- Email: [email protected]
Multiphoton association by two infrared laser pulses
Chemical Physics Letters 433 (2006) 48–53 www.elsevier.com/locate/cplett
Multiphoton association by two infrared laser pulses Emanuel F. de Lima *, Jose´ E.M. Hornos Departamento de Fı´sica e Cieˆncia dos Materiais, Instituto de Fı´sica de Sa˜o Carlos, USP, CP-369, 13560970, Sa˜o Carlos-SP, Brazil Received 17 October 2006; in final form 13 November 2006 Available online 18 November 2006
Abstract In this theoretical work, we consider the photoassociation reaction of the collision pair O + H induced by ultrashort infrared pulses. We analyze a multiphoton transition aimed at populating an intermediate vibrational state directly from the continuum. Two laser pulses of commensurate carrier frequencies are used to coherent control the photoassociation process. In particular, we investigate how the selective association is sensitive to both relative phase and synchronization of the pulses. Ó 2006 Elsevier B.V. All rights reserved.
1. Introduction Interest in photoassociation reactions has grown recently due to the creation of molecules in the ultracold regime [1–4]. In those reactions, two colliding atoms or radicals are bound by the interaction with an external field. However, in most experimental schemes, vibrationally very highly excited molecules are produced, which are not stable with respect to subsequent collisions [5]. Therefore, an important goal is to reveal physical factors leading to the formation of molecules with low vibrational energy. One route to photoassociation is the use of ultrashort laser pulses to pump a transition from the continuum to a bound vibrational level. For instance, the formation of Hg2 by femtosecond pulses in an electronic excited state has been investigated both experimentally and theoretically [6–8]. On the other hand, the use of infrared radiation to produce molecules in the electronic ground state is possible due to the interaction between the laser and the dipole moment associated to the collision process [9]. The first theoretical investigations of this phenomenon by Korolkov et al. [10,11] indicated the possibility of efficient vibrational state-selective association of O + H. Recently, Niu et al. [12] investigated the relations between the collision momentum and vibrational state-selective association of HI. *
Corresponding author. Fax: +55 1633739827. E-mail address: [email protected] (E.F. de Lima).
0009-2614/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.11.066
Those works were concerned only with the formation of highly excited vibrational states. An interesting question is the generalization of such transitions to much lower levels, which are hampered by the weak bound-continuum dipole coupling. Another relevant subject is the dynamics of atoms and molecules driven by two-color fields of commensurate frequencies [13,14]. Several experiments and theoretical calculations have demonstrated the effects of relative phase of two lasers on ionization and dissociation yields [15–19]. The interpretation of those phenomena is based in the context of quantum interference between the different pathways leading to the same objective [20–22]. Of course, it is expected that similar interference effects appear also in the photoassociation dynamics by infrared pulses. Therefore, it would be possible to coherent control the association yield by adjusting the relative phase or the synchronization of multiple laser pulses. In this Letter, we investigate the multiphoton association (MPA) process induced by infrared laser pulses extending the works of Korolkov et al. [10,11] and Niu et al. [12]. This phenomenon is analogous to the multiphoton dissociation in the inverse direction. Our theoretical calculations indicate that intermediate vibrational levels can be efficiently populated through multiphoton transitions, creating favoring conditions for the formation of vibrationally stable molecules. In order to control the MPA reaction, we use two laser pulses with commensurate car-
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
rier frequencies, x and 2x. A similar two-pulse scheme has been successfully applied to the selective multiphoton excitation of bound levels considering lasers of close frequencies [23]. Correspondingly, we show that the MPA can be coherently controlled manipulating the relative phase and the timing of the pulses. 2. Model and methods The initial state of the system is described by a Gaussian wave-packet in the energy continuum, representing a free scattering state of the collision pair: 1=4 h r r i 2 0 Wðr; t ¼ 0Þ ¼ exp ij r ð1Þ 0 pa2 a hj0 is the relative where r0 is the average initial distance, momentum and a is the width of the wave-packet. The wave-packet collides with the Morse potential, which stands for the bond to be formed: V ðrÞ ¼ D exp½2bðr re Þ 2D exp½bðr re Þ
