δ 12. We find a highly accurate analytic description of the functions δ 11 ( δ 0, n)

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1 Coplete-return spectru for a generalied Rosen-Zener two-state ter-crossing odel T.A. Shahverdyan, D.S. Mogilevtsev, V.M. Red kov, and A.M Ishkhanyan 3 Moscow Institute of Physics and Technology, 47 Dolgoprudni, Russia Institute of Physics, NASB, F. Skarina Avenue 68, Minsk 7, Belarus 3 Institute for Physical Research of NAS of Arenia, 3 Ashtarak, Arenia The general seiclassical tie-dependent two-state proble is considered for a specific field configuration referred to as the generalied Rosen-Zener odel. This is a rich faily of pulse aplitude- and phase-odulation functions describing both non-crossing and ter-crossing odels with one or two crossing points. The odel includes the original constant-detuning non-crossing Rosen-Zener odel as a particular case. We show that the governing syste of equations is reduced to a confluent Heun equation. When inspecting the conditions for returning the syste to the initial state at the end of the interaction with the field, we reforulate the proble as an eigenvalue proble for the peak Rabi frequency and apply the Rayleigh-Schrödinger perturbation theory. Further, we develop a generalied approach for finding the higher-order approxiations, which is applicable for the whole variation region of the involved input paraeters of the syste. We exaine the general surface U n = U n ( δ, δ), n = const, in the 3D space of input paraeters, which defines the position of the n -th order return-resonance, and show that for fixed δ the curve in { U n, δ} plane, i.e., the δ = const section of the general surface is accurately approxiated by an ellipse which crosses the U n -axis at the points ± n and δ-axis at the points δ and δ. We find a highly accurate analytic description of the functions δ ( δ, n) and δ ( δ, n) as the eros of a Kuer confluent hypergeoetric function. Fro the point of view of the generality, the analytical description of entioned curve for the whole variation range of all involved paraeters is the ain result of the present paper. The general seiclassical tie-dependent two-state proble [,] is written as a syste of two coupled first-order differential equations for probability aplitudes a ( ) and a ( ) of the first and second states, respectively, containing two arbitrary real functions of t tie, U (t) (aplitude odulation; U > ) and δ (t) (phase odulation): da iδ da +iδ i = Ue a, i = Ue a. () dt dt When discussing the excitation of an effective two-state quantu syste by an optical laser field, the aplitude odulation function U (t) is referred to as the Rabi frequency and the derivative δ ( t) = dδ dt of the phase odulation is the detuning of the laser frequency fro t / the transition frequency. Below, we discuss the following field configuration with constant U >, δ, sech( t), δ ( t) = δ + (sech( )) U ( t) = U t δ t (). We refer to this odel as the generalied Rosen-Zener odel t

2 because it includes the original constant-detuning Rosen-Zener odel [3] as a particular case when δ =. The tie-variation shape of the detuning for several values of the paraeter δ is shown in Fig.. As it is seen, this is a rich faily describing both non-crossing and tercrossing odels with one or two crossing points. The detuning does not cross ero if δ δ or δ δ, while at δ + δ it touches the origin and there exist two / < / > = crossing points located at t = ± arcsech[ δ / δ ] if δ / δ. Finally, we note that < the particular case δ is an exceptional one since then the detuning asyptotically goes to ero as t. = δ = δ = δ =..5 U (t) t Fig.. Generalied Rosen-Zener odel defined by Eq. (): U =. 75, δ =. Syste () is equivalent to the following linear second-order ordinary differential equation d a dt dδ du da + i + U a dt U dt dt =. (3) The transforation of the independent variable = ( + tanh( t)) / reduces this equation to the confluent Heun equation [4] d a d ( + iδ ) / ( iδ ) / da U + iδ + + a =. (4) d ( ) If δ =, this equation is reduced to the Gauss hypergeoetric equation. Accordingly, the solution satisfying the initial conditions a ( t = ), a ( t = ) discussed here is written as = =

