On the detection of closed orbits in time-dependent systems by using double-pulse lasers
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S U Khan,1,2,3, M L Du1,21Institute of Theoretical Physics, Chinese Academy of Sciences. Beijing 100190, China 2University of Chinese Academy of Sciences, Beijing, 100049, China
First author contact:3Author to whom any correspondence should be addressed. Received:2020-03-26Revised:2020-05-6Accepted:2020-06-3Online:2020-08-05
Abstract In the photodetachment of atoms or negative ions by a double-pulse laser, the first pulse of the double-pulse laser generates waves and the delayed second pulse may detect them. The phenomenon of the excitation and detection of waves by a double-pulse laser can be used to identify the closed orbits in the system. We demonstrate this phenomenon with a negative hydrogen ion (H−) by analyzing the total population excited by a double-pulse laser in a time-dependent field for different physical parameters. By analyzing the total excited population using a double-pulse laser, we can uncover all the closed orbits existing in the system. We demonstrate that this can be realized by scanning the first pulse position and the time delay between the two pulses. Keywords：double-pulse laser;detection of closed orbits;photodetachment spectra;time-dependent systems
PDF (1434KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article S U Khan, M L Du. On the detection of closed orbits in time-dependent systems by using double-pulse lasers. Communications in Theoretical Physics, 2020, 72(9): 095503- doi:10.1088/1572-9494/aba23e
The photoabsorption cross-sections of atoms and negative ions exhibit oscillations in the presence of external electric [1–4] and magnetic fields [5, 6] for photon energy near the ionization threshold. These oscillations are connected to closed orbits by semiclassical closed-orbit theory [7–9] which depicts the oscillations as the interference between waves going out from and returning to the atomic (negative ion) source region. Such semiclassical models and also quantum theories [10–13] are in good agreement with the experimental results. Considerable efforts have been made to establish the correspondence between the oscillations in the photodetachment cross-section caused by a continuous laser and closed orbits in the system [8, 9, 14–17].
Pulsed lasers have a finite width and can be used to excite wavepackets. Albert and Zoller [18, 19] noted that wavepackets may provide a link between the classical trajectory of a particle and quantum theory and that the evolution of wavepackets furnishes real-time observations of the atomic or molecular systems. An experiment has been sketched theoretically [18] in which a laser has two pulses, separated by a time delay, which only differ in their relative phase. The first pulse excites the wavepacket and the second pulse is turned on upon the return of this wavepacket to the origin. The relative phases of the two pulses then determine the extent of the constructive or destructive interference between the wavepackets produced by these pulses. Noordam et al [20] examined the effects of the relative phases of the two pulses theoretically and predicted that Ramsey interference fringes would be observed. Moreover, an experimental study of wavepackets in short pulse photoionization in the presence of an electric field was carried out by Broers et al [21]. To observe the effect of the relative phase between the two pulses, Wang and Starace investigated the photodetachment cross-section of H− by a short-single-pulse and a double-pulse laser in the presence of a constant electric field [22] and parallel electric and magnetic fields [13], as the photodetachment of negative ions permits a direct observation of the interplay between quantum interference and classical motion. A formula was later derived for the absorption cross-section from a short-pulse laser for atoms or negative ions in external fields [23]. Zhao et al [24] studied photodetachment cross-sections by the use of a pulse laser in the presence of static parallel electric and magnetic fields. The population generated by a double-pulse laser has been expressed in terms of the oscillator strength from closed-orbit theory which connects the features of double-pulse spectra to the features of closed-orbit theory [25]. It is described there that the number of peaks and their energy positions reflect the structure of the closed orbits in the system.
The study of photoionization of Rydberg atoms in an oscillating radio frequency field has been performed experimentally [26] and the well-established time-independent closed-orbit theory has been extended to time-dependent systems [27]. Recently, Yang and Robicheaux [28] have theoretically investigated the photodetachment dynamics of negative ions in the presence of a single-cycle THz pulsed electric field in detail. The theoretical formulations they made are applicable to any type of time-dependent field. The results from the semiclassical time-dependent closed-orbit theory and the quantum mechanical results calculated numerically were compared and found to be in best agreement. Following Yang and Robicheaux [28], several authors studied the photodetachment cross-section in the presence of different types of external time-dependent configurations such as a constant electric field superposed by a linear time-dependent field, a pure sinusoidal field and a bi-chromatic sinusoidal time-dependent field [29–31]. Very recently, the authors investigated the double-pulse spectra of negative ions in the presence of a constant electric field superposed with a time-dependent oscillatory field [32], applying semiclassical time-dependent closed-orbit theory [27, 28]. A formula was derived for the total population of electrons excited by a double-pulse laser. It was shown there that the transit time of the closed orbit completely coincides with the time delay td between the two pulses at the peak in the total population. The authors, however, have not discussed the correspondence between the structures of closed orbits and double-pulse spectra.
