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'''Keywords:''' momentum shift, nonadiabatic tunnelling ionization, time-resolved interference, subcycle and intercycle interference | '''Keywords:''' momentum shift, nonadiabatic tunnelling ionization, time-resolved interference, subcycle and intercycle interference | ||
| − | =1. Introdution= | + | ='''1. Introdution'''= |
Nonadiabatic tunneling ionization is one of the most fundamental and common dynamic processes in ultrafast ionization, and has always been a hot topic of concern in the study of ultrafast laser physics. Instead of adiabatic tunneling ionization, the barrier formed by the Coulomb field and the laser field can’t be regarded as static any more. We often use the Keldysh parameter to distinguish these two ionization processes [1]. The physical meaning of the Keldysh parameter <math display="inline">\gamma \mbox{=}\frac{w_{laser}}{w_{tunnel}}=\sqrt{\frac{I_p}{2U_p}}</math> is the ratio of the time an electron in ground state takes to tunnel through the entire potential barrier to the time of a single cycle of the external laser field, where <math display="inline">I_p</math> is the ionization energy, <math display="inline">U_p=\frac{E_0^2}{4{\omega }^2}</math> is the pondermotive energy with <math display="inline">E_0</math> being the amplitude of the laser field and <math display="inline">\omega </math> being the angular frequency of the laser field. If <math display="inline">\gamma \ll 1</math> , the ground state electron crossing the potential barrier time is much smaller than the laser field period, and the tunneling ionization process can be regarded as an adiabatic process. When the parameter is higher <math display="inline">\gamma \approx 1</math> , the laser field is no longer quasi-static during electron tunneling, in this condition the nonadiabatic properties of tunneling ionization begin to emerge [3-6]. | Nonadiabatic tunneling ionization is one of the most fundamental and common dynamic processes in ultrafast ionization, and has always been a hot topic of concern in the study of ultrafast laser physics. Instead of adiabatic tunneling ionization, the barrier formed by the Coulomb field and the laser field can’t be regarded as static any more. We often use the Keldysh parameter to distinguish these two ionization processes [1]. The physical meaning of the Keldysh parameter <math display="inline">\gamma \mbox{=}\frac{w_{laser}}{w_{tunnel}}=\sqrt{\frac{I_p}{2U_p}}</math> is the ratio of the time an electron in ground state takes to tunnel through the entire potential barrier to the time of a single cycle of the external laser field, where <math display="inline">I_p</math> is the ionization energy, <math display="inline">U_p=\frac{E_0^2}{4{\omega }^2}</math> is the pondermotive energy with <math display="inline">E_0</math> being the amplitude of the laser field and <math display="inline">\omega </math> being the angular frequency of the laser field. If <math display="inline">\gamma \ll 1</math> , the ground state electron crossing the potential barrier time is much smaller than the laser field period, and the tunneling ionization process can be regarded as an adiabatic process. When the parameter is higher <math display="inline">\gamma \approx 1</math> , the laser field is no longer quasi-static during electron tunneling, in this condition the nonadiabatic properties of tunneling ionization begin to emerge [3-6]. | ||
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In this paper, we numerically simulate the formation of the PMDs of a hydrogen atom ionized by few-cycle laser pulses with the strong-field approximation theory(SFA) [31,32] in time domain. By tracking the relative positions of the maximum of the PMDs and the negative vector potential corresponding to the circularly polarized laser electric-field peak, we find that the momentum shift is formed by the continuous interference of ionization signals over time. It's worth noting that, a drastic change occurs in the evolution obviously. To find the reason of this change, we further analyze the momentum shift induced by orthogonal two-color laser fields with different durations. In such laser fields, interference types are distinguishable. Ultimately, we draw a conclusion that subcycle and intercycle interferences influence the