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Influence of the oscillating electric field on the photodetachment of H− ion in a static electric field
Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
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Influence of the oscillating electric field on the photodetachment of H− ion in a static electric field De-hua Wang School of Physics and Optoelectronic Engineering, Ludong University, Yantai 264025, China
a r t i c l e
i n f o
Article history: Received 1 September 2016 Received in revised form 19 October 2016 Accepted 8 December 2016 Available online 9 December 2016 PACS: 32.80.Gc 31.15.xg Keywords: Time-dependent closed orbit theory Oscillating electric field Photodetachment cross section
a b s t r a c t Using the time-dependent closed orbit theory, we study the photodetachment of H− ion in a timedependent electric field. The photodetachment cross section is specifically studied in the presence of a static electric field plus an oscillating electric field. We find that the photodetachment of negative ion in the time-dependent electric field becomes much more complicated than the case in a static electric field. The oscillating electric field can weaken the photodetachment cross section greatly when the strength of the oscillating electric field is less than the static electric field. However, as the strength of the oscillating electric field is larger than the static electric field, four types of closed orbits are identified for the detached electron, which makes the oscillating amplitude in the photodetachment cross section gets increased again. 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 understanding of the connections between quantum and classical description for the time-dependent Hamiltonian systems and may guide the future experimental research for the photodetachment dynamics in the time-dependent electric field. © 2016 Elsevier B.V. All rights reserved.
Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Hamiltonian and the classical motion of the detached electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 The photodetachment cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1. Introduction In the study of the connections between quantum and classical description of simple Hamiltonian systems, photodetachment of negative ions in applied fields provide prototypes for experiment and theory and have attracted much attention in the past several decades [1–11]. The closed orbit theory has been proven to be a powerful tool for the study of the photodetachment dynamics [12]. Most of the experiment and theory to date have dealt with photodetachment of negative ions in static external fields or near surface, such as in a static electric field, in static electric and magnetic fields [3–10], near elastic or metal surfaces, etc [13–17]. In
E-mail address: [email protected] http://dx.doi.org/10.1016/j.elspec.2016.12.003 0368-2048/© 2016 Elsevier B.V. All rights reserved.
such time-independent systems, the energy of the detached electron is conserved and the closed orbits of detached electron can be easily searched out. As to the time dependent systems, the researches are relatively rare. In 2005, Hu and Han et al. used the time-dependent–wave-packet method to calculate the kinetic energy distribution of the D+ ion resulting from the recollision between an electron and its parent ion [18,19]. Spellmeyer, Haggerty and Delos et al. have studied the recurrence spectra of a Ryderg atom in a static electric field plus a weak oscillating field [20,21]. Our group have studied photodetachment of negative ion in a weak oscillating field near a metal surface [22]. In these previous studies, the oscillating electric field is very weak, so they adopt the first order classical perturbation theory to calculate the photoabsorption rate or photodetachment rate of atom or ion. As to the strong oscillating electric field, the first order classical perturbation
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
theory is not suitable. Recently, Yang and Robicheaux extended the standard closed orbit theory to the time-dependent electric field and studied the photodetachment of negative ion in the presence of a single-cycle terahertz pulse [23,24]. Inspired by these studies, our group studied the photodetachment of negative ion in a timedependent gradient electric field [25]. In this paper, by using the time-dependent closed orbit theory, we study the photodetachment of H− ion in a static electric field plus an oscillating electric field. It is found that the photodetachment dynamics is profoundly changed when the external field is oscillating with time. Since the applied electric field is time-dependent, the classical actions of the returning orbits also vary as a function of the time. In addition, an additional term related to the electron returning momentum will appear in the oscillating amplitude of the photodetahcment cross section. The electron returning momentum corresponds to each closed orbit is usually different from the initially outgoing momentum. Therefore, the photodetachment cross section of negative ion in the time-dependent electric field depends both on the energy and the time. Our study suggests that the detached electron’s closed orbit depends on the strength of the oscillating electric field sensitively. When the strength of the oscillating electric field is less than the static electric field strength, only one closed orbit of the detached electron exists and the oscillating electric field can weaken the photodetachment cross section. As the oscillating electric field strength is larger than the static electric field strength, four detached electron’s closed orbits are found and the photodetachment cross section oscillates in a much more complex way. The photodetachment cross section we put forward is universal no matter the oscillating electric field is weaker or stronger than the static electric field. On account of its generality, the time-dependent closed orbit theory provides a clear physical picture for understanding the photodetachment electron dynamics in the time-dependent electric field. This paper is organized as follows: In Section 2, we study the classical motion of the detached electron in a static electric field plus an oscillating electric field. Different types of the detached electron’s closed orbit are identified. In Section 3, on the basis of the time-dependent closed orbit theory, we derive an analytic formula for the instantaneous and average photodetachment cross section of H− ion in the time-dependent electric field. Section 4 gives the calculation of the photodetachment cross section for different oscillating electric field strength. Finally, some conclusions of this paper are given in Section 5. Atomic unit (which is abbreviated as a.u.) is used throughout this work unless indicated otherwise. 2. Hamiltonian and the classical motion of the detached electron Suppose the H− ion sits at the origin, a time-dependent electric field is along the z axis, which is composed of a static electric field plus an oscillating electric field: F = F0 + F1 sin(ωt)
