Femtosecond pulses and dynamics of molecular photoexcitation: RbCs example
We investigate the dynamics of molecular photoexcitation by unchirped femtosecond laser pulses using RbCs as a model system. This study is motivated by a goal of optimizing a two-color scheme of transferring vibrationally-excited ultracold molecules to their absolute ground state. In this scheme the molecules are initially produced by photoassociation or magnetoassociation in bound vibrational levels close to the first dissociation threshold. We analyze here the first step of the two-color path as a function of pulse intensity from the low-field to the high-field regime. We use two different approaches, a global one, the ’Wavepacket’ method, and a restricted one, the ’Level by Level’ method where the number of vibrational levels is limited to a small subset. The comparison between the results of the two approaches allows one to gain qualitative insights into the complex dynamics of the high-field regime. In particular, we emphasize the non-trivial and important role of far-from-resonance levels which are adiabatically excited through ’vertical’ transitions with a large Franck-Condon factor. We also point out spectacular excitation blockade due to the presence of a quasi-degenerate level in the lower electronic state. We conclude that selective transfer with femtosecond pulses is possible in the low-field regime only. Finally, we extend our single-pulse analysis and examine population transfer induced by coherent trains of low-intensity femtosecond pulses.
pacs:33.80.-b, 34.80.Gs, 31.10.+z, 33.15.-e,
Rb and Cs atoms have been simultaneously trapped and laser cooled in a magneto-optic trap down to ultracold temperature (K). Ultracold RbCs molecules have been formed through photoassociation in excited vibrational levels of the Rb(5)Cs(6) , or symmetries. These molecules decay through spontaneous emission, mainly toward stable levels of the Rb(5)Cs(6) electronic state; the upper of those levels has a binding energy in the range of cm kerman2004a (). The relevant molecular terms are shown in Fig. 1.
In the heteronuclear RbCs molecule, two-step conversion processes from the state (denoted below by ’’) toward the state (denoted below by ’’) are possible by using, as intermediate step, levels of the or symmetries, with a spin-mixed character. As a result, molecules in the absolute ground level Rb(5)Cs(6) are formed. These processes have been recently investigated experimentally sage2005 (); kerman2004a () and theoretically bergeman2004 (); stwalley2004 (); tscherneck2007 ().
Ultracold stable polar molecules in their absolute ground vibrational level have been populated for the first time sage2005 (); kerman2004a () using a two-color incoherent population transfer through a low-lying level of the state. A resonant ‘pump’ laser pulse transfers the population of the metastable, vibrationally excited molecules to an electronically excited level; then a second tunable ‘dump’ laser pulse resonantly drives the population to the absolute ground level. The two laser pulses used in this stimulated transfer have a duration of about 5 ns.
In the KRb molecule, using a Stimulated Raman Adiabatic Passage (STIRAP) with counterintuitive pulses in the microsecond range, Ni et al ni2008 () transferred extremely weakly-bound Feshbach molecules in the electronic state toward the lowest vibrational level either of the stable or of the metastable states using intermediate level with symmetry .
For several years, researchers at Aimé Cotton Laboratory are exploring theoretically, on the example of the Cs and Rb molecules, coherent schemes using chirped laser pulses to form molecules in an excited electronic state through photoassociation of ultracold atoms, and then to stabilize them through stimulated emission luc2009 (); luc2003 (); luc2004 (). The motivation was to fully exploit optical techniques for controlling the formation of cold molecules in the absolute ground level. The studied laser pulses were in the picosecond range, the domain well-adapted to the vibrational dynamics of the wavepackets created by the pulse in the light-coupled electronic states.
However, from a technological point of view, picosecond lasers and corresponding pulse shapers are not yet available. On the other hand, in the femtosecond domain there were important recent developments of efficient laser sources and pulse shapers. Furthermore, coherent trains of pulses, obtained from mode-locked femtosecond lasers cundiff2002 (), permit a transient coherent accumulation of population, manifested by the enhancement of transition probabilities and by a gain in the spectral resolution stowe2006 ().
Our objective here is to analyze the possibilities offered by femtosecond sources in implementing efficient two-color paths for transferring vibrationally-excited ultracold molecules to their absolute ground state. In this scheme the molecules are initially produced by photoassociation or magnetoassociation in bound vibrational levels close to the first dissociation threshold. Numerical analysis is carried out for the RbCs molecule. More precisely, the present paper is devoted to the choice of the optimal pulse for implementing the first step of the two-color paths. Notice that femtosecond pulses have a broad bandwidth and may reach high intensities. Consequently we have to analyze the dynamics of coherent excitation of a large number of vibrational levels, from the low field up to the high field regime.
To solve the time-dependent Schrödinger equation, we first use the ‘Wavepacket’ method (WP), where we calculate globally the evolution of vibrational wavepackets propagating along electronic states coupled by the laser pulse luc2009 (). Using this approach, it appears that, in the high-field regime, the calculated dynamics and the population transfer drastically differ from what is expected from intuitive two-level-system arguments. To understand these surprising results, we compare the WP results to solutions obtained using a small subset of vibrational levels: we refer to this model as the ‘Level by Level’ method (LbyL). In both approaches, the dependence of the wave function on the interatomic distance is obtained from the Mapped Fourier Grid Hamiltonian (MFGH) method kokoouline1999 (); willner2004 ().
By comparing the WP results with the LbyL solutions, we precisely identify vibrational levels critically responsible for the strongly nonlinear dynamics in the high-field regime. In the high-field regime, the dynamics of the photoexcitation process is governed both by nearly-resonant and by far-from-resonance excitations. The adiabaticy of the resonant and non-resonant excitations can be easily analyzed in detail in the simple case of a two-level system. For a multilevel system, we show that, in the high-field regime, the dynamics of time-evolution of the population in nearly-resonant levels is strongly affected by the adiabatic excitation of far-from-resonance levels. For a particular level, the adiabaticity of the excitation by an unshaped Gaussian pulse is found to be simply related to the value of its detuning with respect to the carrier laser frequency. In the photoexcitation process under study, the initial level lies close to the dissociation threshold, in an energy domain where the density of vibrational levels is high. We show that the presence of such a quasi-degenerate group of levels in the ground electronic state leads in the high field regime to a spectacular blockade of the excitation process.
We conclude from the analysis that, while femtosecond laser pulses are concerned, control of the photoexcitation process is possible only in the low field regime. To improve the efficiency of the population transfer, we investigate some schemes using coherent trains of low-intensity femtosecond pulses.
The paper is organized as follows. First we specify the photoexcitation process (Sec. II.1) and also characterize Gaussian pulse (Sec. II.2). Then we briefly describe the two employed approaches (the WP and LbyL methods) to solving the time-dependent Schrödinger equation (Sec. II.3). The photoexcitation dynamics is dramatically affected as the pulse intensity is increased. It’s dependence on the pulse intensity is computed in the WP approach and is described in Sec. III. These results are further analyzed in Sec. IV in the framework of the LbyL method. This framework allows us to identify levels responsible for the observed photoexcitation dynamics (Sec. IV.1). We further exhibit the link between adiabaticity and detuning first in the simple case of a two-level system (Sec. IV.2) and then for the multi-level system under study (Sec. IV.3). The excitation blockade due to the presence of quasi-degenerate group of levels in the ground state is studied in Section IV.4. Finally, we comment on the photoexcitation dynamics induced by coherent trains of low-intensity femtosecond pulses in Section V.
The paper contains several appendices used for recapitulating essential results and to precise notation. Appendix A briefly reviews the Mapped Fourier Grid Hamiltonian (MFGH) employed throughout the paper. The ’Wavepacket’ and the ’Level by Level’ methods are described in the Appendix B. Appendix C recalls the definition of the diabatic and adiabatic bases used in our analysis. A simple model for the blockade of excitation due to the presence of a quasi-degenerate group of levels in the lower electronic state is described in Appendix D, whereas Appendix E lists relevant properties of ultrashort pulse trains.