ð2Þ
where D is the dissociation energy and re is the equilibrium bond length. An external laser field interacts with the dipole moment, l(r), associated with the scattering process. The interaction Hamiltonian is then given in the semiclassical approximation by H int ðr; tÞ ¼ lðrÞEðtÞ
ð3Þ
where E(t) is the electric field component along the dipole moment, which decays exponentially to zero with increasing distance: r lðrÞ ¼ qr exp ð4Þ rd where q is the effective charge and rd gives the range of the interaction. Therefore, the total Hamiltonian of the relative motion is given by H ðr; tÞ ¼
p2 þ V ðrÞ þ H int ðr; tÞ 2m
ð5Þ
In order to solve the corresponding Schro¨dinger equation, we write the wave-function in terms of the eigenstates of the Morse oscillator: Z 1 intðN XÞ Wðr; tÞ ¼ am ðtÞ/m ðrÞ þ aðj; tÞ/ðj; rÞ dj ð6Þ m¼0
0
where the eigenfunctions, /m(r) and /(j,r), can be represented in terms of Kummer functions of first and second kind, respectively [24]. The summation in the above expression refers to the discrete part of the spectrum with energy levels given by Em ¼
2 b2 h 2 ðN mÞ 2m
ð7Þ
49
The integer part of N + 1 gives the number of bound states and it is related to the other molecular constants by (N + 1/ 2)2 = 2mD(h2b2)1. In the continuum sector, the energy is written in terms of j as E(j) = h2b2j2/2m. The equations of motion for the coefficients am(t) and a(j,t) form a finite set of integral–differential equations coupled by the matrix elements of the external field in the unperturbed basis. They can be solved by the continuum expansion method derived in our recent work [25], which provides also exact analytical formulae for all the matrix elements. The technique is based on the expansion of the continuum coefficient a(j): 1 X ~ap ðtÞLp ðjÞ aðj; tÞ ¼ ð8Þ p¼0
where Lp is related to the generalized Laguerre polynomials, Lp, by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p!kkþ1 Lp ðjÞ ¼ ð9Þ jk=2 expðjk=2ÞLkp ðkjÞ Cðk þ p þ 1Þ where k is an arbitrary positive real parameter. The dynamical equations for am(t) and for the new coefficient and a˜p(t) are written as a finite set of ordinary differential equations, if we truncate the summation in Eq. (8) to Nc polynomials. The time-dependent wave-function can then be propagated by algorithms like Runge-Kuta. This technique has proved to be a fast and accurate way to obtain the time-dependent solutions [25]. Following the previous works of Korolkov et al. [10,11], we consider the photoassociation reaction O(3P) + H(1s) ! OH(m). The classical dissociation energy for the OH bond is D = 43769 cm1, which corresponds to ˚, 5.4 eV. The range of the Morse potential, b1, is 0.44 A while the reduced mass is 0.94 of the proton mass. For those parameters, we obtain N = 21.58 and thus there are 22 discrete vibrational states. The effective charge of the dipole is 0.328 of the electronic charge and the range of ˚ . We chose the average initial the interaction, rd, is 0.6 A ˚, distance well outside the interaction region, r0 = 18.77 A ˚. while the width of the wave-packet is set to a = 5.12 A In order to propagate the wave-function, we have set the parameter k to 40.58 and the maximum number of polynomials to Nc = 250. 3. Multiphoton association Multiphoton association can take place when the onephoton transition from the continuum to a top vibrational state, m1, closely match the multiphoton resonance condition to a lower discrete level, m2: Ec Em1 nðEm1 Em2 Þ
ð10Þ
where Ec is the colision energy and n is an integer. The initial average collision energy is chosen to be Ec = 478.76 cm1, which is in the typical order of magnitude of atomic beam experiments. In what follows, we shall concentrate on the population of the states m1 = 17 and
50
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
Table 1 Optimized pulse parameters and final population of the vibrational levels Target
E1 (MV cm1)
x (cm1)
t1 (fs)
m = 17
m = 13
Total
a b c d
190 201 167.7 642
2353 2391 2374 7072
78.6 174.9 106.6 152
0.704 0.037 0.501 0.0
0.087 0.711 0.363 0.195
0.791 0.752 0.865 0.195
m1 = 17 level corresponds roughly to the two-photon transition from m1 = 17 to the m2 = 13 state, that is, n = 2 in Eq. (10). Therefore, three photons with energy of 2363 cm1 can induce a transition from the continuum to the m2 = 13 level. Initially, we consider the electric field given by one fixedshape pulse:
Target-state: (a) m = 17 by one photon; (b) m = 13 by three photons; (c) Total association; (d) m = 13 by one photon.
