Coherent Control of Bond Making
Liat Levin, Wojciech Skomorowski, Leonid Rybak, Ronnie Kosloff, Christiane P. Koch, Zohar Amitay
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Coherent Control of Bond Making
Liat Levin, ∗ Wojciech Skomorowski, ∗ Leonid Rybak, Ronnie Kosloff, Christiane P. Koch, and Zohar Amitay The Shirlee Jacobs Femtosecond Laser Research Laboratory,Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel Theoretische Physik, Universit¨at Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany Fritz Haber Research Centre and The Department of Physical Chemistry, Hebrew University, Jerusalem 91904, Israel (Dated: November 6, 2018)We demonstrate for the first time coherent control of bond making, a milestone on the wayto coherent control of photo-induced bimolecular chemical reactions. In strong-field multiphotonfemtosecond photoassociation experiments, we find the yield of detected magnesium dimer moleculesto be enhanced for positively chirped pulses and suppressed for negatively chirped pulses. Our abinitio model shows that control is achieved by purification via Franck-Condon filtering combined withchirp-dependent Raman transitions. Experimental closed-loop phase optimization using a learningalgorithm yields an improved pulse that utilizes vibrational coherent dynamics in addition to chirp-dependent Raman transitions. Our results show that coherent control of binary photo-reactions isfeasible even under thermal conditions.
PACS numbers: 42.65.Re, 82.50.Nd, 82.53.Eb, 82.53.Kp
A long-standing yet unrealized dream since the earlydays of coherent control, about 30 years ago, is the co-herent control of photo-induced bimolecular chemical re-actions [1, 2]. Realizing this dream will create a newtype of photochemistry with selective control of yieldsand branching ratios [3, 4]. Shaped femtosecond laserpulses act there as special ”photo-catalysts” with a firstpulse inducing and controlling the formation of a chem-ical bond, and a second time-delayed pulse breaking thedesired bonds within the generated molecule. The sec-ond step, photodissociation into target channels with thedesired branching ratios, has been demonstrated earlyon [5–11], once femtosecond lasers and pulse shapingtechnology became available. On the other hand, and instriking contrast, no study has previously demonstratedcoherent control of bond making. The photo-inducedcreation of a chemical bond between the colliding reac-tants, also termed photoassociation, using femtosecondlaser pulses has proven to be much more challenging [12–16]. Particularly at high temperature, a typical situationfor chemical reactions, the starting point for photoas-sociation is rather unfavorable to coherent control sincemany scattering states are incoherently populated. Anecessary requirement for the coherent control of pho-toassociation is thus preparation of quantum states withsome coherence. Key are vibrational coherences in thedesired bond. As we have previously demonstrated withtwo-photon femtosecond photoassociation of hot magne-sium atoms [16, 17], such coherences can be generated byFranck-Condon filtering of quantum correlated states, ex-ploiting correlations between rotational and translationalmotion in the initial incoherent thermal ensemble. Thesecoherences should be amenable to coherent control.Here we demonstrate coherent control of bond-makingin strong-field multiphoton femtosecond photoassociation(PA) of hot magnesium atoms. Our experimental resultsshow the PA yield of detected Mg molecules to be coher- P o t en t i a l ene r g y ( c m − ) Interatomic distance (bohr) S+ S P+ S S’+ S P+ P D+ SX S g+ A S u+ (2) S u+ (1) P u (2) P u1 D u1 P g 1 D g (3) S g+ (a) E nhan c e m en t r e l a t i v e t o t he T L pu l s e Chirp parameter k (10 −2 ps ) (b) I (exp)I (exp)I (theory)I (theory) FIG. 1: (a) The bond-making photoassociation process in-volves excitation of pairs of atoms into bound levels of elec-tronically excited states (bold