Closed-orbit Theory Research Articles

Photodetachment microscopy with double pulse in time-dependent fields

We present the photodetachment microscopy of the singly charged negative ions using the double pulse with a frequency near the ionization threshold in the presence of the time-dependent external electric field. In the presence of time-dependent electric field, the number of photoelectron trajectories contributing to the electron flux at a space–time point at the detector may be more than two. We demonstrate that the relative flight times of the classical trajectories, to reach a point at the detector plane can be calculated by observing the photoelectron flux at a particular time at the detector and scanning the second pulse by varying the time delay between the two coherent laser pulses. Moreover, investigating the photoelectron flux, we have found that the flux distribution can be controlled by the relative phase difference between the two coherent pulses separated by a short time delay, at spatial and temporal points where the classical trajectories associated with photoelectron emerging at different pulses arrive at the detector simultaneously and exhibit interference. The temporal and spatial photoelectron flux distribution oscillates as a function of the relative phase difference between the pulses. The time-dependent quantum propagator approach with stationary phase approximation is linked analytically with the time-dependent closed orbit theory.

Closed-orbit theory for the photodetachment rate in a static magnetic field transversely superimposed to an oscillatory electric field

The closed-orbit theory is used for the study of the photodetachment rate of electrons from single-charged anions in presence of a homogeneous magnetic field that is perpendicular to a time-dependent electric field. The photodetachment process in the near-threshold region is achieved by a weak laser field. We describe in detail the semiclassical method to derive the general formula for the instant photodetachment rate in time-dependent systems. We find that the photodetachment rate is affected by the presence of the static magnetic field since, within the semiclassical picture, it contributes to fold back the electron trajectories back to their emission position: generating closed orbits. For weak magnetic fields, however, the number of closed orbits does not change, then the modulation of the photodetachment rate is virtually unaffected when it is compared to the case with no magnetic field. On the contrary, the modulation is clearly distinctive and complicated for strong magnetic field intensities due to the contribution of a larger number of closed orbits. Despite we provide general formulas, numerical experiments and discussions are focused on the emission of electrons as s-waves and p z -waves.

Manipulating the photodetachment dynamics of the H2− ion by a Gaussian laser field

The photodetachment of the ion in a time-dependent electric field is investigated for the first time. On the basis of time-dependent closed orbit theory, we calculate the photodetachment cross section (PCS) of a ion manipulated by a Gaussian laser field. We use the superposition principle and consider the interaction between two atomic centers in the ion to construct the electron wave function after photodetachment from the ion. Three different types of closed orbits for the photodetached electron in the Gaussian laser field are found. The PCS of the ion can be controlled by varying the Gaussian laser field strength, pulse duration, and final time where the electron moves in the laser field through numerical calculation. The oscillating cross section of the ion is approximately twice that of the H− ion by comparing their PCS in the same Gaussian laser field, thereby suggesting that the interference effect of the two centers in the ion is extremely significant. This study provides a semiclassical method for exploring the photodetachment dynamics of the negative molecular ion in the Gaussian laser field. Furthermore, it serves as guide for future molecular photodetachment microscopy experiments in the time-dependent electric field.

Photodetachment of the H– ion in a forced harmonic potential

The photodetachment of a H– ion in a forced harmonic potential driven by a general time-dependent oscillating electric field has been investigated in the semi-classical closed orbit theory for the first time. It is found that the driven electric field frequency can affect the photodetachment cross-section of this system greatly. If the frequency of the driving electric field is equal to the harmonic frequency, a resonance phenomenon occurs in the classical motion of the detached electron. The interference effect between the returning electron wave travelling along the closed orbit with the initial outgoing wave gets stronger, causing the photodetachment cross-section to oscillate in a complicated manner. When the frequency of the driving electric field is unequal to the harmonic frequency, the driving electric field can weaken or strengthen the oscillatory structure in the photodetachment cross-section. In addition, the strength and initial phase in the driving electric field can also influence the photodetachment dynamics of the system. Our work provides a new method for controlling the photodetachment of negative ions in a harmonic potential and may guide future experimental research for cavity dynamics or in the ion trap.

Semiclassical theory of double-pulse spectra for time dependent systems

A theory of double-pulse photodetachment spectra in the presence of time dependent field has been developed. We derived a general formula for the total population generated by the double-pulse laser applying the time dependent closed orbit theory. We analyzed the double-pulse spectra for H− in the presence of constant electric field superposed with a sinusoidal time dependent field. The effects of the amplitude of time dependent field and laser parameters have been investigated. It has been shown that the population has peak or valley when the time delay between the two pulses of the double-pulse laser is equal to the time of the closed orbit emanating at the peak of the first pulse of the double-pulse laser. We also shown that the laser phase parameter modulates the population.

