Laser Control of Vibrational Transfer Based on Exceptional Points

Oct 26, 2009 - ... of an open quantum system. Raam Uzdin , Roland Lefebvre. Journal of Physics B: Atomic, Molecular and Optical Physics 2010 43, 23500...
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J. Phys. Chem. A 2010, 114, 3031–3037

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Laser Control of Vibrational Transfer Based on Exceptional Points† O. Atabek* and R. Lefebvre‡ Laboratoire de Photophysique Mole´culaire du CNRS, UniVersity Paris-Sud, Baˆtiment 210, Campus d’Orsay 91405, Orsay, France ReceiVed: June 29, 2009; ReVised Manuscript ReceiVed: September 28, 2009

The coalescence of two Floquet resonances with complex quasi-energies in the photodissociation of a molecule can be reached through an appropriate choice of laser parameters (wavelength and intensity). The corresponding point in the parameter plane is called exceptional. We show that a condition for two resonances to yield such a point is that they correspond, respectively, to a shape-like resonance and a Feshbach-like resonance. We examine how the resonances behave along a contour in the parameter plane, which goes around the exceptional point. The resonances exchange their labels. This amounts to a laser control of the vibrational transfer of the undissociated molecules from one field-free state to another. Introduction Laser-induced and -controlled molecular fragmentation dynamics and transition-state observations in chemical reactions have been among the basic concerns of Benoıˆt Soep.1 The interpretations of such processes share the concept of resonance. Moreover, the control scenarios themselves rest on basic mechanisms involving properties of resonances. This article is devoted to a laser control of vibrational state transfer taking advantage of resonance coalescence. The complex resonance energies, which are the poles of the scattering matrix, provide a simple description of fragmentation processes.2 The imaginary part of such energies gives an estimate of the rate at which an initial state is decaying. There has recently been considerable interest in the possibility of producing, by a proper choice of the parameters of the equations governing a system, the degeneracy of two of its complex eigenenergies. Some examples belong to classical physics, such as a pair of coupled damped oscillators3 or the field distribution in a microwave cavity.4 Many examples are found in quantum physics, for atoms,5,6 in electronH2 collisions,7 or for a model of a sequence of coupled rectangular wells.8 Other examples concern many-body systems, such as quantum phase transitions9 or superconductors.10 When a resonance coalescence is achieved, the corresponding point in parameter space is called an exceptional point (EP).11,12 This is to be distinguished from the well-known situation where energy-degenerate states are associated with two different eigenfunctions, as is the case when the degeneracy is due to symmetry. As shown in ref 13, EPs are branch points where a self-orthogonal state is obtained. The concept of self-orthogonality may be surprising, but it must be emphasized that because of the special boundary conditions valid for a resonance, the usual scalar product has to be replaced with the one for biorthogonal functions.14,15 Another consequence of EPs is that by adiabatic variation of the parameters along a closed contour around such a branch point, it is possible to go from one nondegenerate resonance pole of the scattering matrix to another.16,17 †

Part of the “Benoıˆt Soep Festschrift”. * Corresponding author. E-mail: [email protected]. ‡ Also at UFR de Physique Fondamentale et Applique´e, Universite´ Pierre et Marie Curie, 75321 Paris, France.

We discuss here the case of laser-induced photodissociation of a diatomic molecular species which, for a continuous wave (cw) field, can be treated with the Floquet formalism.18 We have already proved that EPs are present in the photodissociation of a molecule and presented some of the consequences.19 We pursue this problem, with emphasis on the nature of the states leading to coalescence for two other resonances of the same system at a different wavelength. This is an opportunity to study some additional properties of EPs in this context. The eigenenergies of the Floquet Hamiltonian are called quasi-energies. They are complex because the system is unstable. Because the applied electric field amplitude and its wavelength are two parameters which can be chosen at will, there is a considerable richness of situations which can be produced. The questions we address are: (1) Can we predict which states can lead to a pole coalescence when a molecule photodissociates as a result of the interaction with a laser field? (2) What are the fingerprints of the coalescence of two photodissociation resonance states? (3) What are the best conditions to produce a transition from one field-free vibrational state to another when varying adiabatically the laser parameters in a closed contour around the EP? After consideration of these different issues, their experimental feasibility is briefly discussed in the conclusions. Floquet Formalism If H(R, t) is the periodic time-dependent Hamiltonian describing a 1D molecular system interacting with a cw field, then the solution of the time-dependent Schro¨dinger equation is given by