arrows) which are monitored byUV emission (dotted arrow). (b) Emitted UV intensity I UV (see text) as a function of the linear chirp parameter (squares: I TL = 7 . × W/cm ; circles: I TL = 5 × W/cm ). ently controlled by linearly chirped pulses: The yield isstrongly enhanced, compared to an unshaped transform-limited pulse, by positively chirping the pulses, and sig-nificantly suppressed for negatively chirped pulses. Themeasured PA yield is further enhanced by performinga closed-loop phase optimization of the best positivelychirped pulse, using a genetic algorithm. Our ab ini-tio model reveals the control mechanism to include pu-rification via Franck-Condon filtering of collision ener-gies and partial waves, chirp-dependent coherent Ramantransitions, and vibrational coherent molecular dynam-ics. Our results prove that coherent control of binaryphoto-reactions is feasible even under thermal conditions.The bond-making excitation scheme for the free-to-bound PA process, Mg+Mg+ hν → Mg ∗ , is shown inFig. 1(a): Pairs of magnesium atoms, part of an en-semble held at a temperature of 1000 K, collide in the X Σ + g ground electronic state and are photoassociatedvia multiphoton transitions by an intense phase-shapednear-infrared femtosecond pulse. The laser pulse is of lin-ear polarization, 840 nm central wavelength, 13 nm band-width, and 70 fs transform-limited (TL) duration. Forthe linearly chirped pulses, the spectral phase of the pulseis of the form Φ( ω ) = k ( ω − ω ) with ω the centralfrequency and k the linear chirp parameter. The PAprocess involves a broadband free-to-bound non-resonanttwo-photon transition from the X Σ + g state to the ex-cited (1) Π g state as well as strong-field dynamics andresonant dipole transitions between the latter and higher-lying electronically excited states. The pulse shape con-trols the PA dynamics and the final populations of thevarious electronically excited states. Within this mani-fold, bound rovibrational levels of the A Σ + u , (2) Σ + u and(1) Π u states emit UV light at 285.5 − P → S line of the Mg atom at 285 . I UV is proportionalto the total population in these molecular states, reflect-ing the corresponding PA yield of Mg molecules. Thispopulation is our control objective.Experimentally, magnesium vapor of about 5 Torr par-tial pressure in a heated cell with argon buffer gas isirradiated by a shaped laser pulse. The shaping wasdone using a liquid-crystal spatial light phase modula-tor [18, 19]. The UV radiation emitted toward the laser-beam entrance to the cell is collected at a small anglefrom the laser-beam axis using a proper optical setup.Coherent control of femtosecond photoassociation isdemonstrated in Fig. 1(b) by plotting the UV signal I UV ,normalized with respect to the signal obtained for theTL pulse ( k = 0), versus the chirp parameter k . Anoverall high degree of chirp control and a strongly asym-metric chirp dependence are observed. In particular,a large enhancement is obtained for positively chirpedpulses whereas negatively chirped pulses lead to strongsuppression. The chirp enhancement ( E ) also exhibitsan intensity dependence, which is a clear indication ofthe strong-field regime: As the pulse energy, or, equiva-lently, the peak TL intensity I T L , increases, the maximalenhancement E max and the corresponding chirp k max be-come larger. We find E max =4.2 at k max =0.004 ps and E max =5.4 at k max =0.006 ps for I T L = 5 . × W/cm and 7 . × W/cm , respectively. Since two linearlychirped pulses, one of positive chirp | k | and the otherof negative chirp −| k | , have identical instantaneous tem-poral intensity but different instantaneous temporal fre-quency and phase, the degree of coherent control is bestreflected by the enhancement ratio E ( k max ) /E ( − k max ).It amounts here to about 40 for I T L = 7 . × W/cm ,i.e., the experimentally observed PA yield is enhanced by this factor for the positively chirped pulse with k = k max = 0 .