formation of momentum shift significantly, reflected in the irregular changes of momentum shift with time. All these in-depth studies of the momentum shift phenomenon can help us further understand the nonadiabatic tunneling process. | In this paper, we numerically simulate the formation of the PMDs of a hydrogen atom ionized by few-cycle laser pulses with the strong-field approximation theory(SFA) [31,32] in time domain. By tracking the relative positions of the maximum of the PMDs and the negative vector potential corresponding to the circularly polarized laser electric-field peak, we find that the momentum shift is formed by the continuous interference of ionization signals over time. It's worth noting that, a drastic change occurs in the evolution obviously. To find the reason of this change, we further analyze the momentum shift induced by orthogonal two-color laser fields with different durations. In such laser fields, interference types are distinguishable. Ultimately, we draw a conclusion that subcycle and intercycle interferences influence the formation of momentum shift significantly, reflected in the irregular changes of momentum shift with time. All these in-depth studies of the momentum shift phenomenon can help us further understand the nonadiabatic tunneling process. | ||
| − | =2. Methods= | + | ='''2. Methods'''= |
We use the SFA to simulate the direct ionization process of a hydrogen atom exposed to ultrafast laser field. In the SFA, the Coulomb potential is neglected and the final continuum wave function is described as a plane wave. The probability amplitude of a photoelectron with momentum <math display="inline">\boldsymbol{p}</math> can be written formally as [atomic units (a.u.) are used throughout the paper unless indicated otherwise]. | We use the SFA to simulate the direct ionization process of a hydrogen atom exposed to ultrafast laser field. In the SFA, the Coulomb potential is neglected and the final continuum wave function is described as a plane wave. The probability amplitude of a photoelectron with momentum <math display="inline">\boldsymbol{p}</math> can be written formally as [atomic units (a.u.) are used throughout the paper unless indicated otherwise]. | ||
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|- | |- | ||
| − | | [ | + | | <math display="inline">\begin{aligned}M\left(\mathbf{p},t\right)&=-i\frac{1}{\left(2\pi\right)^{3/2}}\int_{t_{i}}^{t}dt^{\prime}\mathbf{E}\left(t^{\prime}\right)\exp\Bigl[iS_{p}\left(t^{\prime}\right)\Bigr]\exp\Bigl(iI_{p}t^{\prime}\Bigr)\\&\times\int d\mathbf{r}\left\{\exp(i\mathbf{q}\bullet\mathbf{r})\right\}^{*}\bullet\mathbf{r}\bullet\psi_{0}\left(\mathbf{r}\right)\end{aligned}</math> |
|} | |} | ||
| style="width: 5px;text-align: right;white-space: nowrap;" | (4) | | style="width: 5px;text-align: right;white-space: nowrap;" | (4) | ||
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<span id='_Hlk157456024'></span>In our simulation two types of lasers are used to interact with hydrogen atom, they can be written as: <math display="inline">\boldsymbol{E}_{\boldsymbol{cp}}\left(t\right)=E_0f\left(t\right)\left[{\overset{\rightarrow}{e}}_xcos\left(\omega t\right)+\right. </math><math>\left. {\overset{\rightarrow}{e}}_ysin\left(\omega t\right)\right]</math> is the electric-field strength of the circularly polarized(CP) laser field as shown in Fig. 1(a), <math display="inline">E_{OTC}\left(t\right)=E_0f\left(t\right)\left[{\overset{\rightarrow}{e}}_xcos\left(\omega t\right)+\right. </math><math>\left. {\overset{\rightarrow}{e}}_ysin\left(2\omega t+\right. \right. </math><math>\left. \left. \frac{\pi }{2}\right)\right]</math> is the electric-field strength of the orthogonal two-color laser field(OTC) in Fig. 1(b). The intensities of CP and OTC pulses are <math display="inline">I\mbox{=}1.2\times {10}^{14}W/cm^2</math> and <math display="inline">I\mbox{=}3.5\times {10}^{14}W/cm^2</math> where <math>\left(I\mbox{ in }W/cm^2)\mbox{=}3.51\times {10}^{16}\times (E_0^2\mbox{ in }a.u.\right)</math> . We point out that the laser pulses used here are strong enough to contribute significant ionization, the SFA will give a quite accurate simulation for the parameters we used[33]. Meanwhile, the frequency <math display="inline">\omega \mbox{=}0.057a.u.