Ψ(R, t) ) e-iEFt/pΦF(R, t)

(1)

where ΦF(R, t) is a periodic function of time that satisfies the eigenvalue equation

HF(R, t)ΦF(R, t) ) EFΦF(R, t) with the Floquet Hamiltonian being given by

10.1021/jp9060736  2010 American Chemical Society Published on Web 10/26/2009

(2)

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HF(R, t) ) H(R, t) - ip

∂ ∂t

Atabek and Lefebvre

(3)

EF is the quasi-energy, which, for time-periodic Hamiltonians, plays to some extent the same role as an eigenenergy of a timeindependent Hamiltonian. Our model is 1D and involves two electronic states. H2+ is an illustrative example of this situation18 with a ground electronic state X2Σg+ and an excited state A2Σu+ denoted hereafter as |g〉 and |u〉, respectively. The ground-state potential accommodates 19 bound vibrational levels, whereas the excited-state potential is purely repulsive. The wave function is written

|Ψ(R, t)〉 ) χg(R, t)|g〉 + χu(R, t)|u〉

(4)

R is the nuclear coordinate. We adopt the length-gauge and the long-wavelength approximation for the matter-field interaction. The nuclear wave functions are solutions of the time-dependent Schro¨dinger equation

[H - ip ∂t∂ 1]Ξ(R, t) ) 0

(5)

Ξ is the column vector made of χg and χu. H is the matrix operator

H ) 1TR +

[

Vg(R) d(R)ε(t) d(R)ε(t) Vu(R)

]

(6)

TR )(-p2/(2µH2+))(d2/dR2) is the usual vibrational kinetic energy operator, where µH2+ is the reduced mass. Vg(R) and Vu(R) are the two Born-Oppenheimer potentials. d(R) is the electronic transition moment between states |g〉 and |u〉. ε(t) is the linearly polarized laser electric field amplitude of the form ε0 cos(ωt), with a wavelength λ ) 2πc/ω and intensity I ∝ ε02. Applying the Floquet ansatz to this case consists of writing the vector wave function as

[

]

[

χg(R, t) φg(R, t) ) e-iEFt/p χu(R, t) φu(R, t)

]

(7)

Because φk(R, t) (k ) g, u) is time periodic, it can be Fourier expanded as +∞

φk(R, t) )

∑

einωtφnk (R)

(8)

n)-∞

The channel wave functions φnk (R) (k ) g, u) are solutions of the coupled equations

[T + Vg + npω - EF]φng - V [φn-1 + φn+1 u u ] ) 0

Figure 1. Field-dressed potentials of H2+ involved in the Floquet formalism. The wavelength is λ ) 800 nm. Left panel: six potentials extracted from the infinite series implied in the coupled eq 9. The labels indicate the g or u states, together with the number of absorbed (n < 0) or emitted (n > 0) photons. The two potentials in dotted red describe the single-photon dissociation process. The two potentials above describe two- and three-photon (virtual) emission, whereas the two below are for two- and three-photon absorption. Right panel: the adiabatic potentials obtained from the potentials shown in red dotted lines in the left panel by diagonalization of the matter-field interaction. The intensity is 0.13 × 1013 W/cm2. The two horizontal lines show energies representative of a shape resonance of V- (the lower one) and of a Feshbach resonance of V+ (the upper one).

g, n ) 2, 4,..., or u, n ) 1, 3,..., which correspond to virtually emitted photons, whereas channels g, n ) -2,-4..., and u, n ) -1,-3,..., correspond to photon absorption. The calculations for a given intensity must be carried out with an increasing number of channels until convergency is reached. Such a test is presented below. The solution with null values at the origin (R ) 0) and Siegert asymptotic outgoing wave boundary conditions in the open channels produces a complex quasi-energy of the form EF ) Re(EF) - iΓF/2. In the following, EF ) Re(EF) will be called the energy of the resonance and ΓF will be called its width. The equations are solved with a matching technique based on the Fox-Goodwin propagator20 with exterior complex scaling.21 This implies the choice of a matching point where continuity of outward and inward propagated functions is ensured.22 The calculations are first done with two channels only, that is φg0(R) and φu-1(R), designated briefly as φg(R) and φu(R). The effect of adding further channels is examined later. Because these functions are obtained from the coupled equations involving the field dressed potentials, which cross each other in Figure 1, they are designated as diabatic. It is possible, for each value of the coordinate R, to obtain through a linear transformation22 the adiabatic channel functions associated with the adiabatic potentials V-(R) and V+(R) of the right panel of Figure 1. The linear transformation consists of applying to the diabatic vector solution the matrix diagonalizing the potential matrix. This is an indirect way of obtaining the solutions of the coupled equations in the adiabatic representation, including nonadiabatic kinetic couplings. These solutions are denoted ψ-(R) and ψ+(R). Application to H2+: The Search for an Exceptional Point

