Rydberg state excitation in molecules manipulated by bicircular two-color laser pulses

Abstract. Multiphoton resonant excitation and frustrated tunneling ionization, manifesting the photonic and optical nature of the driving light via direct excitation and electron recapture, respectively, are complementary mechanisms to access Rydberg state excitation (RSE) of atoms and molecules in an intense laser field. However, clear identification and manipulation of their individual contributions in the light-induced RSE process remain experimentally challenging. Here, we bridge this gap by exploring the dissociative and nondissociative RSE of H2 molecules using bicircular two-color laser pulses. Depending on the relative field strength and polarization helicity of the two colors, the RSE probability can be boosted by more than one order of magnitude by exploiting the laser waveform-dependent field effect. The role of the photon effect is readily strengthened with increasing relative strength of the second-harmonic field of the two colors regardless of the polarization helicity. As compared to the nondissociative RSE forming H2  *  , the field effect in producing the dissociative RSE channel of   (  H  +    ,  H  *    )   is moderately suppressed, which is primarily accessed via a three-step sequential process separated by molecular bond stretching. Our work paves the way toward a comprehensive understanding of the interplay of the underlying field and photon effects in the strong-field RSE process, as well as facilitating the generation of Rydberg states optimized with tailored characteristics.


Introduction
Benefiting from their unique properties, such as extremely large electron orbital radius and low electron binding energy, Rydberg atoms or molecules have a huge potential for applications in a wide range of fields, from quantum physics and chemistry to astrophysics. [1][2][3] For instance, they can serve as building blocks for advanced technologies in precision measurements, 4 quantum information processing, 5 quantum nonlinear optics, 6 and long-range many-body interactions. 7,8 In particular, the Rydberg state excitation (RSE) of atoms or molecules driven by intense laser fields has a plethora of exciting applications, including accelerating and de-accelerating neutral particles, 9,10 generating nearthreshold harmonics 11 and coherent extreme-ultraviolet light emission, 12 and realizing multiphoton Rabi oscillations. 13 In analogy to the multiphoton and tunneling mechanisms proposed for photoionization, the strong-field RSE of atoms or molecules can be accessed via either resonant multiphoton excitation [14][15][16][17][18] or electron recapture excitation. [19][20][21][22][23][24][25][26][27][28][29][30][31] These two scenarios illustrate the individual photon and field contributions of the driving light to the RSE via direct photon absorption or electron recapture, termed as the "photon effect" and "field effect," respectively. In the photon point of view, the atoms or molecules are resonantly excited to the Rydberg states by simultaneously absorbing multiple photons, which is primarily governed by the photon energy of the incident light. Alternatively, in the field scenario, the electron tunnels through the laserdressed Coulomb barrier, quivers in the oscillating laser field, and may be recaptured into the Rydberg orbitals of the parent ion after the conclusion of the laser pulse. 20,21 Depending on the laser parameters, e.g., the field intensity, wavelength, and polarization, both the photon and field effects have been observed in the strong-field RSE of atoms [14][15][16][17]21,29 and molecules. 18,19,[22][23][24][25][26][27][28]30 For instance, the RSE accessed via the photon scenario has been observed at short wavelengths and low intensities regardless of the laser polarization. 18,19 In contrast, since the trajectory of the tunneled electron critically depends on the detailed waveform of the laser field, the electron-recapture-induced RSE occurs in a linearly polarized laser field but is suppressed in circular polarization, in which case electrons are driven away from the ionic cores. 21 In principle, the photon and field effects coexist and both contribute to the RSE, even when one of them is dominant. 31,32 While they serve as the cornerstone to understand and control the strong-field RSE of atoms and molecules, an unambiguous identification of the individual contributions of the photon and field effects, however, is experimentally challenging, and has not been achieved so far. Here, we bridge this gap and report an experimental demonstration of disentangling and controlling the photon and field effects in RSE of a H 2 molecule by employing bicircular two-color (BCTC) laser fields.
