The femtosecond laser can regulate the molecular dynamics in real time. A benefit of light manipulation of molecular processes is controlling the evolution of the wave packet. The photoelectron spectra and state populations map the wave packet dynamics information of the excited state. They were studied in multi-level molecular systems, and were found to be sensitive to the parameters of the laser fields.^[1–33]

For an alkali metal dimer, a femtosecond pump–probe pulse can be used for studying the dynamics of the wave packet on the excited state. Braun et al.,^[1] Miao et al.,^[2,3] and Liu et al.^[4] found that the state populations and photoelectron spectra of NaI change with the delay time. Zhao et al.^[5] studied the sensitivity of the predissociation dynamics of NaI to the field-free orientation. Yu et al.^[6] studied the effect of the delay time and laser wavelength on the evolution of the wave packet of NaK. The periodical motion of the wave packet leads to the periodical change of the photoelectron spectra. Maet al.^[7] found that the wave packet on the excited state moves periodically, the state population of NaRb is affected by the laser intensity. Zhang et al.^[8,9] suggested that the laser wavelength affects the oscillation period of the wave packet, and the peak positions and heights of the photoelectron spectra of NaLi vary with the delay time. The efficiency of population transfer through diabatic coupling strength changes with the laser intensity and laser wavelength. The delay time alters the peak positions and heights of the photoelectron spectra. Zhu et al.^[10] presented that the delay time affects the state populations of the CsI molecule. Meng et al.^[11,12] and Wang et al.^[13] studied the sensitivity of the state populations of NO to the laser intensity, laser wavelength, pulse width, and delay time, and presented the periodical change with the laser intensity and pulse width. Cheng et al.^[14] found that the selective excitation of excited states can be achieved by phase modulation. Niu et al.^[15] proposed that the rovibrational population transfer can be controlled by a two-overlapping-pulse scheme. Xie et al.^[16] found that the population exchange can occur due to the driving action of an external field.

The dynamics of the wave packet of a homonuclear diatomic molecule shares some similar characteristics with that of a heteronuclear diatomic molecule. The periodical wave packet motion of the four-level Na₂ was studied in Refs. [17]–[19]. Yuan et al.^[20,21] found that the state populations change with the delay time; the more effective scheme for controlling the population transfer of the three-level Na₂ is the one with a short pulse width and positive delay time, which is explained by the light-induced potentials (LIPs). Hu et al.^[22] and Yan et al.^[23] studied the sensitivity of the population transfer and the angular distribution of population of the four-level Li₂ to the molecular rotation. Han et al.^[24] and Jing et al.^[25] presented that the laser intensity, wavelength, and delay time affect the state population of the four-level Li₂. Liu et al.^[26,27] proposed that the delay time only alters the peaks height, but does not change the peak positions of the photoelectron spectra of Li₂. Miao et al.^[28,29] suggested that the peak positions and heights in the photoelectron spectra of anion change with the delay time. The total photoelectron signal shows a periodical oscillation with the increase of the delay time.

The Autler–Townes (AT) triple splitting and the selective distributions of the dressed states may be attained by altering the laser intensity, wavelength, and pulse envelope.^[30–32] No studies focus on the sensitivity of the wave packet dynamics of the four-level ladder K₂ to the delay time. New data on the delay time dependence of the photoelectron spectra and state populations of the four-level ladder K₂ molecule is obtained in the presence of two control pump fields and one probe field via the time-dependent quantum wave packet method.

Four states (ground state , excited states and , and ionic ground state , abbreviated to ∣X〉, ∣A〉, ∣2〉, and , respectively) are included in the ionization of the four-level ladder K₂,^[30,34,35] as shown in Fig. 1.

Fig. 1.

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	Fig. 1. Potential energy curves of K2 molecule taken from Refs. [30], [34], and [35].

The wave functions of the three-state model are written in the column vector 1 where , , , and are the wave functions of ∣X〉, ∣A〉, ∣2〉, and , respectively. is a continuum state and can be discretized into a band of quasicontinuum states. The is expressed as 2 where N is the number of discrete states of the K₂ ion. The time-dependent Schrödinger equation in the Born–Oppenheimer frame is expressed as 3 where is the potential matrix and is given by 4 where V_X, V_A, V₂, and V_I are the potential energy curves of the four electronic states at zero field. is the emitted photoelectron energy. Other nonzero off-diagonal matrix elements are given by 5 6 7 8 9 10 where , , and are the Rabi frequencies, , , and are the transition dipole moments, e₁, e₂, and e₃ are the amplitudes of the pump and probe laser fields, , , and the angular frequencies, , , and are the envelopes, τ is the pulse width, is the delay time between the pump-1 and pump-2 pulses, and is the delay time between the pump-2 and probe pulses.