(1)
Where F0 is the static electric field, F1 and ω are the amplitude and frequency of the oscillating electric field, respectively. A zpolarized laser field is applied for the photodetachment, which has the following form[21]: fL (t) = 2FL cos(ωL t),
(2)
where ωL is the frequency of the laser light, which is much larger than the oscillating electric field frequency ω. In atomic unit, the frequency of the laser light ωL is equal to the photon energy. The laser field is also along the +z-axis. For the time-dependent system, we expand the phase space by including t and its conjugate momentum pt as two additional dimensions[24]. Therefore, the
21
Hamiltonian governing the electron motion in the expanded phase space is: H(, z, p , pz , t) =
1 2 1 2 p + pz + (F0 + F1 sin(ωt))z + pt + Vb (r), (3) 2 2
here Vb (r) describes the interaction between the active electron and the hydrogen atom, which is a short-ranged potential and can be neglected after the electron is photo-detached from hydrogen atom by the laser light. We introduce an “evolution time” for the electron traveling in the time-dependent electric field, = t − ti . Here, t denotes the real, laboratory time and ti is the initial outgoing time of the electron trajectory. In addition to the standard Hamiltonian canonical equations, two extra motion equations are added: dt/d = 1 and dpt /d = −∂H/∂t. The initial conditions correspond to Hamiltonian canonical motion equations are: ( = 0) = 0, z( = 0) = 0, p ( = 0) = k0√sin i , pz ( = 0) = k0 cos i , t( = 0) = ti , pt ( = 0) = −E. Here, k0 = 2E denotes the initial momentum of the detached electron, i is the initial outgoing angle of the detached electron relative to the +z axis. By solving the Hamiltonian motion equations, we obtain the classical motion equations of the detached electron:
⎧ (t) = k0 sin i (t − ti ) ⎪ ⎪ ⎪ ⎨
z(t) = [k0 cos i + F0 ti −
⎪ ⎪ ⎪ ⎩ + F1 [sin(ωt) − sin(ωt )]
F1 1 cos(ωti )] · (t − ti ) − F0 (t 2 − ti2 ) . (4) ω 2
i
ω2
The physical picture of the photodetachment of negative ion in the external field can be described as follows [10]: After the negative ion is illuminated by the laser field, a steady stream of outgoing electron waves will produce. These waves propagate along the classical trajectories of the electron. Due to the influence of the electric field, some of the electron trajectories will be returned back to the atom. This kind of the electron trajectory is called the closed orbit. The returning wave will interfere with the outgoing wave, and this interference causes the oscillatory structures in the photodetachment cross section. From Eq. (4), we find the motion of the detached electron along the direction is a free motion, then only the electron trajectory emitting along the z-axis can return back to the origin by the external electric field. Therefore, the initial outgoing angle for the closed orbit of the detached electron is i = 0 or i = . Let z(t) = 0, we have: [k0 cos i + F0 ti −
F1 1 F1 cos(ωti )] · (t − ti ) − F0 (t 2 − ti2 ) + 2 [sin(ωt) ω 2 ω
− sin(ωti )] = 0
(5)
From the above equation, we can obtain the initial time ti and the returning time t of each closed orbit. From calculation, we find if F1 ≤ F0 , there is only one closed orbit of the detached electron, which is similar to the case in the static electric field [10]. However, if F1 > F0 , the whole time-dependent electric field can be positive or negative. Under this condition, four types of detached electron’s closed orbits will appear. For simplicity, we call them up orbit, down orbit, up-down orbit and down-up orbit, respectively. The graphic demonstrations of these four types of closed orbit for the detached electron in the time-dependent electric field are shown in Fig. 1. 3. The photodetachment cross section According to the previous studies, the photodetachment cross section is equal to the rate of the production of detached electron R(t)divided by the photon flux density in the laser field [20]: (E, t) =
4Ep R(t) R(t) . = cE × H/(4Ep ) c4FL2 cos2 (ωL t)
(6)
22
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Fig. 1. Graphic demonstration of the four types of closed orbits for the detached electron in the static electric field plus an oscillating electric field. The static electric field strength F0 = 500 kV/cm. The amplitude and frequency of the oscillating electric field are: F1 = 3F0 , ω = 2.42 × 10-5 a.u. The photon energy Ep = 1.0 eV. (a) the up closed orbit; (b) the down closed orbit; (c) the up-down closed orbit; (d) the down-up closed orbit.
After averaging over a cycle of the laser field,we obtain: (E, t) =
2Ep cFL2
R(t),
the external electric field. Therefore, the whole photodetachment rate can be decomposed into two parts: (7)
R(t) = R0 (t) +
Rj (t)
(10)
j
where c is the speed of the light, Ep = E + Eb is the photon energy. E is the energy of the detached electron, Eb = 0.754eV is the binding energy of H− ion. The photodetachment rate R(t) is defined as [24]: R(t) = −2Im < I(t)| (r, t) >,
(8)
in which I(t) is the detached electron wave source function and (r, t) is the electron wave function at each instant time t. The electron wave source function can be written as [20]: I(t) = FL e−iEt Di ,
(9)
where D is the dipole operator, for the z-polarized laser light, D = Z. − −kb r i is the initial wave function of H ion: i (r) = Be /r, with B =
0.31552 andkb = 2Eb Since I(t)is well localized at a small region around the atom center, the detached electron wave function (r, t) in this region can be split into two types. One is the directly outgoing electron wave dir (r, t) and another is the returning wave ret (r, t)driven back by
The first part R0 (t) describes the photodetachment rate without any electric field, which is a smooth background term: R0 (t) = −2Im < I(t)|
dir (r, t)
>.
(11)
The second part is the oscillating part caused by the external fields. Rj (t) is the overlap integral between the returning electron wave traveling along the j-th closed orbit with the electron wave source function: j ret (r, t)
Rj (t) = −2Im < I(t)|
>,
(12)
j
here, ret (r, t) describes the returning electron wave associated with the j-th closed orbit of the detached electron. The direct outgoing wave dir (r, t)at the instant time t can be written as: dir (r, t)
= FL
in which out (r)
out (r)e out (r)
−iEt
,
(13)
can be written as: (1)
= C(k0 )Ylm (, )k0 h1 (k0 r)
(14)
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C(k0 ) is a complex energy-dependent coefficient [24]:
4 3
C(k0 ) = i
4Bk0 (kb2
(15)
2 2
+ k0 )
Ylm (, )is the spherical harmonic function. For the photodetachment of H− ion, the initial wave function of H− ion is an S state, (1) then the outgoing wave is a p-wave with l = 1, m = 0 [10]. h1 (k0 r) is the outgoing spherical Bessel function [10]. By substituting Eq. (13) and (9) into Eq. (11) and carrying out the overlap integral < Di | out (r) >, the smooth background term in the photodetachment rate can be obtained: R0 (t) =
FL2 k0 |C(k0 )|2 .
