Carrier-envelope-phase measurement of sub-cycle UV pulses using angular photofragment distributions

Carrier-envelope-phase (CEP) of sub-cycle ultraviolet (UV) pulse strongly influences the dynamics of quantum systems, but its characterization is not accessible experimentally. Here we investigate photodissociation of a diatomic molecule from its ground-rovibrational state in a linearly polarized weak sub-cycle UV pulse with a controlled CEP. The angular distribution of photofragments shows an asymmetric profile deviating from the well-known cos2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cos }^{2}$$\end{document}- or sin2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sin }^{2}$$\end{document}-like ones, which can be identified as a way to imprint CEP. We unveil that such an effect stems from the temporal neighboring rotational excitation by molecular permanent dipole interaction through the joint contributions between counter-rotating and rotating terms. This in turn, opens different pathways in photodissociation dynamics. Given that the temporal excitation between various states with close energies can be manipulated by CEP of sub-cycle UV pulses, our results pave ways for understanding and manipulating electron, nuclear and their joint dynamics with variation of CEP of attosecond pulses. The carrier-envelope-phase of sub-cycle UV pulses strongly influences the dynamics of quantum systems, but its characterization is not accessible experimentally. Here, an asymmetry in the of angular photofragment distributions of diatomic molecules is identified as a way to imprint carrier-envelope-phase on a measurable quantity.

U ltrashort pulses available at modern light sources, provides new opportunities to study and control ultrafast electronic-nuclear motions in molecules on their natural time scales [1][2][3] . Moreover, the few-cycle attosecond laser pulses with carrier-envelope-phase (CEP) open the possibilities to capture and steer the dynamics at sub-cycle duration 4,5 . Few-cycle CEP-locked infrared (IR) pulses, especially its combination with an isolated attosecond extreme ultraviolet (XUV) pulse, have been explored by many groups and boosted ultrafast science into versatile degrees of freedom 2 , as for example, manipulating electron localization in molecular dissociation ionization 6 , steering charge localization following molecular photoionization 7 , quantum-phase control of absorption profile 8 , and many others about electronic dynamics in photoexcitation, photoionization and photoabsorption processes [9][10][11][12][13][14][15] . From the other side, these dynamics of the quantum systems induced by sub/few-cycle attosecond ultraviolet (UV) pulses can be used for an accurate characterization of these pulses. Reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) 16 and frequency-resolved optical gating for complete reconstruction of attosecond bursts (FROGCRAB) 17 have been documented into the normal methods to characterize the temporal envelope and spectral phase of a pulse. However, measurement of the CEP of a UV/XUV pulse, which is essential in full characterization of the attosecond waveforms, is a difficult task and has not been realized experimentally in spite of that several schemes have been proposed theoretically 5,11,18 .
Direct photodissociation of molecules by pulses could help to investigate instant nuclear wavepacket redistribution and localization that accompanies photoabsorption, photoexcitation, or photoionization processes [19][20][21][22] . For example, the ultrafast molecular photodissociation can be steered through laser-fieldinduced potentials 23 , spatial and angular distribution of photofragments (ADP) can be manipulated with strong laser field 24 or by intense UV pulse, triggering the effect of electron-rotation coupling (R-Ω coupling) [25][26][27][28][29] . However, direct photodissociation of molecules probed by few-cycle pulses for the effect of CEP, especially sub-cycle attosecond UV/XUV pulses, have rarely been explored.
There are enormous progresses in generation of single subcycle pulse in a broad range of wavelengths, as IR 30,31 and UV pulses 32,33 , which gives an optimistic forecast for the CEP-stable sub-cycle UV pulse to be obtained in the near future. As we show in the present study, much attention should be paid to the cooperation between the rotating and counter-rotating channels in one-photon excitation when the sub-cycle UV pulse is used, which is also the fundamental difference from other reported strategies for photoelectron angular distribution to characterize the CEP of isolate attosecond pulse 5,11,18 , and the asymmetry of angular photoelectron has been proved as an efficient way to reconstruct the relative phase between XUV pulses [34][35][36] .
In this work we consider diatomic LiF molecule as a showcase probed from its ground rovibronic state to valence excited dissociative states by linearly polarized sub-cycle attosecond UV pulses with various CEP. Figure 1a illustrates the excitation scheme and shows involved potential energy curves of LiF. Transition to both valence-excited states A 1 Σ and B 1 Π from the ground X 1 Σ state is followed by the dissociation. The time and spectral dependence of the used sub-femtosecond UV pulse are illustrated in Fig. 1b and c, respectively for three different CEPs. As it is illustrated in Fig. 1d-g the ADP alters with the varying of CEP, both in amplitude and the peak position (see Fig. 1h, i. As we will show, the asymmetry of ADP profile is due to neighboring rotational temporal excitation triggered by the highly off-resonant sub-cycle pulses. This allows to use the present scheme in order to perform fine characterization of CEP of an isolated attoesecond sub-cycle pulse.

