1. Introduction

While the laser-atom interaction has been understood in quite some details, increasing interests have already been turned towards more complex systems, such as molecules, interacting with ultra-strong and (or) ultra-short laser pulses. Taking the molecular high-order harmonic generation (HHG) as an example, in the past decades, a lot of research works both in theory and in experiments have tried to understand, control and utilize this process [1–9]. Among, the so-called harmonic spectral minima phenomenon is one of the most interesting and important findings. This phenomenon was discovered by M. Lein et al. [10] and now has been widely investigated for its applications in molecular orbital tomography [4,11] and attosecond dynamics measurements [11], etc. However, there has no an ultimate and established theory to completely clarify its physical origin. In fact, there are two controversial expressions, one is the two-center-destructive-interference (TCDI) which is dependent on molecular structure [10, 12, 13] and the other is the multi-electron orbitals interference which has no dependence on molecular structure [4, 14]. They both agree that it is the electron wave packets interference that leads to the spectral minima, but the former is the two-center interference like Young’s two-slit interference, while the latter is the interference between the electron wave packets which are ionized from different molecular orbits. There also exists contradictions about whether or not the location of the spectral minima moves when changing the driving laser pulse’s parameters: some observed that it moves correspondingly [4, 7, 15, 16], while others argued that it is fixedly located [8, 10–13].

In addition, there has been a recent upsurge in the development of the mid-infrared laser pulse technology and its applications in the strong-field studies. CO₂ lasers with about 10 μm have already used in earlier experiments [17] and revealed the relevance of the tunneling ionization picture for strong-field physics. The surprising observation of an enhancement of yield in the low-energy part of the photoelectron spectrum driven by a mid-infrared laser pulse provided new insights into the ionization dynamics of atoms [18–22]. It is necessary to investigate the harmonic spectral minima by using mid-infrared laser pulses, and naturally, the carrier-envelope phase (CEP) dependence effect is one of the essential topics under the few-cycle mid-infrared pulse mechanism. For the first time, to our knowledge, it demonstrates that such interference effect can be controlled by changing the CEP of the driving laser pulse.

In this paper we will study the interaction process between ${\text{H}}_2^{+}$ and mid-infrared laser pulses with a central wavelength of 2.2 μm. Two models, Born-Oppenheimer (BO) and non-Born-Oppenheimer (NBO) approximations, [23–26] are considered and the laser pulse CEP-dependence is studied in detail. The main difference between the two models is that whether the internuclear separation changes during the evolution of the interaction process. This exactly leads to a difference for the induced high-order harmonic spectra. In fact, the spectral minima position (SMP) is fixed for the BO model and moves for the NBO model with the variation of the CEPs. Of course, simulation based on the NBO model is more realistic for laser- ${\text{H}}_2^{+}$ evolution process. Under this model, temporal analysis will be adopted for the harmonics and the internuclear separation at the time when the minima harmonic emits will be traced. Then we will give a demonstration that, the spectra minima under both the two models, agree with the predictions of the TCDI theory. Finally, the variation law of several quantities on CEP-dependence will be given, including the SMP, the effective potential, the internuclear separation and the electron’s de Broglie wavelength at the time for destructive interference occurring.

2. Theoretical models and methods

First we introduce the NBO model and the corresponding method. In our numerical simulations, the reduced-dimensional model is used for ${\text{H}}_2^{+}$, where only the two most important coordinates are considered, R, the internuclear separation (distance between the two protons), and z, the electron position (distance from the center mass of the nuclei to the electron). The molecular axis is assumed to be parallel to the polarization direction of the driving laser pulse. All these lead to the field-free Hamiltonian given as [Atomic units (a.u.) are used unless otherwise mentioned] [24, 26–28]

The target ${\text{H}}_2^{+}$ is assumed to be in its ground state (1sσ_g) before turning on the laser pulse and this initial state can be obtained by employing the evolution of the field-free Schrödinger equation in the imaginary time domain [29]. The TDSE (3) can be numerically solved by using the splitting-operator method [30] on a two-dimensional grid containing 1024 points on the z axis with step size of 0.2 a.u. and 256 points on the R axis with step size of 0.1 a.u.. The evolution time step size is set to T/8192 = 0.037 a.u.. In order to avoid the spurious reflections of the electron wave packet from the boundaries, a mask function formed as cos^1/8 [31] is employed on both the z- and R-dimension.