(16)
In the following, we aim to calculate the oscillating part Rj (t)in the photodetachment rate. We first draw a small spherical surface of radius R centered at the atomic center. The outgoing wave on this sphere surface is time independent: out (R, i , i )
= C(k0 )
eik0 R Y ( , ) R lm i i
. (17)
After the electron wave emits outwards from this sphere at time ti with initial energy E, it will propagate under the influence of the time-dependent electric field. At the final time t, it will return back to this sphere surface. According to the semiclassical approximation, the returning wave function can be written as [24]: j ret (r, t)
= FL
out (R, i , i )Aj
exp[i(Sj − Eti − j /2)]
(18)
where the subscript j labels the j-th trajectory. Aj is the amplitude, Sj is the action along the j-th trajectory. j is the Maslov index. The amplitude factor Aj is defined as follows: Aj = |
Jj ( = 0) Jj ( = t − ti )
1/2
|
(19)
where J() is the Jacobian at time [22]:
J() = det
∂(, z, t) ∂(ti , i , )
(20)
By substituting the classical motion equations of the detached electron into the Jacobian, after a lengthy derivation, we obtain: Aj =
k0 R | |1/2 k0 (t − ti ) k0 − [F0 + F1 sin(ωti )] cos i (t − ti )
(21)
The classical action Sj along the j-th closed orbit is given as follows:S =
t ti
pdq.
When the electron wave returns back to the small sphere surface around the atomic center, the influence of the external electric field can be neglected. Therefore, the returning wave function can be approximated as a plane wave traveling along j-th closed orbit: j ret (r, t)
where
j = FL e−iEt ˜ ret (r, t)
˜ j (r, t)can ret
˜ j (r, t) ret
=
(22)
be written as:
j C(kret )Nco Ylm (i )e±ikret z
(23)
j
Nco is a matching factor: j
Nco =
Aj R
exp i(S˜ j − j
) 2
(24)
where S˜ j = Sj + E(t − ti ). The symbol “±” in the exponential function is related to the direction of the returning wave. If the returning wave is traveling along the −z axis, we choose “-” in the phase factor; however, for those returning wave traveling along the +z axis, we choose “ + ”.
23
kret is the returning electron momentum, which is usually different from the initial momentum k0 of the detached electron in the time-dependent electric field. j By substituting the returning wave ˜ ret (r, t) and the source wave function I(t) into Eq. (12), after carrying out the overlap intej gral < I(t)| ret (r, t) >, the oscillating part in the photodetachment rate can be derived: Rj (t) = 3gj FL2 C ∗ (k0 )C(kret )
Aj R
exp i(S˜ j − j
) 2
(25)
Where gj is a factor, which is defined as follows:
gj =
+1ifi = ret
(26)
−1ifi = − ret
Therefore, the instantaneous time-dependent photodetachment cross section of H− ion in a static field plus an oscillating electric filed can be written as: (E, t) =
2Ep cFL2
⎛ = 0 × ⎝1 +
R(t)
j
√ 16 2B2 2 E 3/2 3c(Eb +E)3
where 0 = H−
C(kret ) Aj 3gj sin (S˜ j − j ) 2 C(k0 ) k0 R
⎞ ⎠
(27)
is the photodetachment cross section of
ion without any external fields. From the above equation, we find the photodetachment cross section of H− ion in the time-dependent electric field is changed due to the influence of the oscillating electric field, therefore, we average the photodetachment cross section over a cycle of the oscillating electric field to obtain the theoretically predicated average photodetachment cross section, which is only a function of the energy of the detached electron. The average photodetachment cross section is defined as follows: 1 (E) = Trf
Trf
(E, t)dt
(28)
0
Here, Trf = 2/ω is the period of the oscillating electric field. 4. Results and discussions In our calculation, we keep the static electric field strength F0 = 500kV/cm, the frequency of the oscillating electric field ω = 2.42 × 10-5 a.u. Then we study the influence of the oscillating electric field strength on the photodetachment cross section of this system. Firstly, we choose the oscillating electric field strength F1 ≤ F0 , under this condition, the whole electric field is pointing along the +z axis and there is only one closed orbit for the detached electron. The returning time plot for the detached electron’s closed orbit is shown in Fig. 2(a). From this figure, we can get the initial, outgoing time ti and the returning time t for the closed orbit. Fig. 2(b) is the graphic demonstrations of the detached electron’s closed orbit. From this figure, we find when F1 ≤ F0 , the detached electron’s closed orbit is similar to the case in the static electric field. In Fig. 3, we calculate the instantaneous time-dependent photodetachment cross section of H− ion in a static field plus an oscillating electric filed. From this figure, we find oscillatory structures appear in the cross section, which are caused by the interference between the returning electron wave traveling along the closed orbit with the outgoing wave. However, the oscillatory structures are changed greatly with the increase of the oscillating electric field strength. Fig. 3(a) shows the photodetachment cross section with the oscillating electric field strength F1 = 0.05F0 ,
24
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4
(a)
0
i
t (ps)
2
-2 -4 -3.0
z(t)(a.u.)
80
-1.5
0.0
1.5
3.0
(b)
40
0 0.00
0.02
0.04
0.06
t(ps) Fig. 2. (a) Returning time plot for the closed orbit of the detached electron in the static electric field plus a weak oscillating electric field. The static electric field strength F0 = 500 kV/cm. The amplitude and frequency of the oscillating electric field are: F1 = 0.05F0 , ω = 2.42 × 10-5 a.u. The photon energy Ep = 1.0 eV. (b) The closed orbit of the detached electron in the same time-dependent electric field.