Near-threshold photodetachment microscopy in the presence of a transverse magnetic field

Near-threshold photodetachment microscopy of mononegative ions is theoretically studied when a homogeneous transverse magnetic field $\mathrm{B}$ is superimposed to the static electric field $\mathrm{E}$. The electron flux distribution is obtained using two different approaches: the quantum source theory with the energy-Green function evaluated by means of the stationary phase approximation and the closed-orbit theory. Here we describe the physical ideas while developing the theory in detail. Both theoretical methods yield consistent descriptions for all field intensities and angular momentum of the detached electrons we have investigated and also reproduce reported experimental results. From our calculations, we have found that the presence of the transverse magnetic field leads to a global displacement of the recorded electron flux distribution along the $\mathrm{E}\ifmmode\times\else\texttimes\fi{}\mathrm{B}$ direction. Furthermore, we found that the shape of the recorded interference pattern changes from circular to elliptical as the magnetic field increases. We have theoretically characterized and quantified those effects on the interference pattern due to the presence of the transverse magnetic field.

Photodetachment of the H− ion in a quantum well with one expanding wall

The standard closed orbit theory is extended to investigate the photodetachment of a negative ion near a moving wall. The photodetachment cross section of the ${\mathrm{H}}^{\ensuremath{-}}$ ion in a quantum well with one expanding wall is specifically put forward. In contrast to the photodetachment of the negative ion in a static quantum well, the returning kinetic energy of the detached electron is different from its initial value after collision with the moving wall, therefore an additional modulation factor appears in the oscillating cross section, which depends on the electron's initial outgoing and returning momentum. Four different types of closed orbits are found for the detached electron in a quantum well with one expanding wall, and their connections with the oscillating cross section are analyzed quantitatively. The calculation results suggest that the photodetachment cross section of this system depends on the speed of the moving wall and the initial distances from the ion to the walls sensitively. The method used in this paper is universal and can be extended to study the photodetachment of the nonhydrogenic negative ion near a moving wall, whether the wall is moving with constant velocity or oscillatory. The results provide insight into the behavior of photodetachment dynamics of the negative ion in the presence of a moving boundary and may have potential interest for future experimental researches.

Theoretical studies of the effects of a transverse magnetic field in photodetachment microscopy

Photodetachment microscopy of monovalent anions is studied theoretically with a uniform transverse magnetic field superimposed to the electric field . The main experimental results given by Chaibi et al. (EPL, 82 (2008) 20005) have been reproduced using the closed-orbit theory, and confirmed through the quantum source theory with the energy Green function evaluated by means of the stationary phase approximation. Both theoretical methods yield consistent descriptions of the electron flux distribution for all field intensities and angular momentum of electrons we studied. Our calculations show that the presence of the magnetic field leads to a global displacement of the ring patterns along the -direction, confirming the experimental observations under weak magnetic-field conditions. In addition, we found that the shape and size of the interference patterns change from circular to elliptical as the magnetic field increases.

Photodetachment in a cavity: From rectangles to parallel plates

Simple analytic formulas are presented for photodetachment cross sections of H− inside a rectangular cavity using semiclassical closed-orbit theory. The formulas predict oscillations in the spectrum and correlate the oscillations with closed orbits of the system. Explicit fully-quantum-mechanical formulas are also derived. We compare the results from closed orbit theory and quantum-mechanical calculations. We also show how the photodetachment cross section formulas for a rectangular cavity can be reduced to those for a cavity consisting of two plates.

Photodetachment dynamics of H− ion in a harmonic potential plus a time-dependent oscillating electric field**Project supported by the National Natural Science Foundation of China (Grant No. 11374133) and the Taishan Scholars Project of Shandong Province, China (Grant No. ts2015110055).

The photodetachment dynamics of H− ion in a harmonic potential plus an oscillating electric field is studied using the time-dependent closed orbit theory. An analytical formula for calculating the photodetachment cross section of this system is put forward. It is found that the photodetachment cross section of this system is nearly unaffected for the weak oscillating electric field strength, but oscillates complicatedly when the oscillating electric field strength turns strong. In addition, the frequency of the harmonic potential and the oscillating electric field (the frequency of the harmonic potential and the frequency of the oscillating electric field are the same in the paper, unless otherwise stated.) can also affect the photodetachment dynamics of this system. With the increase of the frequency in the harmonic potential and the oscillating electric field, the number of the closed orbits for the detached electrons increased, which makes the oscillatory structure in the photodetachment cross section much more complex. Our study presents an intuitive understanding of the photodetachment dynamics driven by a harmonic potential plus an oscillating electric field from a space and time dependent viewpoint. This study is very useful in guiding the future experimental research for the photodetachment dynamics in the electric field both changing with space and time.

