigenvalues of that Hailtonian is by calculating the eigenstates eigenvalues of Ĥ( Taking into account that the Ĵ i operators can be represented by (j + (j + atrices then by using the for of these atrices we can write the operator Ĥ( as a tridiagonal atrix with these sae diensions of Ĵi whose eleents can be written as Ĥ ( ik = { (i j + ˆk if k = i ; Ω (k (j i +  if k = i + ( with the property Ĥ(q By considering the properties (4 of the bosonic operators the atrix for of the ki ik operator Ĥ( we introduce the following vector for its eigenstates = (Ĥ(q Ψ n = n + n + + n + +j n + j n = ( where the C coefficients are to be deterined by the eigenvalue equation Ĥ ( Ψ n = Ψ n ( which also deterines the eigenvalues Basically the vector Ψ n corresponds to the state of the syste with total angular oentu J = j(j + each vector coponent represents one of j + possible values of J z In other words the coponent n + k of the vector state Ψ n represents the quantu state {n j j = j k} By considering that  k = if k <
Brazilian Journal of Physics vol no 4 Deceber 875 also  k =  k = k! k (k! if k ( (k +! k! k + ˆN k = ( k + k (4 then by substituting Eqs ( ( into Eq ( we obtain the set of (j + coupled algebraic equations which can be written in a copact for as ( δk h(k C + h C + ( δ hk C = E k n+(k k k n+(k kj+ (k+ n+k n C (5 n+(k where k = j + the h-coefficients are defined as h i k = { Θ [n + (k ] [n + (k ] (j k + if i = k Ω (k (j k + [n + (k ]!/ [n + (k ]! if i = k (6 With this set of (j + equations plus the noralization condition for the eigenstate it is possible to obtain the (j + variables that appear in (5 This set of equations can be written in a atrix for as [h I] C = (7 where the tridiagonal h-atrix has its eleents given by Eq (6 with the syetry condition h ki = h ik being C the colun atrix C = + + +j (8 The secular equation associated to Eq (7 gives us the algebraic equation to calculate the j + possible values for the eigenvalues E j+ n + D j E j n + D j E j n + + D E n + D + D = (9 where the D coefficients are related to the h-atrix eleents Finally by using Eqs ( ( ( we have Ĥ Ψ n = Ψ n where = (n + j ω + ( So far we have only discussed about the possibility of having n By considering that n corresponds to the sallest nuber of photons we ight easure in the vector state Ψ n then when n it eans that we can find all values of j fro j to j Therefore when we have Ψ r = C C C C (j+r (j + r j n < the vector state Ψ n corresponds to states in which there are not enough photons to excite the higher j values even if all of the are absorbed In this case the vector state ( that represents this situation will be written as { r null coponents} {(j + r + non zero coponents} (
876 ANF Aleixo AB Balantekin MA Cândido Ribeiro where j r < In particular the case r = j that corresponds to the vacuu cobined with all particles in their lowest state represents the ground state of the syste The deterination of the eigenstates eigenvalues of the Hailtonian ( for these j possible n negative values corresponds to a particular application of the general treatent presented before it will be studied analyzed for each particular j value in the applications below Multi-ato odel with intensitydependent ulti-photon interaction We now introduce an intensity-dependent interaction between atter constituted by N two-level particles radiation via an -photon process by using the following for for the interaction Hailtonian ( Ĥ int = Ω ˆB Ĵ + + ˆB Ĵ ( where the new operators are defined as ( ˆB =  ˆN ˆB = ( ˆN  ( The other ters in the total Hailtonian ( keep the sae fors as used in the previous Section These facts iply that the diagonal eleents of the total Hailtonian Ĥ are still given by Eq ( the non diagonal atrix eleents now are written as Ĥ ik Ĥ( ik = Ω (k (j i + ˆB if k = i + ( with i k = j + with the property Ĥki = Ĥ Again those both ters Ĥ ik Ĥ in the total Hailtonian ( coute with each other Therefore it is possible to use the sae technique as used in the preceding Section to obtain the eigenstates eigenvalues for the total Hailtonian Now by considering