When the time-dependent wave function ψ(z, R, t) is obtained, the time-dependent expectation value of the dipole acceleration can be computed by means of Ehrenfest’s theorem [32]

As for the BO model, the only one thing need to do is fixing the internuclear separation R to a constant. The wave function evolution is obtained by using the same method with that of the NBO model.

3. Results and discussions

First, for the BO model, the fixed internuclear separation R of ${\text{H}}_2^{+}$ is set to its equilibrium distance, 2.63 a.u., which is obtained via computing the potential curve of the ground state 1sσ_g. We note that, the equilibrium distance of ${\text{H}}_2^{+}$ is different for different theoretical models and for different potential parameters. Under the one-dimensional BO model with the selected soft-core potential, it’s about 2.63 a.u. [9, 23]. The laser intensity is 5.0 × 10¹⁴W/cm². Four representative CEPs are considered from zero to 0.75π with an equal interval of 0.25π and the corresponding harmonic spectra are shown in Fig. 1. It clearly shows the appearance of one minima in the plateau region for all the harmonic spectra. The more interesting thing is that, the position of the spectral minima is fixed with CEP changes, which is located around 340th (about 190 eV).

 

Fig. 1 CEP-dependent harmonic spectra of ${\text{H}}_2^{+}$ driven by few-cycle Gaussian mid-infrared laser pulses with a central wavelength of 2.2 μm, FWHM of 7.34 fs, and peak intensity of 5.0 × 10¹⁴W/cm². The dashed vertical line indicates the position of the spectral minima. The Born-Oppenheimer approximation is performed, the internuclear separation R is fixed in the equilibrium distance of 2.63 a.u..

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Since ${\text{H}}_2^{+}$ has only one single electron, interference between multi-electron orbits cannot happen. In addition, the alignment angle θ of molecular axis respect to the laser polarization direction is fixed to zero in our simulations, then interference between contributions from different alignment situations which was recently proposed by Qin et al. [33] also cannot happen. Therefore, only the structural two-center interference could happen during the dynamic laser- ${\text{H}}_2^{+}$ evolution process. As we know that, this kind of structural minima is only dependent on the molecular structure and the alignment angle θ, and exhibits no change with variation of the laser parameters when the molecular structure and the alignment angle is fixed (e.g., product of Rcosθ is fixed for a diatomic molecule) [7,10,11]. Exactly, minima shown in Fig. 1 is a new direct proof of the TCDI theory with CEP changes beyond the previous demonstrations [7,10,11].

Then, what will happen to the harmonic spectrum of ${\text{H}}_2^{+}$ under the NBO model, where the internuclear separation R will be stretched or compressed under the drive of a laser pulse. Now, we use the same laser parameters with those in Fig. 1, and also start evolve ${\text{H}}_2^{+}$ from its ground state. The corresponding harmonic spectra shown in Fig. 2(a) demonstrate that, the SMP is different to those in the harmonic spectra based on the BO model and changes with the variation of the CEPs. How could this happen with the same target ${\text{H}}_2^{+}$ and the same laser parameters? As we know, the most essential difference between the BO and NBO models is that whether the nuclei are movable during the dynamic evolution process. In fact, the nuclei moves and the internuclear separation R will change during the laser- ${\text{H}}_2^{+}$ evolution process [23, 27] in the NBO model, while R is fixed in the BO model. We note that, the R-fixing approximation is reasonable for a weak laser driver, and, this dimension reduction can save much computation time. Fig. 2(b) displays the laser electric field with CEP=0 and the corresponding evolution of R. It shows that, from the sub-sub-peak A during pulse rising time, the R stretches larger and larger from the equilibrium distance of 2.64 a.u. and finally to about 7.9 a.u. when the laser pulse has passed. We also note that as we mentioned, the addition of R dimension makes the equilibrium distance a little different from that of the BO model with an increase by 0.01 a.u. here.

 

Fig. 2 (a) CEP-dependent harmonic spectra of ${\text{H}}_2^{+}$ driven by few-cycle Gaussian mid-infrared laser pulses. The laser parameters are the same with those in Fig. 1. Eight CEPs are considered and five spectra are shown in this figure. Inset are the spectra between harmonic 40th and 80th around the minima with shifts along the vertical direction magnified for a clearer view. The black arrows indicate the spectral minima. (b) The Gaussian laser field with case of CEP=0 and the corresponding time-dependent internuclear separation R(t). The other laser parameters are the same with those in Fig. 2(a). The pulse sub-sub-peak during rising time is indicated with A.