The split-operator Fourier method is used to solve the time-dependent Schrödinger equation.^[29,30] The population on each electronic state is written as^{[2–13,22,24,25,27,29]} 11 The photoelectron spectrum is defined as^{[1,3,4,6,8,22,24,27,29–32]} 12 The potential energy curves are taken from Ref. [37]. The transition dipole moments are taken from Ref. [38]. In the calculation, the energy range spans over 0–1.2 eV and N is taken to be 120.

From the dressed-state theory, the excited state ∣2〉 is dressed by the laser field and splits into three substates ∣α〉, ∣β〉, and ∣γ〉 with energy , V₂, and . The wave function of state ∣2〉, including the influence of the laser field, can be expressed by dressed states ∣α〉, ∣β〉, ∣γ〉 whose energies are dressed in order,^[25,29] 13 14 15 16 where , , , and .

The delay time dependence of the wave packet motion is studied. Figure 2 displays the evolution of the wave packet on excited states ∣A〉 and ∣2〉 for two cases: fs with fs and fs with fs. The wave packet on excited state ∣A〉 moves periodically with roundtrip time 550 fs for the two cases. The periodical motion was not reported before for the four-level ladder K₂, but for other alkali metal dimmers, such as the four-level Na₂,^[17,18] NaI,^[1] NaK,^[6] NaRb,^[7] and NaLi.^[8,9] The wave packet moves from the inner turning point (7 a.u.) towards the outer turning point (11 a.u.) during 0–275 fs, and reaches the outer turning point at 275 fs. Then the wave packet moves back to the inner turning point during 275–550 fs, and reaches the inner turning point at 550 fs. This is a complete motion within an oscillation period, which leads to the periodical variation of the peak position in the photoelectron spectra with delay time, see Fig. 3. The wave packet on excite state ∣2〉 moves periodically with roundtrip time 700 fs only for fs with , and the oscillation amplitude is 7–12 a.u.

Fig. 2.

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	Fig. 2. The evolution of the wave packet on excited states ∣A〉 and ∣2〉 with time and internuclear separation: (a) and (c) , , (b) and (d) , . Other laser parameters are , , , and .

Fig. 3.

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	Fig. 3. The evolution of photoelectron spectra of K2 (a) at different delay time with and (b) at different delay time with . All the other parameters are the same as those in Fig. 2.

The delay time dependence of the photoelectron spectra is studied. Figure 3 displays the evolution of the photoelectron spectra within the first oscillating period (0–550 fs). Three features can be seen in Fig. 3. Firstly, AT triple splitting with symmetric profiles appears in the photoelectron spectra at zero delay time, which results from the sufficient Rabi oscillation within the resonant region.^[39] The symmetric triplet was observed for the four-level Na₂,^[40] Li₂,^[22,25] and K₂.^[30–32] The splitting disappears progressively with the increase of the delay time, and the three peaks transform into double peaks in Fig. 2(a) and into one peak in Fig. 2(b) at 30 fs. The disappearance of AT splitting is due to the non-sufficient Rabi oscillation within the off-resonant region induced by the larger delay time. According to Eq. (13), the probabilities of the central peak and two side peaks are determined by and , respectively. The larger lowers , which results in the disappearance of the central peak, while the larger lowers , which results in the disappearance of the two side peaks. Secondly, the peaks shift to lower energy during 0–275 fs while they shift to higher energy during 275–550 fs. The symmetric center is at about 0.484 eV at zero delay time, which is calculated by ,^[39] where and are the potentials of the vibrational ground state and the ionic state, respectively, and is the photon energy. At zero delay time ( and ), is the potential of the ionic state at the equilibrium distance r₀ of the neutral ground state, i.e., . The two side peaks are at and . As the delay time increases from 0 fs to 275 fs, the difference between the potentials of the excited state ∣2〉 and ionic state increase with the increase of the internuclear separation, leading to the decrease of the photoelectron energy with the increase of the internuclear separation. It is easy to adapt the same procedure to interpret the increase of the photoelectron energy with the decrease of the internuclear separation as the delay time increases from 275 fs to 550 fs, see Fig. 2(b). This was reported before for other alkali metal dimmers four-level Na₂,^[17–19] NaI,^[1] and NaK,^[6] but not for the four-level K₂. Liu et al.^[26,27] found that the delay time only affects the peak height, but does not affect the peak positions in the photoelectron spectra of four-level Li₂. The different results may be induced by the different pulse widths (260 fs in Refs. [26] and [27] versus 30 fs in this study). The wave packet may almost reside stationarily at the equilibrium position of the excited state potential due to the longer pulse width,^[17] which leads to the independence of the peak positions on the delay time. In fact, studies on the periodical variation of the photoelectron spectra are with short pulse width (shorter than 100 fs).^{[1,6,17–19]} Thirdly, the peak height decreases progressively with the increase of the delay time in Fig. 3(a), which indicates the decrease of the amount of emitted photoelectrons with the increase of the delay time (0–550 fs) for . The peak height decreases and then increases with the increase of the delay time in Fig. 3(b), which indicates that the amount of the emitted photoelectrons decreases and then increases with the increase of the delay time (0–550 fs) for .