As mentioned above, a double pulse may be used to excite waves and detect them in real time. The time delay between the two laser pulses may be programmed in such a way that the second pulse passes over the negative ion at about the same instant that the wavepacket produced by the first pulse returns to the ion source. In the case of time-dependent systems, by varying the peak of the first pulse to take place at different times and scanning the second pulse by varying the time delay td between the pulses, it is possible to uncover all the closed orbits existing in a time-dependent system. In the present work, we intend to use this technique to show the correspondence between the closed orbits and double-pulse spectra in time-dependent systems, while analyzing the population of excited-state electrons by a double-pulse laser in a time-dependent field for different physical parameters.
We assume that the double-pulse laser is polarized in the z-direction and is the sum of two Gaussians separated by a time delay td. The double-pulse laser is considered to have a frequency ω and has the following mathematical form,$ \begin{eqnarray}f(t)=A{{\rm{e}}}^{-\tfrac{{\left(t-{t}_{p}\right)}^{2}}{2{\alpha }^{2}}}\cos (\omega t)+A{{\rm{e}}}^{-\tfrac{{\left((t-{t}_{p})-{t}_{d}\right)}^{2}}{2{\alpha }^{2}}}\cos (\omega t+\beta ),\end{eqnarray}$satisfying the condition $\alpha \gg \tfrac{1}{\omega }$ [25, 32]. The constants A, α, β define the amplitude of the laser, the width of the pulse and the relative phase between the two pulses, respectively. The variable tp represents the peak of the first pulse in the double-pulse laser and t is the running laboratory time. The shape of the double laser pulse is shown in figure 1. The study in [32] was limited to the specific case of tp=zero. Here, we generalize the formula to arbitrary values of tp and apply it to the study of the large-scale features of double-pulse spectra and the evolution of closed orbits.
Figure 1.
New window|Download| PPT slide Figure 1.The double-pulse profile, based on equation (1), for the parameters $\alpha =1.065\ast {10}^{3}$ a.u., tp=0, td=5α.
When the laser acts upon a negative ion, it generates an outgoing electron wave which will later move in the time-dependent electric field. As the wave propagates out more and more, the effect of the external field becomes more and more significant. Far from the nucleus, the classical Hamiltonian corresponding to the time-dependent external field polarized in the z-direction controls the dynamics of the electron wave. Since the laser and the electric field are both oriented in the z-direction and the atomic potential of the residual atom is ignored after the detachment of the electron by the laser [15, 33], the system therefore possesses cylindrical symmetry and the azimuth angle φ can be ignored. The motion of the detached electron can be described by two coordinates, z and ρ. The Hamiltonian for the system in the cylindrical coordinate system in augmented phase space can then be described as [27]:$ \begin{eqnarray}H(z,\rho ,t,{p}_{z},{p}_{\rho },{p}_{t})=\displaystyle \frac{{p}_{z}^{2}}{2}+\displaystyle \frac{{p}_{\rho }^{2}}{2}+F(t)z(t)+{p}_{t}.\end{eqnarray}$Accordingly, Hamilton’s standard canonical equations must be augmented by adding the two equations,$ \begin{eqnarray}\begin{array}{rcl} & & \displaystyle \frac{{\rm{d}}t}{{\rm{d}}\tau }=1,\\ & & \displaystyle \frac{{\rm{d}}{p}_{t}}{{\rm{d}}\tau }=-\displaystyle \frac{\partial H}{\partial t},\end{array}\end{eqnarray}$where $\tau =t-{t}_{i}$ is the evolution time of the classical trajectory in the augmented phase space. For convenience, the initial conditions for t and pt are set as t(τ=0)=ti and ${p}_{t}(\tau =0)=-{E}_{0}$. The variable ${p}_{t}=-E(t)$ is the instantaneous energy of the electron while moving in the external field. Using Hamilton’s time-dependent canonical equations, the classical motion of the detached electron in the time-dependent field can be described by the following two equations [28]:$ \begin{eqnarray}\rho (t)={k}_{0}\sin ({\theta }_{i})(t-{t}_{i}),\end{eqnarray}$$ \begin{eqnarray}z(t)=\left({k}_{0}\cos ({\theta }_{i})-A({t}_{i})\right)(t-{t}_{i})+{\int }_{{t}_{i}}^{t}A(t^{\prime} ){\rm{d}}{t}^{\prime} ,\end{eqnarray}$where ${k}_{0}=\sqrt{2{E}_{0}}$ with E0 is the initial outgoing energy of the detached electron, θi is the angle made by the initial momentum with the z-axis and A(t) is the vector potential associated with the time-dependent electric field.