EðtÞ ¼ E1 sin2 ½pðt t1 Þ=ðDtÞ cosðxtÞ
m2 = 13, which are, respectively, 1884 cm1 and 6610 cm1 bellow the dissociation threshold. For the chosen parameters, the one-photon transition from the continuum to the
where E1 is the amplitude, x is the carrier frequency. The pulse start at t1 and has duration of Dt = 550 fs. As the colliding Gaussian wave-packet approaches the potential well, we turn on the laser pulse. The resulting
ð11Þ
Fig. 1. Photoassociation dynamics. (a) One-photon process: E1 = 642 MV cm1, x = 7072 cm1 and t1 = 152 fs. (b) Three-photon process: E1 = 201 MV cm1, x = 2391 cm1 and t1 = 174.9 fs.
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
association probability depends on three pulse parameters: the carrier frequency, x, the amplitude, E1, and the beginning of the interaction, t1. The traditional quasi-Newton method with finite-difference gradient [26] was used allowing variations of the laser intensity up to 1 GV cm1 and frequencies in the interval of 2000 cm1 around the exact resonance conditions. In Table 1, we show the optimization results for different target states. For each case, the laser parameters are given together with the population of the relevant vibrational levels after the interaction with the pulse. In the case (a), the objective is to induce a one-photon transition to the m = 17 state. Approximately 70% of the population reaches the target level, while a small fraction of 8.7% is found in the m = 13 state, resulting in a total association probability of 80%. In the case (b), the objective is to populate the m = 13 state via a three-photon transition. We obtain 71.1% in the target state, while a residual population of 3.7% appears in the m = 17 level, resulting in a total association probability of 75%. An interesting fact can be noted by comparing the optimized pulse parameters for the above cases. The amplitudes and frequencies are nearly the same, while the beginning of the pulses, t1, differ about 100 fs. When the pulse start at t1 = 78.6 fs the state m = 17 is preferentially populated and for t1 = 174.9 fs results in an efficient population of the m = 13 level. In the case (c), the objective is to maximize the overall association probability. We obtain a population of 50% in the m = 17 superposed to 36% in the m = 13 state, resulting in a total association probability of more than 86%. We note again that the amplitude and frequency are not considerably changed and that the pulse start at t1 = 106.6 fs. In contrast to these results, the direct transition by one-photon from the con-
51
tinuum to the m = 13 level is not very efficient. This fact can be seen in the case (d), where we obtain less than 20% in the target state. This is caused by the weak coupling of the m = 13 state with the continuum. The peak value of the discrete-continuum matrix element of the m = 17 state is at least one order of magnitude greater than the corresponding value for m = 13. In Fig. 1, the photoassociation dynamics by one-photon and three-photon transition are compared. The parameters of the fields correspond to the cases (d) and (b) of Table 1, which have the m = 13 level as the target state. In the direct one-photon transition, depicted in Fig. 1a, we see that only a small fraction of roughly 20% were transferred to the m = 13 state. In contrast, the three-photon transition is highly efficient. As can be seen in Fig. 1b, the target population reaches more than 70%. We can also observe a residual population in the m = 17 state that begins to be populated almost 100 fs before the m = 13 state. 4. Two laser pulses In this section, we consider the photoassociation process induced by two laser pulses with commensurate carrier frequencies, x and 2x. Our objective is to determine the sensitive of the multiphoton association to the relative phase and separation of the pulses. The electric field is now given by two superposed parts: pðt t1 Þ EðtÞ ¼ E1 sin cosðxtÞ þ E2 Dt pðt t2 Þ sin2 cosð2xt þ /Þ Dt 2
ð12Þ
Fig. 2. Photoassociation dynamics induced by two optimized pulses. The parameters of the field are: E1 = 177.57 MV cm1, E2 = 122 MV cm1, x = 2387 cm1, t1 = t2 = 175 fs and / = 1.55p.
52
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
Fig. 3. Population of the bound states as a function of the relative phase of two simultaneous pulses. The parameters of the field are: E1 = 177.57 MV cm1, E2 = 122 MV cm1, x = 2387 cm1, t1 = t2 = 175 fs.