</math> makes the ponderomotive energy <math display="inline">U_p</math> close to the ionization potential, hence <math display="inline">\gamma \approx 1</math> . Under this condition, ionization process will exhibit significant nonadiabatic effects. In additional, all the pulses have the same envelop function <math display="inline">f\left(t\right)={cos}^2(\frac{\pi t}{\tau })</math> , the duration <math display="inline">\tau </math> changes from <math display="inline">2T</math> to <math display="inline">3T</math> . Because of the influence of the pulse envelope, the few-cycle CP and OTC laser plulses will show us more pronounced interference structures in the momentum spectrum, including subcycle and intercycle interferences. | <span id='_Hlk157456024'></span>In our simulation two types of lasers are used to interact with hydrogen atom, they can be written as: <math display="inline">\boldsymbol{E}_{\boldsymbol{cp}}\left(t\right)=E_0f\left(t\right)\left[{\overset{\rightarrow}{e}}_xcos\left(\omega t\right)+\right. </math><math>\left. {\overset{\rightarrow}{e}}_ysin\left(\omega t\right)\right]</math> is the electric-field strength of the circularly polarized(CP) laser field as shown in Fig. 1(a), <math display="inline">E_{OTC}\left(t\right)=E_0f\left(t\right)\left[{\overset{\rightarrow}{e}}_xcos\left(\omega t\right)+\right. </math><math>\left. {\overset{\rightarrow}{e}}_ysin\left(2\omega t+\right. \right. </math><math>\left. \left. \frac{\pi }{2}\right)\right]</math> is the electric-field strength of the orthogonal two-color laser field(OTC) in Fig. 1(b). The intensities of CP and OTC pulses are <math display="inline">I\mbox{=}1.2\times {10}^{14}W/cm^2</math> and <math display="inline">I\mbox{=}3.5\times {10}^{14}W/cm^2</math> where <math>\left(I\mbox{ in }W/cm^2)\mbox{=}3.51\times {10}^{16}\times (E_0^2\mbox{ in }a.u.\right)</math> . We point out that the laser pulses used here are strong enough to contribute significant ionization, the SFA will give a quite accurate simulation for the parameters we used[33]. Meanwhile, the frequency <math display="inline">\omega \mbox{=}0.057a.u.</math> makes the ponderomotive energy <math display="inline">U_p</math> close to the ionization potential, hence <math display="inline">\gamma \approx 1</math> . Under this condition, ionization process will exhibit significant nonadiabatic effects. In additional, all the pulses have the same envelop function <math display="inline">f\left(t\right)={cos}^2(\frac{\pi t}{\tau })</math> , the duration <math display="inline">\tau </math> changes from <math display="inline">2T</math> to <math display="inline">3T</math> . Because of the influence of the pulse envelope, the few-cycle CP and OTC laser plulses will show us more pronounced interference structures in the momentum spectrum, including subcycle and intercycle interferences. | ||
| − | =3. Results and discussion= | + | ='''3. Results and discussion'''= |
Firstly，we use a simple CP laser pulse to interact with a hydrogen atom as shown in Fig. 1(a), of which the pulse duration is <math display="inline">N=2</math> . The formation of the PMDs corresponding to different instantaneous moments are tracked by the SFA theory in time domain <math display="inline">-T\leq t\leq T</math> . These time-resolved PMDs showcased the details of the ionization processes, which helps us investigate the nonadiabatic effect in the tunneling ionization. | Firstly，we use a simple CP laser pulse to interact with a hydrogen atom as shown in Fig. 1(a), of which the pulse duration is <math display="inline">N=2</math> . The formation of the PMDs corresponding to different instantaneous moments are tracked by the SFA theory in time domain <math display="inline">-T\leq t\leq T</math> . These time-resolved PMDs showcased the details of the ionization processes, which helps us investigate the nonadiabatic effect in the tunneling ionization. | ||