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Based on the facts mentioned above, it is well understandable that the internuclear separation R cannot be the initial equilibrium distance and should be somehow larger when the ionized electron returns to the nuclei. It means that the molecular structure has been changed for the two-center interference. Therefore, de Broglie wavelength of the returned electron that satisfies the TCDI condition should also change. Consequently, the corresponding generated harmonics would have different energies, namely the SMPs would be finally different. For example, the two SMPs for case of CEP=0 are clearly different for the two models with the same initial state and the same laser parameters (see Fig. 1 and Fig. 2(a)). One more finding is that, the minima position moves with CEP changes even under the same NBO model with the same ${\text{H}}_2^{+}$ initial state and the same other laser parameters. As we know, the CEP can dramatically affect the instantaneous laser field when a short pulse is considered. Since the FWHM of the laser pulse used is only one laser optical period, we think that the electron trajectory should be affected to a large extent when the CEP changes. Therefore, we assume that, the internuclear separation at the time for destructive interference occurring may be different for different CEPs, which eventually results in different SMPs.

Before verifying our assumption, we note that, in the NBO simulations, there are non-clear spectral minima occurring for some CEPs. This non-clearness is attributed to the weak interference effect, probably for the low degree of separation of the two potential wells of ${\text{H}}_2^{+}$. However, the more interesting thing for us is that how the SMP will change when CEP changes. As for this, it is necessary for more clear spectral minima occurring. Thus, we think that the electron interference effect should be more pronounced. As we know that, using the short-range potential, with adjusting the potential parameters, one can make the two potential wells of ${\text{H}}_2^{+}$ more separated or much deeper. This is very much analogous to the Young’s two-slit interference situation and this is helpful to enhance the interference effect and the visibility of interference minima as also. The form of short-range potential used is given as [22, 31, 37]

 

Fig. 3 Four representative CEP-dependent harmonic spectra of ${\text{H}}_2^{+}$ with short-range potential driven by Gaussian mid-infrared laser pulses. The laser peak intensity is 2.0 × 10¹⁵W/cm², other parameters are the same with those in Fig. 1. The red solid lines give the smoothed spectra, and the blue arrows point to where the SMPs are located.

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In order to verify our assumptions, the following four-step procedure for all the eight CEPs will be taken.

First, as to get a precise determination to the SMP, a smoothing to the harmonic spectra is adopted [12],

Second, the time-frequency analysis for harmonics around the SMP will be performed by the wavelet transformation to the time-dependent dipole acceleration 〈d_A(t)〉 [34–36],

 

Fig. 4 (a) The time-frequency analysis of the harmonics around the minima order for CEP=0.250π. (b) The temporal intensity profile of minima harmonic for CEP=0.250π. t_di, the time corresponds to the peak value, is the destructive interference time.

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Third, R(t_di), the accurate internuclear separation when the destructive interference occurs can be determined by checking the time-dependent internuclear separation R(t).

Finally, whether the spectra minima agrees with the prediction based on TCDI theory needs to be confirmed. In fact, we can calculate the de Broglie wavelength of the return electron which is responsible for the minima harmonic emission, by TCDI and TDSE theory, respectively. First, based on the TCDI theory, R(t_di) = (2m + 1)λ_m/2 [m = 0, 1, 2, ···] should be satisfied for target ${\text{H}}_2^{+}$, where λ_m is the de Broglie wavelength of electron which is responsible for the minima harmonic emission. Generally, one can calculate the zero order de Broglie wavelength, λ₀ (m = 0), from this equation for a given R(t_di). Second, based on the TDSE theory, one can calculate the de Broglie wavelength from directly using the SMP, which is given by λ′ = 2π/[2(E_k)]^1/2, where E_k is the kinetic energy of return electron which is responsible for the minima harmonic emission. Now, if the spectral minima exactly results from the TCDI, the electron de Broglie wavelength λ₀ from TCDI theory and λ′ from TDSE theory should be the same. For E_k, previous works presented different treatments, namely, E_k = nω_L [7] or E_k = nω_L − I_P [8], where I_P is the ground state ionization potential when the target is in the equilibrium distance. In fact, the two treatments are based on two different physical pictures–the acceleration of the returning continuum electron by the molecular potential is included or neglected [15, 38].