Fig. 3. The instantaneous photodetachment cross section for the H− ion in the static electric field plus different oscillating electric field. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The photon energy Ep = 1.0 eV. The oscillating electric field strength is as follows: (a) F1 = 0.05F0 ; (b) F1 = 0.1F0 ; (c) F1 = 0.5F0 ; (d) F1 = 1.0F0 .
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
25
Fig. 4. The average photodetachment cross section of H− ion in a static electric filed plus an oscillating electric filed. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The oscillating electric field strengths are given in each plot. The black line denotes the photodetachment cross section in a static electric filed plus an oscillating electric filed, while the dotted line shows the cross section in a static electric field only.
under this condition, the oscillatory structure in the cross section is relatively simple. With the increase of the oscillating electric field strength, the oscillatory structures become much more complex. The reason can be interpreted as follows: with the increase of the electric field strength, the period of the closed orbit becomes shorter. The interference effect of the returning electron wave with the initial outgoing electron wave becomes significant, which makes the oscillatory structure in the time-dependent cross section becomes complex. In Fig. 4, we calculate the average photodetachment cross section of H− ion in a static electric filed plus an oscillating electric filed under the condition F1 ≤ F0 . In order to see the influence of the oscillating electric filed on the photodetachment cross section clearly, we also plot the photodetachment cross section of H− ion in a static electric field[10], which is denoted by the dotted line. In Fig. 4(a), we calculate the photodetachment cross section with the oscillating electric field strength F1 is far less than F0 , F1 = 0.001F0 . Under this condition, the cross section is nearly the same as only the static electric field exists[10], and the influence of the oscillating electric field can be neglected. The correspondence of our method with the result given by Du for the static electric field case suggests the correctness of our time-dependent theoretical calculation. As we increase the strength of the oscillating electric field, its influence becomes apparent. Fig. 4(b) shows the photodetachment cross section with the oscillating electric field strength F1 = 0.03F0 . We find as the photon energy is less than 1.18ev, the influence of the oscillating electric field on the photodetachment cross section is very
small, but with the increase of the photon energy, the oscillating electric field begins to weaken the photodetachment cross section and the oscillating amplitude in the cross section becomes reduced. As we further increase the oscillating electric field strength, the region of the photon energy that can weaken the cross section becomes enlarged and the oscillating amplitude becomes smaller. For example, in Fig. 4(d), F1 = 0.1F0 , the cross section is weakened as Ep > 0.96ev. As F1 ≥ 0.5F0 , the oscillation in the cross section is nearly disappeared, as we show in Fig. 4(e) and (f). Next, we study the influence of the oscillating electric field on the photodetachment cross section with the oscillating electric field strength F1 > F0 . Under this condition, the whole electric field oscillates with the time, and there are four types closed orbit for the detached electron as we demonstrated in Fig. 1. Fig. 5 shows the returning time plots for these four types of closed orbit with F1 = 3F0 , the photon energy Ep = 1.0 eV. From this figure, we can get the initial, outgoing time ti for each possible closed orbit returning back to the atom center at time t. For the up closed orbit, the initial outgoing time ti can be positive or negative, but for the other three types of closed orbits, the initial outgoing time ti should be negative. Considering the contribution of these four types of closed orbits,we calculate the average photodetachment cross section of H− ion in the time-dependent electric field with F1 = 3F0 . The result is given in Fig. 6. Fig. 6(a) shows the total average photodetachment cross section. Compared to the average photodetachment cross section with the oscillating electric field strength F1 is less than
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D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
Fig. 5. Returning time plot for the four types closed orbit of the detached electron in a static electric filed plus an oscillating electric filed. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The oscillating electric field strength F1 = 3F0 . The photon energy Ep = 1.0 eV.
the static electric field strength F0 , F1 ≤ F0 , we find the oscillating amplitude is increased and the oscillatory structure gets complex. In order to show the influence the oscillating electric field on the photodetachment cross section clearly, we calculate the oscillating part in the total photodetachment cross section, which is shown in Fig. 6(b). From this figure, we find that the oscillating photodetachment cross section exhibits a multi-periodic oscillatory structure, in contrast to the single sinusoidal curve for the case in the static electric field. Fig. 6(c–f) shows the contribution of each kind of closed orbit to the oscillating cross section, respectively. Fig. 6(c) corresponds to the contribution of the up closed orbit to the cross section, it is clearly seen that the oscillating amplitude is very large but the oscillating frequency is relatively small. Fig. 6(d) shows the contribution of the down closed orbit to the cross section, the oscillating amplitude is decreased a little but its oscillating frequency gets increased. Fig. 6(e) is the contribution of the up-down closed orbit to the cross section, we find the oscillating amplitude is further decreased. Fig. 6(f) shows the contribution of the down-up closed orbit to the cross section, we find the oscillating amplitude is very small compared to the contribution of the other three types of closed orbits. From this figure we find that the contribution of the up and down closed orbit to the cross section is significant and play the dominate role, however, the contribution of the downup closed orbit to the cross section is relatively small and can be omitted. Finally, we calculate the oscillating photodetachment cross section for different oscillating electric field strength under the condition F1 > F0 . Fig. 7(a) shows the oscillating cross section with
F1 = 2F0 . Under this condition, due to the influence of the four types of closed orbit, the oscillatory structure in the cross section gets complicated, but the oscillating amplitude is relatively small. With the increase of the oscillating electric field strength, the amplitude in the oscillating cross section gets increased. As we can see from Fig. 7(b–c) clearly. 5. Conclusion In summary, according to the time-dependent closed orbit theory, we have studied the photodetachment of H− ion in a static electric field plus an oscillating electric field. Firstly, by solving the time-dependent Hamiltonian canonical equations of motion, we find out all the closed orbit of the detached electron. It is found that the number of the closed orbit is dependent on the oscillating electric field strength sensitively. When the strength of the oscillating electric field is less than the static electric field, there is only one closed orbit of the detached electron, which is similar to the photodetachment in the static electric field. However, as the strength of the oscillating electric field is larger than the static electric field, there are four different types of closed orbit for the detached electron. Next, we derive an analytical formula for calculating the photodetachement cross section of this system, which can be written as a smooth background term plus many oscillating terms. This formula is universal no matter the oscillating electric field strength is less or larger than the static electric field. Finally, we calculate the photodetachment cross section of this system for different oscillating electric field strength. Our results suggest that
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
(a)
27
(b)
(c)
(d)
(e)
(f)
Fig. 6. (a) The average photodetachment cross section of H− ion in a static electric filed plus an oscillating electric filed. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The oscillating electric field strength F1 = 3F0 . (b) The oscillating part in the total photodetachment cross section of this system. The black line denotes the oscillating cross section in a static electric filed plus an oscillating electric filed, while the dotted line shows the oscillating cross section in a static electric field only. (c) The oscillating cross section induced by the up closed orbit. (d) The oscillating cross section induced by the down closed orbit. (e) The oscillating cross section induced by the up-down closed orbit. (f) The oscillating cross section induced by the down-up closed orbit.