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In order to unfold the trend of <math display="inline">\Delta p</math> more accurately, we present the momentum shift <math display="inline">\Delta p</math> with respect to the laser pulse duration in Fig.5. The momentum shift depends sensitively on the pulse duration. We can find that, the trends are the same when the duration varies from <math display="inline">\tau =1.7T</math> to <math display="inline">\tau =2.1T</math> and from <math display="inline">\tau =2.3T</math> to <math display="inline">\tau =2.7T</math> . However, an obvious difference appears between <math display="inline">\tau =2.1T</math> and <math display="inline">\tau =2.3T</math> (red circle), where <math display="inline">\Delta p</math> decreases sharply as had happened in Fig.3. What is more important, when the duration ranges from <math display="inline">\tau =2.1T</math> to <math display="inline">\tau =2.3T</math> , the interference type ICI is emerging significantly as shown in Fig.4. This correspondence further illustrates that the interference type is influencing the momentum shift synchronously. | In order to unfold the trend of <math display="inline">\Delta p</math> more accurately, we present the momentum shift <math display="inline">\Delta p</math> with respect to the laser pulse duration in Fig.5. The momentum shift depends sensitively on the pulse duration. We can find that, the trends are the same when the duration varies from <math display="inline">\tau =1.7T</math> to <math display="inline">\tau =2.1T</math> and from <math display="inline">\tau =2.3T</math> to <math display="inline">\tau =2.7T</math> . However, an obvious difference appears between <math display="inline">\tau =2.1T</math> and <math display="inline">\tau =2.3T</math> (red circle), where <math display="inline">\Delta p</math> decreases sharply as had happened in Fig.3. What is more important, when the duration ranges from <math display="inline">\tau =2.1T</math> to <math display="inline">\tau =2.3T</math> , the interference type ICI is emerging significantly as shown in Fig.4. This correspondence further illustrates that the interference type is influencing the momentum shift synchronously. | ||
| − | =4. Conclusion= | + | ='''4. Conclusion'''= |
In conclusion, the time-resolved nonadiabatic tunnelling ionization PMDs of a hydrogen atom exposed to strong laser fields are studied with the strong-field approximation. Our simulations show that the nonadiabatic momentum shift <math display="inline">\Delta p</math> evolves with time, meanwhile, it will be influenced by the interference type. When the electric wave of a laser pulse oscillates over time periodically, the interference type ICI gradually becomes important. This makes <math display="inline">\Delta p</math> drastic change from the situation that only SCI dominates. Momentum shift is an important nonadiabatic phenomenon, to look into it in time domain provide us a new perspective, more interesting details are presented. | In conclusion, the time-resolved nonadiabatic tunnelling ionization PMDs of a hydrogen atom exposed to strong laser fields are studied with the strong-field approximation. Our simulations show that the nonadiabatic momentum shift <math display="inline">\Delta p</math> evolves with time, meanwhile, it will be influenced by the interference type. When the electric wave of a laser pulse oscillates over time periodically, the interference type ICI gradually becomes important. This makes <math display="inline">\Delta p</math> drastic change from the situation that only SCI dominates. Momentum shift is an important nonadiabatic phenomenon, to look into it in time domain provide us a new perspective, more interesting details are presented. | ||
Revision as of 14:26, 16 May 2024
Abstract: Momentum shift is an important sign of nonadiabatic tunnelling ionization process, to investigate the mechanism of momentum shift, we use the strong-field approximation theory to track the formation of ionization momentum spectra of hydrogen atom under the action of different laser pulses in time domain. By observing the ionization momentum spectra of different structures with time, we find that the momentum shift is formed by the continuous interference and evolution of ionization signals over time. Meanwhile, we further analyze how subcycle and intercycle interference influencing the formation of momentum shift. Before the duration is long enough that intercycle interference emerges, momentum shift grows smoothly. This finding reveals the different intrinsic mechanisms for the formation of momentum shift in many-cycle and few-cycle laser pulses. This work lays foundation for deeper understanding of nonadiabatic tunnelling process and makes the regulation of momentum shift possible.