3 = F U, U ;( + iδ ) / ; ), (5) a RZ ( fro which the known forula by Rosen and Zener [3] for the final transition probability to the second state a (t) is derived: )sech( π δ / )) P RZ = (sin( π U. (6) This forula states the known π - theore [,] according to which the syste returns to the initial state ( a, a ) = (,) if U = n, n =,,... N. Note that if the population inversion (i.e., a total transition to the second state) is discussed, the situation is different. The peculiarity here is that the syste is copletely inverted only at exact resonance, δ =. Furtherore, it is seen that the axial transition to the second state allowed for the given fixed value of the detuning is achieved at U = / + n. It should be noted that the return-tothe-initial-state theore coes already fro the discussion of the Rabi box-odel with constant aplitude and detuning odulations [5]. Furtherore, the solution of the proble for the exact resonance case reveals that for any pulse of duration T the transition probability is given as T p ~ (sin( A)), where A = U ( t) dt, i.e., A is the area under the pulse envelope. This forula shows that, indeed, the transition probability vanishes when A is a ultiple of π and for half-integer ultiples of π we get a coplete population inversion. It was conjectured that this is a general property. However, the close inspection has revealed that in general this is not the case. Both the stateent concerning the inversion and the one regarding the return are violated for asyetric pulses while for syetric pulses the return property survives with a changed value of the pulse area (see, e.g., the discussion of this point in [6]). Qualitatively the sae picture is observed when a quadratic-nonlinear extension of the twostate approxiation is discussed in the fraework of two-ode photo- or agneticassociation of ultracold atos in degenerate quantu ensebles [7]. When inspecting the conditions for returning to the initial state at the end of the interaction with the field, we note that the additional boundary condition a ( t = + ) = together with the initial conditions defines an eigenvalue proble for the peak Rabi frequency U that can be written as Hˆ a da U a = iδ ( ), (7) d a ( t = ), a ( t = ), a ( + ), (8) = = = 3

4 where the operator Ĥ stands for the standard constant-frequency Rosen-Zener odel: H d d ( + iδ i d ( ) + ) / ( δ + ) / ˆ = d. (9) Having at hand the solution of the proble for the Rosen-Zener case with the eigenfunctions of the proble given by equation (5) and eigenvalues U = n one ay apply the Rayleigh-Schrödinger perturbation theory to exaine, approxiately, the behavior of the syste for non-ero δ, at least, for sall δ. Thus, we suppose that δ is sall enough and for the n th-order coplete return case apply the expansion U where, according to Eq. (5), Equation (7) then reads a n RZn ( + = a + iδ ) u + ( iδ ) u..., () n RZn ( + = U + iδ ) U + ( iδ ) U..., () a RZn = F ( n, n;( + iδ ) / ; ) and U RZn = n. () ( ˆ H ( n + ( iδ ) U + ( iδ ) U +..)) = iδ d d ( a + ( iδ ) u ( ) ( a + ( iδ ) u + ( iδ ) u +...) RZn RZn so that equating the ters at equal powers of δ we obtain ( ˆ n ) a RZn = + ( iδ ) u +...) (3) H, (4) ( Hˆ n ) u U a = ( ) darzn RZn, (5) d ( Hˆ n ) u U u U a = ( ) du RZn (6) d and so on for higher orders. Eq. (4) is, of course, autoatically satisfied for any n. Now, to treat Eq. (5) for u, we apply an expansion in ters of functions a RZn : Note now that the functions = u = b a, b = const. (7) RZn RZ a are orthogonal in the interval [,] with the weight function ( iδ ) / ) (+ iδ ) / ( w( ) = : w( ) a a d = C δ, (8) RZ RZn n n 4

5 where δ n is the Kronecker delta and C n is a constant. Substituting then expansion (7) into Eq. (5), ultiplying the resultant equation by [,] we obtain w ) arz ( and integrating over the interval Taking here and for darzn ( n ) bc = UCnδ n + ( ) w( ) arz d. (9) d = n we get the first correction for the Rabi frequency: U = C n (+ iδ ) / ( iδ ) / ( ) da RZn n we deterine the coefficients of expansion (7): b d a RZn d, () = Vn ( n ) C, = darzn Vn ( ) w( ) arz d. () d In the first approxiation, according to Eq. (), the Rabi frequency supporting the return of the syste to the initial state is given as U n = n iδu. Calculation of the integrals involved in Eq. () leads to the following siple final result U n = n δ δ. () 4n Nuerical siulations show that this is a satisfactory approxiation as long as the paraeter δ is sall copared with n U =. However, at large δ and δ, one needs a ore accurate result, especially, for lower-order return resonances. For this reason, we calculate the next correction ter U using Eq. (6). Again, applying for u the expansion = u = d a, d = const (3) RZ and ultiplying the equation by w ) arz ( with = n and integrating we obtain U = bncn U Cn = bvn. (4) Since U C n = V, for the second-order correction to the Rabi frequency we obtain nn U = C Vn n =, n ( n ) C. (5) Unfortunately, the integrals involved in this su are not calculated analytically. However, we will now show that the final su itself can be calculated with rearkably high 5

