Probing multiphoton light-induced molecular potentials

The strong coupling between intense laser fields and valence electrons in molecules causes a distortion of the potential energy hypersurfaces which determine the motion of nuclei in a molecule and influences possible reaction pathways. The coupling strength varies with the angle between the light electric field and valence orbital, and thereby adds another dimension to the effective molecular potential energy surface, allowing for the emergence of light-induced conical intersections. Here, we demonstrate in theory and experiment that the full complexity of such light-induced potential energy surfaces can be uncovered. In H$_2^+$, the simplest of molecules, we observe a strongly modulated angular distribution of protons which has escaped prior observation. These modulations directly result from ultrafast dynamics on the light-induced molecular potentials and can be modified by varying the amplitude, duration and phase of the mid-infrared dressing field. This opens new opportunities for manipulating the dissociation of small molecules using strong laser fields.


Introduction
Potential energy surfaces describe the forces acting on the nuclei of a molecule. Within the Born-Oppenheimer approximation the motion of the nuclei along these potentials is treated independently of the electronic motion. This picture breaks down when the electronic level separation becomes comparable to the kinetic energy of the nuclei. This occurs at specific points in the molecular geometry, which are known as conical intersections and which are a hallmark of polyatomic molecules [1]. Conical intersections play an eminent role in visible and ultraviolet photochemistry [2,3], for example in isomerization [4,5], and electron transfer processes [6]. Moreover, they are strongly implicated in the photostability of DNA by way of allowing radiation-less de-excitation [7].
The single-photon transition between two dipole-coupled electronic states can also create a conical, albeit transient, intersection. Hence, these localized features of the laser-dressed potential energy surface have been dubbed light-induced conical intersections (LICI) [8,9]. Their precise position and the underlying dipole coupling strength are determined by the frequency and intensity of the incident light.
LICIs can also be found in diatomic molecules, since the angle between the light polarization and the molecular axis adds a another degree of freedom to the nuclear motion, [10,11]. The angle-dependent distortion of the molecular potential energy surfaces in a linearly polarized laser field directly affects molecular dissociation [12][13][14][15][16] and has been predicted to cause rotational excitation [17][18][19][20]. Recently, experimental indications of LICI in H 2 + have been found in angle-resolved ion spectra [21].
In ultrashort infrared laser fields, the light intensity can easily exceed the threshold for multiphoton transitions. While LICIs are a consequence of single-photon couplings and therefore the potential energy scales linearly with respect to variations of the laser field strength, multiphoton couplings lead to unique structures of their own. In the case of diatomic molecules these structures become non-linear point intersections of the potential energy surfaces. The one-dimensional treatment of single and multiphoton 3 resonances has led to the prediction of light-induced potentials (LIPs) [21][22][23][24][25][26][27]. However, the consequences of the angle-dependent coupling strength around non-linear point intersections for the dissociation dynamics have so far been largely unexplored.
Here, we show in theory and experiment that non-linear light-induced point intersections can result in strong modulations of the angular ion yield. We choose the simplest molecule, H 2 + , which is widely regarded as a prototype system for the interaction of molecules with light [28]. Due to the sparsity of electronic states, H 2 + can often be described as a two-level system consisting of the two lowest electronic states, 1s g and 2p u . When coupled to intense laser fields these states give rise to intensity-dependent dissociation mechanisms known as bond softening [29] and above threshold dissociation [30]. The possibility to control them using the laser amplitude, frequency, phase and pulse duration has been demonstrated [31][32][33][34][35]. In particular, the opposite parity of the  g and  u states can lead to electron localization [36], giving rise to charge resonance-enhanced ionization [37,38] and symmetry breaking in dissociation [39][40][41] under the influence of two-color laser fields [42,43] or carrier-envelope phase stable few-cycle pulses [44][45][46][47]. Descriptive treatments for most of these phenomena are provided by dressed-state pictures, such as LIPs. Shown are the laser-dressed states  g and  u -1ħω. Along the laser polarization, i.e.,  = 0, the state crossing indicated in Figure 1(b) opens up and turns into an avoided crossing. This necessarily lowers the potential barrier at the avoided crossing permits and permits dissociation of formerly bound molecules (bond softening). Importantly, no such avoided crossing occurs when the laser polarization is 4 perpendicular to the internuclear axis. Therefore a LICI is formed at the internuclear distance where the laser-dressed  g and  u -1 states cross.