+ φn+1 [T + Vu + npω - EF]φnu - V [φn-1 g g ] ) 0

(9) where V stands for 1/2ε0 d(R). The single-photon two-channel approximation involves only φg0 and φu-1, labeled as g, 0 and u,-1. Multiphoton processes are described through channels

The potentials involved in the coupled equations (eq 9) when applying the Floquet formalism to H2+ are presented in the left panel of Figure 1. They are built with the parameters proposed by Bunkin and Tugov.23 The adiabatic potentials (Figure 1) are very useful for the understanding of the resonances. At high

Laser Control of Vibrational Transfer

Figure 2. Real parts of the Floquet quasi-energies (left panel) and the associated widths (right panel) for the resonances issued from a few field-free vibrational states of H2+ (υ ) 10-13) as a function of laser intensity. The wavelength is 800 nm. The avoided crossing between 12 and 13 is at the source of the entire analysis of this article. Also shown in the left panel (red curves) are the first two bound energies (υ+ ) 0, 1) of the upper adiabatic potential.

intensity, we expect the states associated with these adiabatic potentials to be very good zeroth order approximations for the resonances of the full problem. We have previously observed24,25 that the Floquet resonances fall into two categories, those which at high intensity are of Feshbach type, and those which are essentially of shape nature. The first class are precisely those with energies increasing with intensity and finally merging with the levels of the upper adiabatic potential. The others have energies asymptotically decreasing as the barrier of the lower adiabatic potential is lowered. We will call them, in brief, “Feshbach” and “shape”, respectively. We present in Figure 2 the real parts of the Floquet quasi-energies from 10 to 13 when the wavelength is 800 nm (close to the Ti/sapphire laser). The resonances are labeled with the quantum number of the fieldfree vibrational state from which they are issued. The energies labeled 11 and 13 belong to the first class, as shown by comparison with the adiabatic energies (υ+ ) 0, 1). States 10 and 12, on the contrary, have energies decreasing for strong enough field and are to be regarded as being shape. An avoided crossing involving 12 and 13 is visible in the left panel of Figure 2. It involves a pair of states belonging to the two different classes. There is clearly an interchange of their characters (shape versus Feschbach) at an intensity of ∼0.13 × 1013 W/cm2. The right panel of Figure 2 gives the widths as a function of intensity for these same resonances. It is seen that at the intensity producing the avoided crossing between 12 and 13, the widths are comparable so that there is proximity between the complex Floquet energies of these two states. We study in more detail the character interchange between 12 and 13 when going across the avoided crossing by looking at the wave functions of these two resonances. Figures 3 and 4 show the real parts of the adiabatic wave functions ψ-(R) and ψ+(R) for the two resonances υ ) 12 and 13 for two intensities, one before the avoided crossing (I ) 0.05 × 1013 W/cm2) and one after (I ) 0.25 × 1013 W/cm2). For υ ) 12, after the avoided crossing, the dominant component is ψ-(R), whereas it is ψ+(R) for υ ) 13. The latter function has essentially the nodal structure of a bound state function with a quantum number υ+ ) 1. We conclude this part of the analysis by emphasizing that the avoided crossing is, as is generally the case, the signature of a change of nature of the eigensolutions. This dichotomy is

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Figure 3. Real parts of the adiabatic wave functions ψ- (R) (in black) and ψ+ (R) (in red) for the resonance υ ) 12 at two different intensities: 0.05 × 1013 (upper panel) and 0.25 × 1013 W/cm2 (lower panel). This is before and after the avoided crossing depicted in Figure 2. The lower channel is dominant at the higher intensity. This is interpreted as a signature of the shape-like character of this resonance. All wave functions are damped beyond R ) 8 au because of the action of the exterior complex scaling transformation.