The BCTC laser fields have attracted considerable attention in steering two-dimensional directional bond breaking, 33 manipulating the electron-ion rescattering 34 and nonsequential double ionization (NSDI), [35][36][37][38] generating circularly polarized extreme ultraviolet light, [39][40][41] and spatially controlling surface reactions on nanoparticles at the nanoscale. 42 Past studies using BCTC laser fields mostly focused on the atomic and molecular ionization dynamics. [33][34][35][36][37][38][39][40][41][42] Until recently, the excitation dynamics of atoms and molecules via frustrated tunneling ionization 21 were theoretically studied with BCTC laser pulses, which, however, only takes into account the field effect. 43,44 In this article, we use these versatile BCTC laser fields to experimentally investigate the strong-field RSE of H 2 molecules, where the created Rydberg fragments are measured and identified in the coincidence measurements. Studying the yield probabilities of the nondissociative (H 2 þ nℏω → H 2 Ã , denoted as H 2 Ã ) and dissociative [H 2 þ n ω → H þ þ H Ã þ e − , denoted as ðH þ ; H Ã Þ] RSE channels versus the relative field strength and polarization helicity of the two colors, we clearly determine the individual contributions of the field and photon effects. For a given relative field strength of the two colors, the electron recapture process facilitated by the field effect of the driving light is switched on or off by switching the polarization of the BCTC fields from counterrotating to co-rotating, whereas the contribution of the photon effect in the RSE yield remains comparable. The observed yield difference in the counterrotating field as compared to the co-rotating case therefore qualitatively gives the contribution of the field effect in producing the corresponding RSE channels. By finely adjusting the relative field strength of the two colors, we can manipulate the waveform of the laser field and the number of photons participating in the RSE processes and thus the relative contributions of the field and photon effects.

Experimental Methods
In our experiment, as schematically illustrated in Fig. 1(a), the BCTC laser fields were generated by spatiotemporally overlapping two circularly polarized fundamental wave (FW) and second-harmonic (SH) pulses using a Mach-Zehnder interferometer. The SH pulse was created by frequency doubling the linearly polarized FW laser pulse (25 fs, 790 nm, and 10 kHz) in a β-barium borate crystal. Followed by the independent circular polarization shaping in two arms of the interferometer, the FW and SH pulses with the opposite or same polarization helicity were recombined using a dichromatic mirror to produce counterrotating or co-rotating two-color laser fields. The BCTC laser pulses were focused onto the supersonic gas jet of H 2 in the apparatus of cold target recoil ion momentum spectroscopy (COLTRIMS). 45,46 The peak intensities in the focus for the two colors, i.e., I FW and I SH , were calibrated separately by examining the proton momentum spectrum for the FW pulses 47 and tracing the field-intensity-dependent energy shift of the discrete above threshold ionization spectrum of H 2 for SH pulses. 48 The intensities can be adjusted individually via the neutral filters installed in each beam arm before their collinear recombination.
By measuring the electron, ion, and excited neutral Rydberg atom in coincidence, the COLTRIMS apparatus allows us to unambiguously identify the RSE channels of H 2 Ã and ðH þ ; H Ã Þ. As shown in Fig. 1(b), the H 2 Ã channel can be identified in the photoelectron-photoion coincidence (PEPICO) spectrum, 19,29,49 which is measured on the basis of the postpulse field ionization of the laser-induced Rydberg states, 50-52 whereas the ðH þ ; H Ã Þ pair can be discriminated in the photoion-photoion coincidence (PIPICO) spectrum, 53 as shown in Fig. 1(c). We note that the strong-field-induced neutral dissociation of H 2 through superexcited states, 54 accompanied by further ionization of neutral fragments, has a negligible effect on our measurements. More details of the experimental methods and the information on the principle quantum number of the measured Rydberg fragments are presented in the Supplementary Material.