The laser detaches the electrons in the form of outgoing waves which propagate along the classical trajectories of an electron in an external field. Depending on the field, some of the trajectories may return back to the source region. The time-dependent extended semiclassical closed-orbit theory [27, 28] states that the instantaneous excited-state population is the sum of the constant background term representing the instantaneous excited-state population in the absence of an external field and an oscillatory term contributed by all the closed orbits returning to the atomic center at that particular instant. Mathematically, the instantaneous excited-state population can be expressed as:$ \begin{eqnarray}P(t)={P}_{0}(t)+\displaystyle \sum _{\gamma }{P}_{r}^{\gamma }(t).\end{eqnarray}$In [28], the authors developed a time-dependent closed-orbit theory and focused on the control of the photodetachment cross-section by the amplitude of the time-dependent field as the number of closed orbits in the system is changing with the amplitude of the time-dependent field.
Following the time-dependent closed-orbit theory algorithm, the scaled instantaneous double-pulse laser excited-state population for any type of time-dependent field and any type of negative ion source can be written as:$ {l}{P}^{{\rm{S}}}(t)=\left({\rm{e}}^{-\tfrac{{\left(t-{t}_{p}\right)}^{2}}{{\alpha }^{2}}}+{{\rm{e}}}^{-\tfrac{{\left((t-{t}_{p})-{t}_{d}\right)}^{2}}{{\alpha }^{2}}}\right.\\ +\,2{{\rm{e}}}^{\tfrac{-{\left(t-{t}_{p}\right)}^{2}-{\left((t-{t}_{p})-{t}_{d}\right)}^{2}}{2{\alpha }^{2}}}\cos (\beta )\\ +\,\displaystyle \sum _{\gamma }\left[{g}^{l}(2l+1)\left|\displaystyle \frac{C({k}_{\mathrm{ret}})}{C({k}_{0})}\right|\displaystyle \frac{{{ \mathcal A }}_{\gamma }}{{k}_{0}R}\left(({f}_{1}+{f}_{2})\sin \left({\tilde{S}}_{\gamma }-{\mu }_{\gamma }\displaystyle \frac{\pi }{2}\right)\right.\right.\\ \left.\left.\left.+\,{f}_{3}\sin \left({\tilde{S}}_{\gamma }-{\mu }_{\gamma }\displaystyle \frac{\pi }{2}+\beta \right)+{f}_{4}\sin \left({\tilde{S}}_{\gamma }-{\mu }_{\gamma }\displaystyle \frac{\pi }{2}-\beta \right)\right)\right]{\delta }_{m0}\right),$where the scaling factor is ${k}_{0}| C({k}_{0}){| }^{2}$, the energy-dependent factor is $\tfrac{C({k}_{\mathrm{ret}})}{C({k}_{0})}=\tfrac{{k}_{\mathrm{ret}}{\left({k}_{\mathrm{ret}}^{2}+{k}_{b}^{2}\right)}^{2}}{{k}_{0}{\left({k}_{\mathrm{ret}}^{2}+{k}_{b}^{2}\right)}^{2}}$ with ${k}_{b}=\sqrt{2{E}_{b}}$, ${k}_{0}=\sqrt{2{E}_{0}}$, and ${k}_{\mathrm{ret}}=\sqrt{2{E}_{\mathrm{ret}}}$ where Eb, E0, and Eret are the binding energy, the initial outgoing energy, and the returning energy of the photodetached electron, respectively. The extended action $\tilde{S}={S}_{\gamma }+{E}_{0}(t-{t}_{i})$ and the amplitude ${{ \mathcal A }}_{\gamma }$ must be