Non-linear point intersections
While single-photon transitions dominate in moderately intense visible laser fields, multiphoton transitions become relevant already at moderate intensities, when the wavelength is shifted into the mid-infrared [48]. For example, the three-photon transition by 2300 nm light becomes significant already at an intensity of approximately 5• 10 12 W/cm 2 , see Supplementary Figure 1. Several crossings of potential curves that correspond to multiphoton transitions between the  g and  u states of H 2 + at a wavelength of 2300 nm are shown in Figure 1(c). The corresponding LIP energy landscape calculated for an intensity of 3 • 10 13 W/cm 2 is presented in Figure 1

Structured proton angular distribution
In order to experimentally probe the light-induced molecular potentials depicted in Fig. 1(d,e), describing the situation where the mid-IR field induces multiphoton dynamics in the dissociation process, but does not cause ionization, we implement the two-pulse scheme depicted in Fig. 2(a). First, an intense, fewcycle visible laser pulse ionizes neutral H 2 , producing a bound coherent wave packet in H 2 + with a nearly isotropic alignment with respect to the laser polarization [49]. Second, a moderately intense mid-IR pulse creates the LIPs on which dissociation occurs. The LIPs are probed by recording the momentum distribution of protons resulting from the dissociating part of the molecular wave packet. The molecular ions dissociate along their initial alignment direction, unless rotational dynamics occur as predicted e.g.
The intent of the two-pulse scheme is to decouple the production of the molecular wavepacket from the field that generates the LIPs. Thus, scanning the time delay between the laser pulses allows for probing the LIPs at selected times within the mid-IR pulse. Moreover, the use of a shorter wavelength pulse for ionization allows us to reduce the focal volume averaging in the long-wavelength field, which often washes out subtle features in strong-field experiments (e.g., [50]). Finally, choosing a perpendicular relative polarization of the visible and mid-IR pulses is expected to avoid overlap between the signal of interest produced by the mid-IR pulse, and any protons produced by the visible pulse alone. The experimental set-up is described in the Methods section Experiment. W/cm 2 ) mid-IR (top inset in Fig. 2(b)) pulses, suggests that the on-axis features arise from bond softening H 2 + by either the visible or mid-IR pulses alone. Note, that the signal along the mid-IR polarization in the two-color experiment does not arise from dissociative ionization of neutral H 2 by the mid-IR pulse on its own , as no notable ionization of neutral H 2 is obtained at the intensity of 3• 10 13 W/cm 2 . Hence, the comparative mid-IR only data (top inset in Fig. 2(b)) is presented for a higher intensity of 1 • 10 14 W/cm 2 .
The additional spots in the two-color data are tentatively attributed to dynamics caused by the lightinduced structures in the molecular potential energy landscape, cf Fig. 1(d).
Surprisingly, the experimental results presented in Figure 2(b) exhibit a much more pronounced angular structure than the TDSE results for the mid-IR field alone, presented in Figure 1