The pulses have the same duration Dt = 550 fs, the first one starts at t = t1 and the second at t = t2. The photoassociation probability depends on the amplitudes, E1, E2, the frequency, x, the relative phase, /, and the timing of the pulses, DT = t1 t2. Initially, we have optimized the pulse parameters considering two simultaneous pulses, t1 = t2 = 175 fs. As in
the previous section, our target is the m = 13 state. The resulting amplitudes are E1 = 177.57 MV cm1 and E2 = 122 MV cm1. The optimized frequency is x = 2387 cm1 and the phase is / = 1.55p. We show the dynamics for those values of the electric field in Fig. 2. It can be seen that the target level reaches more than 83%, while the m = 17 has only a very small final occupation, less than 1%. Compar-
Fig. 4. Population of the bound states as a function of the relative separation of two pulses. The parameters of the field are: E1 = 177.57 MV cm1, E2 = 122 MV cm1, x = 2387 cm1, / = 0 and t1 = 175 fs.
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
ing to Fig. 1b, we observe that the addition of the second pulse can enhance the association probability and also the state-selectivity. In Fig. 3, we show the influence of the relative phase of the pulses on the photoassociation reaction. We plot the final population of the states m = 13, m = 17 and the total association probability as a function of /. The pulses are simultaneous and the remaining parameters are set to the optimized values given in the previous paragraph. The population of the target grows above 80% near / = 3p/2 and remains near 60% in the interval [0, p/2]. The absolute minimum occupation of the m = 13 level corresponds to a maximum of the m = 17 population. This tuning between those states reflects the essential role of the phase on the selective association. We see in the figure that the total association probability follows the same behavior of the population of the m = 13 state, but with smaller amplitude. We note also that the final bound population is restricted to the subspace formed by the states m = 17 and m = 13. In Fig. 4, we show the final occupation of the relevant states as a function of the timing of the pulses, DT = t1 t2. The beginning of the first pulse is fixed at t1 = 175 fs. The vibrational populations oscillate with period in the scale of 10 fs. As in the case of the phase, the overall population follows the behavior of the m = 13 state with smaller amplitude and the maximum occupation of the level m = 17 implies in the minimum population in the state m = 13. Our calculations have shown that this strong oscillatory behavior extends over a wide interval around t1, ± 400 fs, but with decreasing amplitude. 5. Conclusions The present theoretical letter has investigated the multiphoton association process induced by infrared pulses. In our model, a Gaussian wave-packet in the continuum collides with the Morse potential in the electronic ground state of OH(m). A specific intermediate vibrational level, m = 13, was set as the target while the collision energy and the laser parameters were adjusted to pump the desired transition. We have verified that although the target is not accessible via direct one-photon transition, it could be efficiently populated by a three-photon resonance using only one fixedshape pulse. In addition, it was possible to pass from an one-photon transition to a top state, m = 17, to a three-photon transition to a lower state, m = 13, changing only the starting time of the pulse. We showed that is possible to have extensive control of the target population introducing a second laser pulse with commensurate carrier frequency.
53
In analogy with the corresponding coherent control of the ionization and dissociation yields, the photoassociation is highly sensitive to both relative phase and synchronization of the pulses. These parameters can be adjusted to enhance or suppress the multiphoton transition. The MPA may be the first step to create favoring conditions for the formation of vibrationally stable molecules with infrared radiation. The subsequent stabilization of the molecule can be performed by a sequence of pulses aimed at populating the vibrational ground state. Finally, in order to have a more realistic description of the MPA process, our results should be generalized by taking into account the rotational degree