Coherence revival during the attosecond electronic and nuclear quantum photodynamics of the ozone molecule
Gábor J. Halász, Aurelie Perveaux, Benjamin Lasorne, Mike A. Robb, Fabien Gatti, Ágnes Vibók
CCoherence revival during the attosecond electronic and nuclearquantum photodynamics of the ozone molecule
G. J. Halász, A. Perveaux, B. Lasorne, M. A. Robb, F. Gatti, and Á. Vibók Department of Information Technology, University of Debrecen,H-4010 Debrecen, PO Box 12, Hungary CTMM, Institut Charles Gerhardt Montpellier,Université Montpellier 2, F-34095 Montpellier, France Imperial College London, Department of Chemistry, London SW7 2AZ, UK and Department of Theoretical Physics, University of Debrecen,H-4010 Debrecen, PO Box 5, Hungary ∗ A coherent superposition of two electronic states of ozone (ground and Hartley B)is prepared with a UV pump pulse. Using the multiconfiguration time-dependentHartree approach, we calculate the subsequent time evolution of the two correspond-ing nuclear wave packets and the coherence between them. The resulting wave packetshows an oscillation between the two chemical bonds. Even more interesting, the co-herence between the two electronics states reappears after the laser pulse is switchedoff, which could be observed experimentally with an attosecond probe pulse.
I. INTRODUCTION
The construction of single few-cycle ultrashort laser pulses or trains of ultrashort pulsesenables controlling different photophysical and photochemical processes. Experimentalistscan excite and probe electron dynamics in atoms and molecules in real time [1–10]. Mon-itoring the subfemtosecond motion of valence electrons over a multifemtosecond time spanthat results in taking real-time snapshots of ultrafast transformations of matter. Successfultheoretical and experimental investigations of the electron dynamics of the Kr atom havebeen performed recently [7, 8, 10]. However, extending these techniques to molecules re-mains a challenge. Problems arise because electron dynamics in molecules often are strongly ∗ Electronic address: [email protected] a r X i v : . [ phy s i c s . a t m - c l u s ] M a y coupled to nuclear dynamics.For molecules, various approaches have been developed so far. In most attophysics sim-ulations, only the electron dynamics is treated, and the molecular geometries (nuclear posi-tions) are assumed to be fixed [11–16]. Within this approach an arbitrarily large moleculecan be examined. To achieve this, one needs to use an ultrashort laser pulse during theprobe process. If longer probe laser pulses are applied, the nuclei have time to move. Inthis situation the nuclear dynamics has to be considered as well. For the simplest ion, H +2 ,or molecule, H , it is easily feasible [18–24], but for diatomics containing many electronsor even for polyatomics the problem to be solved is more complex and difficult [25–27]. Inthe first situation (e.g. H +2 or H ) the total time-dependent Schrödinger equation (TDSE)can be solved numerically including explicitly both the electronic and nuclear degrees offreedom. In contrast, the case of many electrons or polyatomics implies to face either theproblem of electron correlation or of a large number of nuclear degrees of freedom [17].Recently, we proposed a nonadiabatic scheme for the description of the coupled electronand nuclear motion in the ozone molecule [28]. An initial coherent nonstationary state wasprepared by two pump pulses. It was a superposition of different weakly-bound states inthe Chappuis band [36] (which are populated by NIR radiation), as well as in the Hartleyband [36] (which is populated by the 3rd harmonic pulse). In this situation neither theelectrons nor the nuclei were in a stationary state, and we used nonadiabatic quantumdynamics simulations. As the transition dipole moments are very different between theground and Hartley states compared to the ground and Chappuis bands we had to applysignificantly different intensities for the two pump