1. Introduction

In the interaction of external laser pulse, electrons in ground state of atoms or molecules can absorb photons of light to overcome itself ionization potential barrier formed by the electron-core Coulomb force and electric field, known as above-threshold ionization (ATI) [1]. However, if the energy absorbed from light is not high enough to prompt the electrons in ground state to escape the atoms or molecules, the electrons are then recaptured into highly-lying excited states, i.e, Rydberg states. Rydberg state excitation (RSE) of atoms and molecules has particular applications in precision measurements [2], quantum nonlinear dynamics [3], and quantum information [4,5]. Driven by strong laser fields, the RSEs have been used to accelerate neutral-particle [6–8], reveal multiphoton Rabi oscillations [9], and the generation of near-threshold harmonics [10]. Manifestation of Rydberg states in the spectra of the liberated electrons had been first observed by Freeman et al. in 1987 [11] and discussed mostly in the multiphoton resonance picture (later denoted "Freeman resonances"). As a pioneer investigation, trapping of significant populations in excited states had been deduced already from the electron spectra in a pump-probe experiment [12], in addition, the formation of Rydberg states is also attributed to a substructure of the high-order ATI peaks [13]. In the ultrastrong laser with intensity of $\sim 10^{15}$ W/cm$^2$, the surprisingly survival of electrons in Rydberg states is observed in experiment, was dubbed "frustrated tunneling ionization" (FTI) process [14–20]. Atomic excitation dynamics underlying RSE are explored by enormous theoretical studies, such as numerically solving time-dependent Schrödinger equation (TDSE) [21–26], simiclassical method [16,27], and quantum model (QM) proposed in our previous works [28–33], in which the atomic excitation dynamics in RSE can be regarded as a coherent recapture process accompanying ATI.

In addition to atomic RSE, molecular RSE (mRSE) has also been observed in abundant experiments [34–39]. For example, fast excited neutral fragments are produced during the Coulomb explosion of H$_2$, D$_2$, and N$_2$ molecules in intense laser fields [34,35,38]. For the heteronuclear molecule exposed to strong laser fields, such as CO, it has been found that the liberated electron tends to be captured by C$^+$ to form C$^*$ rather than by O$^+$ to form O$^*$ [39]. As compared to atoms, in diatomic molecules, wave packets released from the two atomic sites can interfere with each other, causing the double-slit interference in strong field ionization [40–42]. This two-center interference strongly depends on the bond lengths. It is well known that the internuclear distance also plays a crucial role in molecular dissociation [43–47] where the dissociation pathways propagate along the potential energy curves of molecules determined by the internuclear distance, high-order harmonic generation in solids [48], and the RSE of breaking molecules [34,49–51]. For resonant multiphoton excitation, the internuclear separation varies with the laser intensity, it alters the photon energy partition between the ejected electrons and nuclei and thus leads to distinct kinetic energy spectra of the nuclear fragments [50]. However, the physical mechanism of mRSE in strong laser fields is yet a puzzle, specifically for internuclear-distance-resolved excitation dynamics. For theoretical studies, even though numerically solving TDSE can produce well the experimental observations, it is difficult to extract clear physical picture of mRSE process and is also time-consuming job.

In this work, we theoretically investigate the excitation dynamics in mRSE as the internuclear distance varies with employing the molecular QM and TDSE methods. Comparing with TDSE and semiclassical method, our QM method can easily extract the excitation dynamics encoded in the electron’s trajectories and phases. We take molecule H$_2^+$ as an example to calculate the $n$-distribution of the excitation yield as laser intensities varies for different internuclear distances. We find that the electron in ground state prefers to localize at higher-lying Rydberg-state orbitals and the energy distribution of final Rydberg states is broadened for the larger bonding lengths. Based on our electron’s trajectories analysis associated with the Rydberg state density, the internuclear-distance dependence of above phenomenon can be attributed to the interference between the partial electron wave packets detached from different molecular cores. Atomic units (a.u.) are employed unless otherwise indicated.