of freedom and the electronic transitions. However, multiphoton transitions in the high intensity regime are also expected in more complex systems. References [1] M. Mackie, J. Javanainen, Phys. Rev. A 60 (4) (1999) 3174. [2] J. Vala, O. Dulieu, F. Masnou-Seeuws, P. Pillet, R. Kosloff, Phys. Rev. A 63 (2000) 013412. [3] E. Luc-Koenig, R. Kosloff, F. Masnou-Seeuws, M. Vatasescu, Phys. Rev. A 70 (2004) 033414. [4] K.M. Jones, E. Tiesinga, P.D. Lett, P.S. Julienne, Rev. Mod. Phys. 78 (2) (2006) 483. [5] C.P. Koch, J.P. Palao, R. Kosloff, F. Masnou-Seeuws, Phys. Rev. A 70 (2004) 013402. [6] U. Marvet, M. Dantus, Chem. Phys. Lett. 245 (1995) 393. [7] P. Backhaus, J. Manz, B. Schimidt, Adv. Chem. Phys. 101 (1997) 86. [8] P. Backhaus, B. Schimidt, Chem. Phys. 217 (1997) 131. [9] E. Juarros, P. Pellegrini, K. Kirby, R. Coˆte`, Phys. Rev. A 73 (2006) 041403(R). [10] M.V. Korolkov, J. Manz, G.K. Paramonov, B. Schmidt, Chem. Phys. Lett. 260 (1996) 604. [11] M.V. Korolkov, B. Schmidt, Chem. Phys. Lett. 272 (1997) 96. [12] Y. Niu, S. Wang, S. Cong, Chem. Phys. Lett. 428 (2006) 7. [13] H. Han, P. Brumer, Chem. Phys. Lett. 406 (2005) 237. [14] A.I. Pegarkov, Chem. Phys. Lett. 409 (2005) 8. [15] V. Constantoudis, C.A. Nicolaides, J. Chem. Phys. 122 (2005) 084118. [16] E. Charron, A. Giusti-Suzor, F.H. Mies, Phys. Rev. Lett. 71 (5) (1993) 692. [17] D.W. Schumacher, F. Weihe, H.G. Muller, P.H. Bucksbaum, Phys. Rev. Lett. 73 (1994) 1344. [18] L. Sirko, S.A. Zelazny, P.M. Koch, Phys. Rev. Lett. 87 (2001) 043002. [19] L. Sirko, P.M. Koch, Phys. Rev. Lett. 89 (2002) 274101. [20] M. Shapiro, P. Brumer, Rep. Prog. Phys. 66 (6) (2003) 859. [21] H. Rabitz, R. Vivie-Riedle, M. Motzkus, K. Kompa, Science 288 (2000) 824. [22] I. Walmsley, H. Rabitz, Phys. Today 8 (2003) 43. [23] G.K. Paramonov, Chem. Phys. 177 (1993) 169. [24] E.F. de Lima, J.E.M. Hornos, J. Phys. B 38 (2005) 815. [25] E.F. de Lima, J.E.M. Hornos, J. Chem. Phys. 125 (2006) 164110. [26] P.E. Gill, W. Murray, M.H. Wright, Practical Optimization, Academic Press, London, 1992.
Multiphoton association by two infrared laser pulses Emanuel F. de Lima *, Jose´ E.M. Hornos Departamento de Fı´sica e Cieˆncia dos Materiais, Instituto de Fı´sica de Sa˜o Carlos, USP, CP-369, 13560970, Sa˜o Carlos-SP, Brazil Received 17 October 2006; in final form 13 November 2006 Available online 18 November 2006
Abstract In this theoretical work, we consider the photoassociation reaction of the collision pair O + H induced by ultrashort infrared pulses. We analyze a multiphoton transition aimed at populating an intermediate vibrational state directly from the continuum. Two laser pulses of commensurate carrier frequencies are used to coherent control the photoassociation process. In particular, we investigate how the selective association is sensitive to both relative phase and synchronization of the pulses. Ó 2006 Elsevier B.V. All rights reserved.
1. Introduction Interest in photoassociation reactions has grown recently due to the creation of molecules in the ultracold regime [1–4]. In those reactions, two colliding atoms or radicals are bound by the interaction with an external field. However, in most experimental schemes, vibrationally very highly excited molecules are produced, which are not stable with respect to subsequent collisions [5]. Therefore, an important goal is to reveal physical factors leading to the formation of molecules with low vibrational energy. One route to photoassociation is the use of ultrashort laser pulses to pump a transition from the continuum to a bound vibrational level. For instance, the formation of Hg2 by femtosecond pulses in an electronic excited state has been investigated both experimentally and theoretically [6–8]. On the other hand, the use of infrared radiation to produce molecules in the electronic ground state is possible due to the interaction between the laser and the dipole moment associated to the collision process [9]. The first theoretical investigations of this phenomenon by Korolkov et al. [10,11] indicated the possibility of efficient vibrational state-selective association of O + H. Recently, Niu et al. [12] investigated the relations between the collision momentum and vibrational state-selective association of HI. *
Corresponding author. Fax: +55 1633739827. E-mail address: [email protected] (E.F. de Lima).