Figure 4. Same as Figure 3 for the resonance υ ) 13. The dominant channel after the avoided crossing is now the upper one. This Feshbach resonance merges finally with the υ+ ) 1 state of the upper adiabatic potential, as shown by its nodal structure.

expected for any diatomic molecule showing field-dressed potentials with a crossing such as that occurring in Figure 1. To achieve coalescence, we calculate the resonance energies as a function of intensity with various values of the wavelength. Figure 5 shows the widths of two such calculations for λ ) 780 and 788.2 nm. This shows that the latter value of λ reduces the gap between the two widths. It is important to notice that for the two values of the wavelength, 12 is Feshbach, whereas 13 is shape. Our best estimate for the parameters leading to resonance coalescence is λEP ) 788.26 nm and IEP ) 0.1323 × 1013 W/cm2. Figure 6 gives the widths and the real parts of the Floquet quasi-energies for the pair of states near resonance coalescence. The left panel is for λ ) 788.2 nm, slightly below λEP, whereas the right panel is for λ ) 788.3 nm, slightly above λEP. We note the interchange in the characters of the two resonance states when going across λEP. This is an essential feature when selecting the best trajectory in the parameter plane to induce a change of label of a resonance state, as discussed below.

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Figure 5. Widths as a function of intensity on the way to the exceptional point for two wavelengths (thin red for λ ) 780 nm and thick black for λ ) 788.2 nm) close to λEP ) 788.26 nm. Solid lines for resonance 12; dashed lines for resonance 13.

Figure 6. Widths and energies as a function of intensity for λ ) 788.2 nm (left panel) and λ ) 788.3 nm (right panel). λEP is estimated to be 788.26 nm. In the left panel, 12 (solid line) is a Feshbach state with a decreasing width beyond IEP ) 0.1323 × 1013 W/cm2, whereas 13 (dashed line) is shape with an increasing width. In the right panel, the situation is reversed.

Figure 7. Real parts of the diabatic wave functions in the channels g and u for three different intensities and λ ) λEP (788.26 nm). Left column: with no field (I ) 0); middle column, with a field halfway to IEP (I ) 0.061× 1013 W/cm2); right column, with a field close to IEP (I ) 0.132 × 1013 W/cm2). The thick black lines are for the resonance υ ) 12, whereas the thin red lines are for υ ) 13.

There are two possibilities. If the operator ∂HF/∂ε0 commutes with the Hamiltonian, then its action on |ΦF〉 is some constant multiplying |ΦF〉, and the ratio in eq 11 becomes this constant at the EP. In our case, ∂HF/∂ε0 ) d(R) cos(ωt), which does not commute with HF so that only the denominator goes to zero. This shows that when approaching the EP, the intensity derivatives of both the real part of EF (lower panels of Figure 6) and of the width (upper panels of Figure 6) are going to infinity. We also note that if we had taken λ and ε0 as the parameters, we would have two EPs at λEP and (ε0EP. We have already mentioned a very spectacular property of resonance wave functions at an EP: As the poles coalesce, there is in fact a single wave function defined at this point13 that has the property of self-orthogonality. The so-called self-overlap σ of a given resonance solution ΦF(R, t) is evaluated as

σ ) (ΦF(R, t)||ΦF(R, t)) n)+∞

)

∞ ∑ ∫0 dR[(φng(R))2 + (φnu(R))2]

(12)

n)-∞

Self-Orthogonality and Coalescence of Wave Functions In Figure 6, the tangents to the curves giving the widths and the energies near the EPs are vertical with a cusp shape. This is due to the self-orthogonality property of the wave functions at the EP.13 The quasi-energy EF can be written

EF )

(ΦF |HF |ΦF) (ΦF |ΦF)

(10)

The rule for biorthogonal wave functions is used, this meaning that there is no complex conjugation involved in the integrals. This is symbolized by the use of parentheses instead of the usual bracket notation. EF is finite even when (ΦF|ΦF) ) 0 because HFΦF ) EFΦF; however, with the help of the Helmann-Feynman theorem, we can also write26

∂EF ∂EF ∂ε0 (ΦF |∂HF/∂ε0 |ΦF) ∂ε0 ) ) ∂I ∂ε0 ∂I (ΦF |ΦF) ∂I

(11)