that ˆB k = if k < ˆB k = k k! (k! k = k  k if k (4 also ˆB k = (k + (k +! k! k + = k +  k (5 then using Eqs ( ( into Eq ( we obtain a new set of (j + coupled algebraic equations which can be written in a copact for as ( δk h(k C + h C + ( δ hk C = E k n+(k k k n+(k kj+ (k+ n+k n C (6 n+(k where k = j + the new h i k coefficients with i k are related with the previous one (h i k by h i k = n + (k h i k with i = k (7 the syetry property h i k = h k i The diagonal h kk coefficients reain given by Eq (6 Again the above set of equations can be written in a atrix for as given by Eq (7 except that the non diagonal atrix eleents of h appear now substituted by these h i k new eleents In this case to have the spectra we need to resolve Eq (9 where the D coefficients take into account the changes in Eq (6 It is iportant to note that the inclusion of negative values of n for this case can be done in the sae way as indicated in the previous Section However in contrast with what we have for the previous case the effects of the intensity dependence on the eigenstates eigenvalues for n negative values will only be observed for j 4 Applications A Single-particle syste In this case j = / the Hailtonian Ĥ( will be a atrix Applying this Hailtonian to the eigenstate
Brazilian Journal of Physics vol no 4 Deceber 877 equation ( we obtain the atrix equation (7 with ĥ = [ ] h h (4 h h being the h ik coefficients given by Eq (6 Then the syste of equations (5 becoes { hc n + h + = h + h + = + (4 fro Eq (9 we have an algebraic quadratic equation with coefficients D = (h + h D = h h h for the eigenvalues which will be given by E (± n = { Θ [ n + (n + ( ] } 4Ω (n +! ± + [Θ(n + + n! ] (4 Fro Eqs (4 (4 the eigenstate noralization condition we can deterine the C coefficients show that the eigenstates for the total Hailtonian Ĥ can be written as [ ] Ψ (± = n n ( ±γ (± (44 n + n+ + with n = = where γ (± n+ = n+ + δ δ n+ n+ γ (± δ n+ = Θ(n + + Ω (n +!/n! (45 with eigenvalues given by E (n (± = ω + + E (± n n (46 For the case of intensity-dependent interaction we obtain (with the change h ik h ik i k the following eigenvalues of Ĥ( E (± n = Θ [ n + (n + ( ] ± 4Ω (n + (n +! n! + [Θ(n + + ] (47 while the eigenstates for the total Hailtonian Ĥ can be written in the sae for given by Esq (44 (45 but with δ n+ = Θ(n + + Ω (n + (n +!/n! (48 The eigenvalues for the total intensity-dependent Hailtonian will be written as in Eq (46 but now with E (± given n by Eq (47 These results can be reduced to the stard odel to the intensity dependent interaction stard odel results [ 4] when we assue the value = turn off the nonlinearity ter taking Θ = For j = / the only possible negative n value is r = j = that corresponds to the ground state of the syste Considering this fact using Eqs ( (4 (4 it is easy to show that E Ψ = [ = [ ] ( ω ω ] (49 In the case of the intensity dependent interaction we have these sae results for the eigenvalues eigenstates as written in Eqs (49 B Three-particle syste In this case we can have j = / or j = / We have already studied the first possibility For the second j value the Hailtonian Ĥ( is a 4 4 atrix that applying to
878 ANF Aleixo AB Balantekin MA Cândido Ribeiro the eigenstate equation ( give us the atrix equation (7 with h h ĥ = h h h h h 4 (4 h 4 being the h ik coefficients given by Eq (6 Then the syste of equations (5 becoes h + h + = h + h + + h + = + (4 h + + + + h 4 + = + h 4 + + + = + Eq (9 is an algebraic quartic equation with coefficients D = (h + h + + D = h h + h + h + h + h + ( h + h + h 4 D = (h h + h h + h + h (4 + (h + h + (h + h + (h + h h 4 D = h h ( h + h h 44 h + h h h h h 4 4 the eigenvalues E (4 can be obtained by its roots as shown in Appendix II Fro Eqs (68 (4 ( the n eigenstate noralization condition we can deterine the C coefficients show that the eigenstates for the total Hailtonian can be written as where = n + ( γ (k n + (γ (k Ψ (k n+ ( + γ (k n+ γ (k n γ (k n n + n+ n + γ (k n+ n + (4 with k = 4 γ (k n = Γ (k γ (k n+ = Γ (k γ (k n+ Γ (k Γ (k γ(k n+ = Γ(k 4 (44 Γ (k = h E (k n h Γ (k = its associated