calculated by the time-dependent closed-orbit theory algorithm. The mathematical expressions for the action Sγ and the amplitude ${{ \mathcal A }}_{\gamma }$ are given as follows:$ \begin{eqnarray}{S}_{\gamma }=\displaystyle \frac{1}{2}{\int }_{{t}_{i}}^{t}{p}_{\rho }^{2}{\rm{d}}t+\displaystyle \frac{1}{2}{\int }_{{t}_{i}}^{t}{p}_{z}^{2}{\rm{d}}t-{\int }_{{t}_{i}}^{t}F(t)z{\rm{d}}t,\end{eqnarray}$and$ \begin{eqnarray}{{ \mathcal A }}_{\gamma }=\displaystyle \frac{1}{{k}_{0}(t-{t}_{i})}{\left|\displaystyle \frac{{k}_{0}{R}^{2}}{\left({k}_{0}-F({t}_{i})\cos ({\theta }_{i})(t-{t}_{i})\right)}\right|}^{\tfrac{1}{2}}.\end{eqnarray}$For a particular time-dependent field, the action Sγ and the amplitude ${{ \mathcal A }}_{\gamma }$ may easily be calculated using equations (8) and (9). The amplitude ${{ \mathcal A }}_{\gamma }$ can also be written as [28],$ \begin{eqnarray}\displaystyle \frac{{{ \mathcal A }}_{\gamma }}{R}=\displaystyle \frac{1}{{k}_{0}(t-{t}_{i})}{\left|\displaystyle \frac{{p}_{z}({t}_{i}){\rm{d}}{t}_{i}}{{p}_{z}(t){\rm{d}}t}\right|}^{\tfrac{1}{2}}.\end{eqnarray}$The Maslov index μγ characterizes the geometric properties of the trajectory and determines the number of singularities along the trajectory including the returning points, caustics, and foci etc [33]. The constant g may be equal to +one or −one, depending on whether the outgoing and returning directions of the closed orbit are the same or opposite. The value of l depends upon the specific negative ion source. The terms fs are the exponentials with s=1, 2, 3, 4, given as $ \begin{eqnarray*}\begin{array}{rcl}{f}_{1} & = & {{\rm{e}}}^{-\tfrac{{\left(t-{t}_{p}\right)}^{2}}{2{\alpha }^{2}}-\tfrac{{\left({t}_{i}-{t}_{p}\right)}^{2}}{2{\alpha }^{2}}},\\ {f}_{2} & = & {{\rm{e}}}^{-\tfrac{{\left((t-{t}_{p})-{t}_{d}\right)}^{2}}{2{\alpha }^{2}}-\tfrac{{\left(({t}_{i}-{t}_{p})-{t}_{d}\right)}^{2}}{2{\alpha }^{2}}},\\ {f}_{3} & = & {{\rm{e}}}^{-\tfrac{{\left(t-{t}_{p}\right)}^{2}}{2{\alpha }^{2}}-\tfrac{{\left(({t}_{i}-{t}_{p})-{t}_{d}\right)}^{2}}{2{\alpha }^{2}}},\\ {f}_{4} & = & {{\rm{e}}}^{-\tfrac{{\left((t-{t}_{p})-{t}_{d}\right)}^{2}}{2{\alpha }^{2}}-\tfrac{{\left({t}_{i}-{t}_{p}\right)}^{2}}{2{\alpha }^{2}}},\end{array}\end{eqnarray*}$ where t=ti is the originating time of the closed orbits, which may or may not be equal to tp. Each term in equation (7) has its own physical meaning. For example, the first and the second terms represent the instantaneous excited-state population from the first and second pulses, as they are not coherent. The third term describes the interference of the instantaneous population due to the two pulses after a time delay td. Finally, each term in the summation describes the recurrence of the wavepacket. The detailed derivation of equation (7) is given in the appendix.