The delay time dependence of the state populations is studied. Figure 4 shows the evolution of the state populations at different delay time for two cases: –2100 fs with (case 1) and –2100 fs with (case 2). Three features are seen in Fig. 4. Firstly, for case 1, the populations on the ground state and excited state ∣AA〉A take the Rabi oscillation with the pump-1 pulse on. The Rabi oscillation stops when the pump-1 pulse is off. The population on the ground state remains unchanged after the pump-1 pulse disappears. The population on excited state ∣A〉 remains unchanged after the pump-1 pulse disappears and before the pump-2 pulse arrives. There is no population on excited state ∣2〉 before the pump-2 pulse arrives. The populations on excited states ∣A〉 and ∣2〉 take Rabi oscillation with the pump-2 pulse on. There is no population on the ionic state before the probe pulse arrives. For case 2, the populations on the ground state and excited states take Rabi oscillation with the pump pulses on. The Rabi oscillation stops when the pump pulses disappear. The populations on the ground state and excited state ∣A〉 remain unchanged after the pump pulses disappear. The population on excited state ∣2〉 remains unchanged after the pump pulses disappear and before the probe pulse arrives. There is no population on the ionic state before the probe pulse arrives. Secondly, the Rabi oscillation becomes weaker as the delay time increases. The disappearance of AT splitting in the photoelectron spectra results from the nonsufficient Rabi oscillation, see Fig. 3. Thirdly, for case 1, the populations on the ground state and excited state ∣A〉 present the periodical change with the increase of the delay time. For case 2, the populations on the ground state and two excited states present the periodical change with the increase of the delay time. The periodical change of the population with the delay time was not studied before for the four-level K₂, and can be interpreted by the coupling effect between the two pulses. The coupling effect is weakened by the larger delay time. Accordingly, the intensity of the Rabi oscillation is weakened. The different Rabi oscillation results in the different population left on each electronic state.^[9]

Fig. 4.

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	Fig. 4. (color online) The evolution of state populations (a)–(f) at different delay time with and (g)–(l) at different delay time with . The solid, dash, dot, and dash dot represent ground state ∣X〉, excited state ∣A〉, excited state ∣2〉 and ionic state , respectively. Other laser parameters are the same as those in Fig. 2.

The effect of the delay time (case 1: –2100 fs and , case 2: –2100 fs and ) on the photoelectron spectra and state populations of the four-level ladder K₂ molecule by a pump1–pump2–probe pulse is firstly investigated via the time-dependent wave packet approach. The wave packet on excited state ∣A〉 moves periodically with roundtrip time 550 fs and oscillation amplitude 7–11 a.u., while the wave packet on excite state ∣2〉 moves periodically with roundtrip time 700 fs and oscillation amplitude 7–12 a.u. only for case 2. The periodical motion of the wave packet leads to the periodical variation of the photoelectron spectra. The photoelectron spectra show AT triple splitting with symmetric profiles at zero delay time. As the delay time increases, the three peaks become double peaks for case 1 and one peak for case 2. The coupling effect is weakened by the larger delay time. Accordingly, the intensity of the Rabi oscillation is weakened. The disappearance of AT splitting in the photoelectron spectra at larger delay time results from the non-sufficient Rabi oscillation. The different Rabi oscillation leads to the different population left in each electronic state, which leads to the periodical change of the populations in the ground state and excited states with the increase of the delay time. The results illustrate that by controlling the delay time, the needed population in the excited state of interest can be obtained, which provides the foundation for realizing the light manipulation of molecular processes.