due to the interference effect between the returning electron wave traveling along the closed orbit with the initially outgoing electron wave, oscillatory structures appear in the photodetachment cross section. Compared with the case in the static electric field, the number of the closed orbit for the detached electron in the
time-dependent oscillating electric field is increased, which makes the oscillatory structures in the photodetachment cross section becomes much more complex. It is found when the strength of the oscillating electric field is less than the static electric field, the oscillating electric field can weaken the photodetachment cross section
28
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
Fig. 7. The oscillating photodetachment cross section of H− ion in a static electric filed plus different oscillating electric filed. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The oscillating electric field strength is as follows: (a)F1 = 2F0 ; (b)F1 = 5F0 ; (c)F1 = 8F0 .
greatly. However, as the strength of the oscillating electric field is larger than the static electric field, the oscillating amplitude in the photodetachment cross section gets increased again. In this work, in order to compare our result with the photodetachment of H− ion in the static electric field, we only consider the influence of the z-polarized laser field on the photodetached dynamics of the electrons. As to the influence of the laser light field polarized along other directions, we will discuss in our future work. This study provides a clear and intuitive picture for the photodetachment processes of negative ion in the presence of a time-dependent electric field. We hope that our work will be useful in guiding the future experimental research for the photodetachment dynamics in the time-dependent electric field. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 11374133, and Taishan scholars project of Shandong province (ts2015110055). References [1] H.C. Bryant, A. Mohagheghi, J.E. Stewart, J.B. Donahue, et al., Phys. Rev. Lett. 58 (1987) 2412.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
A.R.P. Rau, Hin-Yiu Wong, Phys. Rev. A 37 (1988) 632. M.L. Du, J.B. Delos, Phys. Rev. A 38 (1988) 5609. M.L. Du, Phys. Rev. A 40 (1989) 1330. A.D. Peters, C. Jaffé, J.B. Delos, Phys. Rev. A 56 (331) (1997). A.D. Peters, C. Jaffé, J.B. Delos, Phys. Rev. Lett. 73 (1994) 2825. A.D. Peters, J.B. Delos, Phys. Rev. A 47 (1993) 3020. A.D. Peters, J.B. Delos, Phys. Rev. A 47 (1993) 3036. Z.Y. Liu, D.H. Wang, Phys. Rev. A 56 (1997) 4605, 56, 2670 (1997). M.L. Du, Phys. Rev. A 70 (2004) 055402. G.C. Yang, J.M. Mao, M.L. Du, Phys. Rev. A 59 (1999) 2053. M. L. Du, J. B. Delos, Phys. Rev. A 38, 1896 (1988); Phys. Rev. A 38, 1913 (1988). G.C. Yang, Y.Z. Zheng, X.X. Chi, Phys. Rev. A 73 (2006) 043413. H.J. Zhao, M.L. Du, Phys. Rev. A 79 (2009) 023408. D.H. Wang, S.S. Li, Y.H. Wang, H.F. Mu, J. Phys. Soc. Jpn. 81 (2012) 114301. T.T. Tang, D.H. Wang, J. Phys. Chem. C (2011) 20529. H.J. Zhao, M.L. Du, Phys. Rev. E 84 (2011) 016217. J. Hu, K.L. Han, G.Z. He, Phys. Rev. Lett. 95 (2005) 123001. T.S. Chu, Y. Zhang, K.L. Han, Int. Rev. Phys. Chem. 25 (2006) 201. N. Spellmeyer, D. Kleppner, M.R. Haggerty, V. Kondratovich, J.B. Delos, J. Gao, Phys. Rev. Lett. 79 (1650) (1997). M.R. Haggerty, J.B. Delos, Phys. Rev. A 61 (2000) 053406. D.H. Wang, S.S. Wang, T.T. Tang, J. Phys. Soc. Jpn. 80 (2011) 094301. B.C. Yang, F. Robicheaux, Phys. Rev. A 92 (2015) 063410. B.C. Yang, F. Robicheaux, Phys. Rev. A 93 (2016) 053413. D.H. Wang, Z.H. Chen, S.H. Cheng, J. Phys. B 49 (2016) 205001.