0009-2614/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.11.066
Those works were concerned only with the formation of highly excited vibrational states. An interesting question is the generalization of such transitions to much lower levels, which are hampered by the weak bound-continuum dipole coupling. Another relevant subject is the dynamics of atoms and molecules driven by two-color fields of commensurate frequencies [13,14]. Several experiments and theoretical calculations have demonstrated the effects of relative phase of two lasers on ionization and dissociation yields [15–19]. The interpretation of those phenomena is based in the context of quantum interference between the different pathways leading to the same objective [20–22]. Of course, it is expected that similar interference effects appear also in the photoassociation dynamics by infrared pulses. Therefore, it would be possible to coherent control the association yield by adjusting the relative phase or the synchronization of multiple laser pulses. In this Letter, we investigate the multiphoton association (MPA) process induced by infrared laser pulses extending the works of Korolkov et al. [10,11] and Niu et al. [12]. This phenomenon is analogous to the multiphoton dissociation in the inverse direction. Our theoretical calculations indicate that intermediate vibrational levels can be efficiently populated through multiphoton transitions, creating favoring conditions for the formation of vibrationally stable molecules. In order to control the MPA reaction, we use two laser pulses with commensurate car-
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
rier frequencies, x and 2x. A similar two-pulse scheme has been successfully applied to the selective multiphoton excitation of bound levels considering lasers of close frequencies [23]. Correspondingly, we show that the MPA can be coherently controlled manipulating the relative phase and the timing of the pulses. 2. Model and methods The initial state of the system is described by a Gaussian wave-packet in the energy continuum, representing a free scattering state of the collision pair: 1=4 h r r i 2 0 Wðr; t ¼ 0Þ ¼ exp ij r ð1Þ 0 pa2 a hj0 is the relative where r0 is the average initial distance, momentum and a is the width of the wave-packet. The wave-packet collides with the Morse potential, which stands for the bond to be formed: V ðrÞ ¼ D exp½2bðr re Þ 2D exp½bðr re Þ
ð2Þ
where D is the dissociation energy and re is the equilibrium bond length. An external laser field interacts with the dipole moment, l(r), associated with the scattering process. The interaction Hamiltonian is then given in the semiclassical approximation by H int ðr; tÞ ¼ lðrÞEðtÞ
ð3Þ
where E(t) is the electric field component along the dipole moment, which decays exponentially to zero with increasing distance: r lðrÞ ¼ qr exp ð4Þ rd where q is the effective charge and rd gives the range of the interaction. Therefore, the total Hamiltonian of the relative motion is given by H ðr; tÞ ¼
p2 þ V ðrÞ þ H int ðr; tÞ 2m
ð5Þ
In order to solve the corresponding Schro¨dinger equation, we write the wave-function in terms of the eigenstates of the Morse oscillator: Z 1 intðN XÞ Wðr; tÞ ¼ am ðtÞ/m ðrÞ þ aðj; tÞ/ðj; rÞ dj ð6Þ m¼0
0
where the eigenfunctions, /m(r) and /(j,r), can be represented in terms of Kummer functions of first and second kind, respectively [24]. The summation in the above expression refers to the discrete part of the spectrum with energy levels given by Em ¼
2 b2 h 2 ðN mÞ 2m
ð7Þ
49
The integer part of N + 1 gives the number of bound states and it is related to the other molecular constants by (N + 1/ 2)2 = 2mD(h2b2)1. In the continuum sector, the energy is written in terms of j as E(j) = h2b2j2/2m. The equations of motion for the coefficients am(t) and a(j,t) form a finite set of integral–differential equations coupled by the matrix elements of the external field in the unperturbed basis. They can be solved by the continuum expansion method derived in our recent work [25], which provides also exact analytical formulae for all the matrix elements. The technique is based on the expansion of the continuum coefficient a(j): 1 X ~ap ðtÞLp ðjÞ aðj; tÞ ¼ ð8Þ p¼0
where Lp is related to the generalized Laguerre polynomials, Lp, by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p!kkþ1 Lp ðjÞ ¼ ð9Þ jk=2 expðjk=2ÞLkp ðkjÞ Cðk þ p þ 1Þ where k is an arbitrary positive real parameter. The dynamical equations for am(t) and for the new coefficient and a˜p(t) are written as a finite set of ordinary differential equations, if we truncate the summation in Eq. (8) to Nc polynomials. The time-dependent wave-function can then be propagated by algorithms like Runge-Kuta. This technique has proved to be a fast and accurate way to obtain the time-dependent solutions [25]. Following the previous works of Korolkov et al. [10,11], we consider the photoassociation reaction O(3P) + H(1s) ! OH(m). The classical dissociation energy for the OH bond is D = 43769 cm1, which corresponds to ˚, 5.4 eV. The range of the Morse potential, b1, is 0.44 A while the reduced mass is 0.94 of the proton mass. For those parameters, we obtain N = 21.58 and thus there are 22 discrete vibrational states. The effective charge of the dipole is 0.328 of the electronic charge and the range of ˚ . We chose the average initial the interaction, rd, is 0.6 A ˚, distance well outside the interaction region, r0 = 18.77 A ˚. while the width of the wave-packet is set to a = 5.12 A In order to propagate the wave-function, we have set the parameter k to 40.58 and the maximum number of polynomials to Nc = 250. 3. Multiphoton association Multiphoton association can take place when the onephoton transition from the continuum to a top vibrational state, m1, closely match the multiphoton resonance condition to a lower discrete level, m2: Ec Em1 nðEm1 Em2 Þ