pulses not to obtain differences betweenthe populations of the Hartley and the Chappuis states larger than one order of magnitude.Consequently, we used × and W/cm intensities to populate the Hartley andChappuis states, respectively, which is not trivial to achieve experimentally while furtherprobing the system with an attosecond XUV pulse.However, opportunities arise to reasonably simplify the task. As we excite only the Bstate of the Hartley band with a much larger intensity pump pulse than in our previouswork, the population obtained in this state is more pronounced. The non-stationary stateis a coherent superposition of these two (ground and B) electronic states, and the motion ofthe electronic wave packet can thus be probed assuming much less complicated experimentalsetups than in the previous situation.Our original motivation was to perform a numerical simulation for an experimentallyeasier situation. An interesting phenomenon emerged from this investigation: the revival ofthe electronic coherence after the pump pulse is off, which could also be probed experimen-tally. The main aim of the present paper is to report this uncommon finding that can beexplained because we only coupled the X and B electronic states, between which there is nononadiabatic coupling and no conical intersection.As in our previous work, the nuclear wave packets, the electronic populations, the relativeelectronic coherence between the ground X and B electronic states and the electron wavepacket dynamics were calculated. The time evolution of the electronic motion was plottedin the Franck-Condon (FC) region only due to the localization of the nuclear wave packetaround this point during the first − fs. The electron density shows a fast oscillationpattern between both chemical bonds, which we expect could be observed by an attosecondprobe pulse.The paper is organized as follows. Sec. II gives some insights into the formalism andmethods used here. Results and their discussions are presented in Sec. III. Sec. IV is devotedto conclusions. Some useful remarks are provided in appendix about the electronic-structureresults. II. METHODS AND FORMALISM
In this section a short summary is given about the methods and formalism used in oursimulations. For more details we refer to our former paper [28].
A. Time-dependent molecular Schrödinger equation
In the adiabatic partition (beyond Born-Oppenheimer [29]), the total molecular wavefunction Ψ tot ( (cid:126)r el , (cid:126)R, t ) can be assumed as a sum of products of electronic wave functions, ψ kel ( (cid:126)r el ; (cid:126)R ) , and nuclear wave packets, Ψ knuc ( (cid:126)R, t ) : Ψ tot ( (cid:126)r el , (cid:126)R, t ) = n (cid:88) k =1 Ψ knuc ( (cid:126)R, t ) ψ kel ( (cid:126)r el ; (cid:126)R ) . (1)Here k denotes the k − th adiabatic electronic state, (cid:126)r el and (cid:126)R are the electronic andthe nuclear coordinates, respectively. We are interested in solving the coupled evolutionof the nuclear wave packets, Ψ knuc ( (cid:126)R, t ) , by inserting the product ansatz (1) into the time-dependent Schrödinger equation of the full molecular Hamiltonian. Integrating over theelectronic coordinates one obtains the coupled nuclear Schrödinger equations: i (cid:126) ∂∂t Ψ knuc ( (cid:126)R, t ) = (cid:88) l =1 ,n H k,l Ψ lnuc ( (cid:126)R, t ) . (2)Here H k,l is the matrix element of the vibronic Hamiltonian, which reads, e.g., for n = 2 , H = T nuc + V k K k,l − K k,l T nuc + V l , (3)where T nuc is the nuclear kinetic energy, V k ( k = 1 , ...n ) is the k − th adiabatic potential energyand K k,l with k (cid:54) = l is the vibronic coupling term between the ( k, l ) − th electronic states.The latter contains the nonadiabatic coupling term (NACT). In the presence of an externalelectric field the light-matter interaction, − (cid:126)µ ( k, l ) · −→ E ( t ) (electric dipole approximation),where (cid:126)E ( t ) is an external field resonant between the k − th and the l − th states and (cid:126)µ ( k, l ) is the (cid:126)R − dependent transition dipole moment, is also included in this coupling term. Inthe present situation, there is no significant nonadiabatic coupling between the ground andHartley state, therefore K k,l denotes only the light-matter interaction.One has to solve the time-dependent nuclear Schrödinger equation given by Eq. (2). Oneof the most efficient approaches for this is the MCTDH (multiconfiguration time-dependentHartree) method [30–33].The MCTDH nuclear wave packets, Ψ knuc ( (cid:126)R, t ) , contain all the information about therelative phases between the electronic states. Therefore Ψ knuc ( (cid:126)R, t ) can also be written as: Ψ knuc ( (cid:126)R, t ) = exp( − iW k ( (cid:126)R ) t/ (cid:126) ) a k ( (cid:126)R, t ) . (4)Here, W k ( (cid:126)R ) is the potential energy of the k − th state. The first part of this wave functionis the phase factor, ( exp( − iW k ( (cid:126)R ) t/ (cid:126) ) ), of the k − th state, which oscillates very fast. B. Density Matrix
Here we define the working formulas that are used in the next section. Calculating themonly requires the knowledge of the nuclear wave packets.The two-dimensional nuclear density function (depending on R and R , the two bondlengths, and integrated over θ , the bond angle) is: (cid:12)(cid:12) Ψ inuc ( R , R , t ) (cid:12)(cid:12) = ˆ Ψ inuc ( R , R , θ, t )Ψ i ∗ nuc ( R , R , θ, t ) sin θdθ. (5)The total density matrix of the molecule is defined as: ρ ii (cid:48) ( (cid:126)R, (cid:126)R (cid:48) , t ) = (cid:68) ψ iel ( (cid:126)r el ; (cid:126)R ) (cid:12)(cid:12)(cid:12) Ψ tot ( (cid:126)r el , (cid:126)R, t ) (cid:69) (cid:68) Ψ tot ( (cid:126)r el , (cid:126)R (cid:48) , t ) (cid:12)(cid:12)(cid:12) ψ i (cid:48) el ( (cid:126)r el ; (cid:126)R (cid:48) ) (cid:69) (6) = Ψ inuc ( (cid:126)R, t )Ψ i (cid:48) ∗ nuc ( (cid:126)R (cid:48) , t ) , where brackets denote integration over the electronic coordinates only.The electronic population function of the i − th state is: P i ( t ) = ˆ ρ ii ( −→ R , −→ R , t ) d −→ R . (7)Analogously, the electronic relative coherence between the i − th and i (cid:48) − th electronicstates can be approximated as: C ii (cid:48) ( t ) = ˆ ρ ii (cid:48) ( −→ R , −→ R , t ) d −→ R / (cid:113) P i ( t ) P i (cid:48) ( t ) . (8) C. Electronic Structure Treatment
Here we briefly review the represention used for the electronic wave packet. We consideronly two (ground and Hartley B) electronic states. At the FC geometry, each electronicstate can be represented by its charge density in the three-dimensional space, ρ i ( (cid:126)r, (cid:126)R F C ) = N ˆ N ( spin ) dσ dσ . . . dσ N ˆ N − space ) dτ . . . dτ N (9) (cid:12)(cid:12)(cid:12) ψ iel ( (cid:126)r = (cid:126)r, σ , (cid:126)r , σ , ..., (cid:126)r N , σ N ; (cid:126)R F C ) (cid:12)(cid:12)(cid:12) . Here i = X or B . It is often called the one-electron density, although rigorously, it is Ntimes the one-electron density summed over both spin states of electron 1. It is defined asthe density of probability of finding one among N electrons in any spin state (up or down) atpoint (cid:126)r ≡ ( x, y, z ) and time t for the molecule in state X and B, respectively, and geometry (cid:126)R F C .The transition density between states X and B is defined in the three-dimensional spaceas: γ XB ( (cid:126)r ; (cid:126)R F C ) = N ˆ N ( spin ) dσ dσ . . . dσ N ˆ N − space ) dτ . . . dτ N (10) × ψ X ∗ el ( (cid:126)r = (cid:126)r, σ , (cid:126)r , σ , ..., (cid:126)r N , σ N ; (cid:126)R F C ) × ψ Bel ( (cid:126)r = (cid:126)r, σ , (cid:126)r , σ , ..., (cid:126)r N , σ N ; (cid:126)R F C ) . It is a measure of the interference between both states. The total molecular wave packetobserved at a fixed geometry, here at the FC point, is a coherent mixture of both electronicstates, whereby the time-dependent coefficients are the nuclear wave packets at the FC point: Ψ mol ( (cid:126)r , σ , (cid:126)r , σ , ..., (cid:126)r N , σ N ; (cid:126)R F C , t ) = (11) Ψ Xnuc ( (cid:126)R F C , t ) ψ Xel ( (cid:126)r , σ , (cid:126)r , σ , ..., (cid:126)r N , σ N ; (cid:126)R F C )+ Ψ