6 accuracy using a different approach. The crucial observation here is that the result of the second-order approxiation can be rewritten as a bilinear for with respect to U n and δ : U n δ = δ, (6) n δ δ where the functions δ = δ ( δ, n) and δ = δ ( δ, n) are the roots of the equation n + ( iδ ) U + ( iδ) U =. For a fixed δ Eq. (6) defines a second-degree curve in the U, } -plane, which in our case ay be an ellipse or parabola. In order to find the { n δ functions δ and δ which define the actual for of the curve, we explore the liit U of the proble for vanishing but non-ero U. Note that the axis U n corresponds to the non-physical case of an interaction without a laser field. Matheatically, it is seen that the axis U presents a degenerate case when each point of the axis satisfies n = the return condition a ( + ) because the solution of the proble in this case is a ( t). = It is for this reason that one should explore the vanishing liit U for non-ero U. Considering the last ter in Eq. (4) as a perturbation and applying the ethod of variation of constants, we find that the solution of the proble in the liit U satisfying the applied initial conditions in the first approxiation is written as a F( ) y F( ) y = y d + y d W, (7) W where F ( ) = U /( ( )) and W is the Wronskian of fundaental solutions y =, y = e iδ (+ iδ )/ ( ) ( iδ )/ d (8) of the hoogeneous equation, i.e., Eq. (4) when U is put equal to ero. Note that W = d y / d. The first integral involved in Eq. (7) is syetric with respect to the change, hence it vanishes in the liit ( t + ). Further, it can be shown that for = the last ter in Eq. (7) is written in ters of the Kuer confluent hypergeoetric functions. The final result reads ( t + ) = π U (sech( π δ / )) F (( iδ ) / ;; iδ) a = +. (9) Fro here, we conclude that in the liit of infinitesially sall but non-ero U coplete return occurs if 6

7 F (( + iδ ) / ;; iδ). (3) = Hence, the functions δ = δ ( δ, n) and δ = δ ( δ, n) are the roots of this equation. Note that though we have used a perturbation approach, however, due to the applied liiting procedure the result, Eq. (3), is exact. The eros of the hypergeoetric function of Eq. (3) have been studied by any authors. A review on this subject is presented in [8]. For a fixed δ there exists infinite sequence of pairs δ, } each of which corresponds to a particular return-resonance of soe order { δ n, n N. And for fixed δ and n the pair { δ, δ} is unique. An iportant property of the eros is that for any n holds δ δ, n) = δ ( δ, ) so that the two roots, ( n δ and δ always have different signs. With this observation, we conclude that for fixed finite δ and n the curve in { U n, δ} plane defined by Eq. (6) is an ellipse, which crosses U n -axis at the points ± n and δ-axis at the points δ and δ. The upper half-ellipses for several values of δ for the first resonance n = are shown in Fig.. These curves are the δ = const sections of the general surface U = U δ, ) defining the position of the first ( δ return-resonance. This surface is shown in Fig. 3. Fro the point of view of the generality, the analytical description of this surface for the whole variation range of all involved paraeters is the ain result of the present paper. Fig.. The ellipses defined by Eq. (6) for n = and several δ. 7

8 Fig.3. The surface U = U δ, ) defined by Eq. (6) for n =. n n ( δ Acknowledgents This research has been conducted within the scope of the International Associated Laboratory (CNRS-France & SCS-Arenia) IRMAS. The research has received funding fro the European Union Seventh Fraework Prograe (FP7/7-3) under grant agreeent No. 55 IPERA. The work has been supported by the Arenian State Coittee of Science (SCS Grant No. RB-6). References. B.W. Shore, The Theory of Coherent Atoic Excitation (Wiley, New York, 99).. L. Allen, J.H. Eberly, Optical Resonance and Two-Level Atos (Wiley, New York, 975). 3. N. Rosen and C. Zener. Phys. Rev., 4, 5 (93). 4. A. Ronveaux, Heun s Differential Equations (Oxford University Press, London, 995). 5. I.I. Rabi, Phys. Rev. 5, 65 (937). 6. A. Babini and P.R. Beran, Phys. Rev. A 3, 496 (98). 7. A. Ishkhanyan, R. Sokhoyan, B. Joulakian, and K.-A. Suoinen, Optics Coun. 8, 8 (9). 8. G.N. Georgiev and M.N. Georgieva-Grosse, J. Telecoun. Inf. Techn. 3, (5). 8