Numerical results
In a first step to understand the dynamics producing the structured proton angular distribution observed in cross-polarized fields, we solve the 2D TDSE for H 2 + , taking both laser pulses into account, see Methods for details. For the visible few-cycle pulse, we also consider a weak pulse pedestal at 5% of the peak intensity. The initial alignment of the molecular axis with respect to the laser polarization is assumed to be istotropic.
It has been recognized in the literature that angular modulations in the proton spectra can arise from rotational dynamics in the vicinity of the LICI [18][19][20][21]; more specifically, simulations that include rotational motion in the dissociation dynamics show angular modulations that are absent when the rotational degrees of freedom are frozen. These modulations can be connected to rotational scattering of the dissociating wavepacket from the LICI [18][19][20][21]. A first candidate for the physical mechanism underlying the appearance of a structured proton angular distribution is therefore the formation of a high-order rotational wavepacket in the dissociating molecular cation. In order to test the role of rotational dynamics, we perform a first set of calculations where rotational transitions are artificially switched off and present the results in Fig. 3(a). Evidently, strong modulations in the angular distribution are obtained, even without the inclusion of wavepacket rotation. This suggests that rotational dynamics are not the primary physical mechanism underlying the angular modulation in the proton momentum distribution. Therefore, it will be important to identify how angular modulations arise already within a 1D treatment.
The second set of calculations (Figure 3 Figure 3a and 3b, we conclude that rotational dynamics play a significant but secondary role in defining the final momentum distribution; in contrast with the previously considered pure LICI case, the addition of rotations is not the sole cause of the angular structures in our experiment. In order to identify the essential mechanism creating the angular structure in the absence of rotations, we employ two-color Floquet theory (see Methods). We calculate the angle-dependent field-dressed states of H 2 + using a two-color laser field, ⃗ ( ) = √ VIS cos( VIS + VIS )̂ + √ IR cos( IR + IR )̂. ⃗⃗⃗ The field consists of a moderately intense mid-infrared field ( IR = 2280 nm, I IR = 3×10 13 Wcm -2 ) and a weak visible field (I VIS = 1×10 13 Wcm -2 ), corresponding to the pulse pedestal used in the TDSE calculations.
We take  VIS =  IR / 3 to ensure the periodicity of the laser fields required by Floquet theory. The resulting light-induced potential energy landscape depends on the relative optical phase, ∆ = VIS − IR .
A detailed analysis and discussion of the results from Floquet theory is presented in Supplementary Note 4. In brief, we find that the Floquet states represent a conclusive basis for understanding the emergence of the angular structure in the proton momentum distribution, even without rotational dynamics taken into account. The following picture can be invoked.
As the alignment angle of the molecular axis in the polarization plane (see Fig. 3), , is increased, the field components parallel to the molecular axis vary as E VIS () = E 0,vis cos(), and E IR () = E 0,IR sin(). This leads to a pronounced angle dependence of the field-dressed potential energy curves (see Figure 3c), where several Floquet state crossings open up and close again as  is varied. Specifically, at = 0°, i.e. for alignment of the molecular axis perpendicular to the mid-IR polarization, the effect of the mid-IR field is insignificant and dissociation proceeds as in the single-color case (seen in Fig. 2(b)). Specifically, the wavepacket dissociates on the purple surface in Fig. 3(c). As is increased, a new dissociation channel 11 due to one-photon coupling by the mid-IR field opens up. The new channel competes with the original one, which moves population to the pronounced feature at = 10°, making the on-axis feature much narrower than in the single-color case. This new dissociation channel corresponds to dissociation on the red surface in Fig. 3(c). As is further increased, the width of the avoided crossing reaches 2 IR , which closes the dissociation channel and gives rise to a LICI at ~ 30°, clearly visible in Fig. 3(c).
Notably, the computational results obtained without (Fig. 3(a)) and with ( Fig. 3(b)) rotations strongly differ at ~ 20°. We attribute this to the presence of the LICI which promotes strong rotational dynamics, as the nuclear wave packet propagates around the cone in the LIP landscape. In a similar manner, the splitting of the narrow feature at 0° in Fig. 3(a) into the double peak structure in Fig. 3(b) is attributed to the point intersections at  = 0°.

Delay dependence
Scanning the time delay between the visible and mid-IR pulses in our experiment allows us to probe the variations in the LIPs throughout the mid-IR pulse. The time delay controls the time of ionization, which determines the (i) strength, (ii) duration, and (iii) phase of the mid-IR field at the time it interacts with the molecular ion. In Figure 4, we analyze the fragment momentum distribution for overlap of the ionizing visible pulse with the rising edge, the maximum and the falling edge of the mid-IR laser pulse.
Each of the presented spectra are integrated over two mid-IR cycles and therefore not expected to be sensitive to the mid-IR phase. For earlier delays, when ionization occurs on the rising edge of the mid-IR pulse (Figure 4(b)), the weaker signal at 90° indicates that dissociation occurs before the molecular ion interacts with the center of the 14 mid-IR pulse. Similar observations are made for the feature at intermediate angles (around ~40° in Fig.   3(a)). Moreover, its angular position also varies from ~30° in Fig. 4(b) towards ~40° in Fig. 4(c).
Contrary to the non-linear features, the feature at =10°, i.e. close to the visible polarization axis, exhibits little delay dependence. As discussed above, this feature can be understood as a consequence of the single-photon couplings by both, the visible and the mid-IR fields. The absence of non-linearity in this process explains the insensitivity of the 10° feature to the mid-IR intensity.