Bnuc ( (cid:126)R F C , t ) ψ Bel ( (cid:126)r , σ , (cid:126)r , σ , ..., (cid:126)r N , σ N ; (cid:126)R F C ) Thus, the corresponding total time-dependent charge density reads: ρ tot ( (cid:126)r, t ; (cid:126)R F C ) = | Ψ Xnuc ( (cid:126)R F C , t ) | ρ X ( (cid:126)r ; (cid:126)R F C ) + | Ψ Bnuc ( (cid:126)R F C , t ) | ρ B ( (cid:126)r ; (cid:126)R F C )+ 2 Re Ψ X ∗ nuc ( (cid:126)R F C , t )Ψ Bnuc ( (cid:126)R F C , t ) γ XB ( (cid:126)r ; (cid:126)R F C ) . (12)Now, we define the excited-state differential charge density at the FC point as the differ-ence of the total charge density between the excited state B and the ground state [40]: ∆ ρ B ( (cid:126)r, t ; (cid:126)R F C ) = ρ tot ( (cid:126)r, t ; (cid:126)R F C ) − [ | Ψ Xnuc ( (cid:126)R F C , t ) | + | Ψ Bnuc ( (cid:126)R F C , t ) | ] ρ X ( (cid:126)r ; (cid:126)R F C )= | Ψ Bnuc ( (cid:126)R F C , t ) | [ ρ B ( (cid:126)r ; (cid:126)R F C ) − ρ X ( (cid:126)r ; (cid:126)R F C )] + 2 Re Ψ X ∗ nuc ( (cid:126)R F C , t )Ψ Bnuc ( (cid:126)R F C , t ) γ XB ( (cid:126)r ; (cid:126)R F C )= | Ψ Bnuc ( (cid:126)R F C , t ) | (cid:52) ρ B ( (cid:126)r ; (cid:126)R F C ) + 2 Re Ψ X ∗ nuc ( (cid:126)R F C , t )Ψ Bnuc ( (cid:126)R F C , t ) γ XB ( (cid:126)r ; (cid:126)R F C ) , (13)where (cid:52) ρ B ( (cid:126)r ; (cid:126)R F C ) = ρ B ( (cid:126)r ; (cid:126)R F C ) − ρ X ( (cid:126)r ; (cid:126)R F C ) . E x c it a ti on e n e r gy ( e V ) R (a.u.) Figure 1: (Color online) The potential energy surfaces of ozone as functions of the dissociationcoordinate: ground state (X, solid line) and Hartley state (B, dashed line), the arrow denotesexcitation of the B state.
III. RESULTS AND DISCUSSION
In our present work only two electronic states of ozone are involved in the numericalsimulations. The gound state X with A symmetry and the highly-excited B state in theHartley band with B symmetry. In Fig. 1 we show a one-dimensional cut along the O- O bond through the potential energy surfaces (PESs) of both electronic states. We notehere, as there is no nonadiabatic coupling between these two states, that the adiabatic anddiabatic energies are identical. A UV linearly-polarized Gaussian laser pump pulse was usedto prepare a coherent superposition of the two stationary - the ground X and the populatedB - electronic states. The center wavelength and the intensity of the pulse are nm and W/cm , respectively. The FWHM is fs. The PESs and (cid:126)R -dependent dipole momentsoccurring in the radiative coupling terms were taken from Refs. [34, 36, 37].The FC point has C v symmetry. As a consequence, only the y -component ( B ) ofthe transition dipole between the ground state X ( A ) and Hartley B ( B ) is nonzero.Therefore the only effective polarization of the electric field is y (see upper panel on Fig.2). E x E y Time E l ec t r i c F i e l d P opu l a ti on -4 -2 0 2 4 6 8 10 Time (fs)
Ground stateHartley
Figure 2: (Color online) Upper panel: The applied electric field. Lower panel: Time evolution ofthe diabatic populations on the ground ( X ) and diabatic excited ( B ) states. -1.0-0.50.00.51.0 C oh e r e n ce -10 -5 0 5 10 15 20 25 Time (fs)
Real partImaginary partAbsolute value
Figure 3: (Color online) Relative electronic coherence as a function of time. The real, the imaginaryparts and the absolute value of the relative electronic coherence between the ground ( X ) and Hartley( B ) states. -80-60-40-20020406080 ( a . u . ) Time (fs) i=i’=Hartleyi=Ground state ; i’=Hartley R e ρ ii (cid:18) − → R F C , − → R F C , t (cid:19) Figure 5: (Color online) Local population density for state B (black) and real part of the interference(last) term in Eq. (13) (dashed green) at the FC point as functions of time.(a) (b) (c) (d)(e) (f) (g) (h)(i) (j) (k) (l)Figure 4:
Snapshots of the time evolution of the nuclear wavepacket density along both O - O bonds.