006 ps as compared to the negatively chirpedpulse. This striking evidence of phase control calls for anexplanation in terms of the underlying quantum molecu-lar dynamics.Our first principles modeling of the multiphoton PAprocess utilizes the theoretical framework of Ref. [17],combining ab initio electronic structure theory withquantum molecular dynamics for the Mg molecule inthe presence of a strong laser field and thermal averag-ing based on random phase wavefunctions. Here, we ex-tend the model of Ref. [17] and explicitly account for allelectronic states shown in Fig. 1(a), in order to improvethe treatment of the Stark shifts for the electronicallyexcited states [22]. The UV emission signal I UV is calcu-lated from the final populations of the appropriate states, A Σ + u , (2) Σ + u and (1) Π u , via their Einstein coefficients.These electronic states have a significant electronic tran-sition dipole moment to the X Σ + g state and rovibra-tional levels that are located below the P + S atomicthreshold, giving rise to emission at wavelengths largerthan 285.3 nm.As Fig. 1(b) shows, our theoretical model clearly re-produces the main features of the experimental results– enhancement of the signal for positive chirp and sup-pression for negative chirp. The dependence on intensity,i.e., the larger values of E max and k max for larger inten-sity, is also predicted qualitatively correctly by the calcu-lations. Quantitatively, the simulations show a slightlysmaller peak enhancement, E max =4.5 instead of 5.4 forthe larger intensity; and the maximum is located at largerchirps compared to the experimental data. The discrep-ancy with respect to E max can easily be resolved by asmall scaling of the (1) Π g Stark shift. For example, scal-ing this Stark shift by a factor of 0.95, well within theestimated error bounds of the calculated polarizabilities,increases E max from 4.5 to 5.8. On the other hand, theshift in k max is most likely linked to the relative slopesof the potentials of the (1) Π g state and all highly ex-cited states that are accessed from it. Due to the num-ber of electronic states that are involved, it is not possi-ble to identify a single or few parameters whose changewould result in an improved model. The inaccuracy ofthe highly excited states of Mg in our model is confirmedby recent spectroscopy [20] which revealed the well depthof the adiabatic (1) Π u state to be larger by nearly 50%than the original ab initio result [17]. This inaccuracy isnot surprising: Potential energy curves of highly excitedstates are more prone to error than lower ones since theyoften originate from the interaction between two open-shell excited-state atoms. Such interactions may lead tomolecular electronic states which are very different fromthe reference ground state and thus demand an even morecorrelated approach than the coupled cluster methodwith single and double excitations that was employed inRef. [17]. Moreover, the high density of electronic states E nhan c e m en t r e l a t i v e t o t he T L pu l s e Chirp parameter k (10 −2 ps )(a) 1−P X A S u+ (2) S u+1 P u P opu l a t i on ( a r b . un i t s ) (b) TL pulsek = 0.006 ps k = −0.006 ps k = 0.015 ps k = 0.025 ps P opu l a t i on ( a r b . un i t s ) Binding energy (10 cm −1 )(c) TL pulsek = 0.006 ps k = −0.006 ps k = 0.015 ps k = 0.025 ps FIG. 2: Theoretical results ( I TL = 7 . × W/cm ) for:(a) Chirp dependence of the total photoassociated population(1 − P X ) and the final populations in the states of the probedUV-emitting band. (b) Final vibrational distribution in theintermediate (1) Π g state for various values of the chirp pa-rameter k . (c) Same as in (b) but obtained within a reducedmodel comprising only the X Σ + g and (1) Π g states. and the occurrence of possibly numerous avoided cross-ings between them result in a multi-reference nature ofthe electronic problem in the experimentally probed en-ergy window which cannot be accurately described by amodel based on a single-determinant assumption. Thesefacts together stretch the capabilities of state of the art abinitio methods. Therefore, more detailed spectroscopicdata would be necessary to improve all relevant potentialenergy curves and allow for full quantitative agreementbetween theory and experiment.The qualitative agreement observed in Fig. 1(b) is,however, certainly sufficient to examine the theoreticalresults in view of the mechanism that underlies the chirpcontrol. To this end, Fig. 2(a) displays the chirp depen-dence of all the population that is photoassociated, givenby 1 − P X , with P X the final population in the X Σ + g ground electronic state, comparing it to the chirp depen-dence of the final population in the probed UV-emittingstates: Whereas almost no chirp dependence is observedin the total PA yield (1 − P X ), a clear chirp dependenceis seen in the final populations of the emitting states, inparticular a large asymmetry in