Results
Photodissociation by sub-cycle UV pulses with CEP. We consider photodissociation of LiF molecule induced by the interaction with a sub-cycle attosecond UV pulse (fwhm = 0.22 fs, ω = 7.07 eV), whose electric field time and spectral dependence for various CEP are shown in Fig. 1b, c, respectively; the broadband pulse can excite both close lying 1 Π and 1 Σ valence excited states simultaneously. Note that for both few-cycle and sub-cycle pulses, the electric field should be defined from the vector potential to guarantee zero time-integrated electric field 37,38 . In the case of relatively weak (I = 10 13 W cm −2 ) and moderate (I = 5 × 10 13 W cm −2 ) intensities, the ground state is depopulated by about 0.05% and 0.25%, respectively. Due to this, we could safely suppose that off-resonant excitations to higher electronic states and ionization are negligibly small for the effect discussed here and disregarded in the present study. In the weak UV field, the second UV transition from the dissociative state could happen, but such a transition only perturbs the dissociative yield and does not affect the ADP profile 39 . Figure 1d-g shows the ADP profiles for the two electronics, 1 Σ (d, e) and 1 Π (f, g), and the two intensities, weak (d, f) and moderate (e, g). These figures demonstrate clearly a strong dependence of the ADP on CEP of the sub-cycle pulse. Indeed, when CEP φ = π/2, the profiles follow the well-understood behavior as cos 2 β and sin 2 β for XΣ → Σ and XΣ → Π transitions, respectively 19 . However, when CEP goes away from π/2, the ADP profiles experience sufficient peak shift and amplitude modulation depending of CEP φ. This phenomena become already visible at the low field intensity (Fig. 1d, f), yet increase drastically with the increasing of laser intensity (Fig. 1e, g). Using shown asymmetric spectral profiles S(β), we define the amplitude modulation (ratio as Sðβ¼0ÞÀSðβ¼πÞ