eigenvalues can be written as h E (k n h Γ (k = h E (k n h Γ (k 4 = h E (k n (45 E (k n = ( n + ω + E (k n (46 Again the results for the intensity-dependent interaction are obtained with the change h ik h ik with i k Basically this change iplies in a odification in the D D D coefficients of the algebraic quartic equation for the eigenvalues in the Γ-ters for the eigenstates For this value of j the possible n negative values are r = j+ = r = j+ = r = j = the last one being the ground state of the syste First using Eqs ( (4 for r = we find the following syste of equations C + Ω! C = E C Ω! C + [ Θ( + ] C + Ω (! Ω! C + [ Θ( + ] C = E C (!! C = E C (47 fro Eq (9 we have an algebraic cubic equation with coefficients
Brazilian Journal of Physics vol no 4 Deceber 879 D = [ Θ(5 + ] D = { Θ( [Θ( + ] 4 Ω [4! + (!/!] } D = { 4Ω! + [ Θ( + ]} [ Θ( + ] Ω (!/! (48 for the eigenvalues E ( that can be obtained by using the recipe presented in Appendix I Fro Eqs (47 the noraliza- tion condition with the eigenvalues E ( Hailtonian Ĥ in this case can be written as where γ (k = Ψ (k Ω! E (k + with k = the eigenvalues given by we can calculate the C coefficients show that the eigenstates for the total = + ( γ (k ( + γ (k γ (k γ (k (49 γ (k = Ω (!/! E (k [ Θ( + ] (4 E (k = (k ( ω + E (4 The odifications required to get the results for intensity dependent interaction case can be reduced to the change of the factors! by! (!/! by 6 (!/! in Eqs (47 (48 (4 For the case of r = after taking the sae procedure used before we find the following syste of equations { C + Ω! C = E C Ω! C + [ Θ( + ] C = E C (4 giving for the coefficients of the algebraic quadratic equation for the eigenvalues (9 { D = [Θ( + ] D = [ Θ( + ] Ω! So we find (4 E (± = { } [Θ( + ] ± [Θ( + ] + Ω! (44 Again fro Eqs (44 (4 the eigenstates noralization condition we can deterine the C coefficients show that the eigenstates for the total Hailtonian Ĥ for this case can be written as where Ψ (± = + ( γ (± ±γ (± γ (± = + δ δ (45 δ being its eigenvalues given by = Θ( +! Ω (46 E (± = (± ( 4 ω + E (47 Now the results for the intensity dependent interaction case can be obtained with the change of the factor! by! in Eqs (4 (4 (44 (46 Finally using Eqs ( (4 (4 for r = we find E = ( 6 ω + E
88 ANF Aleixo AB Balantekin MA Cândido Ribeiro with E = (48 Ψ = (49 For this last case we do not observe the effects of the intensity dependent interaction on the results for the eigenstates eigenvalues of the total Hailtonian After these applications we ephasize that is just a atter of siple nuerical work to find the eigenstates the energy spectra for any higher value of j It is straightforward to do an analytical calculation up to five particle syste j ax = 5/ when we have a resonant situation ( = we do not consider the presence of the nonlinearity ter (Θ = 5 Nuerical results discussion In this section we show soe exaples of exact nuerical applications of the foralis developed in the previous sections In tables we show the results for energy eigenvalues for the cases of one-particle three-particle systes; j = / j = / / respectively The energies are given in units of ω The used values of the constants Ω Θ are chosen in a such way that the odel approxiation is preserved The bosonic quantu nuber n is chosen as j n nuber of photons taking part in each process as = Results in bold face st for the intensity-dependent interaction case otherwise for non intensity-dependent one As shown in the tables for all values of the nuber of levels increases fro to j + as n goes fro j to n = Since the ground-state is a photon free state the results for the intensity-dependent non intensity-dependent cases for the ground states are equal for each value of For each value of j the first excited state for = (labeled by n = j + also has a sae value for the intensitydependent non intensity-dependent situations However now this is because of the dependence on n shown by the eigenvalues For each value of j as increases the splitting aong energy levels also increases This effect is stressed in the case of intensity-dependent interaction