the population of the A Σ + u state. This suggests that the observed chirp de-pendence does not originate from the non-resonant X Σ + g to (1) Π g transition, but rather results mainly from thestrong-field dynamics on the (1) Π g and higher lying elec-tronically excited states. If one assumes the last photonthat excites into the UV-emitting states to constitute aweak probe, it is the shape of the vibrational distributionin the intermediate (1) Π g state that should be respon-sible for the chirp dependence of the signal in Fig. 1(b). Indeed, the final vibrational distribution in the (1) Π g state, plotted for various chirps in Fig. 2(b), shows aclear dependence on both sign and magnitude of thechirp parameter. The analysis is further supported byFig. 2(c) which presents, for comparison, the final vibra-tional distribution in the (1) Π g state, obtained withina reduced two-state model. It comprises only the X Σ + g and (1) Π g states rather than all the electronic states ofthe full model. The results of the two-state model dif-fer both qualitatively and quantitatively from those ofthe full model and show, in particular, no dependence onthe sign of the chirp. The chirp dependence in the fullmodel can then be rationalized in terms of resonant Ra-man transitions between the (1) Π g state and the higherlying u -states: The time dependence of the instantaneousfrequency of the chirped pulse leads to an up (down) shiftof the (1) Π g vibrational distribution for negative (pos-itive) chirp. The magnitude of the up- or down-shiftdepends on the absolute value of the chirp parameter.Down-shifting the (1) Π g vibrational distribution resultsin an enhanced UV emission signal I UV because it favorstransitions into bound levels of the A Σ + u state in theprobed UV-emitting band, whereas an up-shifted (1) Π g vibrational distribution is predominantly excited into dis-sociative states which do not contribute to the molecularemission signal. Our picture of a perturbative final probephoton is confirmed by comparing the calculated final vi-brational distributions in the UV emitting states to theFranck-Condon projection of the final (1) Π g distribu-tion onto these states. The narrowing of the vibrationaldistribution due to the chirp, observed in both Fig. 2(b)and (c), is readily understood in terms of a competitionbetween non-resonant Stark shifts and chirp. The chirplowers the peak intensity and thus the Stark shift suchthat less power broadening of the vibrational distribu-tion is induced by the chirped pulses as compared to theTL pulse. Figure 2(b) also indicates why a chirp rate of k = 0 .
015 ps is optimal: For even larger chirps, we donot observe a further down-shift of the (1) Π g vibrationaldistribution. This is attributed to both the limited band-width of the pulse and the significantly reduced peak in-tensity for larger chirps. Thus the optimal chirp resultsfrom a competition between sufficient intensity for theRaman transitions and down-shifting of the vibrationaldistribution. This interpretation is also supported by thelarger peak enhancement observed for the larger intensityin Fig. 1(b).Our understanding of the control being facilitated byshaping the vibrational (1) Π g distribution suggests thatfurther enhancement of the PA signal should be possibleby exploiting the (1) Π g vibrational dynamics, in addi-tion to the Raman transitions. To explore this possibility,we have experimentally carried out an automated closed-loop phase optimization using a learning algorithm [6],with the measured PA enhancement relative to the TLpulse as the performance criterion. Each generation of (a)(cid:13) (b)(cid:13) (cid:13) (cid:13) E nhan c e m en t r e l a t i v e t o t he T L P u l s e (cid:13) Generation(cid:13) T e m po r a l I n t en s i t y & F r eq . ( a r b . un i t s ) (cid:13) (cid:13) (cid:13) Time (fs)(cid:13)
Optimized Pulse(cid:13) (cid:13) Intensity (cid:13) (cid:13) Freq.(cid:13)Best Chirped Pulse(cid:13) (cid:13) Intensity (cid:13) (cid:13) Freq.(cid:13)
FIG. 3: Results for experimental closed-loop phase optimiza-tion using a learning algorithm, starting from the best linearlychirped pulse: (a) Enhancement as a function of generation;improvement is very fast. (b) The optimized pulse comparedto the best linearly chirped pulse ( k = 0 .
006 ps ). c oh e r e n ce -400 0 400 800 1200 0240510 e l ec t r i c f i e l d ( GV / m ) optimized pulsek = 0.006 ps k = 0.015 ps -400 -200 0 200 400 600 800 1000 1200 1400 time (fs) a u t o c o rr e l a ti on k = 0.006 ps optimizedpulse k = 0.015 ps (a)(b) FIG. 4: Role of vibrational coherence and dynamics in the(1) Π g state – vibrational coherence measure (a) and auto-correlation function (b). the learning algorithm contains 24 members (chromo-somes), where each member is a spectral phase patternapplied to the pulse. The first generation of the optimiza-tion includes the five best linearly chirped pulses, with k = 0 .