Ii Photoexcitation of RC
ii.1 Photoexcitation process
In the RbCs molecule, it has been shown that the two-color path is very efficient in transferring to the absolute ground level the molecules obtained in the level after photoassociation followed by spontaneous radiative decay londono2009 (). The symmetry results from the coupling through the spin-orbit interaction of the singlet and the triplet electronic states. The level is a mix of vibrational levels (52.7%) with and of levels (47.3%) with . In the first step of the two-color path, only the components of the coupled wave functions can be excited; we have shown that the excitation probabilities and level are very similar. Therefore, in this paper, we restrict the analysis of the photoexcitation dynamics to the study of the transition. The rotational structure of the vibrational levels as well as the hyperfine structure are ignored.
We consider excitation by a Gaussian laser pulse with a duration and a carrier frequency resonant with the transition between the vibrational levels and ,
where and are absolute energies of the two levels.
The initial level has a binding energy of only cm and it lies very close to the Rb(5)Cs(6) dissociation limit. The excited level with binding energy cm with respect to the Rb(5)Cs(6 dissociation limit is tightly bound (Fig. 1). There are substantial differences in the two vibrational wave functions. The wave function of the initial level extends from 9 to 27 ( denotes the Bohr radius) and the wave function of the resonant level is located at much smaller internuclear distance, 7 to 11 . As a result, the Franck-Condon factor is relatively small ().
In the same Fig. 1 we also show the wave function, in the Hund’s case representation, of the spin-orbit-mixed vibrational level , which has an energy close to the energy of the pure Hund’s case resonant level . One should notice the similarity between the vibrational component in the triplet state of the wave function and the vibrational wave function of the pure level for , that is in the -range where the overlap of both wave functions is the largest.
The wave function of the level, strongly off-resonant with the studied laser pulse but connected to the level through a ’vertical’ transition (the outer turning points of both wave functions are located at ), is also reported in Fig. 1. The corresponding Franck-Condon overlap, , is much larger than that one of the resonant transition.
ii.2 Characteristics of the laser pulse
The laser pulse is assumed to have a Gaussian profile and to be Fourier-transform-limited, with a time-independent carrier frequency fixed to . We do not consider chirped pulses because the mechanism of adiabatic population transfer occurring during excitation with chirped pulses has been previously extensively analyzed and optimized cao1998 (); cao2000 (); luc2003 (); luc2009 (). The motivation of the present work is to investigate a completely different excitation mechanism, resulting from the use of ultrashort unchirped pulses, and to interpret in detail its dynamics.
The laser pulse is described by an electric field with an amplitude varying with time as:
where is the maximum amplitude and denotes the complex time-dependent amplitude. The Gaussian envelope , with maximum , is given by
The instantaneous intensity of this pulse illuminating an area , is equal to
where ( is the velocity of light, the vacuum permittivity). has a full width at half maximum (FWHM) equal to . The pulse duration and the energy of the pulse satisfy:
In the spectral domain, the electric field is obtained from the Fourier transform of the complex time-dependent electric field ,
For the pulse of duration fs considered here, the bandwith , defined by the FWHM of , is of the order of cm.
ii.3 Photoexcitation dynamics: ’Wavepacket’ and ’Level by Level’ descriptions
To analyze the dynamics of the photoexcitation process (Eq. (1)), we consider the time-dependent Schrödinger equation describing the internuclear dynamics of the Rb and Cs atoms
where denotes the molecular Hamiltonian in the Born-Oppenheimer approximation and where the coupling between the laser and the molecule, written in the dipole approximation, is expressed in terms of the dipole moment operator . The electric field of the laser pulse with polarization reads .
In the excitation process, we focus on the redistribution of the population between the vibrational levels, disregarding rotational components of the wavepackets . This approximation is justified because the centrifugal energy is negligible and thereby vibrational wavepackets do not depend on value of the total angular momentum . All our calculations were carried out for a fixed value of , and below we do not identify it explicitly.