ð10Þ
where Ec is the colision energy and n is an integer. The initial average collision energy is chosen to be Ec = 478.76 cm1, which is in the typical order of magnitude of atomic beam experiments. In what follows, we shall concentrate on the population of the states m1 = 17 and
50
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
Table 1 Optimized pulse parameters and final population of the vibrational levels Target
E1 (MV cm1)
x (cm1)
t1 (fs)
m = 17
m = 13
Total
a b c d
190 201 167.7 642
2353 2391 2374 7072
78.6 174.9 106.6 152
0.704 0.037 0.501 0.0
0.087 0.711 0.363 0.195
0.791 0.752 0.865 0.195
m1 = 17 level corresponds roughly to the two-photon transition from m1 = 17 to the m2 = 13 state, that is, n = 2 in Eq. (10). Therefore, three photons with energy of 2363 cm1 can induce a transition from the continuum to the m2 = 13 level. Initially, we consider the electric field given by one fixedshape pulse:
Target-state: (a) m = 17 by one photon; (b) m = 13 by three photons; (c) Total association; (d) m = 13 by one photon.
EðtÞ ¼ E1 sin2 ½pðt t1 Þ=ðDtÞ cosðxtÞ
m2 = 13, which are, respectively, 1884 cm1 and 6610 cm1 bellow the dissociation threshold. For the chosen parameters, the one-photon transition from the continuum to the
where E1 is the amplitude, x is the carrier frequency. The pulse start at t1 and has duration of Dt = 550 fs. As the colliding Gaussian wave-packet approaches the potential well, we turn on the laser pulse. The resulting
ð11Þ
Fig. 1. Photoassociation dynamics. (a) One-photon process: E1 = 642 MV cm1, x = 7072 cm1 and t1 = 152 fs. (b) Three-photon process: E1 = 201 MV cm1, x = 2391 cm1 and t1 = 174.9 fs.
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
association probability depends on three pulse parameters: the carrier frequency, x, the amplitude, E1, and the beginning of the interaction, t1. The traditional quasi-Newton method with finite-difference gradient [26] was used allowing variations of the laser intensity up to 1 GV cm1 and frequencies in the interval of 2000 cm1 around the exact resonance conditions. In Table 1, we show the optimization results for different target states. For each case, the laser parameters are given together with the population of the relevant vibrational levels after the interaction with the pulse. In the case (a), the objective is to induce a one-photon transition to the m = 17 state. Approximately 70% of the population reaches the target level, while a small fraction of 8.7% is found in the m = 13 state, resulting in a total association probability of 80%. In the case (b), the objective is to populate the m = 13 state via a three-photon transition. We obtain 71.1% in the target state, while a residual population of 3.7% appears in the m = 17 level, resulting in a total association probability of 75%. An interesting fact can be noted by comparing the optimized pulse parameters for the above cases. The amplitudes and frequencies are nearly the same, while the beginning of the pulses, t1, differ about 100 fs. When the pulse start at t1 = 78.6 fs the state m = 17 is preferentially populated and for t1 = 174.9 fs results in an efficient population of the m = 13 level. In the case (c), the objective is to maximize the overall association probability. We obtain a population of 50% in the m = 17 superposed to 36% in the m = 13 state, resulting in a total association probability of more than 86%. We note again that the amplitude and frequency are not considerably changed and that the pulse start at t1 = 106.6 fs. In contrast to these results, the direct transition by one-photon from the con-
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tinuum to the m = 13 level is not very efficient. This fact can be seen in the case (d), where we obtain less than 20% in the target state. This is caused by the weak coupling of the m = 13 state with the continuum. The peak value of the discrete-continuum matrix element of the m = 17 state is at least one order of magnitude greater than the corresponding value for m = 13. In Fig. 1, the photoassociation dynamics by one-photon and three-photon transition are compared. The parameters of the fields correspond to the cases (d) and (b) of Table 1, which have the m = 13 level as the target state. In the direct one-photon transition, depicted in Fig. 1a, we see that only a small fraction of roughly 20% were transferred to the m = 13 state. In contrast, the three-photon transition is highly efficient. As can be seen in Fig. 1b, the target population reaches more than 70%. We can also observe a residual population in the m = 17 state that begins to be populated almost 100 fs before the m = 13 state. 4. Two laser pulses In this section, we consider the photoassociation process induced by two laser pulses with commensurate carrier frequencies, x and 2x. Our objective is to determine the sensitive of the multiphoton association to the relative phase and separation of the pulses. The electric field is now given by two superposed parts: pðt t1 Þ EðtÞ ¼ E1 sin cosðxtÞ þ E2 Dt pðt t2 Þ sin2 cosð2xt þ /Þ Dt 2
ð12Þ
Fig. 2. Photoassociation dynamics induced by two optimized pulses. The parameters of the field are: E1 = 177.57 MV cm1, E2 = 122 MV cm1, x = 2387 cm1, t1 = t2 = 175 fs and / = 1.55p.
