Goldilocks Zone for Enhanced Ionization in Strong Laser Fields
M. Möller, A. M. Sayler, P. Wustelt, L. Yue, S. Gräfe, G. G. Paulus
GGoldilocks Zone for Enhanced Ionization in Strong Laser Fields
M. M¨oller,
1, 2
A. M. Sayler,
1, 2
P. Wustelt,
1, 2
L. Yue, S. Gr¨afe, and G. G. Paulus
1, 2 Institute of Optics and Quantum Electronics, Abbe Center of Photonics,Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany Helmholtz Institut Jena, Fr¨obelstieg 3, 07743 Jena, Germany Institute of Physical Chemistry and Abbe Center of Photonics,Friedrich Schiller University Jena, Helmholtzweg 4, 07743 Jena, Germany
Utilizing a benchmark measurement of laser-induced ionization of an H +2 molecular ion beamtarget at infrared wavelength around 2 µ m, we show that the characteristic two-peak structurepredicted for laser-induced enhanced ionization of H +2 and diatomic molecules in general, is a phe-nomenon which is confined to a small laser parameter space — a Goldilocks Zone. Further, wecontrol the effect experimentally and measure its imprint on the electron momentum. We replicatethe behavior with simulations, which reproduce the measured kinetic-energy release as well as thecorrelated-electron spectra. Based on this, a model, which both maps out the Goldilocks Zone andillustrates why enhanced ionization has proven so elusive in H +2 , is derived. Since its first complete quantum description in the1920’s [1], the H +2 bond has served as the prototype forall molecular systems [2]. This is particularly true forthe attosecond dynamics of molecular bonding in strongfields, where the insights gained from H +2 have servedas a foundation for the understanding of more complexbonds. For example, the concepts of bond hardening,bond softening, above-threshold dissociation and above-threshold ionization were first determined from H +2 andare now ubiquitous in the descriptions of laser-inducedmolecular dynamics [2].However, another foundational process — enhancedionization (EI) — continues to be evasive and contentious[3, 4], partially due to the difficulty of direct measure-ments of H +2 . The EI process was predicted by early time-dependent Schr¨odinger equation (TDSE) calculations forH +2 with fixed internuclear distances and showed thationization, i.e. H +2 → p + + p + + e − , is enhanced for spe-cific internuclear distances [5]. Since, EI has frequentlybeen used as one of the processes invoked to explain andpredict laser-induced molecular dynamics for small andmore complex molecules alike [6–12].In a common picture of EI in H +2 , depicted in Fig. 1,there are two distinct mechanisms leading to ionization.If the nuclear separation, R , is relatively small and thelaser field is relatively large, then the electronic wave-function will follow the laser, moving to the downhill sideof the potential with each optical half cycle and tunnelout with sufficient laser intensity around the peaks of thefield. Alternatively, if the laser field ramps up slowly, themolecule stretches and the potential barrier between thetwo protons grows. This can effectively trap a portion ofthe electron wavepacket on the uphill proton and allowionization from the upper potential over the deformedinner barrier [12]. Thus, ionization is enhanced near R -values corresponding to the aforementioned cases and acharacteristic double peak structure should emerge in the R - or KER-dependent electron yield [5].Although this is a logical and straightforward expla- nation and the double-peak structure is obvious in thecalculated static-field ionization rate of fixed-nuclei H +2 molecules [13, 14], many studies of H +2 over the past twodecades could not clearly identify EI, which has fueled thedebate over the relevance of the concept [4, 15–19]. More-over, experimental effects; such as the initial vibrationalstate distribution of a H +2 target, the imprint of the pre-requisite ionization of a H target, depletion, intensity-volume effects, conversion from the measured KER to theinferred R -values and the coupled electron-nuclear dy-namics; make the process and interpretation much morecomplex [3, 4]. Measurements, which use neutral H as atarget, often leave uncertainty about which observed ef-fects are, at least partially, due to prerequisite ionization.For example, indications of EI have only been clearlyseen when creating a traveling nuclear wavepacket froma ionization from a neutral target, H → H +2 + e − , andprobing the resulting dissociative state with time-delayedfew-cycle pulses [4]. However, this leaves many questionabout the prevalence and importance of the phenomenonfor typical laser fields.To settle these long-standing