The total population or the total number of excited electrons, once the double pulse passes through the system, can be calculated by integrating equation (7) over the width of the double pulse in temporal space,$ \begin{eqnarray}{P}_{{\rm{tot}}}^{{\rm{S}}}=\int {P}^{{\rm{S}}}(t){\rm{d}}t.\end{eqnarray}$
Time-dependent double-pulse photoabsorption spectra may be used to investigate closed orbits in time-dependent systems. Applying the formula in equation (11), the total excited population from the double-pulse laser can be calculated for any type of time-dependent field. By fixing the peak of the first pulse of the double-pulse laser at some time t=tp and scanning the time delay td between the pulses while calculating the total excited population, there may be some peaks in the total population for values of time delay td. The number of peaks and their positions at different td reflects the structure of the closed orbits originated at the peak of the first pulse of the double-pulse laser. Repeating these calculations for the peak of the first pulse at different times t=tp, one may probe all the closed orbits existing in the time-dependent system.
We demonstrate the probing of closed orbits from double-pulse photoabsorption spectra for H− in a time-dependent electric field composed of a constant part and a time-dependent oscillatory part i.e. $F(t)={F}_{0}+{F}_{1}\sin ({\rm{\Omega }}t)$ in figure 2(b) for some fixed parameters. The peak of the first pulse of the double-pulse laser is at $t={t}_{p}=-9746$ a.u. and the total population is calculated for different values of time delay td between the pulses. The total population has three peaks at different td. It was shown by the authors in [32] that the time delay td corresponding to the peak in the total population matches the transit time of the closed orbit originated at the peak of the first pulse. Hence, in this case, the number of peaks shows that there are three closed orbits originated at the peak of the first pulse i.e $t={t}_{p}=-9746$ a.u., It depicts that there is a one-to-one correspondence between the number of peaks in the total population and the closed orbits originated at the peak of the first pulse. Noting all three values of td corresponding to the peaks in the total population and adding it to the time at the peak of the first pulse tp=−9746 a.u., one gets the laboratory frame return time of the closed orbits originated at the peak of the first pulse. The geometrical shapes of the closed orbits can be found by substituting these times, the initial time tp=−9746 a.u. and the returning time ${t}_{{\rm{ret}}}={t}_{d}+{t}_{p}$ in the trajectory equations (5).
Figure 2.
New window|Download| PPT slide Figure 2.(a) The time period T of the closed orbits from the classical trajectory equation, (5), is plotted against the originating time of the closed orbits ‘ti’ for the fixed parameters Eph=0.9 eV, β=0, α=106.53 a.u., F0=100 kV cm−1, ${F}_{1}=3{F}_{0}$ and ${\rm{\Omega }}=2.9489\,\ast {10}^{-4}$ a.u., (b) A plot of the total population based on equation (11) against the time delay td between the two pulses of the double-pulse laser for the peak of the first pulse fixed at tp=−9746 a.u. The total population has peaks at different td. The peaks show the number and structure of the closed orbits originated at the peak of the first pulse. For demonstration, we also show in figure. (a) that the classical periods of the closed orbits originated at the peak of the first pulse i.e. t=tp=−9746 a.u., of the double-pulse laser match the peak position in td.
Applying this procedure, the orbits corresponding to the peaks in the total population in figure 2(b) are shown in figure 3. The positions and strengths of the peaks show the structures of the orbits. The orbits which go a long distance spread more, so a small part of the wavefunction may be reflected in the source region which results in a weak recurrence in the total population. This can also be attributed to the phase accumulated along the orbit.
Figure 3.
New window|Download| PPT slide Figure 3.The orbits corresponding to the peaks in the total population in figure 2(b). Note the time delay td corresponding to the peaks in the total population and added to tp to get the laboratory return time t. The orbits can then be plotted by inserting these values into the trajectory equations from (5). The strengths of the peaks in the total population reflect the structure of the orbits. The orbits which go further spread more and a small part of the wavefunction reflects in the source region which may result in a small peak in the total population and vice versa.