Sðβ¼0Þ
) and peak shift (relative shift of peak position to β = π/2) with respect to CEP, which are presented in Fig. 1h and Fig. 1i for states A 1 Σ and B 1 Π, respectively. They reveal nearly linear dependence of CEP for both states and two different pulse intensities, which can be considered as a versatile scheme for CEP characterization of a single attosecond pulse.
Such CEP-dependent behavior for the ADP curves is rather unexpected. Indeed, excitation of a molecule in its ground rovibronic state with ν j i ¼ 0 and JMΩ j i¼ 000 j iin the weak field limit results in the population of only J = 1 rotational state in the valence excited dissociative states bringing about the well-known sin 2 -or cos 2 -like ADP profiles. As shown in Supplementary Note 2, the asymmetric profiles with respect to CEP are similar and universal even considering non-zero initial rotational temperature for the ensemble. The deviation from this conventional behavior tells that already a weak sub-cycle pulse with asymmetric CEP may open other pathways leading to excitation of more excited rotational states.
Perturbation and numerical analysis. To verify this assumption, we consider a simplified two-level model with transition energy ω 21 excited by a weak UV pulse with frequency ω in a perturbation limit, when the wavefunction in the excited state can be expressed as (see Supplementary Note 3) À ie Àiφ e Àiω 21 t e Àiθ À ðtÞ R À ðtÞe À where Δ ± 21 ¼ ω 21 ± ω and τ 2 = fwhm 2 /4ln2, the first and second term are the counter-rotating term and rotating term, obtained by a one-photon transition operator D + (t) and D − (t) applied to the same state, respectively. The total transition operator is and Ω the Rabi frequency. R ± (t) and θ ± (t) are the amplitude and argument, respectively, and θ ± (t → ∞) = 0. Note that using Floquet representation 40 Roudnev et al. 41 have also showed that the CEP can be separated from the wavefunction. In a near resonant for a short pulse, resulting in a great domination of rotating term in ψ 2 . The nontrivial phase of ψ 2 (excluding the where Δθ comes from the cooperation between counter-rotating term and rotating term. This phase relation becomes 'exact' 3 2 π À φ for a long pulse, when the counter-rotating term is usually neglected in the rotating-wave approximation. While in a far-off-resonant is significant and the excitation can be temporally opened from either D + (t) or D − (t), so that the counter-rotating and rotating terms contribute together to the excitation.
The above analysis unveils the fact that, besides the electronic excitation by the transition dipole interaction, a weak sub-cycle UV pulse can also temporally open the neighboring rotational excitation of the same electronic state by permanent dipole interaction. Such a scheme of excitation by weak sub-cycle UV pulses is illustrated in Fig. 2a: 'path1' illustrates a conventional way of electronic excitation in a weak field by interaction D 12 , while 'path2' shows additional process involving neighboring rotational excitations before and after electronic transition (in the initial and valence-excited electronic states) by D 11 and D 22 , respectively. The two vertical lines between electronic states represents the rotating term and counter-rotating term, and the thick one indicates the predominant contribution of the rotating term. While the two arrow-curves represent comparable contributions of both rotating term and counter-rotating term to the rotational excitation. Note that in 'path2' for the electronic excitation XΣ → AΣ, transitions to both rotational states J = 0 and J = 2 are allowed, while for the XΣ → BΠ excitation, only state J = 2 is allowed. Excitations to higher rotational states are too small and neglected in the present illustration.
Furthermore, the phase difference between different excited rotational states from 'path1' and 'path2' is quite important for the discussion of their interference. Figure 2b shows the CEPdependent nontrivial phase ϕ J¼1   increases with 3 2 π À φ, which is in consistency with the two-level perturbation analysis, while ψ J¼0=2 2 varies only slightly around a constant value with CEP changes. The phase difference Δϕ ¼ ϕ J¼0=2 2 À ϕ J¼1 2 shows not much difference from −π + φ, and Δϕ = − π/2 and 0 (π) when CEP equals to π/2 and −π (0), respectively. Thus one can conclude that 'path1' and 'path2' do not interfere coherently when φ = π/2 and that when CEP reaches 0 or π the interference becomes largest. The above discussion based on simplified models show clearly the reasons for the CEP-dependence of the ADP profiles shown in Fig. 1d-g, where the ADP is symmetric for CEP=π/2 and follows well as cos 2 β or sin 2 β, while it turns into significant asymmetry as CEP approaches 0 or π. The negative amplitude modulation for state AΣ (and positive peak shift for state BΠ) for CEP less than π/2, is exactly the destructive and constrictive results of 'path1' and 'path2' in forward and backward angles, respectively, with phase difference Δϕ in [−π, −π/2), and vice versa.
Photodissociation by few-cycle UV pulses. In order to show its unique role of sub-cycle pulse (fwhm = 0.22 fs), the ADP for about one-cycle (fwhm = 0.44 fs) and two-cycle pulses (fwhm = 0.88 fs) with weak intensity (10 13 W cm −2 ) were also computed and presented in Fig. 3a-f for CEP = 0. Here the full simulations (labeled 'path1'+'path2'), which include both channels and their interference are compared against a model simulation 'path1', where all interactions with the permanent dipole moment are neglected. As it shows, in the model simulations the ADP follows nearly exactly to the conventional cos 2 β or sin 2 β shape, while the full calculations deviate more and more from conventional ADP profile with the decreasing of pulse duration towards sub-cycle pulse. It is worthwhile to note, that there is no observable differences for two-cycle and longer pulses. This observation is consistent with the result revealed in the two-level perturbation model, showing that 'path2' will be exponentially suppressed with the increasing of pulse duration. Figure 3g shows the population ratio of particular vibrational J state in the excited electronic state relative to J = 1. As one can see clearly, the contributions of J = 0/2 states decrease from about 10 −3 for half-cycle pulse to 10 −4 and < 10 −6 for one-cycle and two-cycle pulses, respectively. Note that the absolute excitation population of J = 1 state is about or less then 0.1% of the ground state population, so that values of~10 −6 obtained with two-cycle pulses indicate that these states are hardly excited, or J = 1 state could be the only excited rotational state when interaction with a weak long pulse is considered.