questions and clearlydemonstrate the EI effect in H +2 , here, we implement,the first to our knowledge, intensity-dependent measure-ment of the short-wave infrared (SWIR) laser-inducedionization of a H +2 molecular-ion-beam target, which cap-tures both the momenta of the nuclear fragments andthe correlated electron momentum. In this largely un-explored territory of wavelength and intensity, we con-clusively observe the two-peak structure characteristic ofEI and are able to control it with the intensity enve-lope of the laser pulse. Next, using specifically developednuclear-wavepacket propagation calculations that includeionization, which are further validated by the correlatedKER and electron momenta spectra, we determine theGoldilocks Zone, i.e. the limited laser parameter space,where EI is visible. Finally, this data is used to formulatea clear-cut model with great explanatory and predictivepower. a r X i v : . [ phy s i c s . a t o m - ph ] D ec l o c a l i z e d e - p + Potential r - e le c tr o n e f f i c i e n t e - t r a n s f e r time R lo w -K E Rio n iz a tio nh ig h -K E Rio n iz a tio n l a r g e - R ( l o w - K E R ) i o n i z a t i o n( s l o w l y i n c r e a s i n g i n t e n s i t y )s m a l l - R ( h i g h - K E R ) i o n i z a t i o n( q u i c k l y i n c r e a s i n g i n t e n s i t y ) e - b a r r i e r p a r t i a l l yt r a p p i n g e - p + R FIG. 1. Schematic of enhanced ionization (EI). (top) Time-and r -dependent potential with projection. Stretching of theinternuclear distance, R , is initiated early in the pulse and theelectron, e − , is driven back and forth between the two nucleiby the laser field (dashed arrows). For electric fields thatincrease quickly in time, after a few cycles the electron canionize from the lower potential (bottom left) resulting in largekinetic-energy release (KER). For electric fields that increaseslowly in time, the field is not large enough to ionize at small R and the electron continues (dotted arrows) to oscillate, asthe molecule stretches, until the internuclear distance is largeenough to create an internuclear potential barrier (bottomright) partially localizing the electron and facilitating ioniza-tion from the upper well. This enhances the ionization prob-ability resulting in small KER. The experimental challenges arise from the dilute ion-beam target and conditions needed to measure the mo-menta of both protons and the electron [20–22], see sup-plemental material for details [23]. The electron momen-tum, p e , is determined using the sum momentum of theproton, i.e. without directly detecting the electron. Thisrequires an extremely high experimental precision andwell-collimated ion beam. Further, despite the signif-icantly more complex laser setup needed, we chose touse SWIR pulses to increase the electron momentum, | p e | ∝ λ . Although our measurement of the electronmomentum is blurred with the momentum distributionthat arises from the temperature of the H +2 molecular ion (a) (b) (a) (b) FIG. 2. (a) Measured intensity- and KER-dependent ioniza-tion yield of H +2 for a 65 fs 2 µ m pulse. The shape of theKER spectra is emphasized by normalization of the spectrumwithin each intensity bin of the 2D plot. (b) Logarithm ofthe measured joint electron-nuclear energy distribution (JED)for I ≈ (0 . ± . × W/cm . The KER spectrum(grey line) and the mean electron energy are overlaid (orangedots). Note the KER-dependent modulation of the widthof the photoelectron spectrum, which has minima where theKER-dependent yield peaks. See text for details. beam, compared to a direct measurement of the electronmomentum [16, 24], our approach reduces the experimen-tal cost and complexity significantly and is, to the bestof our knowledge, the first application of this techniqueto a molecular ion beam laser interaction.The measured intensity- and KER-dependent laser-induced ionization yield for H +2 with 65 fs, 2 µ m pulsesis shown in Fig. 2(a). Here one sees that at low inten-sity ( I < ∼ ) the yield is peaked near 3.5 eV( R ’ . I > ∼ ) the yield is peaked near 5 eV ( R ’ . R s,where the electron wavepacket is effectively transferredto the lower potential well each half cycle. This depletesthe dissociative nuclear wavepacket before it reaches R thereby reducing the peak at lower KER. The character-istic double-peak structure we are in search of only occursin the narrow overlapping transition intensity range ( I ≈ ± ), where both processes can occur.Although this interpretation, based on the measure-ment of the nuclear fragments, tells us a great dealabout the underlying dynamics, it is only half the pic-ture. To gain