2. Theoretical framework

2.1 Molecular QM method

The excitation amplitude of the mRSE process with extension on the atomic quantum model for different internuclear distances $\mathbf {R}$ can be expressed as

Equation (1) can be divided into the partial part and time-dependent part, which can be rewritten as

Here the vector potential $\textbf {A}(t)=\frac {E_0}{\omega }\cos \omega t\mathbf {e_z}$ is linearly polarized along the $z$ axis, and the electric field is defined as $\textbf {E}(t)$=$-\partial \textbf {A}(t)/\partial t$. $E_0$, $\omega$, and $E_n=-Z/(2n^2)$ represent the peak electric field, the angular frequency, and the Rydberg-state energy level, respectively. In the simulations, the pulse duration of 8 optical cycles is used. The integrations over time ($t, t'$) and intermediate momentum $\mathbf {p}$ can be solved by saddle-point approximation as descried in Ref. [28]. The probability of final Rydberg states can be obtained by incoherently summing over all excited states, i.e., $P$=$\sum _{n,l,m}|M_{nlm}|^2$.

In our simulations, the initial state of H$_2^+$ can be expressed as $\phi _{1\sigma _g}(\mathbf {r'},\mathbf {R})$=$\big [\phi _{1s}(\mathbf {r'}\!\!-\mathbf {R}/2) \!+\phi _{1s}(\mathbf {r'}\!\!+\!\mathbf {R}/2)\big ]/\sqrt {2[1+S_{1s}(R)]}$ for small internuclear distances (R=2, 5 a.u.) [53]. However, for the larger bond lengths, the ground state and the first-excited state are almost degenerate and they are strongly coupled even by a weak external field. The modification in the SFA due to this strong coupling introduces a factor $e^{\pm i\mathbf {A}(t)\cdot \mathbf {R}/2}$ in the wave function of ground state for a very large $R$ [54]. Therefore, for $R=10$ a.u., $\phi _{1\sigma _g}(\mathbf {r'},\mathbf {R})$=$\big [\phi _{1s}(\mathbf {r'}\!-\!\mathbf {R}/2)e^{i\frac {\textbf {R}}{2}\cdot \textbf {A}(t)} +\phi _{1s}(\mathbf {r'}\!+\!\mathbf {R}/2)e^{-i\frac {\textbf {R}}{2}\cdot \textbf {A}(t)}\big ]/\sqrt {2[1+S_{1s}(R)]}$, where R denotes the internuclear distance and $S_{1s}(R)$ indicates the atomic orbital overlap integral. The 1$s$ wave function is expressed as $\phi _{1s}(\mathbf {r'})$=$\frac {1}{\sqrt {\pi }}\kappa ^{3/2}e^{-\kappa \mathbf {r'}}$ with $\kappa$=$\sqrt {2I_p}$. In length gauge, the Fourier transform of the initial state gives rise to the interference term $\cos (\mathbf {M}\cdot \mathbf {R}/2)$ with $\mathbf {M}$=$\mathbf {p}+\mathbf {A}(t')$ for molecules possessing small bonding lengths ($R=$ 2 and 5 a.u.) and with $\mathbf {M}$=$\mathbf {p}$ for molecules possessing the large internuclear distance ($R=$ 10 a.u.) in Eq. (4). To rule out the influence of ionization energy on excitation dynamics, the ionization energy $I_p$ are chosen as 0.5 a.u. for all bonding lengths. Since the laser field is linearly polarized parallel to the molecular axis, the magnetic quantum number is assumed to be $m=0$ in the simulation. The laser pulse duration is ten optical cycles with the wavelength $\lambda =800$ nm ($\omega =1.55$ eV).

2.2 TDSE calculations

Within the dipole approximation and fixed nuclear approximation, the time-dependent Schrödinger equation of the single-electron model molecules exposed to the laser fields in the velocity gauge is written as [37,55]

Here the model potential is given by [56]

In the present paper, the truncated radius is $r_{\mathrm {max}}$=1800 a.u., 1200 B-splines are adopted in the radial direction, and 45 B-splines are employed in the angular direction. The magnetic quantum number is $m$=0, and the time step is $\Delta t$=0.02 a.u.. Convergence of numerical simulations is reached with the above settings.