Discussion
The photodissociation dynamics induced by an isolated attosecond sub-cycle UV pulse have been investigated theoretically. We show that the angular distribution of the dissociation fragments becomes asymmetric when the carrier-envelope phase of the attosecond UV pulse φ differs from π/2. The phenomenon was explained with the help of additional pathways leading to the rotational excitation without necessity of an electronic transition. With the help of numerical simulations based on LiF molecule as a showcase we show that the UV field interaction with the permanent dipole moment in the ground and valence excited electronic states may lead to the population of the adjacent rotational levels. In spite of rather weak population, the effect opens up additional excitation channels on the electron-rotational transitions and thus makes the ADP profile depend on the CEP of sub-cycle pulse. With the help of a model based on perturbation theory, we show that both counter-rotating term and rotating term play important roles in the far off-resonant electronic excitation, resulting in breaking down of the rotating-wave approximation.
As a result of opening various pathways to the rotational levels of the dissociative electronic states with sufficient phase difference, the ADP pattern becomes asymmetric and thus differs from the well-known cos 2 β (for XΣ → Σ excitation) and sin 2 β (for XΣ → Π excitation) profiles. It was also shown that the asymmetry can be enhanced using an oriented or aligned molecular ensemble (Supplementary Note 2). Moreover, the asymmetry is controlled by variation of CEP of the attosecond UV pulse. From another hand, the measurement of the ADP profiles would allow to get a fine measurements of the CEP of an isolated attosecond UV pulse, providing a feasible tool for complete reconstruction of attosecond waveforms. The obtained results are applicable to all molecules with permanent dipole moments. Indeed, since UV frequency is always much larger than the rotational transitions, the temporal excitation of neighboring rotational states can always be manipulated by CEP. Molecules with the large permanent dipole moment are the favorable candidates suggested for measurements.
The sub-cycle pulses, thanks to their broad energy bandwidth, are playing an increasingly important role in many different studies. In the present study we have shown that CEP of the attosecond pulse can be used as a fine tuner for manipulation of the temporal excitation and the followed dynamics. The present results can be further extended and applied for investigations of ultrafast dynamics in atoms and molecules, when electron The two arrow-curves between neighboring rotational levels indicate the comparable importance of rotating and counter-rotating terms to the rotational excitation. The excitation to higher rotational states are too tiny and not illustrated. b Nontrivial phase ϕ J 2 of rotational state J in electronic state 2 excited by both pathway, and the phase difference Δϕ ¼ ϕ 2 between the excited rotational states by`path1' and`path2'. Note that the two-pathway can produce constructive or destructive interference results only when the phase difference Δϕ ≠ ± π/2, and such interference becomes largest when Δϕ reaches 0 or π. excitation and ionization, as well as nuclear dynamics and coupled electron-nuclear dynamics can be effectively controlled by CEP of sub-cycle attosecond UV and XUV pulses.

Methods
The nuclear Hamiltonian for the laser-molecule interactions with rovibronic contribution reads Hðq; tÞ ¼ T 0 ðqÞ þ V 0 ðqÞ À Vðq; tÞ; where T 0 (q) being the total (vibrational and rotational) kinetic operator, V 0 (q) being the involved potential energy curves and the radial-couplings between X 1 Σ and A 1 Σ. The matrix element V ji (q, t) = E(t)μ ji (q)κ ji (βγ) (i, j = X, A, B) represents the laser-dipole interactions on the transition from i to j state with the (transition) dipole moment μ ji (q). We consider a linearly polarized radiation EðtÞ ¼ À dAðtÞ dt , and the vector potential AðtÞ ¼ ffi ffi I p ω gðtÞ sinðωt þ φÞ with the polarization in the spacefixed (SF) Z-axis and peak field intensity I; gðtÞ ¼ expðÀ4 ln 2 Á t 2 =fwhm 2 Þ is the envelope of the vector potential with full-width at half-maximum fwhm; ω and φ are the central frequency and CEP, respectively; κ ji ðβγÞ ¼ sin β e iγ = ffiffi ffi 2 p (ΔΛ ji = 1) or κ ji ðβγÞ ¼ cos β (ΔΛ ji = 0) are the angular parts in laser-dipole interactions 25,26,28 , where Λ i is the projection of total orbital angular moment (L i ) of electronic state i onto the molecular-fixed (MF) z-axis, and β is the Euler angle between MF z-axis and SF Z-axis, γ is the Euler angle representing rotations around the MF z-axis 19 .
The present rovibronic Schr€ odinger equation is efficiently solved with the help of Heidelberg package of multiconfigurational time-dependent Hartree method 42 , which can also be easily applied for the future studies for polyatomic molecules. The sin-DVR(discrete variable representation) and L 2 -normalized Wigner D-functions JMΩðαβγÞ = 2Jþ1 8π 2 À Á 1=2 D J M;Ω ðαβγÞ were used as basis for the vibrational and rotational degrees of freedom 43 , respectively. Here D J M;Ω ðαβγÞ is the Wigner D-function, M and Ω are the projections of total angular moment J onto the SF Z-axis and MF z-axis, respectively. Further details of theory and numerical simulations are given in Supplementary Note 1.

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Code availability
The codes associated with this manuscript are available from the corresponding author on reasonable request.