full access to the dynamics at play, wesimultaneously measure the nuclear fragment momentato produce the joint electron-nuclear energy distribution(JED) shown in Fig. 2(b). Here we see that reduc-tions in the width of the electron spectrum are corre-lated with increases in the yield. Although calculatedJEDs, where the electron and the nuclear dynamics aretreated in equal footing [17, 18, 25], show hints of thisbehavior, that work typically focuses on diagonal energy-conserving lines, which have also been measured, e.g. byWu et al. [26]. In contrast, here we are focused on thelarge-scale behavior.Unlike some of the more complex models of H +2 ion-ization, which look at details of the R -dependent timingof the liberated electron wave-packet [15, 27], the mea-surement behavior here has a relatively straightforwardqualitative explanation. Namely, in addition to increas-ing the yield, enhancing the ionization rate at certain R s lowers the average intensity required for ionization.This, in turn, reduces the photoelectron energy, whichscales with the intensity [28]. Thus, peaks in the KER-dependent ionization yield should overlap with minimain the width of the correlated electron spectrum as ob-served.To further understand the experimental results andextend the control of EI to other laser parameters,e.g. wavelength and pulse duration, we implement a two-surface time-dependent Schr¨odinger calculations [29] andaugmented them for ionization [30] including the corre-lated electron energy, see supplemental material. Usingthe measured laser parameters, this results in the spec-tra shown in Fig. 3. Here we see that the enhancedtwo-surface model is a good qualitative match to themeasured data and accurately predicts the double-peakstructure for roughly the same narrow intensity range.Moreover, the corresponding KER-dependent electronmomentum also follows the measured trend, which con-firms that the model is capturing the relevant underlyingdynamics. The minor differences between measurementand theory, i.e. slightly different peak positions, are likelydue to several imperfections of the model detailed in thesupplemental material.To identify the dynamics at play we examine the calcu-lated nuclear dynamics for the three characteristic situa-tions noted above in Fig. 4(a)–(c). In Fig. 4(a), when theintensity is too small , the leading edge of the laser pulsebegins the dissociation process, i.e. stretching in R of thenuclear wavepacket, and the intensity only becomes suffi- (a) (b) (a) (b) FIG. 3. (a) Calculated intensity- and KER-dependent ion-ization yield of H +2 for a 65 fs 2 µ m pulse. The shape ofthe KER spectra is emphasized by normalization of the spec-trum within each intensity bin of the 2D plot and jitter hasbeen added by simulating the measurement statistics. (b)Calculated h E e i and the KER spectrum at a peak intensityof I = 0 . × W/cm . All calculation results are av-eraged over the intensity distribution in the focal volume aswell as over the relevant vibrational states. cient to ionize after the molecule has stretched to R ’ ’ too big , the intensity ramps up quickly enough that ion-ization depletes the nuclear wavepacket at R ’ ’ just right , the intensity ishigh enough to ionize near R , but low enough to allowpart of the wavepacket to survive and stretch to R beforeionization, which results in the double peak shown.This leads to an intuitive model for predicting whenthe characteristic double-peak structure of EI will be visi-ble. Assuming that stretching of the molecule is initiatedby the laser field at relatively low intensities, I diss , andsmall internuclear distances, R diss , then ionization willoccur at a later time, t ion = t diss + ∆ t , after the moleculehas stretched to R ion and the intensity has increased suf-ficiently to ionize the molecule, I ion . Further, if there are |Y ( R , t ) | ( c ) I = 1 . 2 P W / c m S m a l l R ( L a r g e K E R ) t i m e ( f s ) ( b ) I = 0 . 3 P W / c m D o u b l e P e a k t i m e ( f s ) t i m e ( f s )
R (a.u.) D t I io n I d is s I (PW/cm2) t i m e ( f s ) t o o s t e e p t o o s h a l l o w j u s t r i g h t j u s t r i g h tt d is s t io n ( d ) ( e ) T D S E F i t M e a s u r e m e n tP u l s e D u r a t i o n ( F W H M f s )
I0 (PW/cm2) ( a ) I = 0 . 1 P W / c m L a r g e R ( S m a l l K E R ) S m a l l R ( H i g h K E R )I o n i z a t i o n L a r g e R ( S m a l l K E R )I o n i z a t i o n | D t p - D t f | ( f s ) KER (eV) P io n ( R , t ) FIG. 4. (a)–(c) In color, calculated probability density, | Ψ( R, t ) | , as a function of time, t , and internuclear distance, R , forthree different intensities, I = 0 .