In the lower panel of Fig. 2 the total populations against time, see Eq. (7), are displayedin the ground and Hartley B states up to t= fs (note that they stay constant up to theend of the simulation, at t= fs). The Hartley B state absorbs very strongly due to the0 Figure 6: (Color online) Time evolution of the excited differential electronic charge density, Eq.(13), at the FC geometry (side view). Dark (blue): hole; light (yellow): electron. large value of the transition dipole moment with the ground state [35]. Between the ( − , ) fs interval the population grows continuously, then reaches its maximum and remains atthis value throughout the studied time period. The B state is populated with a yield ofabout 40%. The laser intensity ( W/cm ) is thus large enough to transfer near half ofthe ammount of the wave packet from the ground state to the B state.Fig. 3 shows the electronic relative coherence, Eq. (8), between the ground and B states.In the first time period the coherence increases very fast and reaches its maximum. It retainsthis value for - fs, which is approximately equivalent to the duration of the laser pulseand then it decays during the next - fs. However, this is not the end of the process: afew femtoseconds later ( ∼ fs), the coherence reappears in contrast with what was observedin Ref. [28]. This revival of coherence proves that we have created, to some extent, a "true"coherent superposition in that it is not forced by the presence of an external field. Thisphenomenon could certanly be enhanced experimentally by optimizing the parameters ofthe laser pulse.This revival of electronic coherence is interesting because the pump pulse is already off.This implies that the wave packet oscillates in the B state and then goes back to the FCregion where it is still coherent with the part left in the ground state. To understandthis more deeply we have analysed the nuclear density function, Eq. (5). Results areillustrated in Fig. 4 with snapshots from the structure of the nuclear wave packet density1 | Ψ inuc ( R , R , t ) | at different times. It is seen that a part of the nuclear wave packet staystrapped on the symmetric ridge of the B potential energy surface, where both O - O bondsincrease synchronously. A valley-ridge inflection point occurs, where the nuclear wave packetsplits into three components. One part is bound to come back to the FC region, while therest dissociates along either of both equivalent channels.The local population of the Hartley B state at the FC point (see Fig. 5) has also beencomputed. We are again in the same situation as in Ref. [28], namely, state B is populatedsignificantly only during the first ∼ fs time interval over which the molecule remains aroundthe FC region (at least approximately). However, in this case one part of the nuclear wavepacket returns back here again later on.The total differential charge density at the FC point, Eq. (13), was obtained from elec-tronic wave functions calculated at the SA-3-CAS(18,12)/STO-3G level of theory using adevelopment version of the Gaussian program [39]. We observed no qualitative differenceof these when increasing the basis set to aug-cc-pVQZ or when adding dynamic electroncorrelation at the MRCI level of theory using the Molpro program [38].We limited again our discussion of the electron dynamics to the FC region only due tothe localization of the nuclear wave packet around this point during the first − fs. We seeon Fig. 6 the oscillation of the electronic charge density from one bond to another with aperiod of 0.8 fs. The resulting electronic wave packet is thus a coherent superposition of twochemical structures, O · · · O and O · · · O, each having an excess or lack of electron densitieson one or the other bond. The subfemtosecond oscillation between both structures at theFC geometry prefigures that the dissociation of ozone could be controlled by modulatingthe electron density on the attosecond time scale.