Photodetachment dynamics in a time-dependent oscillating electric field

Using the time-dependent form of closed orbit theory, as developed by Haggerty and Delos [M.R. Haggerty, J.B. Delos, Phys. Rev. A 61, 053406 (2000)], and by Yang and Robicheaux [B.C. Yang, F. Robicheaux, Phys. Rev. A 93, 053413 (2016)], we study the photodetachment dynamics of a hydrogen negative ion in a time-dependent oscillating electric field. Compared to the photodetachment in a static electric field, the photodetachment dynamics of a negative ion in the time-dependent oscillating electric field become much more complicated but more interesting. Since the applied electric field is oscillating with time, the photodetachment cross section of the negative ion in the oscillating electric field is time-dependent. In a time-dependent framework, we put forward an analytical formula for calculating the instantaneous photodetachment cross section of this system. Our study suggests that the instantaneous photodetachment cross section exhibits oscillatory structure, which depends sensitively on the frequency of the oscillating electric field. With increasing frequency of the oscillating electric field, the number of closed orbits increases and the oscillatory structure in the photodetachment cross section becomes much more complicated. The connection between the detached electron’s closed orbit with the oscillating cross section is analyzed quantitatively. This study provides a clear and intuitive picture for the photodetachment processes of a negative ion in the presence of an oscillating electric field. We hope that our work will be useful in guiding future experimental research.

Investigation of the photodetached electronic wave packet dynamics in a magnetic field near a surface

The electronic wave packet dynamics photodetached from H− ion in a magnetic field near an elastic surface has been studied by using the time-dependent perturbation theory combined with the semiclassical closed orbit theory for the first time. Firstly, we put forward an analytic formula for calculating the autocorrelation function of this system. Then we calculate and analyze the autocorrelation function in great detail. It is demonstrated that the quantum wave packet revival phenomenon is significant when the laser pulse width is far less than the period of the detached electron’s closed orbit. As the pulse width is close to the period of the detached electron’s closed orbit, the quantum wave packet revival phenomenon becomes weakened. When the laser pulse width is bigger than the period of the closed orbit of the detached electron, the adjacent revival peaks in the autocorrelation function begin to merge and the quantum revival phenomenon disappears. In addition, the magnetic field strength can also affect the autocorrelation function of this system. As the magnetic field strength is relatively small, the quantum wave packet revival phenomenon is weak. With the increase of the magnetic field strength, the number of the reviving peaks in the autocorrelation function becomes increased and the quantum wave packet revival phenomenon becomes significant. Therefore, we can control the quantum wave packet revival in the autocorrelation function of this system by changing the laser pulse width and the external magnetic field strength. This study can guide the future experimental research on the wave packet dynamics of atoms or ions in the external fields or surfaces.

Photodetachment of the H− ion in a linear time-dependent electric field

Using the time-dependent closed orbit theory, we study the photodetachment of the H− ion in a linear time-dependent electric field for the first time. An analytical formula for calculating the time-dependent photodetachment cross section of this system has been put forward. It is found when the external electric field changes very slowly with time, there is only one closed orbit of the detached electron and the photodetachment cross section is quite stable. However, when the electric field changes quickly with time, three different types of closed orbits are found and the photodetachment cross section oscillates in a much more complex way. The connection of each type of closed orbit with the oscillatory structure in the photodetachment cross section is analyzed quantitatively. In addition, the photon energy and the laser field parameters can also have great influence on the time-dependent photodetachment cross section of this system. This study provides a clear and intuitive picture for the photodetachment dynamics of a negative ion in the presence of a time-dependent electric field and may guide future experimental studies exploring the quantum effect in the photodetachment dynamics of negative ions from a time-dependent viewpoint.

Photodetachment Spectrum of Hydrogen Negative Ion Near a Partially Reflecting Spherical Surface

Photodetachment of negative ions near surfaces is of great interest in view of its fundamental significance and technological applications. We reinvestigate the dynamics of photoelectrons in H− photodetachment near a partially reflecting spherical surface by the semiclassical closed-orbit theory. Reflection parameter R and curvature K is used to observe inelastic and spherical effects of the surface, respectively. The classical action is evaluated from the photodetached electron trajectories incident normally at the surface, arising simultaneously from the source and its image. The derived analytical formula of photodetachment cross section correctly recovers the results of reflective spherical surface published recently based on theoretical imaging method.