1, 0 .
29 and 1 . , respectively. This is intensity averaged over the experimental 2Dtarget geometry and comprised of the incoherent sum of vibrational states. The expectation value of R , h R i , is shown as agray line to guide the eye. The t - and R -dependent ionization probability are displayed as contours with projections on thebottom and left axes, respectively. For reference, the laser field, including its envelope, is shown in red at the top. Note thelegends apply to all three pannels. (d) Examples of pulses, I ( I , FWHM; t ), that fulfill the requirements to result in a doublepeak in the KER-dependent ionization yield (bold lines) and those that do not (thin lines). (e) The position of the measuredKER-dependent double peak with the line marking the region where the double peak is visible (red). Additionally, the positionsof the most predominant KER-dependent double peak from the calculations (white circles), with lines marking the region wherethe double peak is visible. In color, a map of the time delay, ∆ t pulse ( I diss , I ion ; FWHM , I ), between I diss and I ion for a givenpulse, with respect to the travel time determined from the fit parameters, i.e. | ∆ t pulse ( I diss , I ion ; FWHM , I ) − ∆ t fit | . See textfor details. two preferred internuclear distances for ionization, R ion1 and R ion2 , then the double-peaked structure will be lost,if the laser intensity does not ramp up in the very par-ticular way described above. This places constraints onthe intensity envelope of the laser pulse. For example, ifone wishes to maintain the same timing while increasingthe intensity, the pulse length must be increased, see thethick dashed and solid lines in Fig. 4(d).To determine these parameters, we first map outthis region with our measurements and wavepacket-propagation calculations. Specifically, to parameterizethe double peak structure, we fit a double Gaussian tothe calculated data over a large range of laser parameters.Then we find the ratio of the peaks, α = A < /A > , where A < is the amplitude of the lower of the two peaks and A > is the the amplitude of the higher of the two peaks. Thisallows one to determine the Goldilocks Zone where thedouble peak occurs and to what extent it is visible. Theintensity where the maximum value of this ratio, α max ,occurs is plotted in Fig. 4(e) (circles) as a function of the FWHM pulse duration for the calculations at λ = 2 µ mand α > . I diss ’ . · W/cm , I ion ’ . · W/cm and∆ t ’
20 fs, see line in Fig. 4(e). Here the positionsof the calculated maximum ratio between the two peaksfit nicely to the simple model and the fitted values areconsistent with existing measurements and calculations.Therefore, this remarkably simple model can serve as aguide to control EI by balancing the pulse length and in-tensity of the laser. Additionally, the time for the pulseto ramp up from I diss to I ion relative to ∆ t is plotted infalse color to illustrate why EI is so elusive, particularlyfor the short pulses typically used in strong-field physics.In conclusion, we have measured the intensity- andKER-dependent ionization yield, along with the elec-tron momentum, for the benchmark molecular ion, H +2 ,starting directly from a molecular ion beam in the rel-atively unexplored short-wave infrared (SWIR) regime.We demonstrate that the characteristic double-peak fea-ture of enhanced ionization (EI) can only be observedin a very limited laser parameter space — the GoldilocksZone — where