Contents lists available at ScienceDirect
Journal of Electron Spectroscopy and Related Phenomena journal homepage: www.elsevier.com/locate/elspec
Influence of the oscillating electric field on the photodetachment of H− ion in a static electric field De-hua Wang School of Physics and Optoelectronic Engineering, Ludong University, Yantai 264025, China
a r t i c l e
i n f o
Article history: Received 1 September 2016 Received in revised form 19 October 2016 Accepted 8 December 2016 Available online 9 December 2016 PACS: 32.80.Gc 31.15.xg Keywords: Time-dependent closed orbit theory Oscillating electric field Photodetachment cross section
a b s t r a c t Using the time-dependent closed orbit theory, we study the photodetachment of H− ion in a timedependent electric field. The photodetachment cross section is specifically studied in the presence of a static electric field plus an oscillating electric field. We find that the photodetachment of negative ion in the time-dependent electric field becomes much more complicated than the case in a static electric field. The oscillating electric field can weaken the photodetachment cross section greatly when the strength of the oscillating electric field is less than the static electric field. However, as the strength of the oscillating electric field is larger than the static electric field, four types of closed orbits are identified for the detached electron, which makes the oscillating amplitude in the photodetachment cross section gets increased again. 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 understanding of the connections between quantum and classical description for the time-dependent Hamiltonian systems and may guide the future experimental research for the photodetachment dynamics in the time-dependent electric field. © 2016 Elsevier B.V. All rights reserved.
Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Hamiltonian and the classical motion of the detached electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 The photodetachment cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1. Introduction In the study of the connections between quantum and classical description of simple Hamiltonian systems, photodetachment of negative ions in applied fields provide prototypes for experiment and theory and have attracted much attention in the past several decades [1–11]. The closed orbit theory has been proven to be a powerful tool for the study of the photodetachment dynamics [12]. Most of the experiment and theory to date have dealt with photodetachment of negative ions in static external fields or near surface, such as in a static electric field, in static electric and magnetic fields [3–10], near elastic or metal surfaces, etc [13–17]. In
E-mail address: [email protected] http://dx.doi.org/10.1016/j.elspec.2016.12.003 0368-2048/© 2016 Elsevier B.V. All rights reserved.
such time-independent systems, the energy of the detached electron is conserved and the closed orbits of detached electron can be easily searched out. As to the time dependent systems, the researches are relatively rare. In 2005, Hu and Han et al. used the time-dependent–wave-packet method to calculate the kinetic energy distribution of the D+ ion resulting from the recollision between an electron and its parent ion [18,19]. Spellmeyer, Haggerty and Delos et al. have studied the recurrence spectra of a Ryderg atom in a static electric field plus a weak oscillating field [20,21]. Our group have studied photodetachment of negative ion in a weak oscillating field near a metal surface [22]. In these previous studies, the oscillating electric field is very weak, so they adopt the first order classical perturbation theory to calculate the photoabsorption rate or photodetachment rate of atom or ion. As to the strong oscillating electric field, the first order classical perturbation
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
theory is not suitable. Recently, Yang and Robicheaux extended the standard closed orbit theory to the time-dependent electric field and studied the photodetachment of negative ion in the presence of a single-cycle terahertz pulse [23,24]. Inspired by these studies, our group studied the photodetachment of negative ion in a timedependent gradient electric field [25]. In this paper, by using the time-dependent closed orbit theory, we study the photodetachment of H− ion in a static electric field plus an oscillating electric field. It is found that the photodetachment dynamics is profoundly changed when the external field is oscillating with time. Since the applied electric field is time-dependent, the classical actions of the returning orbits also vary as a function of the time. In addition, an additional term related to the electron returning momentum will appear in the oscillating amplitude of the photodetahcment cross section. The electron returning momentum corresponds to each closed orbit is usually different from the initially outgoing momentum. Therefore, the photodetachment cross section of negative ion in the time-dependent electric field depends both on the energy and the time. Our study suggests that the detached electron’s closed orbit depends on the strength of the oscillating electric field sensitively. When the strength of the oscillating electric field is less than the static electric field strength, only one closed orbit of the detached electron exists and the oscillating electric field can weaken the photodetachment cross section. As the oscillating electric field strength is larger than the static electric field strength, four detached electron’s closed orbits are found and the photodetachment cross section oscillates in a much more complex way. The photodetachment cross section we put forward is universal no matter the oscillating electric field is weaker or stronger than the static electric field. On account of its generality, the time-dependent closed orbit theory provides a clear physical picture for understanding the photodetachment electron dynamics in the time-dependent electric field. This paper is organized as follows: In Section 2, we study the classical motion of the detached electron in a static electric field plus an oscillating electric field. Different types of the detached electron’s closed orbit are identified. In Section 3, on the basis of the time-dependent closed orbit theory, we derive an analytic formula for the instantaneous and average photodetachment cross section of H− ion in the time-dependent electric field. Section 4 gives the calculation of the photodetachment cross section for different oscillating electric field strength. Finally, some conclusions of this paper are given in Section 5. Atomic unit (which is abbreviated as a.u.) is used throughout this work unless indicated otherwise. 2. Hamiltonian and the classical motion of the detached electron Suppose the H− ion sits at the origin, a time-dependent electric field is along the z axis, which is composed of a static electric field plus an oscillating electric field: F = F0 + F1 sin(ωt)