Results and Discussion
Figure 2(a) shows the measured yields of the H 2 Ã and ðH þ ; H Ã Þ channels in the counterrotating and co-rotating BCTC fields as a function of the relative-field strength E SH ∕E FW of the two colors. In the measurements, the intensity of the combined laser fields, i.e., I c ¼ ð p I SH þ p I FW Þ 2 , was kept constant ∼6.0 × 10 14 W∕cm 2 for various field ratios. This allows one to minimize the intensity-dependent photon spin effects originating from different field helicities on the ionization and excitation processes 55 and also to obtain a proper event rate for data acquisition. This is confirmed by the nearly flat and helicityindependent single-ionization yields of H 2 þ (see Supplementary Material). Each data point is normalized per laser shot over more than 10 8 shots. The relative collection and detection efficiencies for different fragmental species have been taken into account. 53,56 As shown in Fig. 2(a), noticeable yields of the H 2

Ã
(red open circles) and ðH þ ; H Ã Þ (blue open squares) channels are clearly observed in the co-rotating bicircular fields, which are mostly accessed via the multiphoton excitation process, since the electron recapture process is suppressed here. The yields of H 2 Ã and ðH þ ; H Ã Þ both increase rapidly when the field ratio E SH ∕E FW is increased from 0.5 to 2.0, followed by a broad plateau. This indicates that the contribution of photon effect in the RSE is boosted with the increase of the relative strength of the SH field. The reason for this enhancement is that the photon energy of the SH field is 2 times that of the FW field. Therefore, for a given RSE channel, fewer photons are required with an increasing ratio E SH ∕E FW , leading to increased accessibility of the multiphoton excitation process. When multiphoton excitation already dominates RSE, a further increase in the relative strength of the SH field only leads to a minor rise in the excitation probability, leading to the observed nearly flat H 2 of the SH portion in the total laser field does not increase much beyond this field ratio.
By switching the polarization helicity of the BCTC field from co-rotating to counterrotating, as shown in Fig. 2(a), the yields of the H 2 Ã (red solid circles) and ðH þ ; H Ã Þ (blue solid squares) channels are significantly enhanced (by more than one order of magnitude for H 2 Ã ). In the counterrotating field, both the photon and field effects contribute, and thus the RSE yields are overall higher than that of the co-rotating case, where the field effect is significantly suppressed. For a given E SH ∕E FW ratio, the contribution of the photon effect is comparable in the counterrotating and co-rotating fields; the strikingly enhanced RSE probability in the counterrotating field is thus attributed to the contribution of the field effect. To increase the visibility of the enhancement and quantify the contribution of the field effect as compared to that of the photon effect, in Fig. 2(b), we plot the measured yield ratios of the RSE channels between the counterrotating and co-rotating cases. Noticeable enhancements of both the H 2 Ã and ðH þ ; H Ã Þ channels are observed around E SH ∕E FW ¼ 1.0, with increment factors of ∼28 and ∼5, respectively, suggesting a crucial role of the field effect in boosting the RSE in the counterrotating BCTC laser fields.
The enhancement observed here is similar to the previous observations of the field-ratio-dependent NSDI probability of atoms and molecules in the sense of the control offered by the BCTC laser fields, where the enhanced double ionization is dominated by the electron recollision. [38][39][40] Owing to the peculiar waveform of the counterrotating two-color fields, the tunneled electron will initially be driven away but later be driven back to the parent ion. The two-dimensional close loop trajectory results in a much higher probability for the electron to re-encounter the parent ion, giving rise to the NSDI. The situation is a bit different for the RSE here. In contrast to NSDI, the trajectory of the electron does not have to return to the parent ion, but needs to have a small velocity after the conclusion of the laser field, such that the electron can be captured into a Rydberg orbital. As illustrated in Fig. 1(d), in the counterrotating bicircular laser field, the magnitude of the vector potential features a local minimum (red dot) where the electric field is at a local maximum (green dot) of its magnitude, so that the electron with a maximal tunneling probability will end up with a small final velocity. Such a property of the counterrotating bicircular field leads to a prominent field effect in generating RSE. This observation is again in line with the properties of the counterrotating field, since the local minimum in the magnitude of the vector potential can be lowered by optimizing the field ratio, as plotted in Fig. 2(c) (green solid stars). In contrast, for the co-rotating field, as shown in Fig. 1(e), the vector potential has a local maximum (red dot) in its magnitude where the electric field peaks (green dot), which strongly suppresses electron recapture.
The much higher increment of the H 2 Ã channel compared to that of the ðH þ ; H Ã Þ channel, as shown in Fig. 2(b), implies that the field effect plays a more dominant role in producing the H 2 Ã as compared to the ðH þ ; H Ã Þ channel. To illuminate the difference between the relative roles of the field and photon effects in producing the H 2 Ã and ðH þ ; H Ã Þ channels, we turn to inspect the accessing dynamics of these two channels.