Figure (a shows the behavior of the energy level diagra of the syste for = j = / with the energies given in units of ω In this figure we can see the split of the energy levels when we gradually turn on each coponent of the total Hailtonian Ĥ process that is represented by the coluns (A (B (C (D The colun (A shows only the contribution of the radiation for the energy of the syste since that the other coponents of the total Hailtonian are turned off The colun (B shows the syetric splitting in the original energy levels caused by the inclusion of the atter-radiation interaction with a strength paraeter Ω = ω The colun (C shows the effects of the inclusion of the nonlinear ter with a strength paraeter Θ = ω in the total Hailtonian We can see that the net effect in this case is a shifting up the increasing of the energy gap between the priary shifted levels when n j + Finally the last colun (D presents the increasing of the energy gap between the priary shifted energy levels when we turn on the detuning of the syste with = ω Note that the ground state energy is only odified by the presence of the nonresonant ter Ĵ in the total Hailtonian This is because the ground state is a no photon state It is iportant to note that for j = / case the introduction of the nonlinear ter in the Hailtonian does not affect the results for n negative values This is because this nonlinear ter takes into account a change of two photons thus it is necessary to have eigenstates corresponding to bosonic states coponents k with k > Only for higher j values we can observe the effects of the nonlinearity on the eigenstates eigenvalues for n negatives values Figure (b is the intensity-dependent radiation version of Fig (a If we copare these figures we conclude that the action of the intensity-dependence in the atter-radiation interaction is to generate an aplification in the energy gap of the levels which are sensitive to this dependence (n j + E E E E - (A (B (C (D j = / = (a 4 5 5 5 5-5 - (A (B (C (D j = / = Figure a The behavior of the energy level diagra of the syste for j = / with the energies given in units of ω We can follow the split of the energy levels when we gradually turn on each coponent of the total Hailtonian Ĥ a process that is represented by coluns (A (B (C (D The set of strength used is given in the text Figure (b is the the radiation intensity-dependent interaction version of Figure (a Figures (a (b are the version of Figs (a (b for = j = / recpectively We observe the sae behavior presented for j = / case in the splitting of the energy levels when we turn on each coponent of the Hailtonian except for the splited energy levels nuber the increasing in the energy gap between the levels E E E E - (b 4 5 5 5 5-5 -
Brazilian Journal of Physics vol no 4 Deceber 88 Table Values of for j = / (single particle three particle systes ( ω j = / Ω = ω Θ = = ω n - -4-4 6 6 6 6 84 84 68 5 85 647 86 86 7 87 5 55 8 9 585 47 78 486 7 65 45 566 694 579 7 887 48 698 67 4 4 59 675 87 Table Values of for j = / (three-particle syste ( ω j = / Ω = ω Θ = = ω n - - - 8 8 - -89-89 66 8 847 987 9 9 94 48 475 56 - -87-465 94-554 544-77 7 6 4 96 875 64 784 65 5756 7458 58 6 45-4 44 - -864-59 9 6 89 485 4 78 7 56 558 57 95 886 69 4 957 5 7999 5688 496 64 5-457 - -5 68 6 488 87 58 94 4 46 8 85 4 64 5846 76 4658 565 4688 67459 9 6 58-8 -47-448 445 89 645 4496 658 97 6 6477 99 74 764 96 8454 8 8758 6576 4975 8488 Figure (a shows the behavior of the energy level diagra of the syste for j = / n = = ; with energies given in units of ω In this Figure we can see the split of the energy levels when we gradually turn on each coponent of the total Hailtonian Ĥ process that is represented by coluns (A (B (C (D The leftost colun (A shows only the contribution of the radiation for the energy of the syste since the other coponents of the total Hailtonian are turned off The next colun (B shows the syetric splitting in the original energy levels caused by the inclusion of the atter-radiation interaction with a strength paraeter Ω = ω The third colun (C shows the effects