(1)
Where F0 is the static electric field, F1 and ω are the amplitude and frequency of the oscillating electric field, respectively. A zpolarized laser field is applied for the photodetachment, which has the following form[21]: fL (t) = 2FL cos(ωL t),
(2)
where ωL is the frequency of the laser light, which is much larger than the oscillating electric field frequency ω. In atomic unit, the frequency of the laser light ωL is equal to the photon energy. The laser field is also along the +z-axis. For the time-dependent system, we expand the phase space by including t and its conjugate momentum pt as two additional dimensions[24]. Therefore, the
21
Hamiltonian governing the electron motion in the expanded phase space is: H(, z, p , pz , t) =
1 2 1 2 p + pz + (F0 + F1 sin(ωt))z + pt + Vb (r), (3) 2 2
here Vb (r) describes the interaction between the active electron and the hydrogen atom, which is a short-ranged potential and can be neglected after the electron is photo-detached from hydrogen atom by the laser light. We introduce an “evolution time” for the electron traveling in the time-dependent electric field, = t − ti . Here, t denotes the real, laboratory time and ti is the initial outgoing time of the electron trajectory. In addition to the standard Hamiltonian canonical equations, two extra motion equations are added: dt/d = 1 and dpt /d = −∂H/∂t. The initial conditions correspond to Hamiltonian canonical motion equations are: ( = 0) = 0, z( = 0) = 0, p ( = 0) = k0√sin i , pz ( = 0) = k0 cos i , t( = 0) = ti , pt ( = 0) = −E. Here, k0 = 2E denotes the initial momentum of the detached electron, i is the initial outgoing angle of the detached electron relative to the +z axis. By solving the Hamiltonian motion equations, we obtain the classical motion equations of the detached electron:
⎧ (t) = k0 sin i (t − ti ) ⎪ ⎪ ⎪ ⎨
z(t) = [k0 cos i + F0 ti −
⎪ ⎪ ⎪ ⎩ + F1 [sin(ωt) − sin(ωt )]
F1 1 cos(ωti )] · (t − ti ) − F0 (t 2 − ti2 ) . (4) ω 2
i
ω2
The physical picture of the photodetachment of negative ion in the external field can be described as follows [10]: After the negative ion is illuminated by the laser field, a steady stream of outgoing electron waves will produce. These waves propagate along the classical trajectories of the electron. Due to the influence of the electric field, some of the electron trajectories will be returned back to the atom. This kind of the electron trajectory is called the closed orbit. The returning wave will interfere with the outgoing wave, and this interference causes the oscillatory structures in the photodetachment cross section. From Eq. (4), we find the motion of the detached electron along the direction is a free motion, then only the electron trajectory emitting along the z-axis can return back to the origin by the external electric field. Therefore, the initial outgoing angle for the closed orbit of the detached electron is i = 0 or i = . Let z(t) = 0, we have: [k0 cos i + F0 ti −
F1 1 F1 cos(ωti )] · (t − ti ) − F0 (t 2 − ti2 ) + 2 [sin(ωt) ω 2 ω
− sin(ωti )] = 0
(5)
From the above equation, we can obtain the initial time ti and the returning time t of each closed orbit. From calculation, we find if F1 ≤ F0 , there is only one closed orbit of the detached electron, which is similar to the case in the static electric field [10]. However, if F1 > F0 , the whole time-dependent electric field can be positive or negative. Under this condition, four types of detached electron’s closed orbits will appear. For simplicity, we call them up orbit, down orbit, up-down orbit and down-up orbit, respectively. The graphic demonstrations of these four types of closed orbit for the detached electron in the time-dependent electric field are shown in Fig. 1. 3. The photodetachment cross section According to the previous studies, the photodetachment cross section is equal to the rate of the production of detached electron R(t)divided by the photon flux density in the laser field [20]: (E, t) =
4Ep R(t) R(t) . = cE × H/(4Ep ) c4FL2 cos2 (ωL t)
(6)
22
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
Fig. 1. Graphic demonstration of the four types of closed orbits for the detached electron in the static electric field plus an oscillating electric field. The static electric field strength F0 = 500 kV/cm. The amplitude and frequency of the oscillating electric field are: F1 = 3F0 , ω = 2.42 × 10-5 a.u. The photon energy Ep = 1.0 eV. (a) the up closed orbit; (b) the down closed orbit; (c) the up-down closed orbit; (d) the down-up closed orbit.
After averaging over a cycle of the laser field,we obtain: (E, t) =
2Ep cFL2
R(t),
the external electric field. Therefore, the whole photodetachment rate can be decomposed into two parts: (7)
R(t) = R0 (t) +
Rj (t)
(10)
j
where c is the speed of the light, Ep = E + Eb is the photon energy. E is the energy of the detached electron, Eb = 0.754eV is the binding energy of H− ion. The photodetachment rate R(t) is defined as [24]: R(t) = −2Im < I(t)| (r, t) >,
(8)
in which I(t) is the detached electron wave source function and (r, t) is the electron wave function at each instant time t. The electron wave source function can be written as [20]: I(t) = FL e−iEt Di ,
(9)
where D is the dipole operator, for the z-polarized laser light, D = Z. − −kb r i is the initial wave function of H ion: i (r) = Be /r, with B =
0.31552 andkb = 2Eb Since I(t)is well localized at a small region around the atom center, the detached electron wave function (r, t) in this region can be split into two types. One is the directly outgoing electron wave dir (r, t) and another is the returning wave ret (r, t)driven back by
The first part R0 (t) describes the photodetachment rate without any electric field, which is a smooth background term: R0 (t) = −2Im < I(t)|
dir (r, t)
>.
(11)
The second part is the oscillating part caused by the external fields. Rj (t) is the overlap integral between the returning electron wave traveling along the j-th closed orbit with the electron wave source function: j ret (r, t)
Rj (t) = −2Im < I(t)|
>,
(12)
j
here, ret (r, t) describes the returning electron wave associated with the j-th closed orbit of the detached electron. The direct outgoing wave dir (r, t)at the instant time t can be written as: dir (r, t)
= FL
in which out (r)
out (r)e out (r)
−iEt
,
(13)
can be written as: (1)
= C(k0 )Ylm (, )k0 h1 (k0 r)
(14)
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
C(k0 ) is a complex energy-dependent coefficient [24]:
4 3
C(k0 ) = i
4Bk0 (kb2
(15)
2 2
+ k0 )
Ylm (, )is the spherical harmonic function. For the photodetachment of H− ion, the initial wave function of H− ion is an S state, (1) then the outgoing wave is a p-wave with l = 1, m = 0 [10]. h1 (k0 r) is the outgoing spherical Bessel function [10]. By substituting Eq. (13) and (9) into Eq. (11) and carrying out the overlap integral < Di | out (r) >, the smooth background term in the photodetachment rate can be obtained: R0 (t) =
FL2 k0 |C(k0 )|2 .
