Vibrationally-induced electronic population inversion with strong femtosecond pulses
VVibrationally-induced electronic population inversion with strong femtosecond pulses
Pablo Sampedro, Bo Y. Chang, and Ignacio R. Sola Departamento de Qu´ımica F´ısica, Universidad Complutense, 28040 Madrid, Spain ∗ School of Chemistry (BK21), Seoul National University, Seoul 151-747, Republic of Korea
We discover a new mechanism of electronic population inversion using strong femtosecond pulses,where the transfer is mediated by vibrational motion on a light-induced potential. The process canbe achieved with a single pulse tuning its frequency to the red of the Franck-Condon window. Weshow the determinant role that the sign of the slope of the transition dipole moment can play onthe dynamics, and extend the method to multiphoton processes with odd number of pulses. As anexample, we show how the scheme can be applied to population inversion in Na . The development of femtosecond pulse technology hasallowed the first experimental observation of transitionstructures that are fundamental to study chemical reac-tivity and kinetics[1]; ultrafast imaging and structuraldynamics are currently providing a wealth of spectro-scopic and structural information concerning the dynam-ics of molecules in real time[2, 3]. Additionally, as thelaser acts on the time scale of the nuclear dynamics, ul-trashort and strong pulses can be used not only to moni-tor but also to control the movement of atoms, modifyingthe yield and rate of chemical reactions[4–6]. The use ofclosed-loop learning techniques has allowed a fast fullyexperimental approach to control quantum processes inmolecules with strong fields[7, 8]. However, the exper-imental observation alone is often not enough to ratio-nalize the outcome of the experiments. In many cases,understanding the underlying processes requires the useof models and theoretical simulations to describe the keyaspects of the dynamics. This is the approach followedin this work.To maximize the transition probability between twoquantum states coupled by an external field is the fun-damental challenge underlying most quantum controlschemes. When the interaction is coherent, it is possibleto achieve full population inversion through Rabi floppingby controlling the pulse area[9]. In molecules, however,the density of states forces the use of very long pulses,which make energy relaxation and decoherence a majorconcern. Using shorter (and hence stronger) pulses ren-ders a new problem due to the Autler-Townes splittinggenerated by the accessible yet unpopulated vibrationallevels in the ground state[10, 11]. This can be beaten byworking with very short pulses, demanding even strongerpulses[12].Under strong fields, a plethora of new phenomena oc-cur in molecules. Aside from ionization or other multi-photon processes, the potential energy surfaces changedue to dynamic Stark shifts, dramatically affecting theresonances and photophysics of the molecule[13, 14].One can modulate these laser-induced potential energysurfaces[15] (LIPs) as drivers of the dynamics[16]. Sev-eral photodissociation reactions were controlled in suchway, where under certain conditions the number of pho- tons is conserved, that is, the laser acts as a catalyst[17–19].On the other hand, population inversion is typicallycontrolled by chirping the laser, with the frequency of thepulse sweeping across the absorption band, although bymodulating the field the pulse duration is stretched[20–23]. It is possible however to achieve adiabatic passagewith transformed-limited pulses, as in the APLIP (adia-batic passage by light-induced potentials) scheme[24, 25].Several APLIP scenarios have been proposed[26–28], butall of them require the use of at least two time-delayedpulses non-resonantly coupled to an intermediate state.The key of the method lies in the presence of this inter-mediate electronic state that assists in modulating theLIP to guide the wave packet motion from one electronicstate to the other.If the pulses are long enough, this “motion” is in factadiabatic or quasi-static, by which the wave packet al-ways remains at the bottom of the LIP preserving the ini-tial vibrational quanta[26, 28]. However, less-adiabatic orultrafast APLIP are also possible[12]. In addition, it hasbeen proven that the APLIP principle can be extendedto work for any system with an odd number of potentials( N ), simply by using N − V to that at V , only to re-vert the transition when the pulse is switched off. The a r X i v : . [ phy s i c s . a t o m - ph ] A p r -4 -2 0 2 4R (a ) -0.0300.030.06 E ne r g y ( H a r t r ee ) -4 -2 0 2 4R (a ) -0.0800.080.16 E ne r g y ( H a r t r ee ) (a) (b)V V U U V V U U FIG. 1: Electronic potentials and LIPs generated by a strongfield slightly off-resonant from the absorption band. In (a)the equilibrium geometries are more separated than in (b), sothat the ground state wave function (shown) overlaps excitedstate configurations in the latter case. In (b) the LIPs arecalculated when the transition dipole depends linearly with R . control mechanism that we propose in this work relies onthe correlation between the vibrational motion and theelectronic state for which we term the scheme the Vibra-tionally Induced Electronic Transition in a Light-InducedPotential or VIETLIP.Fig.1 reveals such effect for the simplest system formedby two harmonic oscillators, V and V , coupled by afield. For illustration purposes we have chosen the re-duced mass of the system to be that of Na and the fun-damental harmonic frequency ω to roughly correspondto that of its ground electronic state. The origin of theinternuclear distance is chosen in between the equilib-rium geometries of both potentials. In the first case[Fig.1(a)] we choose the excited state to be exactly asthe ground state but shifted to a new equilibrium ge-ometry, R − R = δ , where δ = 2 a (approximately thedisplacement of the equilibrium geometries of the groundand first excited electronic states of Na ). We assume aconstant transition dipole. Because we are not excitingat the Franck-Condon region, the transition from V to V is hindered by an energy barrier V b that the nuclearwave packet, initially in V , must overcome.A strong field generates LIPs, U and U : U correlateswith V at short R and with V at large R . Since it doesnot have any internal barrier, a nuclear wave packet ini-tially prepared in U will freely move from V to V . U shows the opposite correlation. Fig.2(a) shows the elec-tronic population in V at the end of the excitation as afunction of the pulse duration (plateau of the pulse), ob-tained after integrating the time-dependent Schr¨odingerequation in a grid[30] in the rotating wave approximation(RWA)[9]. We use plateau pulses with a relatively fastsine square switch on/off of 60 fs and a plateau of dura-tion τ . The electronic population follows a slow squared-type oscillation with a period that depends on the pe- A b s o r p t i on y i e l d τ (fs) (a)(b)(c) FIG. 2: Absorption yield as a function of the pulse dura-tion (plateau, τ ) using pulses of different peak amplitude: 0 . .