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E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
Fig. 3. Population of the bound states as a function of the relative phase of two simultaneous pulses. The parameters of the field are: E1 = 177.57 MV cm1, E2 = 122 MV cm1, x = 2387 cm1, t1 = t2 = 175 fs.
The pulses have the same duration Dt = 550 fs, the first one starts at t = t1 and the second at t = t2. The photoassociation probability depends on the amplitudes, E1, E2, the frequency, x, the relative phase, /, and the timing of the pulses, DT = t1 t2. Initially, we have optimized the pulse parameters considering two simultaneous pulses, t1 = t2 = 175 fs. As in
the previous section, our target is the m = 13 state. The resulting amplitudes are E1 = 177.57 MV cm1 and E2 = 122 MV cm1. The optimized frequency is x = 2387 cm1 and the phase is / = 1.55p. We show the dynamics for those values of the electric field in Fig. 2. It can be seen that the target level reaches more than 83%, while the m = 17 has only a very small final occupation, less than 1%. Compar-
Fig. 4. Population of the bound states as a function of the relative separation of two pulses. The parameters of the field are: E1 = 177.57 MV cm1, E2 = 122 MV cm1, x = 2387 cm1, / = 0 and t1 = 175 fs.
E.F. de Lima, J.E.M. Hornos / Chemical Physics Letters 433 (2006) 48–53
ing to Fig. 1b, we observe that the addition of the second pulse can enhance the association probability and also the state-selectivity. In Fig. 3, we show the influence of the relative phase of the pulses on the photoassociation reaction. We plot the final population of the states m = 13, m = 17 and the total association probability as a function of /. The pulses are simultaneous and the remaining parameters are set to the optimized values given in the previous paragraph. The population of the target grows above 80% near / = 3p/2 and remains near 60% in the interval [0, p/2]. The absolute minimum occupation of the m = 13 level corresponds to a maximum of the m = 17 population. This tuning between those states reflects the essential role of the phase on the selective association. We see in the figure that the total association probability follows the same behavior of the population of the m = 13 state, but with smaller amplitude. We note also that the final bound population is restricted to the subspace formed by the states m = 17 and m = 13. In Fig. 4, we show the final occupation of the relevant states as a function of the timing of the pulses, DT = t1 t2. The beginning of the first pulse is fixed at t1 = 175 fs. The vibrational populations oscillate with period in the scale of 10 fs. As in the case of the phase, the overall population follows the behavior of the m = 13 state with smaller amplitude and the maximum occupation of the level m = 17 implies in the minimum population in the state m = 13. Our calculations have shown that this strong oscillatory behavior extends over a wide interval around t1, ± 400 fs, but with decreasing amplitude. 5. Conclusions The present theoretical letter has investigated the multiphoton association process induced by infrared pulses. In our model, a Gaussian wave-packet in the continuum collides with the Morse potential in the electronic ground state of OH(m). A specific intermediate vibrational level, m = 13, was set as the target while the collision energy and the laser parameters were adjusted to pump the desired transition. We have verified that although the target is not accessible via direct one-photon transition, it could be efficiently populated by a three-photon resonance using only one fixedshape pulse. In addition, it was possible to pass from an one-photon transition to a top state, m = 17, to a three-photon transition to a lower state, m = 13, changing only the starting time of the pulse. We showed that is possible to have extensive control of the target population introducing a second laser pulse with commensurate carrier frequency.
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