To probe all the closed orbits in the system for particular parameters, one needs to vary the peak of the first pulse of the double-pulse laser at different times t=tp and scan the second pulse by varying the time delay td between the pulses, while computing the total excited population. We performed this procedure and the results are shown in figure 4. The closed orbits existing in the system for the parameters mentioned in the caption of figure 4 are shown in figure 5. The closed orbit in figure 5(a) goes out with an initial angle ${\theta }_{i}=\mathrm{zero}$ and is reflected by the external field to the residual atom with an angle θf=π. The orbit in figure 5(b) goes downwards initially relative to the $+z$-axis i.e θi=π and returns back to the atomic region with an angle of ${\theta }_{f}=\mathrm{zero}$. Figure 5(c) shows an orbit which goes upwards initially with an angle of θi=zero, reflected by the field towards the source region. Upon reaching the atomic center, it continues its motion in the downward direction and is eventually reflected back towards the center and approaches the source with a return angle of θf=zero. Another type of orbit is shown in figure 5(d) which starts with an initial outgoing angle of ${\theta }_{i}=\mathrm{zero}$ and which returns to the source region with an angle of θf=π. In this case, the orbit returns twice to the atomic center. The first return is called a soft return [28, 34] as the orbit returns to the atomic center with a momentum pz(t)=zero. The orbit shown in figure 5(e) goes out with an angle of θi=zero and returns with an angle of θf=π. An orbit going out and returning with the same angle θi=θf=π is shown in figure 5(f). The closed orbits that were searched for here from double spectra also appear in other systems where the photodetachment of negative ions in the presence of different types of time-dependent electric field has been studied [29–31]. The different closed orbits that exist in the system depend on the system parameters, such as the energy of the laser photons, the strength of the time-dependent field and the frequency of the time-dependent field. A special feature of the double-pulse laser is that the first pulse launches the orbits and the second pulse selects the orbits originated at the peak of the first pulse i.e ti=tp. Hence, by varying tp for different times and scanning the second pulse by varying td, all the closed orbits in the system for particular parameters may be probed.
Figure 4.
New window|Download| PPT slide Figure 4.The horizontal thin curves show the plots of total population from equation (11) against the time delay td between the pulses of a double-pulse laser while changing the peak of the first pulse to be at different times $t={t}_{p}$. The remaining fixed parameters are the same as in figure 2. The total population has different positions and numbers of peaks for different times tp. Noting all the time delays td, corresponding to the peaks in the total population for each time tp, one may extract all the closed orbits existing in the system. The thick curve is the classical period of the closed orbits originating at different times based on equation (5). The closed orbits corresponding to the peaks are plotted in figure 5.
Figure 5.
New window|Download| PPT slide Figure 5.The geometrical structures of the closed orbits from equation (5), corresponding to the peaks in the total population from the double-pulse laser in figure 4. The fixed parameters used here are Eph=0.9 eV, F0=100 kV cm−1, F1=3F0 and ${\rm{\Omega }}=2.9489\,\ast {10}^{-4}$ a.u., The closed orbits shown here correspond to peaks marked by arrows in figure 4. Among all other peaks, each peak corresponds to an orbit having the geometrical shape of one of those shown here.
The system is analyzed by using the prescribed method for different values of the phase parameter β between the pulses of the double-pulse laser. We plotted the total population against the time delay td, while changing the peak of the first pulse to different times tp for different values of β in figure 6. Figures 6(a)–(b) depict that by changing the phase parameter β, while the other parameters are kept fixed, the position of the peaks in the total population is not changed. This reflects that the number of closed orbits existing in the system and their periods do not depend upon the relative phase between the pulses of the double-pulse laser. The modulation of the total population as the relative phase of the two pulses is clear from figures 6(a)–(b). It can be seen that figure 6(a) is the mirrored spectrum of figure 6(b).
Figure 6.
New window|Download| PPT slide Figure 6.The horizontal thin curves show the total population from equation (11) against the time delay td between the pulses of the double-pulse laser for the laser phase parameter (a) β=zero and (b) β=π with the peak of the first pulse at different times t=tp. Other fixed parameters are Eph=0.9 eV, α=106.53 a.u., F0=100 kV cm−1, F1=3F0 and ${\rm{\Omega }}=1.4744\ast {10}^{-4}$ a.u., The number and positions of peaks in the total population does not change with changes of the laser phase parameter β. This shows that while keeping the other parameters fixed, the number of orbits or the periods of the orbits do not change with the laser phase parameter β. The laser phase parameter β just modulates the peaks, as the spectrum in figure 6(b) is the mirror copy of the spectrum in figure 6(a).The thick curve is the classical period of the closed orbits originating at different times, based on equation (2).