In a simple model restricted to the ground and excited electronic states, the two radial components and of the wavepacket are solutions of the coupled system:
where and denote the potentials in the ground and excited states. The coupling of the two electronic states can be written in terms of
where denotes the electronic dipole transition moment resulting from the integration of over the electronic wave functions of the ground and excited electronic states. We disregard the -dependence of the electronic transition dipole, which is taken equal to its asymptotic value . Finally, , the maximum strength of the coupling, is proportional to the square root of the maximum intensity .
The radial part of the wavepackets (resp. ) is a coherent superposition of the stationary vibrational wave functions, eigenstates with energy (resp. and ) of the time-independent Schrödinger equation involving the potential (resp. ). Numerically the radial dependences of all functions are described by using the Mapped Fourier Grid Method (MFGH) kokoouline1999 (); willner2004 (). Let us emphasized that for a single potential, the eigenstates consist of bound levels and discretized scattering levels, which are automatically included in the decomposition of the wavepacket (see Appendix A). A spatial grid of length with mesh points is used for each potential yielding a quasi-complete set of eigenfunctions (see Ref. londono2011 ()).
Two methods are used to solve the time-dependent Schrödinger equation in the rotating wave approximation (RWA). The first method, the Wavepacket description, consists in determining directly the vibrational wavepackets and created by the laser pulse on both electronic states and . Studying the excitation from the vibrational level (Fig. 1), the initial state is chosen to be this initial vibrational level: and . Details on the numerical methods, presented in Refs luc2003 (); luc2004 (), are summarized in Appendix B.1. The time-dependent Schrödinger equation is solved by expanding the evolution operator in Chebyschev polynomials kosloff1994 (). With the MFGH method being used to represent the radial dependence of the wavepackets, the WP method is a global approach which automatically incorporates contributions of the complete set of vibrational levels and with .
The second approach, the Level by Level description, analyzes the coupling by the laser pulse of some beforehand selected subsets of vibrational levels , , with and being numbers of levels in the ground and excited state vibrational subsets respectively.
The chosen levels result in the formation of the ground and excited wavepackets written, in the ’interaction representation’ bookCohen2 (), as:
where the phase factor accounts for the ’free evolution’ of the stationary vibrational levels. In the RWA approximation, the instantaneous probability amplitudes and are determined by solving a system of coupled first-order differential equations (Eq. (LABEL:ch5:eq:eq-coupl-RWA)) presented in the Appendix B.2. For the initial state of the system, the probability amplitude of the level is set to unity: and for all the considered values. The relevant molecular structure data are the relative energies for the ground levels (resp. for the excited levels) with respect to the resonant level (resp. ),
and the overlap integrals
Notice that because of the resonance condition, the energy spacings and may be expressed in terms of the detunings (Eq. (34)).
The WP and LbyL methods are compared in Appendix B.3. The WP/MFGH approach allows one to expand the wavepackets and over the complete set of vibrational levels of the and electronic states:
The evolution of the total population in the two electronic states may be found as
More detailed information is provided by decomposing the wavepackets in the basis of unperturbed vibrational levels or of both electronic states or ,
which gives the instantaneous population of each stationary vibrational level.
Naturally the LbyL approach is equivalent to the WP description if and only if the sets and encompass complete sets with levels, that is all bound levels and all levels of the discretized continua (Appendix A). We emphasize that the WP description automatically takes advantage of the completeness of the set of eigenfunctions provided by the spatial representation of the Hamiltonian on a grid. Furthermore, the description of the dynamics does not depend on the choice of the grid, provided that a sufficiently wide domain of energy is covered by the eigenvalues obtained in the MFGH diagonalization of the Hamiltonian matrix.