Summary and Outlook
In summary, we have demonstrated a powerful approach for probing light-induced molecular potentials.
We observed strongly modulated proton angular distributions in experiments were H 2 + ions produced by a linearly polarized, few-cycle, visible laser pulse, are dissociated by a cross-polarized mid-IR laser field.
We have shown that the modulations can be understood as signatures of complex light-induced potential energy landscapes that are shaped by both single-photon and multiphoton transitions.
Specifically, the modulations arise from a combination of two effects: First, dissociation pathways for a given mid-IR laser intensity open and close as a function of alignment angle; second, rotational motion around light induced point intersections, such as LICIs, shape the modulated angular ion yield.
Probing the LIPs produced by the mid-IR dressing field on its own may be improved by using a shorter pulse for preparation of the bound wave packet, such as a few-cycle UV or attosecond pulse. Previous experiments along these lines (e.g., [41] [53]) were conducted in the single-photon dressing regime and did not study the influence of the light induced potential surfaces on the angular dependence of dissociation.
Our approach allows us to follow the variation of the LIPs throughout the dressing laser field. On the timescale of the mid-IR pulse envelope we observe the opening and closing of dissociation pathways as the dressing field strength changes. On shorter time scales, the propagation of the dissociating wave packet will become accessible with sub-femtosecond time resolution by monitoring the electron localization on either fragment. More generally, we have shown how complex LIP energy landscapes determine the outcome of molecular dissociation, using H 2 as an example. Our approach will allow for elucidating the reaction dynamics of more complex molecules in the presence of LICIs and higher-order point intersections. 16

Methods Experiment
The employed experimental technique is a variant of Ref. [51]. The output of a commercial Ti:Sa chirped pulse amplification (CPA) laser (Coherent Elite, 10 kHz, 2 mJ) is split into two parts. The stronger part (85%) is used to pump an optical parametric amplifier, in order to obtain carrier-envelope phase (CEP) stable idler pulses at 2.3 µm. The second part of the CPA output is focused into an argon-filled hollow core fiber to obtain broadband laser pulses, which are subsequently compressed to a pulse duration of ~5 fs. The laser pulses are recombined using a polished Si mirror (thickness 2.2 mm) at 60° angle of incidence.
After recombination, the pulses are focused in the center of a Cold Target Recoil Ion Momentum Spectroscometer (COLTRIMS) [54]. The intensity of the mid-IR pulse is weak enough to not cause any

Time-dependent Schrödinger equation
For the dynamics in the H 2 + cation, we solve a two-dimensional (one angle and one bondlength) Schrödinger equation that includes dipole coupling between the two relevant electronic states 2  g + and where ⃗ = ( , ) are the bondlength and angle between the laser field and the molecular axis. ( ) is the electric field of the laser that couples the two electronic states. The form of the electronic potential energy curves and , as well as the transition dipole , are taken from Bunkin and Tugov [55].
Equation (1) was solved numerically using the Fourier split-operator method.
In our experiment, the H 2 + system is created starting from the H 2 neutral through strong field ionization.
The initial state of the wave function of the ionic simulations assumes a vertical transition from the ground electronic ( 1  g ) and ground vibrational state of the H 2 neutral to the ground electronic state of the ion. The ground vibrational state on the 1  g of the neutral is modeled as Morse oscillator state using Morse parameters derived from Herzberg [56]. The rotational degree of freedom was initialized to a thermal rotational distribution, with temperature chosen to be low enough such that only the |J=0> rotational ground state is populated.
The laser field used in the calculations presented in Fig. 3 can be expressed as where is the carrier-envelope phase (CEP).

Floquet states
For each molecular alignment angle , the R-dependent Floquet energy curves are calculated for a field ( , ) = √ cos( ) sin + √ cos(3 + ) cos , where is the relative phase of the two fields. The potential energy landscape presented in Fig. 3(b) is for the relative phase = 0.
At each point along R, the Floquet states were constructed as follows. First, the one-period