In the electron recapture scenario, driven by the field effect of the counterrotating bicircular field, H 2 Ã can be formed if the tunneled electron with negligible kinetic energy is recaptured by the parent ion H 2 þ , while for the generation of the ðH þ ; H Ã Þ channel, 53,57,58 the ionization-created nuclear wave packet of H 2 þ needs to stretch to a critical internuclear distance (R ∼ 10 a.u.), where the charge-resonance-enhanced ionization occurs to boost the tunneling of the second electron. 59 The tunneled electron is subsequently recaptured by one of the outgoing H þ ions of the Coulomb explosion double ionization channel, i.e., By considering the kinetic energy release (KER) of the observed ðH þ ; H Ã Þ channel, we estimate a bond stretching time of about 25 fs for the nuclear wave packet propagating from R ∼ 1.4 a.u. (the equilibrium internuclear distance of H 2 ) to ∼10 a.u. on the potential energy curves of H þ 2 . Therefore, the tunneling of the second electron at the critical internuclear distance mostly occurs at the trailing edge of the bicircular laser pulse where the field strength is moderately weak. It results in a lower probability in producing the ðH þ ; H Ã Þ channel as compared to that of the H 2 Ã channel, in which the electron tunneling mostly occurs around the peak of the laser pulse. It explains the much larger increment of the yield of H 2 Ã than that of the ðH þ ; H Ã Þ channel, as shown in Fig. 2(b), where the enhancement is facilitated by the electron recapture process with a maximal efficiency around field ratio of E SH ∕E FW ∼ 1 in the counterrotating BCTC laser pulse.
In contrast, in the multiphoton excitation scenario, the RSE is accessed by directly absorbing multiple photons from the laser field, which will be considerably boosted if multiphoton resonance occurs. As compared to the nondissociative H 2 Ã , the resonance excitation of the stretching H 2 þ to the ðH þ ; H Ã Þ can always be fulfilled with the energy of a certain number of photons matching the potential energy gap between the 2pσ u state and the repulsive Rydberg states at proper internuclear distances. As discussed above, the contribution of the photon effect increases with the rising field ratio E SH ∕E FW for a constant combined laser intensity. This is more obvious in the co-rotating bicircular field, where the electron recapture process is suppressed. As a result, as shown in Fig. 2(a), the yields of the ðH þ ; H Ã Þ channel exceed that of the H 2 Ã channel when the field ratio E SH ∕E FW > 2.5 and 1.75 for the counterrotating and corotating fields, respectively. At the first glance, it seems counterintuitive, since many more photons are required to access the ðH þ ; H Ã Þ than H 2 Ã , which further confirms the importance of the resonance in the multiphoton excitation process. The fact that the yield of ðH þ ; H Ã Þ surpasses that of H 2 Ã with increasing field ratio E SH ∕E FW is consistent with our measurement using a single circularly polarized SH field, which corresponds to the limiting case of E SH ∕E FW → ∞.
As indicated by the thick arrows near the right ordinate of Fig. 2(a), driven by a single circularly polarized SH pulse with an intensity of ∼6.0 × 10 14 W∕cm 2 , the yield of ðH þ ; H Ã Þ (2.0 × 10 −5 counts∕laser shot, blue arrow) is one order of magnitude higher than that of the H 2 Ã (7.4 × 10 −6 counts∕laser shot, red arrow). In this scenario, the photon effect dominates ionization, 18,19 and the ðH þ ; H Ã Þ channel is accessed via the following process: H 2 þ created upon photoionization first stretches on the 1sσ g potential energy curve and transits to the 2pσ u state by absorbing one SH photon via a 1ω SH pathway, followed by a resonant excitation to the repulsive Rydberg state at an internuclear distance around 10 a.u., which eventually fragmentizes into a ðH þ ; H Ã Þ pair with a KER ∼ 4.0 eV, as indicated by the blue arrow pointing to the black dashed curve in Fig. 3(b). Alternatively, the stretching H 2 þ can also dissociate into a hydrogen atom and a proton, i.e., H 2 þ þ nℏω → H þ þ H þ e − [denoted as ðH þ ; HÞ], via a comparable 1ω SH pathway. As shown in Figs. 3(a) and 3(b), for the case using the co-rotating bicircular fields, the yields of the 1ω SH pathway of the ðH þ ; HÞ channel (KER ∼ 1.2 eV) and of the ðH þ ; H Ã Þ channel (KER ∼ 4.0 eV) increase with rising SH field strength. It is consistent with the observed steep enhancement in the ðH þ ; H Ã Þ yield, as shown in Fig. 2(a), as well as the ðH þ ; HÞ and ðH þ ; H þ Þ channels, as shown in Fig. S2(a) in the Supplementary Material, when the field ratio E SH ∕E FW increases from 0.5 to 2. Accordingly, as shown in Fig. 3(a), the ðH þ ; HÞ channel with KER smaller than 0.5 eV produced via the 1ω SH − 1ω FW and 1ω FW pathways is suppressed as the FW field strength decreases [see Fig. S3(a) in the Supplementary Material for the counterrotating case].