Iii Wave Packet description: from low field toward -pulse
iii.1 -pulse condition
Our goal is to find a pulse which yields a population transfer as large as possible from the initially populated vibrational level toward the vibrational level . As mentioned above, we consider only the case of an unchirped transform-limited Gaussian pulse, resonant with the transition , with a duration in the femtosecond domain. The chosen duration is fs, much smaller than the vibrational period ps for the initial level . It is only 6 times smaller than the vibrational period ps in the excited state. Consequently, in the excited electronic state, there are only 6 nearly-resonant levels lying within the bandwidth cm au of the pulse, the levels with detuning respectively equal to -92.0, -46.0, 0, +45.8, +91.5 cm.
The pulse is characterized by the electric field amplitude or, equivalently, by the pulse intensity or by the parameter (Eq. (9)). Given a pair of levels (say and ) we may also introduce the accumulate pulse area bookTannor () as
where denotes the overlap integral of the resonant transition (Eq. (12)). The total pulse area of a Gaussian pulse is
In a two-level system, the angle fully determines the probability amplitudes of the lower level and of the resonantly-excited (i.e., when ) level as bookTannor ()
The -pulse for a resonantly-driven two-level system is defined as ,
where is the Rabi coupling (see Eq. (20)) for the resonant transition at the pulse maximum .
Accounting for the overlap integral and for the pulse duration ps au, the -pulse condition is satisfied when
This large value of intensity is due to the small value of the overlap integral and to the short pulse duration.
iii.2 Low field excitation
We first consider a weak pulse, au, with a pulse area , corresponding to an intensity at the maximum of the pulse cm. The initial population in the level is set equal to unity. The evolution with time of the total population in the excited electronic state and in the resonant level is reported in Fig. 2a,b. The considered populations increase monotonously during the pulse and the total transfer is very small (0.000343), with half population (0.000161) in the resonant level . For the and levels, which have a detuning with respect to the central laser frequency smaller than , the population at the end of the pulse is respectively 0.000076 and 0.000088. There is almost no population in the levels or .
In the perturbative limit, the amplitude of population of the initial level is almost not modified during the pulse. After the end of the pulse, for , the population of the level in the excited electronic state is equal to:
where is the detuning of the excitation of the level from the level and where (Eq. (6)) is the spectral density of the pulse.
In this limit, the population transferred from the level toward the level is proportional to the Franck-Condon factor and to spectral density of the pulse at the excitation frequency shapiro2003 (). As a result, for the weak perturbative pulses, only the nearly-resonant levels, such as , are excited.
The population distribution in the vibrational levels is presented in the left column of Fig. 3 for the excited electronic state (panel a) and for the lowest electronic state (panel b), either at the maximum of the pulse () or after the end of the pulse ( ps). The population of the excited vibrational levels always remains smaller than that of the nearly-resonant levels , and, at the end of the pulse, only these levels remain populated. In the low-field limit, the dynamics of the excitation process involves almost only the nearly-resonant levels (Figs. (2) and (3)).
iii.3 Increasing the field strength
Now we vary the laser coupling and explore the population transferred to the excited levels with . The results of our WP calculations are shown in Fig. 4a. In the low-field limit, the populations increase proportionally to , and, as already noticed, only the levels , and are significantly populated. However, when the pulse area/intensity are increased, the population in the levels with or becomes comparable to the population in the nearly-resonant levels. The population in the resonant level at the end of the pulse, , first increases with increasing and reaches, for au, a relatively small maximum, . This coupling corresponds for the resonant transition to an ’effective’ pulse area of , still in the low-field regime. As is increased further, the oscillates with a period roughly equal to au. Notice that as a function of , the values of the population maxima decrease after two oscillations. This behavior strongly differs from what one would expect intuitively for the resonantly-excited two-level system : in that case, the population would oscillate between the values of 0 and 1, with a period equal to , the value of 1 being reached at au.