Keywords: momentum shift, nonadiabatic tunnelling ionization, time-resolved interference, subcycle and intercycle interference
1. Introdution
Nonadiabatic tunneling ionization is one of the most fundamental and common dynamic processes in ultrafast ionization, and has always been a hot topic of concern in the study of ultrafast laser physics. Instead of adiabatic tunneling ionization, the barrier formed by the Coulomb field and the laser field can’t be regarded as static any more. We often use the Keldysh parameter to distinguish these two ionization processes [1]. The physical meaning of the Keldysh parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma \mbox{=}\frac{w_{laser}}{w_{tunnel}}=\sqrt{\frac{I_p}{2U_p}}}
is the ratio of the time an electron in ground state takes to tunnel through the entire potential barrier to the time of a single cycle of the external laser field, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I_p}
is the ionization energy, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U_p=\frac{E_0^2}{4{\omega }^2}}
is the pondermotive energy with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_0}
being the amplitude of the laser field and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega }
being the angular frequency of the laser field. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma \ll 1}
, the ground state electron crossing the potential barrier time is much smaller than the laser field period, and the tunneling ionization process can be regarded as an adiabatic process. When the parameter is higher Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma \approx 1}
, the laser field is no longer quasi-static during electron tunneling, in this condition the nonadiabatic properties of tunneling ionization begin to emerge [3-6].
Recently, many interesting works have revolved around the nonadiabatic properties of tunneling ionization, such as the non-zero ionization initial momentum [7-8], distorted tunneling exit positions [9-10]，and the tunneling delay time phenomenon [11-15]. The existence of the above nonadiabatic properties leads to a fact that the photoelectrons will not be distributed near the negative vector potential of the laser field as the adiabatic tunneling photoelectrons. There is a difference between the maximum of the photoelectron momentum distributions(PMDs) and the negative vector potentials corresponding to the laser electric-field peaks, leading to the phenomenon of nonadiabatic momentum shift [16-22].
Numerous studies on the formation mechanism of ionization momentum spectra have shown that the formation of ionization momentum spectra is not instantaneous, but evolves over time. During the ionization process, subsequent ionized electrons continuously interfere with previously ionized electrons, ultimately preserving the signal that satisfies the Feynman path integral to form the momentum spectrum [23-24]. During the formation of ionization momentum spectra, subcycle interference(SCI)[25-28] and intercycle interference(ICI) [29,30] are two important types of interferences. They ultimately determined the interference structure of the PMDs. In order to further explore the internal mechanism of momentum shift formation and whether the formation of momentum shift also follows the law of time evolution, it is necessary to analyze the phenomenon in time domain.
In this paper, we numerically simulate the formation of the PMDs of a hydrogen atom ionized by few-cycle laser pulses with the strong-field approximation theory(SFA) [31,32] in time domain. By tracking the relative positions of the maximum of the PMDs and the negative vector potential corresponding to the circularly polarized laser electric-field peak, we find that the momentum shift is formed by the continuous interference of ionization signals over time. It's worth noting that, a drastic change occurs in the evolution obviously. To find the reason of this change, we further analyze the momentum shift induced by orthogonal two-color laser fields with different durations. In such laser fields, interference types are distinguishable. Ultimately, we draw a conclusion that subcycle and intercycle interferences influence the formation of momentum shift significantly, reflected in the irregular changes of momentum shift with time. All these in-depth studies of the momentum shift phenomenon can help us further understand the nonadiabatic tunneling process.