pulse duration and laser intensity are care-fully balanced and the interplay between nuclear stretch-ing dynamics and ionization allows for ionization from abroad nuclear wave packet. This directly address a long-standing debate, explains the elusive nature of enhancedionization, and serves as a guide for how to manipulatelaser parameters to coherently control the phenomenon.Moreover, as the behavior of H +2 serves as the prototypefor all molecular systems and EI is generally believed toplay a decisive role in more complex systems, this hasbroad ramifications for strong-field physics in general.We acknowledge helpful discussions with F. Gross-mann. This work was supported by grants PA730/5 andGR4482/2 of the German Research Foundation (DFG)as well as by laserlab europe. [1] Ø. Burrau, The Science of Nature (Naturwissenschaften) , 16 (1927).[2] H. Ibrahim, C. Lefebvre, A. D. Bandrauk, A. Staudte,and F. L´egar´e, J. Phys. B: At. Mol. Opt. Phys. , 042002(2018).[3] I. Ben-Itzhak, P. Q. Wang, A. M. Sayler, K. D. Carnes,M. Leonard, B. D. Esry, A. S. Alnaser, B. Ulrich, X. M.Tong, I. V. Litvinyuk, C. M. Maharjan, P. Ranitovic,T. Osipov, S. Ghimire, Z. Chang, and C. L. 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1, 2
A. M. Sayler,
1, 2
P. Wustelt,
1, 2
L. Yue, S. Gr¨afe, and G. G. Paulus
1, 2 Institute of Optics and Quantum Electronics, Abbe Center of Photonics,Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany Helmholtz Institut Jena, Fr¨obelstieg 3, 07743 Jena, Germany Institute of Physical Chemistry and Abbe Center of Photonics,Friedrich Schiller University Jena, Helmholtzweg 4, 07743 Jena, Germany
EXPERIMENTAL DETAILS
The measurements here were done with a highly automated ion-beam target coincidence 3D momentum imagingsetup [1–3]. The H +2 ion beam is created in a duoplasmatron ion source, accelerated to 9 keV, velocity and massselected with a Wien filter and collimated using a series of Einzel lenses, adjustable apertures and electrostaticdeflectors. Additionally, a longitudinal electric field in the interaction region is used to distinguish charged andneutral fragments and allow coincidence detection of all fragmentation channels using a time- and position-sensitivemulti-channel plate (MCP) delay-line detector (RoentDek DLD).Here we choose to work in the SWIR ( λ = 2 µ m), not the much more common Ti:Sapphire IR regime ( λ ∼
800 nm),to enable determination of the electron momentum, p e , from the shift in the center-of-mass of the detected protons,i.e. p e = − ( p H + + p H + ). In addition to a Ti:Sapphire tabletop system (Femtopower Compact Pro, Femtolasers)and a high power booster amplification stage (Thales), this requires operation of a high-energy, high-average-poweroptic parametric amplifier (HE-TOPAS OPA, Light Conversion) to achieve the 65 fs (full-width at half-maximum inintensity), 2 µ m, 750 µ J pulses. The pulses are linearly polarized using an acute angle reflection off a Germaniumplate and focused with an f = 150 mm, 90 ◦ off-axis parabolic mirror to peak intensities up to 1.3 PW/cm at 1 kHz.In addition, we continuously scan the intensity and tag each fragmentation event with that information, which allowsfor subsequent sorting and a detailed investigation of the intensity dependence. WAVEPACKET PROPAGATION