In conclusion, we analyzed the photodetachment spectra from a double-pulse laser in a time-dependent field for different physical parameters. We illustrated that there is a one-to-one correspondence between the peaks in the total excited population against the time delay td and the number of closed orbits originating at the peak of the first pulse. Based on this feature of double-pulse spectra, we devised a method for the extraction of closed orbits in time-dependent systems, that may be used in the case of any type of time-dependent field. As a particular system, we analyzed the photodetachment spectra in a constant field superposed by an oscillatory field and extracted the closed orbits existing in the system for the particular parameters. Our results may help experiments to be performed studying the photodetachment spectra from double-pulse lasers in time-dependent external fields.
Acknowledgments
S U K and M L D acknowledge the CAS-TWAS president fellowship program of UCAS for financial assistance and partial support by the National Natural Science Foundation of China (NSFC), grants No. 11 474 079 and No. 11 421 063.
Appendix. The derivation of equation (7)
The double-pulse laser acts upon a negative hydrogen ion (H−) and generates an outgoing electron wave. This outgoing wave may be considered initially as a wave in the absence of an electric field, as the influence of external electric fields is negligible compared to the atomic core potential. Therefore, the initial outgoing wave ψ0(R, θi, φi, ti) can be written as:$ \begin{eqnarray}{{\rm{\Psi }}}_{0}(R,{\theta }_{i},{\phi }_{i},{t}_{i})={f}_{L}({t}_{i}){\psi }_{\mathrm{out}}(R,{\theta }_{i},{\phi }_{i}){{\rm{e}}}^{-{{\rm{i}}{E}}_{0}{t}_{i}},\end{eqnarray}$satisfying the following Schrödinger equation:$ \begin{eqnarray}\left({E}_{0}+\displaystyle \frac{1}{2}{{\rm{\nabla }}}^{2}-{V}_{p}(r)\right){{\rm{\Psi }}}_{0}(R,{\theta }_{i},{\phi }_{i},{t}_{i})={f}_{L}({t}_{i})D{\psi }_{i},\end{eqnarray}$where ${\psi }_{\mathrm{out}}(R,{\theta }_{i},{\phi }_{i})=C({k}_{0}){{\rm{Y}}}_{{lm}}({\theta }_{i},{\phi }_{i})\tfrac{{{\rm{e}}}^{{\rm{i}}{k}_{0}R}}{R}$ is the time-independent part of the wavefunction. C(k0) is the complex energy-dependent function which is:$ \begin{eqnarray}C({k}_{0})={\rm{i}}\sqrt{\displaystyle \frac{4\pi }{3}}\displaystyle \frac{4{{Bk}}_{0}}{{\left({k}_{b}^{2}+{k}_{0}^{2}\right)}^{2}},\end{eqnarray}$where B=0.31552 and Ylm(θi, φi) is the spherical harmonic function that gives the angular distribution of the detached electron [28]. The time-dependent factor fL(t),$ \begin{eqnarray}{f}_{L}(t)=A{{\rm{e}}}^{-\tfrac{{\left(t-{t}_{p}\right)}^{2}}{2{\alpha }^{2}}}+A{{\rm{e}}}^{-\tfrac{{\left((t-{t}_{p})-{t}_{d}\right)}^{2}}{2{\alpha }^{2}}}{{\rm{e}}}^{-{\rm{i}}\beta }\end{eqnarray}$is the laser profile and is the reduced form of equation (1) after using a rotating wave approximation.