of the inclusion of the nonlinear ter with a strength paraeter Θ = ω in the total Hailtonian We can see that the net effect in this case is a shifting up the increasing of the energy gap between the priary shifted levels The last colun (D presents the increasing of the energy gap between the priary shifted energy levels when we turn on the detuning of the syste with = ω Finally we note that as increases the gap between levels also increases This kind of behavior of the energy levels is present for all excited states Figure (b is the intensity-dependent version of Fig (a If we copare these two set of figures we conclude that the action of the intensity-dependence in the atterradiation interaction is to generate an aplification in the energy gap of the levels
88 ANF Aleixo AB Balantekin MA Cândido Ribeiro Non Intensity-Dependent Intensity-Dependent (A (B (C (D (A (B (C (D 8 = 54 ω j = / n = = 8 = -8 ω j = / n = = j = / = 9 j = / = 9 6 6 8 8 4 4 7 6 7 6 E 5 4 E 5 4 8 = 64 ω j = / n = = 8 = 45 ω j = / n = = E E 6 6 E E - E E - 4 4 E - - E - - E - (a - E - (b - P n+k 8 = ω j = / n = = n+k P 8 = 74 ω j = / n = = 6 6 Figure a The j = / version of Figs (a (b 4 4 = 876 ω j = / n = = = 658 ω j = / n = = 8 8 6 6 4 4 8 6 4 - -4-6 8 6 4 - -4-6 = = = (A (B (C (D (A (B (C (D (A (B (C (D (a = = = (B (A (B (C (D (A (B (C (D (A (C (D (b Figure a Soe of energy levels for a j = / syste with n = = as each coponent of the total Hailtonian Ĥ is progressively added a process that is represented by coluns (A (B (C (D Figure (b is the radiation intensity-dependent version of Figure (a k (4 Figure 4 Weight of each coponent P n+k = C in the n+k coposition of the state vectors Ψ (4 for j = / n n = = The left colun is for a non intensity-dependent interaction the right colun is for an intensity-dependent one Figure (4 shows the weight of each coponent P n+k = C with k = in the coposition of the state vectors Ψ (4 for j = / n+k n n = = The values of paraeters Ω Θ are the sae as in the tables In this Figure are shown the j + states fro the less energetic state in the top to the ore one in the botto the size of the vertical bars are proportional to the weight of the correspondent coponent The colun to the left is for the non intensity-dependent situation that one to the right is for the intensity-dependent case Clearly in the non intensity-dependent situation for the lowest energy eigenvalue (in the top the coponents ore iportant in the coposition of the eigenstate correspond to the lower bosonic states As the energy eigenvalue increases (going down in the figure this behavior gradually change in a such way that for the highest energy eigenvalue (in the botto we observe that the coponents with ore iportance in the coposition of the eigenstate correspond to the higher bosonic states For the intensitydependent situation for the positive energy eigenvalues we observe the sae behavior as describe above for the relative iportance of the coponents in the coposition of the eigenstate However for the only one eigenstate with a negative energy eigenvalue (in the top we see that intere- k
Brazilian Journal of Physics vol no 4 Deceber 88 diate coponents present a higher relative iportance than the coponents in the extrees A non observed vertical bar in the figures does not ean that the respective coponent is null but only that in the figure scale its relative value can not be represented Figure (5 is the sae as for Figure (4 with n = = As we can see for negative (top of each colun positive energy eigenvalues the general behavior for the relative iportance of the coponents is the sae as describe before P n+k 8 6 4 8 6 4 8 6 4 8 6 4 Non Intensity-Dependent = - ω j = / n = = = 87 ω j = / n = = = 99 ω j = / n = = = 4985 ω j = / n = = k (5 n+k P 8 6 4 8 6 4 8 6 4 8 6 4 Intensity-Dependent = -594 ω j = / n = = = 4 ω j = / n = = = 678 ω j = / n = = = 9 ω j = / n = = Figure 5 The version of Fig (4 for n = = k 6 Conclusions In this article we introduced a odel for interaction between the atter consisting of N two-level particles the radiation via an -photon process