Wei et al. [15] have experimentally demonstrated that the molecular potential takes an effective value $I_{\text{P}}^{\text{eff}}$ between zero and I_P and is dependent on the laser intensity at the electron recombination time. For each CEP, the calculated λ₀ exactly located between the two values of λ′ obtained by the two treatments in which ionization potential is treated as zero or I_P. This result agree well with Wei’s conclusion. Therefore, the TCDI theory is naturally verified. The effective potential $I_{\text{P}}^{\text{eff}}$ can be calculated according to equation ${\lambda }_0=2\pi /{\left[2\left(n{\omega }_{\text{L}}-I_{\text{P}}^{\text{eff}}\right)\right]}^{1/2}$. The numerically obtained values of $I_{\text{P}}^{\text{eff}}$ and other physical quantities following the above verification procedure including the SMP in order, the corresponding internuclear separation R(t_di) and electron de Broglie wavelength λ₀ are shown in Table 1 for each CEP from zero to π. It clearly verified our assumption that, the internuclear separation R(t_di) indeed changes for different CEPs.

Table 1. The CEP-dependent physical quantities, including the SMP in order, the corresponding internuclear separation R(t_di), the corresponding electron de Broglie wavelength λ₀ and the effective potential $I_{\text{P}}^{\text{eff}}$.

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One should point that, as for a molecule, different to an atom, its different structure (potential surface) will introduce different ionization potentials. For target ${\text{H}}_2^{+}$, the internuclear separation is no longer equilibrium distance when return electron recombines to the ground state under the NBO model. In this case, the ionization potential will change to be a value between zero and I_P. We have shown the CEP-dependence of R(t_di), it is not surprise for $I_{\text{P}}^{\text{eff}}$ to have a CEP-dependence because of its close relationship with R(t_di). On the other hand, $I_{\text{P}}^{\text{eff}}$ is related to the laser intensity [15] (that is the carrier wave). As for a few-cycle pulse driving, when CEP changes, the instantaneous carrier wave would change and naturally $I_{\text{P}}^{\text{eff}}$ would change.

Now, the further demonstration to the CEP-dependence law for above mentioned quantities is shown in Fig. 5 where the corresponding fitting curves are also included. In fact, the internuclear separation R(t_di) and the corresponding electron de Broglie wavelength λ₀ have very good fittings of sine patterns with the same oscillating period of π [Figs. 5(a) and 5(b)] while the SMP and the effective potential $I_{\text{P}}^{\text{eff}}$ have quite good fittings of segment sine patterns with oscillating period of 2π [Figs. 5(c) and 5(d)]. The fitting functions can be given as form of

 

Fig. 5 Four CEP-dependent physical quantities and their fitting curves from zero to 2π (one laser optical period), (a) the internuclear separation R(t_di) and (b) the electron de Broglie wavelength λ₀ at time when the destructive interference occurs, (c) the SMP in order and (d) the effective potential $I_{\text{P}}^{\text{eff}}$.

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4. Conclusions

In this paper, we investigated in detail the carrier-envelope-phase-dependence of the harmonic spectral minima of ${\text{H}}_2^{+}$ driven by few-cycle mid-infrared laser pulses based on the Born-Oppenheimer (BO) and non-Born-Oppenheimer (NBO) approximations. The numerical results shows that, with the variation of the carrier-envelope phase (CEP), the spectral minima position is fixed for the BO model, however it changes for the NBO model. Interestingly, the CEP-dependent position change can be fitted as a sine pattern with an oscillating period of π. With the aid of time-frequency analysis to the harmonics around the spectral minima, it demonstrates that all the spectral minima positions agree well with the prediction from the two-center destructive-interference theory. The laser pulses with different CEPs stretch the two nuclei differently, and drive the electrons fly along different trajectories, then the returned electrons which satisfy to generate minima harmonics would have different kinetic energies. Therefore, the spectral minima positions are different for different CEPs. Other physical quantities including the internuclear separation, the electron de Broglie wavelength, and the effective potential are also CEP-dependent and can also be fitted as a sine pattern with the same oscillating period of π as for the spectral minima position.