004 ps to 0 .
008 ps , while all other members arerandom. Figure 3(a) shows the maximally obtained PAenhancement, achieved within each generation, as a func-tion of the generation number. A fast increase of about35% in the maximum enhancement is observed, from avalue of 5.4 at the first generation to a value of about 7.4after 130 generations. The two corresponding pulses, i.e.,the best linearly chirped pulse and the optimized pulse,are shown in Fig. 3(b). While the optimization essentiallykeeps the positive linear chirp, the main change consistsin a temporal splitting of the optimized pulse into twosub-pulses with a time delay of 130 fs. This time delaycorresponds to the vibrational period of the (1) Π g levelsin the excitation region. It indicates that the optimizedpulse utilizes the vibrational dynamics for improving thePA enhancement. When testing the experimentally optimized pulse inour theoretical model, an enhancement of 7.1 is obtained,surprisingly close to the experimental value. Compar-ing the dynamics under the experimentally optimizedpulse to that of the best linearly chirped pulse revealsthe optimized pulse to populate a significantly broadervibrational band in the (1) Π g state, with more pop-ulation in the lower levels that can directly be excitedby a one-photon transition into the probed UV-emittingband. The optimized pulse is thus shaped to enhancethe transition probability in the resonant Raman transi-tions to these lower (1) Π g levels. The role of coherentvibrational dynamics in the (1) Π g state is further ana-lyzed in Fig. 4 which displays the vibrational coherencemeasure [21], C ( t ) = P i = j | ρ Π g ij ( t ) | , and the autocorrela-tion function, A ( t ) = tr[ ρ Π g (0) ρ Π g ( t )], of the normalized(1) Π g density. The values of C ( t ) in Fig. 4(a) need to becompared to the upper bound of the maximally coherentstate, ( d − /
2, where d is the number of levels, about70 in our case. We thus find a substantial amount ofvibrational coherence in the (1) Π g state. In particularat intermediate times, when the dynamics in the (1) Π g state is relevant, the coherence measure is larger for theoptimized pulse than for the chirped pulses. The secondsub-pulse of the optimized pulse increases the autocor-relation function, reflecting the synchronization of thepulse delay with the vibrational dynamics. These obser-vations confirm that the optimized pulse outperforms thechirped pulses by utilizing coherent vibrational dynamicsin the (1) Π g state, in addition to the chirp-dependentRaman transitions.In summary, we observe strong-field coherent controlof bond formation in the femtosecond photoassociationof thermally hot magnesium atoms using phase-shapedlaser pulses. Our modeling from first principles has al-lowed us to identify a combination of Franck-Condon fil-tering in the free-to-bound non-resonant two-photon stepwith chirp-dependent resonant Raman transitions andcoherent vibrational dynamics in an intermediate elec-tronic state to be responsible for the control. Whereasthe purpose of the FC filtering is mainly purification inorder to allow the generation of molecular coherence, theRaman transitions and vibrational dynamics serve to re-alize phase control. Indeed, the quantum purity in theintermediate state and final UV-emitting states differ byonly 25% for the experimentally optimal linear-chirp of k = 0 .
006 ps . Our demonstration of coherent controlof bond-making under thermal conditions points the waytoward controlling transition probabilities and branch-ing ratios to different target states. For photo-inducedchemical reactions with several product channels, suit-able target states would be those that serve as a gatewayto a different product channel. A feasible route to thecoherent control of photo-induced bimolecular chemicalreactions is now open.We would like to thank Daniel Reich for technicalhelp at an initial stage of this work. This research wassupported by The Israel Science Foundation (Grant No.1450/10) and the Alexander von Humboldt Foundation(WS). ∗ These authors contributed equally.[1] D. Tannor and S. A. Rice, J. Chem. Phys. , 5013(1985).[2] R. Kosloff, S. Rice, P. Gaspard, S. Tersigni, and D. Tan-nor, Chem. Phys. , 201 (1989).[3] S. A. Rice and M. Zhao, Optical control of moleculardynamics (John Wiley & Sons, 2000).[4] P. Brumer and M. Shapiro,