3. Results and discussions

In Figs. 1(a)–1(c), the energy distributions of Rydberg-state probability of molecules calculated by QM in length gauge for different bonding lengths as laser intensity varies from $1$ to $3\times 10^{14}$ W/cm$^2$ are depicted, where the RSE population of a specific principal quantum number $n$ are obtained by summing over angular quantum numbers $l$ ($0\leqslant l \leqslant n-1$) incoherently. The most conspicuous feature is that several stripes are separated by an intensity interval of about 26 TW/cm$^2$, which corresponds to a shift of the ponderomotive energy $\Delta U_p=\hbar \omega =1.55$ eV [$U_p=I_0/(4\omega ^2)$ is the ponderomotive energy], due to the channel closing effect just as in atomic RSE case [21,22,28]. For more delicate inspection, it is also found that the intensity corresponding to the excitation enhancement for each stripe decreases as $n$ increases. This characteristic is more obvious for lower $n$ than that for higher $n$, which can be attributed to energy conservation $j\omega =U_p+I_p+E_n$, where $j$ is the number of absorbed photon and $E_n=-1/n^2$. For a specific stripe, the absorbed photon number $j$ is fixed, hence, the intensity $I_0$ corresponding to excitation enhancement decreases with increasing $n$. Due to the energy variation of Rydberg states scales with $n$ by $n^{-2}$, the change of intensity $I_0$ corresponding to excitation enhancement becomes unclear as $n$ increases.

 

Fig. 1. $n$-distribution of probabilities of Rydberg states as a function of laser intensity calculated by QM in length gauge for molecules possessing different internuclear distances (a) $R=2$ a.u., (b) $R=5$ a.u., (c) $R=10$ a.u. The excitation yield is obtained by summing over all $l$ ($0\leqslant l \leqslant n-1$) incoherently without volume averaging. The vertical white-dashed lines indicate the two intensities $1.68\times 10^{14}$ and $2.72\times 10^{14}$ W/cm$^2$, respectively, corresponding to two channel closings. (d)-(f): Ratios of populations of each $n$ to that of the sum of $n$=4-15 for different internuclear distances by summing over all laser intensities.

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To exclude the intensity effect, we display the intensity-averaged $n$-distribution of RSE probability by summing over all intensities in Figs. 1(d)–1(f). It is demonstrated that for internuclear distance $R=2$ a.u. the RSE probability of $n=6$ dominates remarkably as compared to other Rydberg states. However, for $R=5$ a.u., the most populated state corresponds to the Rydberg state of $n=5$ and the populations of neighboring Rydberg states becomes comparable. Similarly, for the larger internuclear distance $R=10$ a.u., although the RSE probability of $n=5$ still dominates, the difference between the populations of neighboring Rydberg states is narrowing. To validate this $R$-dependence of $n$-distribution, we also stimulate the population of Rydberg states by QM in velocity gauge and by numerically solving the TDSE. The corresponding comparisons for different internuclear distances are plotted in Fig. 2. As we can see from Fig. 2, the results calculated by QM and TDSE are in good agreement, which demonstrate that the FWHM of $n$-distribution $\Delta \nu$ increases with the increasing internuclear distance. Note that in Fig. 2(c) the small deviation of populations calculated by QM from TDSE for low-lying Rydberg states can be attributed to the inappropriate approximation that the molecular Rydberg states are simplified to be atomic for the large bond length. It can be conclude evidently from Fig. 2 that, as the bonding lengths are stretched, the $n$-distribution is broadened due to the increasing contribution of lower-lying Rydberg states.

 

Fig. 2. Comparison between the normalized excitation populations calculated by QM ( in length gauge and velocity gauge) and TDSE as principal quantum number $n$ varies for the internuclear distance (a) $R=2$ a.u., (b) $R=5$ a.u., and (c) $R=10$ a.u. $\Delta \nu$ is the full width at half maximum (FWHM) of $n$-distribution.

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In order to explore the internuclear-distance dependence of $n$-distribution, the two intensities associating with the channel closing, i.e., 1.68 and 2.72$\times 10^{14}$ W/cm$^2$ are chosen for analysis. The above laser intensities correspond to the dominated Rydberg state $n=6$ with angular quantum number $l=5$ for $R=2$ a.u., and the dominant Rydberg state with $n=5, l=3$ for larger internuclear-distance $R=5$ and 10 a.u., respectively. In Figs. 3(a) and 3(b), we plot the normalized populations of Rydberg states as a function of ionization instant for different internuclear distances at two intensities. It can be clearly seen that for the bonding lengths $R=2$, 5, and 10 a.u. with laser intensity $I_0=1.68\times 10^{14}$ W/cm$^2$, the ionization times corresponding to the maximum excitation probability are $0.243\,T, 0.246\,T,$ and $0.248\,T$, respectively [Fig. 3(a)], where $T$ is the laser period. Similarly, for higher intensity $I_0=2.72\times 10^{14}$ W/cm$^2$, the corresponding ionization times are $0.244\,T, 0.247\,T,$ and $0.249\,T$ for $R=$ 2, 5, and 10 a.u., respectively [Fig. 3(b)]. Therefore, in mRSE process, the electron corresponding to the most probable capture is ionized at the rising edge of electric field, and as internuclear distance increases the ionization time of maximal excitation probability moves closer to the peak of electric field.