Closed-orbit theory for photodetachment in a time-dependent electric field

The standard closed-orbit theory is extended for the photodetachment of negative ions in a time-dependent electric field. The time-dependent photodetachment rate is specifically studied in the presence of a single-cycle terahertz pulse, based on exact quantum simulations and semiclassical analysis. We find that the photodetachment rate is unaffected by a weak terahertz field, but oscillates complicatedly when the terahertz pulse gets strong enough. Three types of closed classical orbits are identified for the photoelectron motion in a strong single-cycle terahertz pulse, and their connections with the oscillatory photodetachment rate are established quantitatively by generalizing the standard closed-orbit theory to a time-dependent form. By comparing the negative hydrogen and fluorine ions, both the in-phase and antiphase oscillations can be observed, depending on a simple geometry of the contributed closed classical orbits. On account of its generality, the presented theory provides an intuitive understanding from a time-dependent viewpoint for the photodetachment dynamics driven by an external electric field oscillating at low frequency.

Re-examining Photodetachment of H− near a Spherical Surface using Closed-Orbit Theory**Supported by National Natural Science Foundation of China under Grant Nos. 11474079 and 11421063

Photodeachment of H− near a reflective spherical surface was studied by Haneef et al. [J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 195004] using a theoretical imaging method. The total cross section displays interesting oscillations. Here we re-examine the total photodetachment cross section of this system by directly applying the standard closed-orbit theory. Our result for the total cross section differs from the result obtained by Haneef et al. The difference between the two results vanishes in the limit of large radius of the reflective sphere. We argue that the theoretical imaging method developed originally for photodetachment near a flat surface can not be directly applied to the present system.

Scaled-energy spectroscopy for alkali-earth rydberg atom in external fields

The alkali-earth Rydberg atom in external fields is theoretically investigated by scaled-energy spectroscopy incorporating energy-dependent quantum defects. Closed-orbit theory is successful in reproducing the experimental observations for hydrogen and hydrogen-like alkali atoms in external fields. However, for the alkali-earth atoms such as barium, even the highly excited Rydberg electron can not be taken as isolated from the spectator valence electron(s) for the electron correlations. Closed-orbit theory cannot give a satisfactory explanation since the electron correlations are angular momentum and energy-dependent. In this paper, we perform a quantum calculation incorporating implicitly the quantum defects varying with energy where the electron correlation information is subsumed. The results recover the experimental observations for barium in magnetic or electric fields very well.

Photodetachment spectrum of hydrogen negative ion near a spherical surface

The semi-classical closed orbit theory is applied to study photodetachment of H− near an elastic spherical surface for a z-polarized laser light. It is assumed that similar to the outgoing detached-electron waves from the source, waves propagate from an image of the source behind the surface. We then calculate the classical action for those trajectories that are perpendicular to the surface. The spherical effects in total photodetachment cross section are controlled by curvature κ of the surface. For zero curvature, our results match with the plane wall case while for a large curvature the results become the asymptotic value of the cross section recently published.

Quantum and semiclassical studies on photodetachment cross sections of H− in a harmonic potential**Project supported by the National Natural Science Foundation of China (Grant Nos. 11421063 and 11474079), the Natural Science Foundation of Shanxi Province, China (Grant No. 2009011004), and the Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi Province, China.

The photodetachment cross section of H− in a linear harmonic oscillator potential is investigated. This system provides a rare example that can be studied analytically by both quantum and semiclassical methods with some approximations. The formulas of the cross section for different laser polarization directions are explicitly derived by both the traditional quantum approach and closed-orbit theory. In the traditional quantum approach, we calculate the cross sections in coordinate representation and momentum representation, and get the same formulas. We compare the quantum formulas with closed-orbit theory formulas, and find that when the detachment electron energy is larger than , where ω is the frequency of the oscillator potential, the quantum results are shown to be in good agreement with the semiclassical results.

Quantum and Closed-Orbit Theory Calculations for the Photodetachment cross Sections of $H^-$ near Two Perpendicular Elastic Planes

The photodetachment cross section of H− near two perpendicular elastic planes is investigated in this paper. Both the traditional quantum approach and closed orbit theory are applied to explicitly derive the formulas of the cross section for different laser polarization direction. We first compared the quantum formulas with closed orbit theory formulas, and then found that the quantum results are shown to be in good agreement with the semiclassical results. Further more, we also found that the cross section depends strongly on the direction of the laser polarization. When the polarization is parallel to the closed orbit, the corresponding oscillation in the cross section is very obvious. However, When the polarization is perpendicular to the closed orbit, the corresponding oscillation is too small for closed-orbit theory formula to describe. PACS: 32.80.Gc, 34.35.+a, 03.65.Ta, 31.15.xg