21 GV/cm (dashed red) and 0 .
43 GV/cm(green). In (a) δ = 2 a while in (b) δ = a . In both caseswe assume a constant dipole. In (c) δ = a but we assumea linear dipole. In dotted lines we show the result when thedependence of the dipole changes sign for the higher intensefield. riod of motion of the nuclear wave packet in the LIP.The period depends weakly on the pulse amplitude be-cause the curvature of the LIP (and hence the harmonicfrequency) depends on the strength of the coupling. Forweaker pulses the period will be larger. In any case theperiod is of the order of hundreds of femtoseconds, anorder of magnitude (or more) larger than the Rabi pe-riod which depends on the pulse amplitude. For strongpulses the Rabi frequency is comparable to an electronictransition energy.The VIETLIP scheme works as long as the equilibriumgeometries of the electronic potentials V and V are sep-arated enough, but if δ is too large, E b will be large too,and the scheme will require very strong pulses to removethe energy barrier in the LIP. On the other hand, if δ is smaller than the de Broglie wavelength of the initialnuclear wave function, then part of this wave functioncorrelates with U in the adiabatic representation (for R < U (for R > δ = 1 a . The population fully oscillates at a Rabiperiod that depends exclusively on the Rabi frequency. τ (fs)00.20.40.60.81 A b s op r t i on y i e l d red detuningblue detuning alternate detuning FIG. 3: Absorption yield as a function of the pulse dura-tion (plateau, τ ) for a 3-photon process involving 3 differentpulses. The results depend on the choice of the laser frequen-cies: the red-detuning configuration (intermediate potentialsabove V , V or ∆ > < For strong fields this implies electronic beatings in theorder of the femtosecond.However, it is interesting to see that the presence of acoordinate-dependent transition dipole can compensatethis effect. In Fig.1(b) we show the LIPs when δ = 1 a but the dipole depends linearly with the internuclear dis-tance, µ = R . The effect of the dipole is to separate theequilibrium geometries of the LIPs, allowing to preparethe initial wave function in a single LIP. The slope ofthe dipole (positive or negative) decides the the shapeof the LIPs. If the coupling is − R × (cid:15) (positive slope;larger dipole at large R ) where (cid:15) is the field, then, asin Fig.1(b) U moves the wave packet towards V . Thedynamics is relatively similar (although more complex)to that encountered in the first case. The oscillations inthe yield as a function of the pulse duration show a longperiod corresponding to the nuclear motion, not to theRabi oscillation. However, if the coupling is R × (cid:15) (nega-tive slope; larger dipole at small R ) then the equilibriumgeometry of U is at V . The wave packet is relativelytrapped at the ground electronic state, the trapping in-creasing with the pulse intensity. The yield of absorptionis therefore quite smaller, as shown in Fig.2(c) and theabsorption bands anti-correlate with those obtained withopposite dipole.One of the most interesting aspects of populationtransfer through diabatic wave packet motion on a LIPis that the method can be extended to multiphoton tran-sitions, but only with an odd number of pulses, contraryto the APLIP scheme, which works with an even numberof pulses. Moreover the passage depends on the choice ofthe laser frequencies: for some arrangements the passageis more protected than for others. In Fig.3 we show theresults of population transfer between 4 potentials, V to V , where we assume that there are only sequential cou- A b s o r p t i on y i e l d τ (fs)00.51 A b s o r p t i on y i e l d (a)(b) vibrational periodwave packetshifts to V wave packetreturns to V FIG. 4: Absorption yield for the A band of Na using0 . .