The (||) notation stands for integration over both R and t.21 Expression 12 is used below to study the self-orthogonality phenomenon that occurs whenever two photodissociation resonances coalesce. The fact that the squares instead of the square moduli of the wave function components appear in eq 12 is a consequence of the biorthogonal properties of the wave functions.14,15 Calculations of the wave functions φg and φu are done for three field intensities and λEP. Their real parts are shown in Figure 7. All wave functions are calculated with the convention that the real part of φg(R) at the matching point is unity. The left column corresponds to a null field for which the two energies are, respectively, -2675 cm-1 for υ ) 12 and -1923 cm-1 for υ ) 13 cm-1. The nodal structure in φg is easily recognizable. The middle column is for I ) 0.066 × 1013 W/cm2, halfway to the EP. The energies are, respectively, EF ) -2434 - i 89 cm-1 and EF ) -1787 - i 159 cm-1. The wave function components for the two resonance states are clearly different. For I ) 0.332 × 1013 W/cm2, close to the EP, the two resonances can still be labeled unequivocally as υ ) 12 and 13 with energies -2042 - i 276 cm-1 and -1937 - i

Laser Control of Vibrational Transfer

Figure 8. Test of the self-orthogonality property of the resonance wave functions. The wavelength is 788.89 nm. The absolute value of the self-overlap σ (eq 12) is calculated as a function of intensity for the resonance wave functions υ ) 12 and 13. The minimum occurs when I passes close to IEP ) 0.1323 W/cm2. It should be zero at the exceptional point.

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Figure 10. Trajectories in the energy plane when the system is exposed to a chirped pulse. Upper panel: the two possibilities, with the initial state being either υ ) 12 (solid curve) or 13 (dashed curve). The lower panel shows the pulse in the parameter plane. Also shown are the position of the energy calculated with the parameters of the exceptional point (open circle in the upper panel) and the position of the exceptional point in the parameter plane (full circle in the lower panel).

crossing the line extending from the EP to the right in the parameter plane, there is a change of identity (12 to 13 or vice versa) but no change of character (shape or Feshbach). Transfer around the Exceptional Point

Figure 9. Resonance energies υ ) 12 and 13 for λ+ ) 788.89 nm, just above λEP ) 788.26 nm, or λ- ) 787.63 nm, just below λEP, as a function of intensity. The left panel gives 12 with λ+, which is shape, and 13 with λ-, which is also shape. The right panel gives 12 with λ(Feshbach) and 13 with λ+ (also Feshbach). The energies approach each other in both cases when I approaches IEP and then follow each other closely beyond the EP. The maximum intensity is 0.2 × 1013 W/cm2. The open circle EP denotes the energy with the parameters of the exceptional point.

304 cm-1, respectively. The wave functions are almost undistinguishable, as shown in the right column. All wave functions are calculated with the convention that the real part of φg(R) at the matching point is unity. The values of |σ| ) |(φg|φg) +(φu|φu)| are displayed in Figure 8 as a function of intensity for υ ) 12 and 13. Close to the EP, a considerable reduction is observed. The integration is performed only up to the point where exterior scaling is introduced (R ) 8 au). Figure 9 is displaying another interesting feature of the resonances. We show the resonance energies 12 and 13 as a function of intensity for two wavelengths: either for a wavelength slightly above λEP, denoted λ+, or for a wavelength slightly below λEP, denoted λ-. As expected the energies become close when the intensity approaches IEP. Beyond this value, they remain close, however, with two distinct regimes. Twelve with λ+ and 13 with λ- are both shape with high values of the widths. For 13 with λ+ and 12 with λ-, the widths are smaller and typical of Feshbach resonances. These proximities show that when

Identification of an EP should have other consequences. According to the analysis presented, for example, in ref 16, it should be possible to devise a contour in parameter plane to go continuously from one pole to another. Such a contour should go around the EP, which in fact is a branch point defining a branch cut. It should be desirable, for a control issue, to go from one field-free state (i.e., a pole on the real axis) to another such pole by applying adiabatically a chirped pulse. This, of course, requires the dissociation not to be complete along the path. We have built such contours with an intensity and a wavelength obeying the laws

I ) Imax sin(φ/2);

λ ) λ0 + δλ sin(φ)

(13)

φ takes a set of values from 0 to 2π. Figure 10 shows the loop in parameter plane (lower panel) and the resulting trajectories in the energy plane (upper panel). The trajectories 12 to 13 or 13 to 12 involve widely different widths, in conformity with the previous analysis about Feshbach and shape resonance states. We have observed that if λ0 is 788.26 nm, that is, the wavelength for the EP of the pair 12-13, with Imax ) 0.5 × 1013 W/cm2 and δλ ) 10 nm, then the label change takes place precisely when λ ) λEP for an intensity I > IEP. This shows that a cut starts at λ ) λEP with I g IEP. We may also mention that there is a geometric phase going along with loops around an EP.12,27 This phase is (π after two loops around the EP. To obtain a well-defined estimate of the fraction of undissociated molecules left after completion of a loop, we shall employ the formalism of adiabatic Floquet theory.25,28 This assumes that the chirped laser pulse envelope and its frequency vary sufficiently slowly with time so that the overall fraction of nondissociated molecules PND is given as