The population distribution among the levels of the excited and initial electronic states after the pulse is presented in Fig. 4b,c for three values of the coupling . These couplings correspond to the first three maxima in the variation of as a function of (see the vertical arrows at the top of Fig. 4a). For au, only three nearly-resonant levels are populated and no significant redistribution of population occurs in the levels. For au, more levels, with , are populated and the population is recycled back to levels of the initial state with . For au, a still larger number of and levels is involved in the redistribution of population.
iii.4 pulse: resonant and far-from-resonance excitation
The time-evolution of the total population transferred to the excited electronic state during the excitation by a pulse with a large coupling strength is presented in Fig. 2d. Population maximum (0.094) is attained at the maximum of the pulse ; it becomes smaller when the pulse intensity decreases. The final value, equal to 0.019, is much smaller than unity. The evolution of the population of the resonant level is shown in Fig. 2e. This population does not increase monotonically, as one would expect for a -pulse in a two-level system, but exhibits several () oscillations and the transfer is low (0.00064). A similar behavior is observed for the nearly-resonant levels and with final populations of 0.00043 and 0.00059, respectively. Figure 3 shows the population distribution over various levels of the excited (panel c) and of the lowest (panel d) electronic states at two times ps and at ps. We find that at the end of the pulse a significant fraction of the population is transferred to a large number of strongly-bound levels, mainly to the levels with binding energies in the range of -5200 to -3800 cm. The most populated levels, and , with respective detunings cm and cm, have a population , equal to twice the population of the resonant level . Population is also redistributed within bound and scattering levels of the ground state, in particular within levels (population 0.005). The difference in the energies of these levels with respect to the initially populated level (Eq. (11)) lies in the range cm cm. Only 76% of the population remains in the initial level.
At the maximum of the pulse, there are many levels of the excited electronic state, , which have a population larger by a factor of at least 10 than the population in the nearly-resonant levels ( cm). These strongly-populated levels are such as cm, so they lie far outside the pulse bandwidth and correspond to highly-far-from-resonance excitations. Because of their high population during the pulse, these levels contribute significantly to the excitation dynamics. The time evolution of the population of the level, is reported in Fig. 2f. This is the most populated level in the excited electronic potential with a population reaching 0.0134 at the maximum of the pulse. The time-dependence of this population follows that of the envelope of the pulse intensity, (Eq. (3)). Note that in the low-field case (Fig. 2c), the population of the level is always negligible ().
It is to be emphasized that this behavior can not be explained as Rabi cycling, contrarily to what could be intuitively expected considering the large value of the instantaneous Rabi coupling arising from the large value of the detuning . We recall the definition of the instantaneous Rabi coupling at time for non-resonant transition :
where denotes the Franck-Condon factor. The level is the level of the excited electronic state possessing the largest population at the maximum of the pulse. This can be understood by reminding that this level is excited from the initial level through a vertical transition (see Sec. II.1).
The importance, in the strong field regime, of off-resonant excitation of levels strongly favored by high Franck-Condon factors but lying energetically above the spectral bandwith of the pulse has been experimentally observed in the photoassociation of ultracold atoms with shaped femtosecond pulses salzmann2008 (); mccabe2009 ().
Iv Analysis of the -pulse dynamics: Level by Level description
The WP results demonstrate that, for the high coupling strength , the dynamics of the excitation process involves a large number of vibrational levels, both in the ground and in the excited electronic states. To better understand the dynamics of the population of these levels, we performed LbyL calculations, with various subsets and of bound and quasi-continuum (scattering) levels. These subsets are simply denoted as: .
iv.1 Levels involved in the dynamics
LbyL basis set reproducing the WP dynamics
In the first step we try to reproduce, by optimizing the restricted LbyL basis set, the time-evolution of the total population transferred to the excited electronic state by -pulse (). Some representative results are displayed in the left column of Fig. 5, where the following basis sets are considered: set : , set : and set : . These levels are either bound or discretized scattering vibrational levels in the or electronic states. Let us remark, that, with the mesh grid used in the MFGH approach, only a small energy range ( cm) is described by ’physical’ scattering levels (see Appendix A).
The relatively large set includes, in the lower state, bound levels lying close to the initial one, , and, in the excited state, levels located in the vicinity of the resonantly excited level or in the vicinity of the far-from-resonance level corresponding to the vertical transition. For this set, the total population at the maximum of the pulse is larger by a factor 2 than the population obtained by using the WP approach. At the end of the pulse, a too large population () remains in the excited state. For the set , which includes all the bound levels in the excited state and, in the lower state, a smaller number of levels located in the vicinity of the initially populated one, a similar behavior is obtained, yielding the same final population transfer, but a slightly smaller maximum value at .