 

Fig. 3. (a): Normalized probability of Rydberg states as a function of ionization time $t'$ for different internuclear distances at the chosen laser intensities. For $R = 2$ a.u., the dominant final Rydberg state 650 ($n=6, l=5, m=0$) is chosen. For $R=5$ and 10 a.u., the corresponding Rydberg state is 530 ($n=5, l=3, m=0$). (b): The same as panel (a) but for higher laser intensity. The vertical dashed lines represent the ionization instants corresponding to the maximum of the excitation populations. (c): Simple-man trajectories with different ionization times extracted from (a). (d): Simple-man trajectories with different ionization times extracted from (b). The two upper (below) colorful rectangle denotes capture region of $z<0$ where the electron is captured into Rydberg state 530 (650), respectively. The black and green rectangles denote the capture time ranges of respective electron trajectory. Time dependence of electron’s kinetic energy for laser intensity (e) 1.68$\times 10^{14}$ and (f) 2.72$\times 10^{14}$ W/cm$^2$ with zero initial velocity and ionization time $t'=0.248 T$. The horizontal dashed lines denote a small energy of 0.05 a.u. for illustrating the capture time ranges.

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In our quantum model, the physical picture of RSE can be understood as following: the electron in ground state ionizes with the action by the external electric field, followed by the classical motion driven by electric field in the Volkov state. During the oscillation of electronic motion, the electron can be captured into the Rydberg state provided that two criteria are satisfied (i) the electron reaches the spatial region where the wave functions $\phi _{nlm}(\mathbf {r})$ are concentrated around ($|\phi |^2>0.8|\phi |^2_{\textrm {max}}$ is used in this work, where $|\phi |^2$ denotes the electron probability density of the Rydberg state), as shown in Fig. 4; (ii) the kinetic energy of electron is small enough due to the negative total energy. In Figs. 4(a) and 4(b), we plot the distribution of electron’s probability density of Rydberg states 530 and 650, respectively. For the first criterion with the linearly polarized pulse, the spatial regions for 530 and 650 states are confined in the ranges 6.0 a.u. $<|z|<$12.0 a.u. and 23.0 a.u. $<|z|<$42.0 a.u. in the light polarization direction, respectively. These specific regions are also represented by horizontal rectangles in Figs. 3(c) and 3(d). According to the simple-man model [58], if the electron is only driven by the laser field, then the equation of motion is determined by $\ddot {\mathbf {z}}(t)=-\mathbf {E}(t)$. Here, we assume that the electron’s trajectory starts from $z=0$ at ionization time $t'$ with zero longitudinal velocity. If integrating the equation of motion over the time $t$ twice, we can obtain the trajectory of electron $z(t)=-E_0/\omega (t-t')\cos \omega t'+E_0/\omega ^2(\sin \omega t-\sin \omega t')$. In Fig. 3(c) and Fig. 3(d), we present the electron classical trajectories for different ionization moments extracted from Fig. 3(a) and Fig. 3(b), respectively. The upper (below) colorful rectangle indicates the capture region of $z<0$ where the electron is trapped into Rydberg state 530 (650). Given above-mentioned capture regions, the ranges of the distance from the ion core for 530 and 650 states are 6.0 a.u. $<|z|<$12.0 a.u. and 23.0 a.u. $<|z|<$42.0 a.u., respectively, which are obtained by the different charge distributions of Rydberg-state electron in Fig. 4. In Fig. 3(e) and 3(f), we show the evolution of the kinetic energy of the electron under the given initial conditions and parameters. It can be seen that there are two time intervals in each optical cycle where the kinetic energy is small enough for capture, i. e., $E_{\mathrm {kin}}\leq 0.05$ a.u., as defined above. It can be clearly seen that the ionized electron tends to captured into the 650 state when its ionization occurs away from the peak electric field [see black rectangles in Figs. 3(c) and 3(d)], while electron has much more chances to be captured into the 530 state when its ionization time is much closer to the peak electric field [see green rectangles in Figs. 3(c) and 3(d)].

 

Fig. 4. Distributions of the electron’s probability density of Rydberg states (a): 530 and (b): 650 in the $x-z$ plane. The dashed rectangles denote the spatial capture regions.