IV. CONCLUSIONS
In summary, we have performed numerical simulations of the coupled electron and nuclearmotion in the ozone molecule on the attosecond time scale. An initial coherent nonstationarystate was created as a coherent superposition of the ground and excited Hartley B states.The MCTDH approach was applied to solve the dynamical Schrödinger equation for thenuclei in the framework of the time-dependent adiabatic partition including the light-matterinteraction (electric dipole approximation).2A reasonably large electronic coherence has been obtained between the ground and Hart-ley B states during a short fs time interval. However after this time an interestingphenomenon emerges. After the coherence decays within a certain period of time, a fewfemtosecond later, it appears again. Nuclear wave packet calculations support that we arepresently in a situation where bifurcating reaction paths and valley-ridge inflection pointsare explored on the excited-state potential energy surface. The electronic motion during thefirst − fs shows an oscillation of the electronic charge density from one bond to anotherwith a period of . fs. It is to be expected that this motion can be probed experimentallyby an attosecond XUV pulse. Acknowledgements
The authors would like to thank F. Krausz, R. Kienberger and M. Jobst for support andfor fruitful discussions. We acknowledge R. Schinke for providing the potential energy sur-faces and the transition dipole moment and H.-D. Meyer for fruitful discussions. The authorsalso acknowledge the TÁMOP 4.2.2.C-11/1/KONV-2012-0001 project. Á.V. acknowledgesthe OTKA (NN103251). Financial support by the CNRS-MTA is greatfully acknowledged.
V. APPENDIX
Starting from Eq. (6) and performing further integration over the coordinates of the“last” electron and over the coordinates of the nuclei leads to ˆ ( (cid:126)r ) ˆ ( (cid:126)R ) ρ tot ( (cid:126)r, t, (cid:126)R ) dτ dV (cid:124) (cid:123)(cid:122) (cid:125) =1 = ˆ ( (cid:126)R ) | Ψ Xnuc ( (cid:126)R, t ) | dV (cid:124) (cid:123)(cid:122) (cid:125) P X ( t ) ˆ ( (cid:126)r ) ρ X ( (cid:126)r ; (cid:126)R ) dτ (cid:124) (cid:123)(cid:122) (cid:125) =1 + ˆ ( (cid:126)R ) | Ψ Bnuc ( (cid:126)R, t ) | dV (cid:124) (cid:123)(cid:122) (cid:125) P B ( t ) ˆ ( (cid:126)r ) ρ B ( (cid:126)r ; (cid:126)R ) dτ (cid:124) (cid:123)(cid:122) (cid:125) =1 + 2 Re ˆ ( (cid:126)R ) Ψ X ∗ nuc ( (cid:126)R, t )Ψ Bnuc ( (cid:126)R, t ) dV (cid:124) (cid:123)(cid:122) (cid:125) S XB ( t ) ˆ ( (cid:126)r ) γ XB ( (cid:126)r ; (cid:126)R ) dτ (cid:124) (cid:123)(cid:122) (cid:125) =0 , where P X ( t ) and P B ( t ) are the populations of states X and B, respectively, at time t. S XB ( t ) , the overlap of the nuclear wave packets on states X and B, is a measure of the3global coherence between states X and B for all geometries. This shows that the interferenceterm (involving the coherence and the transition density) does not directly contribute to theprobability of finding the molecule in a given state (it does indirectly though, by having aneffect on the time evolution of the populations).Now, let us turn to Eq. (13). Assuming that the effect of the coupling with the laser pumppulse affects only the electrons for the duration of the observation, then there is no transferof local population density from (cid:126)R F C to other values of (cid:126)R . As long as this approximationholds, then | Ψ Xnuc ( (cid:126)R F C , t < | = | Ψ Bnuc ( (cid:126)R F C , t > | + | Ψ Xnuc ( (cid:126)R F C , t > | (where the pulseis switched on at t=0) and ∆ ρ B ( (cid:126)r, t > (cid:126)R F C ) = ρ tot ( (cid:126)r, t > (cid:126)R F C ) − ρ tot ( (cid:126)r, t < (cid:126)R F C ) ,which thus is a measure of the change of charge density due to the pulse.We note here: ( i ) At the FC point the symmetry point group is C v . By construction,charge densities are A (totally symmetric). 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