To reproduce in the LbyL approach the results obtained in the WP approach, we have found that it is necessary to employ the set which includes all bound vibrational levels in the excited state and a very large number of levels (205) in the lower state, i.e. all bound levels () and discretized scattering levels in a large energy range, with an energy up to cm, described by physical or unphysical levels londono2011 (). In this LbyL calculation, the time-evolution of the total population in the excited state reproduces the one from the WP approach, in particular the low value of the population () transferred at the end of the pulse. Furthermore, the time-dependence of the populations in the resonant level or in the level and also the variation of the total population in the bound levels, represented in the right panel of Fig. 5, reproduce perfectly the variations calculated directly in the WP approach (Fig. 2). In the following, the set is called the ’optimal’ LbyL basis set.
The wide energy range covered by the levels involved in the dynamics is not negligible compared to the frequency of the pulse cm. Therefore the validity of the RWA approximation is questionable. Indeed, for the pairs of levels included in the basis set, the frequencies of the ’rotating’ contributions are not always negligible compared to the frequencies of the neglected ’counter-rotating’ contributions (Appendix B.2). Further investigation would be needed to check that the introduction of the counter-rotating terms does not change the main conclusions of the present analysis.
Two types of dynamics in the excited electronic state
To go further in the analysis of the dynamics, we separate the excited levels of the optimal set into two different groups, according to the time-evolution of their individual population .
For levels with a detuning varying in the range cm cm, the dynamics of population is very similar to that of the resonant level . During the pulse the population exhibits a small number of oscillations of a relatively small amplitude and some population remains in these levels after the pulse. The sum of the population in this group of levels grows almost monotonically during the pulse and reaches the final value (Fig. 6).
As we move further off-resonant and consider bound levels , we find that the evolution of the population is similar to that of the level , i.e., traces time-variation of the pulse intensity . The total population transferred at the maximum of the pulse is very high and it is larger than the population present in the group of levels close to the resonance. Yet no population remains after the end of the pulse.
Thus it appears that two types of dynamics are observed for the levels of the excited electronic state. Levels with a not-too-large detuning remain populated after the laser pulse. Taking into account that the pulse is symmetrical, Gaussian and unchirped, their evolution is necessarily non-adiabatic. Conversely, levels corresponding to highly-off-resonant excitation possess the largest population at the maximum of the pulse, but they do not retain their population after the pulse: such dynamics has thus a quasi-adiabatic character. Below we present a qualitative description which emphasizes a relation between detuning and adiabaticity.
Adiabaticity of the evolution of a system is naturally expressed in the basis of instantaneous eigenvectors of the Hamiltonian, the so-called adiabatic basic bookTannor (); bookMessiah () (see Appendix C). For a system with more than two levels, there is no general way to construct the instantaneous adiabatic basis and thus no general expression of the adiabatic theorem bookMessiah (). In fact, the relationship between adiabaticity, detuning, laser width and coupling strength can be perfectly illustrated in the case of a two-level system , where the instantaneous adiabatic levels can be explicitly constructed. The unperturbed vibrational levels and , define the diabatic basis (see Appendix C). The time-dependent wave function can be decomposed on the diabatic levels . We assume that only the level is initially populated. The levels are coupled by a Gaussian pulse , with bandwidth au, as described in Section II.2. In the RWA approximation, the time-dependent coupling is .
Below we study six different cases, labeled to ; these differ by overlap integrals and detunings (see Table 1). For the overlap integral, we choose values corresponding either to the resonant transition (systems to ) or to the vertical transition (systems and ). The amplitude of the electric field and the dipole transition moment are chosen such as, for , the pulse condition, or , is satisfied except for the cases and , where . Therefore the maximum coupling is either smaller,