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The ionization energies $I_p$ of model molecules discussed in this work are the same, and the effect of the internuclear distances on the molecular excitation mainly depends on the interference term $\cos ^2(\textbf {M}\cdot \textbf {R}/2)$, so we plot the distribution of $\cos ^2(\textbf {M}\cdot \textbf {R}/2)$ as a function of momentum of the emitted electron $\textbf {M}$ and the internuclear distance $\textbf {R}$ in Fig. 5(a). It is demonstrated that the distribution of $\mathbf {M}$ which contributes to constructive interference becomes narrower with increasing $\textbf {R}$. In order to gain physical insights into this, the momentum distributions of emitted electrons for $1s$ initial state and for $1s\sigma _g$ initial state with different bonding lengths are displayed in Figs. 5(b)–5(d). In Fig. 5(b), we see that for $1s$ initial state the distribution of dominating $\textbf {M}$ for each $n$ is similar to that of $1s\sigma _g$ initial state with $\mathbf {R}=2$ a.u., since the range of $\mathbf {M}$ which contributes to constructive interference stays almost unchanged when $\mathbf {R}$ increases from 0 to 2 a.u., and $\mathbf {M}$ ranges from -0.2 a.u. to 0. As bonding length increases, the distribution of dominating $\textbf {M}$ becomes narrower, and the weight of the electron with lower energy on excitation probability increases, so the ionization moment corresponding to the maximum excitation populations is closer to the instant of the peak electric field as shown in Figs. 3(a) and 3(b). Moreover, for $\textbf {R}$=2 a.u., the distribution of dominating $\textbf {M}$ which contributes to $n=6$ states is much broader than that of $n=5$ states. As the bonding length increases, the difference of the distribution of dominating $\textbf {M}$ between $n=5$ and $n=6$ states decreases, giving rise to the much broader $n$-distribution for larger bonding length as shown in Figs. 2(b) and 2(c). Apparently, the interference effect originating from the factor $\cos (\textbf {M}\cdot \textbf {R}/2)$ is responsible for the shift of the ionization instants corresponding to peak excitation probabilities and the change of FWHM of the $n$-distribution of molecules with different bonding lengths.

 

Fig. 5. (a) Distributions of $\cos ^2(\textbf {M}\cdot \textbf {R}/2)$ as a function of momentum of the emitted electron $\textbf {M}$ and the internuclear distance $\textbf {R}$. (b): Momentum distributions of emitted electrons for Rydberg states $n=5,l=3$ and $n=6,l=5$ with $\mathbf {R}=2$ a.u. (initial state $1s\sigma _g$, solid lines) and with $\mathbf {R}= 0$ (initial state $1s$, dashed lines), the ionization time $t'=0.243T$ and the laser intensity is 1.68$\times 10^{14}$ W/cm$^2$. (c): The same as panel (b) only for $1s\sigma _g$ initial state except $\mathbf {R}=5$ a.u. and the ionization time $t'=0.246T$. (d): The same as panel (b) only for $1s\sigma _g$ initial state except $\mathbf {R}=10$ a.u. and the ionization time $t'=0.248T$. The vertical dashed lines denote the respective range of dominant momentum. The dominant momentum ranges are defined by the excitation probability $P_n(t',\mathbf {M})\geq 0.2P_n(t',\mathbf {M})_{\mathrm {max}}$.

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4. Conclusions

In summary, we have theoretically studied the Rydberg-state excitation of molecules with different bonding lengths in strong laser fields by adopting the quantum model simulations and the TDSE calculations. It is found that the excitation enhancements are separated by an intensity space of about 26 TW/cm$^2$ for the Rydberg state with a specific principal quantum number, which is ascribed to the interference of electron wave packets ionized during different optical cycles. Moreover, it is clearly demonstrated that, as the model molecules are stretched, the molecules prefer to be captured into lower-lying Rydberg states, and the above-mentioned distinct feature results from the shift of the ionization instants corresponding to peak excitation populations associated with the analysis of the semiclassical calculations. In addition, the FWHM of $n$-distribution becomes broader for molecules with increasing internuclear distances, since the difference of momentum distribution for adjacent Rydberg states decreases. The shift of the ionization moments corresponding to the maximum excitation and the decreasing difference of momentum distribution of sideward excited states for molecules with stretching bonding lengths originate from the distinct interferences of electronic wave packets detached from different centers. In experiments, the above intriguing characteristic may be observed by investigating the time-delay effect on the molecular excitation in pump-probe laser pulses.