054 GV/cm (red, dashed) and 0 . plings between nearest neighbors so that V is coupledto V by Ω ( t ), V with V by Ω ( t ) and V with V byΩ ( t ). As in multiphoton APLIP, the intermediate po-tentials must be off-resonance. In our model we choose V and V as harmonic oscillators centered at the equi-librium geometries of V and V respectively, althoughthe results are not very sensitive to these parameters.However, they are sensitive with respect to the choiceof detuning: V and V are displaced ± ∆ in the verti-cal axis. For the results in Fig.3 we fixed the peak Rabifrequency as 0 .
05 a.u. and ∆ = 0 .
08 a.u. When V and V are above V and V (red detuning) full populationtransfer by VIETLIP is possible. The results are worsein the blue-detuning configuration ( V and V below V and V ) since the Stark effect shifts the V and V po-tentials to higher energies. Robust population transferis still possible, particularly with stronger fields. How-ever, in the alternate configuration (when one potentiallies above and the other below V and V ) the passageis clearly worse and less robust. In principle, the sameideas can be generalized to any multiphoton transitionwith odd number of pulses.Although the concept of LIPs is essential in under-standing many processes under strong fields, it has beenexperimentally difficult to show evidence of wave packettransfer through LIPs. One of the main difficulties forAPLIP is to isolate the desired process from other com-peting routes. As more pulses act on the system theRWA is typically violated breaking the theoretical re-quirements for the transfer. Here we show that the VI-ETLIP scheme can be applied in realistic conditions. Asan example, we show simulations for the dynamics inNa . We use ab initio electronic potentials and dipoles forthe transition between the ground state Σ g (3 s ) and thefirst excited state Σ u (3 p ) that gives rise to the A band. FIG. 5: Wavepacket dynamics in Na when the pulse durationleads to population inversion (A) or transparency (B). Interestingly, the transition dipole is approximately lin-ear in the internuclear distance and has the appropriatesign. Choosing the frequency at the Franck-Condon win-dow (with maximal overlap with the vibrational levelsin A ) leads to saturation[11]. However, if we tune thelaser below the resonance ( ω = 1 . . − GV/cm peak amplitude. If the wave packethas time to move from V to V through the LIP (firstoscillation) we have maximum absorption. Doubling thetime the wave packet returns to that part of the LIP thatcorrelates to V as the pulse is switched off.Comparing the yield of absorption in Na with themodel results of Fig.2(a) shows some differences. A strik-ing difference is the decay of the second band of maxi-mum absorption (particularly with stronger fields) andthe weak dependence of the position of the bands onthe pulse amplitude. This is an effect due to the dipole.As observed in Fig.1(b) and Fig.2(c) the coordinate de-pendent dipole deforms the LIPs affecting the diabaticmotion of the wave packet in the LIP, which is mostlytrapped around the equilibrium geometry of the LIP forlarge fields (because the R(cid:15) term increases with the dis-tance). For constant dipoles we fully recover the regularbehavior, as shown in Fig.4(b). The simulations in thiscase where performed in the Na potentials with an aver-age constant dipole of 3 .
65 ea . Although the dynamics in the real Na (with coordinate-dependent dipole) aremore complex, the dipole in fact makes population in-version more robust, as evidenced by comparing the sizeof the absorption band as a function of the pulse durationin both cases.In summary, we have proposed a new robust scheme ofpopulation inversion between two electronic states withdisplaced equilibrium geometries. The scheme sharesmany features with APLIP, as the mechanism of pop-ulation transfer is mediated by motion in a LIP. In VI-ETLIP, however, only one strong pulse is needed, thatmust be tuned to the red of the absorption band, andshorter pulses can be used. The process cannot be com-pleted in a fully adiabatic way, so that the pulse durationmust be approximately synchronized to the vibrationalperiod. Moreover the scheme can be extended to anymultiphoton process with odd number of pulses. Finally,we have found an intriguing dependence to the dipolefunction. Obviously, a dipole going to zero for some in-ternuclear distance has always strong implications in adi-abatic passage[31]. In this work we report for the firsttime how the sign of its slope determines the outcome ofthe transition, placing the molecular complexity in theforefront of the control process. Acknowledgments
Financial support by the Spanish MICINN projectCTQ2012-36184 and the Korean International coopera-tion program (NRF-2013K2A1A2054518) and Basic Sci-ence Research program (NRF-2013R1A1A2061898) isgratefully acknowledged. ∗ Electronic address: [email protected][1] A. H. Zewail, Science , 1645 (1988).[2] B. J. Siwick, J. R. Dwyer, R. E. Jordan, and R. J. D.Miller, Science , 1382 (2003).[3] C.-Y. Ruan, V. A. Lobastov, F. Vigliotti, S. Chen, andA. H. Zewail, Science , 80 (2004).[4] S. A. Rice and M. Zhao,
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