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PND ) exp[-

∫0t ΓF(t) dt] f

Atabek and Lefebvre

(14)

Here ΓF(t) is associated with the relevant Floquet quasi-energy eigenvalue calculated using the instantaneous field parameters at time t, and the symbol tf stands for the duration of the light pulse. Between the angle φ determining the loop in the parameter plane and time, we use the relation t ) φtf/2π. We present in Figure 11 PND for two cases: either starting from 12 to go to 13 or the reverse process. The pulse duration tf is 56 fs. This has been previously shown,25 by a comparison with a wave packet propagation to lead to an adiabatic transfer. The results differ drastically. The explanation is to be found again on the nature of the instantaneous resonance states, which are created during the pulse. In the first case (12 to 13), during the first half of the pulse with λ g λEP, the resonance state is shape with high values of the widths. In the second half of the pulse, it matches with the resonance 13, again with large widths, because this resonance is also shape. For the transfer 13 to 12, it is the reverse: the Feshbach resonances of the first half of the pulse match with the Feshbach resonances of the second half after the change of label when going across λEP. There is therefore a better chance of preserving the molecules from dissociating when the transfer amounts to a decrease in the vibrational quantum number. In the latter case, 20% of the molecules survive the dissociation. This is of general occurrence, as shown in ref 19 for the pair 8-9 at quite different wavelengths (λ ≈ 440 nm). Multichannel Case Because all of the calculations presented here are so far with two channels only, it is necessary to check if the transfer from a field-free vibrational state to another persists when additional channels are introduced until convergency is achieved. Our calculations presented in Figure 12 show that there is no significant change in the trajectory when up to six Floquet channels are introduced (the situation depicted in Figure 1). Convergence is reached with this number of channels. The transfers presented here are not linked to the two-channel approximation but are a true property of the Floquet formalism applied to this problem. The EP still exists when further channels are introduced. For the converged calculation, the EP has coordinates λEP ) 789.2 nm and IEP ) 0.125 W/cm2 within the loop used for the two-channel calculation. This observation is important because taking into account the rotational degree of

Figure 12. Check of convergence with respect to the number of channels for the two transitions 12 to 13 (left panel) and 13 to 12 (right panel). N is the total number of channels, whereas M is the number of closed blocks above the reference block. The calculation with N ) 8, M ) 1 gives the same result as N ) 6, M ) 1.

freedom is also made through the introduction of additional channels.25 The existence of the EP and the role it plays for the transfers studied here are not linked to the two-channel approximation but are true properties of the Floquet formalism. The introduction of the rotational degree of freedom will, of course, slightly affect the position of the EP in the parameter plane. However, a sufficiently wide loop, such as the one estimated from two-channels frozen rotational approximation, is expected to be convenient for observing the transfer. Conclusions In conclusion, we have seen that pole coalescence can be easily produced in the laser-induced photodissociation of a molecular system. This is illustrated for the case of H2+ with realistic laser parameters. This should be transposable for other molecular species because the wavelength and the amplitude of the applied electric field are two parameters that give considerable freedom to manipulate the quasi-energies. A timeresolved pump-probe experiment could reveal the adiabatic transition driving the system around the resonance energy of the EP. A chirped pump laser with time-dependent intensities and wavelengths adiabatically following a loop such as the one illustrated in Figure 10 (lower panel) would convey the population of a vibrational state υ onto υ′. If υ is initially much more populated than υ′, then at the end of the pump pulse, υ′ will recover the essential of the nondissociated population. This could be tested by probe lasers tuned to induce fluorescence from both states. The efficiency of the proposed control scenario to change the vibrational state of a molecule at will remains to be checked with the time-dependent approach (wave packet evolution). Another outcome of this work is a clear identification of the states that are possible candidates for pole coalescence in the neighborhood of a given wavelength. Acknowledgment. We thank Pr. N. Moiseyev for helping us clarify the self-orthogonality concept and for use of the Hellmann-Feynman theorem and Prs. A. Mondrago´n and A. Keller for fruitful discussions.

Figure 11. Probability PND for a molecule not to dissociate after being exposed to a chirped pulse. Two different transitions are tested: either from 12 to 13 or from 13 to 12. The duration of the pulse is 56 fs.

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