To further illuminate the photon and field effects in the strong-field RSE process, we take the simplest hydrogen atom to model the RSE process in the BCTC laser pulses by performing a classical trajectory Monte Carlo (CTMC) simulation, which accounts only for the field effect. More details of the simulation are presented in the Supplementary Material. As shown in Fig. 2(c), the experimentally observed field effect in accessing the strong-field RSE process is qualitatively reproduced (blue solid and open circles). It has a similar tendency to (the inverse of) the magnitude of the vector potential [−AðtÞ, green solid and open stars], where the electric field amplitude E maximizes [right ordinate of Fig. 2(c)], as expected for the field effect, which requires the final velocity to be small so that the tunneled electron can be recaptured. We note that due to the lack of photon effect in the CTMC simulation, a much higher yield ratio (∼608) between the counterrotating and corotating cases is observed in Fig. 2(d) (blue solid circles). While a more sophisticated theoretical method (say, full threedimensional time-dependent Schrödinger equation or the strongfield approximation generalized to apply to Rydberg excitation) is necessary to fully account for the photon effect, here we readily include the photon effect based on the simple physical intuition that the photon excitation probability should scale with the individual intensity of the two-color laser field to the power corresponding to the number of photons involved. With this in mind, we arrive at the yield of H Ã with both field and photon effects, shown as orange solid and open squares in Figs. 2(c) and 2(d). For the counterrotating case, the field effect dominates, and adding the additional photon effect does not change the yield much. For the co-rotating case, the field effect is suppressed such that the contribution of the photon effect drastically enhances the yield, and reduces the ratio between the counterrotating and co-rotating cases to be around 32 [orange solid squares in Fig. 2(d)], arriving at a much better agreement with the experimental results of the H 2 Ã channel. This additional contribution illustrates the importance of photon effects in a strongfield RSE process. In contrast, the large ratio between counterrotating and co-rotating cases, as shown in Fig. 2(d), confirms our conclusion that the field effect plays a crucial role in the underlying dynamics of the strong-field RSE process.

Conclusions
We have demonstrated that the bicircular laser fields of tunable relative strength and polarization helicity of the two colors not only support a precise control of the strong-field RSE dynamics, but also allow for an unambiguous discrimination of the individual contributions of the field and photon effects. Our findings show that both the field and photon effects play an important role in accessing the strong-field dissociative and nondissociative RSE of H 2 molecules, while the field effect is crucial in boosting the RSE in the counterrotating BCTC laser fields enabled by the tailored field shape. Our work with BCTC laser fields provides novel insights into the underlying mechanisms, i.e., the field effect versus the photon effect, of the RSE process and offers a promising route for manipulating the output of the RSE of atoms and molecules in intense laser fields. Fig. 3 Field-ratio-resolved KER spectra of the fragmentation channels in the co-rotating case. (a), (b) Measured KER distributions of the nuclear fragments of (a) the ðH þ ; HÞ channel and (b) the ðH þ ; H Ã Þ channel using co-rotating BCTC laser fields at various E SH ∕E FW (colored solid curves) as well as applying single SH pulses (black dashed curves) with an intensity of ∼6.0 × 10 14 W∕cm 2 .