Far from the nucleus, the wavefunction is semiclassical following the classical trajectories. The semiclassical scheme may be exploited to construct the quantum wave by computing the classical trajectories of the detached electron in the external electric field [15]. Therefore, the wavefunction associated with each trajectory is presented as:$ \begin{eqnarray}{\rm{\Psi }}(t)={f}_{L}({t}_{i}){\psi }_{\mathrm{out}}(R,{\theta }_{i},{\phi }_{i}){{ \mathcal A }}_{\gamma }{{\rm{e}}}^{{\rm{i}}({S}_{\gamma }-{E}_{0}{t}_{i}-{\mu }_{\gamma }\pi /2)}.\end{eqnarray}$Depending on the physical conditions, some of the trajectories may return to the source region; these are called closed orbits. To calculate the excited-state population, one must find all the possible closed orbits of the detached electron by using the classical trajectory equations (5). Upon returning to the source region, the wave may be considered as a plane wave [15] and for each closed orbit, the returning wave can be approximated as:$ \begin{eqnarray}{{\rm{\Psi }}}_{\mathrm{ret}}(t)={f}_{L}({t}_{i}){\tilde{{\rm{\Psi }}}}_{\mathrm{ret}},\end{eqnarray}$where ${\tilde{{\rm{\Psi }}}}_{{ret}}$ is the reduced function which is:$ \begin{eqnarray}{\tilde{{\rm{\Psi }}}}_{\mathrm{ret}}=C({k}_{0}){G}_{\mathrm{co}}{{\rm{Y}}}_{{lm}}({\theta }_{i}){{\rm{e}}}^{\pm {\rm{i}}{k}_{\mathrm{ret}}z},\end{eqnarray}$and the factor ${G}_{\mathrm{co}}$ is given as:$ \begin{eqnarray}{G}_{{co}}=\displaystyle \frac{{{ \mathcal A }}_{\gamma }}{R}{{\rm{e}}}^{{\rm{i}}\left({S}_{\gamma }-{E}_{0}{t}_{i}-{\mu }_{\gamma }\tfrac{\pi }{2}\right)}.\end{eqnarray}$The±sign in equation (A7) denotes whether the wave is returning in the positive or negative z-direction. The functions Sγ and ${{ \mathcal A }}_{\gamma }$ represent the phase accumulated by the wavefunction while propagating along a classical trajectory and the amplitude factor of the returning wavefunction, respectively. The detailed calculation of Sγ and ${{ \mathcal A }}_{\gamma }$ can be seen in [32].
According to the standard closed-orbit theory [8, 9, 27], the instantaneous excited-state population can be expressed as:$ \begin{eqnarray}P(t)={P}_{0}(t)+\displaystyle \sum _{\gamma }{P}_{r}^{\gamma }(t),\end{eqnarray}$where$ \begin{eqnarray}{P}_{0}(t)=-2{\rm{Im}}\langle {\rm{I}}(t)| {{\rm{\Psi }}}_{0}(t)\rangle \end{eqnarray}$is the constant background part representing the instantaneous population for the double-pulse laser without any external field. The term ${\rm{I}}(t)={f}_{L}(t){{\rm{e}}}^{-{{\rm{i}}{E}}_{0}t}{\rm{D}}{\psi }_{i}$ represents the source, with D being the dipole operator and ${\psi }_{i}={{B}{\rm{e}}}^{-{k}_{b}r}/r$ is the bound state wavefunction.
Each term ${P}_{r}^{\gamma }$ in the second part of equation (A9) is the oscillatory contribution to the instantaneous excited-state population from a closed orbit returning at time t in the source region and is written as:$ \begin{eqnarray}{P}_{r}^{\gamma }=-2{\rm{Im}} \langle {\rm{I}}(t)| {{\rm{\Psi }}}_{{\rm{ret}}}^{\gamma }(t) \rangle .\end{eqnarray}$The integrals in equations (A10) and (A11) can easily be calculated by using the procedure given in [16]. The final results for the integrals are$ \begin{eqnarray}{P}_{0}(t)={k}_{0}| C({k}_{0}){| }^{2}{| {f}_{L}(t)| }^{2},\end{eqnarray}$and$ \begin{eqnarray}\begin{array}{rcl}{P}_{r}^{\gamma } & = & -{g}^{l}(2l+1){C}^{\ast }({k}_{0})C({k}_{{\rm{ret}}})\displaystyle \frac{{{ \mathcal A }}_{\gamma }}{R}\\ & & \times {\rm{Im}}\left[{f}_{L}^{\ast }(t){f}_{L}({t}_{i}){{\rm{e}}}^{-{\rm{i}}\left({S}_{\gamma }+{E}_{0}(t-{t}_{i})-{\mu }_{\gamma }\tfrac{\pi }{2}\right)}\right].\end{array}\end{eqnarray}$
After performing some simple algebra one may arrive at equation (7).
,1,2,3, M L Du1,21Institute of Theoretical Physics, Chinese Academy of Sciences. Beijing 100190, 
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