2. Methods
We use the SFA to simulate the direct ionization process of a hydrogen atom exposed to ultrafast laser field. In the SFA, the Coulomb potential is neglected and the final continuum wave function is described as a plane wave. The probability amplitude of a photoelectron with momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{p}}
can be written formally as [atomic units (a.u.) are used throughout the paper unless indicated otherwise].
|
(1) |
in which, the ground state wave function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\psi }_0\left(\boldsymbol{r},t\right)=e^{iI_pt}{\psi }_0\left(\boldsymbol{r}\right)}
, for hydrogen atom Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\psi }_0\left(\boldsymbol{r}\right)=\frac{1}{\sqrt{\pi }}e^{-r}}
. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W\left(t\right)=\boldsymbol{r}\cdot \boldsymbol{E}\left(t\right)}
is electron-laser interaction operator under dipole approximation (length gauge), with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{r}}
the electronic coordinate, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{E}\left(\boldsymbol{t}\right)}
the electric field. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_i}
is the moment the laser pulse starts to play.
For the scattering state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \vert {\psi }_p^{\left(V\right)}\left(\boldsymbol{r},t\right)\rangle }
with canonical momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{p}}
can be written as a Gordon–Volkov state
|
(2) |
with the Volkov phase
|
(3) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A\left(t\right)}
is the vector potential, from which the laser field can be derived Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E\left(t\right)=-\frac{d}{dt}A\left(t\right)}
.
Finally, formula (1) can be rewritten in detail as
|
(4) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I_p\mbox{=0}\mbox{.5 }a.u.}
is the ionization potential, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{q}=\boldsymbol{p}+\boldsymbol{A}\left(t\right)}
is the mechanical momentum.
Figure 1. (Color online) (a) The electric field of a 2-cycle (full width) circularly polarized laser pulse with the photon energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega \mbox{=0}\mbox{.057}a.u.}
and the intensity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I\mbox{=}1.2\times {10}^{14}W/cm^2}
. (b) The red solid line represents the electric field of OTC field comprising three 800-nm cycles, the peak intensity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I\mbox{=3}\mbox{.5}\times {10}^{14}W/cm^2}
and the relative phase is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\pi }{2}}
. Both pulses have the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {cos}^2}
envelope.
In our simulation two types of lasers are used to interact with hydrogen atom, they can be written as: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{E}_{\boldsymbol{cp}}\left(t\right)=E_0f\left(t\right)\left[{\overset{\rightarrow}{e}}_xcos\left(\omega t\right)+\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\overset{\rightarrow}{e}}_ysin\left(\omega t\right)\right]
is the electric-field strength of the circularly polarized(CP) laser field as shown in Fig. 1(a), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_{OTC}\left(t\right)=E_0f\left(t\right)\left[{\overset{\rightarrow}{e}}_xcos\left(\omega t\right)+\right. }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\overset{\rightarrow}{e}}_ysin\left(2\omega t+\right. \right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \left. \frac{\pi }{2}\right)\right]
is the electric-field strength of the orthogonal two-color laser field(OTC) in Fig. 1(b). The intensities of CP and OTC pulses are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I\mbox{=}1.2\times {10}^{14}W/cm^2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I\mbox{=}3.5\times {10}^{14}W/cm^2}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left(I\mbox{ in }W/cm^2)\mbox{=}3.51\times {10}^{16}\times (E_0^2\mbox{ in }a.u.\right)
. We point out that the laser pulses used here are strong enough to contribute significant ionization, the SFA will give a quite accurate simulation for the parameters we used[33]. Meanwhile, the frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega \mbox{=}0.057a.u.}
makes the ponderomotive energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U_p}
close to the ionization potential, hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma \approx 1}
. Under this condition, ionization process will exhibit significant nonadiabatic effects. In additional, all the pulses have the same envelop function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left(t\right)={cos}^2(\frac{\pi t}{\tau })}
, the duration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau }
changes from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2T}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3T}
. Because of the influence of the pulse envelope, the few-cycle CP and OTC laser plulses will show us more pronounced interference structures in the momentum spectrum, including subcycle and intercycle interferences.
3. Results and discussion
Firstly，we use a simple CP laser pulse to interact with a hydrogen atom as shown in Fig. 1(a), of which the pulse duration is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N=2}
. The formation of the PMDs corresponding to different instantaneous moments are tracked by the SFA theory in time domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -T\leq t\leq T}
. These time-resolved PMDs showcased the details of the ionization processes, which helps us investigate the nonadiabatic effect in the tunneling ionization.