The bound electron-nuclear wavefunction of H +2 is approximated using solutions to the time-dependent Schr¨odingerequation for the two lowest field-free Born-Oppenheimer (BO) electronic states [4, 5]. The external laser field, E ( t ), couples the electronic states with the transition dipole moment, d gu ( R ), and facilitates the laser-driven nucleardynamics, i.e. stretching of the internuclear distance due to transitions of the nuclear wavepacket between the 1 sσ g and 2 pσ u states. i ddt (cid:20) Ψ sσ g ( R, t )Ψ pσ u ( R, t ) (cid:21) = " V sσ g ( R ) + p R µ − d gu ( R ) · E ( t ) − d gu ( R ) · E ( t ) V pσ u ( R ) + p R µ Ψ sσ g ( R, t )Ψ pσ u ( R, t ) (cid:21) (1)Ionization, i.e. the transition to the 1 /R potential curve from the perspective of the nuclei, is incorporated at each time-step by the use of R - and field-strength-dependent ionization rates for both of the electronic states, i.e. Γ g ( R, | E ( t ) | )and Γ u ( R, | E ( t ) | ) [6, 7]. Namely, since the probability decays as | Ψ | = | Ψ | exp( − Γ dt ) (cid:20) Ψ sσ g ( R, t + dt )Ψ pσ u ( R, t + dt ) (cid:21) = " exp( − Γ g ( R, | E ( t ) | ) · dt ) 00 exp( − Γ u ( R, | E ( t ) | ) · dt ) Ψ sσ g ( R, t )Ψ pσ u ( R, t ) (cid:21) (2)The calculation is done using the split-step method to solve the TDSE, Eq. 1, within a box defined from R = 0 to50 a.u. with a spatial grid of dR = 0 .
025 a.u. and a time step of dt = 0 . envelope of the electricfield. We have checked that the nuclear momentum and ionization probabilities converge for these parameters.This method allows one to track the nuclear wavefunction and determine the R - and t -dependent ionization yield, P ion ( R, t ), i.e. P ion ( R, t ) = ( | Ψ sσ g ( R, t ) | − | Ψ sσ g ( R, t + dt ) | ) + ( | Ψ pσ u ( R, t ) | − | Ψ pσ u ( R, t + dt ) | ) , (3) a r X i v : . [ phy s i c s . a t o m - ph ] D ec using the values from Eq. 2. This can then be converted into a nuclear-energy- and electron-energy-dependentprobability, i.e. P (KER , E e ), using the approximate nuclear energy, E KER i ≈ /R i , and the approximate electronenergy, p e i ≈ − A ( t i ), based on electric field’s vector potential, A ( t ) = − R t −∞ E ( t ) dt . Additionally, to match theexperimental conditions, the theoretical results are averaged over the intensity distribution in the focal volume andthe incoherent sum of vibrational states, ν = 0 – 15, described by the Franck-Condon approximation, is used. INTENSITY ENVELOPE OPTIMIZATION
To produce the characteristic double-peak structure of EI, the laser intensity must ramp up in a very particular way,see Fig. 4(a)–(c). Namely, the molecular dissociation is initiated by the laser field at relatively low intensities, I diss ,then ionization will occur at a later time, t ion = t diss + ∆ t , after the molecule has stretched to R ion and the intensityhas increased sufficiently to ionize the molecule, I ion . Thus, for a cos pulse, I ( t, FWHM) = I cos (cid:0) t π · FWHM (cid:1) , theoptimal intensity-dependent pulse duration for a given peak intensity, I , isFWHM(∆ t, I ion , I dis ; I ) = ∆ t π (cid:2) cos − ( p I diss /I ) − cos − ( p I ion /I ) (cid:3) (4)To determine the optimal parameters, one can use the measured and calculated position of the peak contrast values,see Fig. 4(e) (circles), to fit Eq. 4 and determine the parameters: ∆ t , I ion , and I dis . For the intensity averaged data,this yields the fit parameters: I diss ’ . · W/cm , I ion ’ . · W/cm and ∆ t ’
20 fs, see Fig. ?? (e) (line).For the single intensity calculations, this yields the fit parameters: I diss ’ . · W/cm , I ion ’ . · W/cm and ∆ t ’
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