with = The odel includes the nonlinear effects related with the nonlinear polarisability in a nonlinear ediu nonlinear effects due an intensity-dependent radiation coupling We present the exact results for the eigenstates eigenvalues for soe different values of j in ters of the photon nuber We copare the different spectra to evaluate the effects of the photon nuber involved in the interaction the different sources of nonlinearity also discuss soe aspects of the coposition of the eigenstates of the spectra obtained in these cases We ephasize that our general odel akes possible to resolve siultaneously probles with different Hailtonians; as we set Θ = = our odel reduces to a bare resonant odel for a ulti-ato syste interacting with an electroagnetic radiation via a ulti-photon process; if we put Θ = we get a bare ulti-ato -photon non resonant odel if we set Θ = we have a ulti-ato -photon resonant odel with a nonlinearity We showed that it is straightforward to generalize all these situations for an intensity-dependent interaction A related quantu echanical odel for the propagation of light through a nonlinear ediu taking into account the nonlinear polarisability of radiation was presented in Ref [5] Our Hailtonian can be considered a generalization of this odel Indeed for = using the Holstein-Priakoff boson apping of the angular oentu algebra: Ĵ + = j ˆb ( jˆb ˆb Ĵ = ( ˆb j jˆb ˆb Ĵ = + ˆb ˆb (6 introducing the scaled atter-radiation interaction paraeter Ω = Ω j (6 one sees that in the liit j substitution of Eq (6 into Eqs ( ( ( yields the Hailtonian ( Ĥ = ωâ â + ω ˆb ˆb + Ω âˆb + â ˆb + Θ â â (6 up to a constant Eq (6 is the Hailtonian studied in Ref [5] Our Hailtonian can also be considered as an exactly-solvable coupled-channels proble As any interesting probles in areas ranging fro nuclear physics [6] to the study of transitions fro etastable states [7] have coupled-channels for our Hailtonian can be a starting point for other odels Appendix I The roots of the cubic algebraic equation
884 ANF Aleixo AB Balantekin MA Cândido Ribeiro E qn + D E qn + D E qn + D = (64 can be obtained by[8] with { ( p = 9 D D u = 6 (D D D 7 D (66 E ( qn = (E + + E D E ( qn = (E + + E D + i (E + E E ( qn = (E + + E D i (E + E where E + E that ust be coplex conjugate are given by E ± = [ u ± p + u ] / (65 Appendix II The roots of the quartic algebraic equation E 4 + D qn E + D qn E + D E + D = (67 qn qn can be obtained by[9] E ( = qn 4 (D + B a ± 6 (D + B a (B k + B b E (4 = qn 4 (D B a ± 6 (D B a (B k B b (68 where B a = B k D + 4 D B b = (B k D D /(B a B k = B + + B + 6 D with B ± = [ u ± u + p ] / (69 { [ ] p = 6 (D D 4D D u = 6 D [ ( ] 48 D (D D D + D 8D D (6 Acknowledgents This work was supported in part by the US National Science Foundation Grants No INT-7889 PHY- 76 at the University of Wisconsin in part by the University of Wisconsin Research Coittee with funds granted by the Wisconsin Aluni Research Foundation MACR acknowledges the support of Fundação de Aparo à Pesquisa do Estado de São Paulo - FAPESP (Contract No 98/7- ANFA MACR acknowledge the support of Conselho Nacional de Pesquisa (CNPq (Contract No 94/99- ANFA MACR thank to the Nuclear Theory Group at University of Wisconsin for their very kind hospitality References [] E T Jaynes F W Cuings Proc IEEE 5 89 (96 [] B Buck C V Sukuar Phys Lett A 8 (98 [] C C Gerry Phys Rev A 7 68 (988 [4] M Chaichian D Ellinas P Kulish Phys Rev Lett 65 98 (99 [5] B Deb D S Ray Phys Rev A 48 9 (99 [6] V Buzěk Phys Rev A 9 96 (989 [7] W R Mallory Phys Rev 88 976 (969; I R Senitzky Phys Rev A 4 (97 [8] P D Druond D F Walls J Phys A 75 (98 [9] A Rybin G Miroshnichenko I Vadeiko J Tionen J Phys A: Math Gen 879 (999 [] C Flytzanis C L Tang Phys Rev Lett 45 44 (98 [] N Ioto H A Haus Y Yaaoto Phys Rev A 87 (985 [] A N Aleixo A B Balantekin M A Cândido Ribeiro J Phys A 7 ( [quant-ph/49] [] A N Aleixo A B Balantekin M A Cândido Ribeiro J Phys A 4 9 ( [quant-ph/4] [4] W H Louisell Quantu Statistical Properties of Radiation chap 5 J Wiley - 97
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