Figure 2. (Color online) Instantaneous PMD of hydrogen atom exposed to a 2-cycle circularly polarized laser field at the moment (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=-\frac{1}{2}T}
, (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=\frac{1}{4}T}
, (c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=\frac{1}{2}T}
and (d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=T}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T\mbox{=}\frac{2\pi }{\omega }}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega \mbox{=0}\mbox{.057}a.u.}
and the intensity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I\mbox{=}1.2\times {10}^{14}W/cm^2}
. The black closed curves are negative vector potentials. White dots mark the most probable momentum corresponding to the peak electric field and black triangles mark the most probable momentum corresponding to the maximum ionization rate. The white dotted lines indicate the magnitude of the momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
. The small image in the upper right corner of (c) and (d) shows an enlargement of the marked area with a white dashed box.
As above, Fig. 2 shows four chosen PMDs at the moments: (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=-\frac{1}{2}T}
, (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=\frac{1}{4}T}
, (c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=\frac{1}{2}T}
and (d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=T}
. More details in figure 2(a-d), the black closed curve is the negative vector potential of the CP laser pulse, the white dots denote the negative vector potential corresponding to the electric-field peak intensity, the black triangles denote the maximum ionization rate of the PMDs. Among them, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
is the offset difference between the white dots and the black triangles, as known as the momentum shift in nonadiabatic tunneling ionization[34].
We can see that, at the beginning of the interaction, there is no complex interference between electron wave packets, as shown in Fig.2(a). As time evolves, the instantaneous ionization events add coherently, making the ionization momentum spectrum becoming increasingly complex as shown in Fig.2(b-c). Finally, at the end of the interaction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=T}
, the electron wave packets has already obtained all informations of the laser field, the ionization momentum spectrum gradually forms in Fig.2(d)[23]. At the same time, one can see that the momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
also varies with time as the interference structure.
Due to the theoretical derivation with imag-time theory[35,36], we know that, circularly polarized laser pulse does not cause a non-zero initial transverse momentum
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_z}
, but rather a non-zero initial longitudinal momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{\perp }}
. Therefore, the nonadiabatic momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
will only have the longitudinal momentum component Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{\perp }}
. However, with time evolves, the momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
is composed of lateral momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_z}
and longitudinal momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{\perp }}
in Fig.2(a-c). Until the last moment, only Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{\perp }}
is preserved in Fig.2(d). Which indicates that, the nonadiabatic term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
is time-resolved, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_z}
is disappeared through interference during the evolution.
Figure 3. (Color online) The momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta P}
with respect to the moment as mentioned in Fig. 2. The red circle indicates a rapidly changing of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
corresponding to the interference type changes from SCI to SCI+ICI.
To understand the law of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
varing over time more deeply, we record the magnitude of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
at different times. In Fig.3, seven black squares guided by a solid line from left to right represent the momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
corresponding to seven moments from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=-T}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=T}
. Unsurprisingly, the magnitude of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
changs with time as expected. However, something unusual has happened here. We can see that, in Fig.3 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
changes smoothly overall except the area indicated with a red circle, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
sharply decreases from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=\frac{1}{4}T}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=\frac{1}{2}T}
. This notable phenomenon interests us and the evolution of the interference structure in Fig.2 prompts us.
Correspondingly, the PMDs at these two moments are presented in Fig.2(b) and Fig.2(c). In Fig.2(b), we can see many crescent interference structures, which is corresponding to subcycle interference(SCI) with electron wave packets(EWPs) from different half cycle within one optical cycle. Differently, for the situation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=\frac{1}{2}T}
(Fig.2(c)), many ATI(above-threshold ionization)-like structures emerge, this indicates that intercycle interference (ICI) with EWPs released from different laser optical cycles starts to dominate[37]. So far we can say that, the time-resolved changing of the interference type from only SCI to SCI+ICI for a 2-cycle CP laser pulse results in the drastic decrease of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
in Fig.3.
To further validate our conclusion, we will use more complex few-cycle OTC laser pulses as shown in Fig.1(b) with different durations to repeat the simulations discussed above. When interacting with hydrogen atom, the few-cycle OTC laser pulse will give more details of the interference type in the PMD.
Figure 4. (Color online) Final PMD of hydrogen atom exposed to OTC laser pulse when the pulse duration is (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =1.8T}
, (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.1T}
, (c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.3T}
and (d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.6T}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T\mbox{=}\frac{2\pi }{\omega }}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega \mbox{=0}\mbox{.057}a.u.}
and the intensity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I\mbox{=3}\mbox{.5}\times {10}^{14}W/cm^2}
. The white dots mark the most probable momentum corresponding to the peak electric field and black triangles mark the most probable momentum corresponding to the maximum ionization rate. The white dotted lines indicate the magnitude of the momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta P}
. The small image in the lower right corner of (c) and (d) shows an enlargement of the marked area with a white dashed box.
In Fig.4, final PMDs of hydrogen atom exposed to OTC laser pulses are presented, when the pulse durations are (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =1.8T}
, (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.1T}
, (c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.3T}
and (d) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.6T}
respectively. The photon energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega \mbox{=0}\mbox{.057}a.u.}
, the intensity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I\mbox{=3}\mbox{.5}\times {10}^{14}W/cm^2}
and the keldysh parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma \approx 1}
, hence, nonadiabatic tunneling ionization occurs. Meanwhile, the momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
is marked as the same way as in Fig.2.
Obviously, the interference fringes depend strongly on the duration of an OTC laser pulse. When the duration is short enough, the momentum spectrum only involves simple crescent structures as shown in Fig.4(a) and Fig.4(b), telling us that subcycle interference is in effect. As the number of cycles increases, the structures of the PMDs become more and more complex. In Fig.4(c), we can slightly find the ATI-like cyclic structures upon the crescent structures. While in Fig.4(d), ATI-like cyclic structures are already very obvious. This evolution can be attributed to the gradual emerging of intercycle interference. At the same time, it can be observed that, the momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
also evolves with the duration.
Figure 5. (Color line) The momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta P}
with respect to the laser pulse duration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau }
as mentioned in Fig. 4 with units of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T\mbox{=}\frac{2\pi }{\omega }}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega \mbox{=0}\mbox{.057}a.u.}
. The red circle indicates a rapidly changing of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta P}
corresponding to the emerging of ICI.
In order to unfold the trend of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
more accurately, we present the momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
with respect to the laser pulse duration in Fig.5. The momentum shift depends sensitively on the pulse duration. We can find that, the trends are the same when the duration varies from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =1.7T}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.1T}
and from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.3T}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.7T}
. However, an obvious difference appears between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.1T}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.3T}
(red circle), where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
decreases sharply as had happened in Fig.3. What is more important, when the duration ranges from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.1T}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =2.3T}
, the interference type ICI is emerging significantly as shown in Fig.4. This correspondence further illustrates that the interference type is influencing the momentum shift synchronously.
4. Conclusion
In conclusion, the time-resolved nonadiabatic tunnelling ionization PMDs of a hydrogen atom exposed to strong laser fields are studied with the strong-field approximation. Our simulations show that the nonadiabatic momentum shift Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
evolves with time, meanwhile, it will be influenced by the interference type. When the electric wave of a laser pulse oscillates over time periodically, the interference type ICI gradually becomes important. This makes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta p}
drastic change from the situation that only SCI dominates. Momentum shift is an important nonadiabatic phenomenon, to look into it in time domain provide us a new perspective, more interesting details are presented.
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Document information
Published on 18/06/24
Accepted on 27/05/24
Submitted on 15/05/24
Volume 40, Issue 2, 2024
DOI: 10.23967/j.rimni.2024.05.015
Licence: CC BY-NC-SA license
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