(16)
In the following, we aim to calculate the oscillating part Rj (t)in the photodetachment rate. We first draw a small spherical surface of radius R centered at the atomic center. The outgoing wave on this sphere surface is time independent: out (R, i , i )
= C(k0 )
eik0 R Y ( , ) R lm i i
. (17)
After the electron wave emits outwards from this sphere at time ti with initial energy E, it will propagate under the influence of the time-dependent electric field. At the final time t, it will return back to this sphere surface. According to the semiclassical approximation, the returning wave function can be written as [24]: j ret (r, t)
= FL
out (R, i , i )Aj
exp[i(Sj − Eti − j /2)]
(18)
where the subscript j labels the j-th trajectory. Aj is the amplitude, Sj is the action along the j-th trajectory. j is the Maslov index. The amplitude factor Aj is defined as follows: Aj = |
Jj ( = 0) Jj ( = t − ti )
1/2
|
(19)
where J() is the Jacobian at time [22]:
J() = det
∂(, z, t) ∂(ti , i , )
(20)
By substituting the classical motion equations of the detached electron into the Jacobian, after a lengthy derivation, we obtain: Aj =
k0 R | |1/2 k0 (t − ti ) k0 − [F0 + F1 sin(ωti )] cos i (t − ti )
(21)
The classical action Sj along the j-th closed orbit is given as follows:S =
t ti
pdq.
When the electron wave returns back to the small sphere surface around the atomic center, the influence of the external electric field can be neglected. Therefore, the returning wave function can be approximated as a plane wave traveling along j-th closed orbit: j ret (r, t)
where
j = FL e−iEt ˜ ret (r, t)
˜ j (r, t)can ret
˜ j (r, t) ret
=
(22)
be written as:
j C(kret )Nco Ylm (i )e±ikret z
(23)
j
Nco is a matching factor: j
Nco =
Aj R
exp i(S˜ j − j
) 2
(24)
where S˜ j = Sj + E(t − ti ). The symbol “±” in the exponential function is related to the direction of the returning wave. If the returning wave is traveling along the −z axis, we choose “-” in the phase factor; however, for those returning wave traveling along the +z axis, we choose “ + ”.
23
kret is the returning electron momentum, which is usually different from the initial momentum k0 of the detached electron in the time-dependent electric field. j By substituting the returning wave ˜ ret (r, t) and the source wave function I(t) into Eq. (12), after carrying out the overlap intej gral < I(t)| ret (r, t) >, the oscillating part in the photodetachment rate can be derived: Rj (t) = 3gj FL2 C ∗ (k0 )C(kret )
Aj R
exp i(S˜ j − j
) 2
(25)
Where gj is a factor, which is defined as follows:
gj =
+1ifi = ret
(26)
−1ifi = − ret
Therefore, the instantaneous time-dependent photodetachment cross section of H− ion in a static field plus an oscillating electric filed can be written as: (E, t) =
2Ep cFL2
⎛ = 0 × ⎝1 +
R(t)
j
√ 16 2B2 2 E 3/2 3c(Eb +E)3
where 0 = H−
C(kret ) Aj 3gj sin (S˜ j − j ) 2 C(k0 ) k0 R
⎞ ⎠
(27)
is the photodetachment cross section of
ion without any external fields. From the above equation, we find the photodetachment cross section of H− ion in the time-dependent electric field is changed due to the influence of the oscillating electric field, therefore, we average the photodetachment cross section over a cycle of the oscillating electric field to obtain the theoretically predicated average photodetachment cross section, which is only a function of the energy of the detached electron. The average photodetachment cross section is defined as follows: 1 (E) = Trf
Trf
(E, t)dt
(28)
0
Here, Trf = 2/ω is the period of the oscillating electric field. 4. Results and discussions In our calculation, we keep the static electric field strength F0 = 500kV/cm, the frequency of the oscillating electric field ω = 2.42 × 10-5 a.u. Then we study the influence of the oscillating electric field strength on the photodetachment cross section of this system. Firstly, we choose the oscillating electric field strength F1 ≤ F0 , under this condition, the whole electric field is pointing along the +z axis and there is only one closed orbit for the detached electron. The returning time plot for the detached electron’s closed orbit is shown in Fig. 2(a). From this figure, we can get the initial, outgoing time ti and the returning time t for the closed orbit. Fig. 2(b) is the graphic demonstrations of the detached electron’s closed orbit. From this figure, we find when F1 ≤ F0 , the detached electron’s closed orbit is similar to the case in the static electric field. In Fig. 3, we calculate the instantaneous time-dependent photodetachment cross section of H− ion in a static field plus an oscillating electric filed. From this figure, we find oscillatory structures appear in the cross section, which are caused by the interference between the returning electron wave traveling along the closed orbit with the outgoing wave. However, the oscillatory structures are changed greatly with the increase of the oscillating electric field strength. Fig. 3(a) shows the photodetachment cross section with the oscillating electric field strength F1 = 0.05F0 ,
24
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
4
(a)
0
i
t (ps)
2
-2 -4 -3.0
z(t)(a.u.)
80
-1.5
0.0
1.5
3.0
(b)
40
0 0.00
0.02
0.04
0.06
t(ps) Fig. 2. (a) Returning time plot for the closed orbit of the detached electron in the static electric field plus a weak oscillating electric field. The static electric field strength F0 = 500 kV/cm. The amplitude and frequency of the oscillating electric field are: F1 = 0.05F0 , ω = 2.42 × 10-5 a.u. The photon energy Ep = 1.0 eV. (b) The closed orbit of the detached electron in the same time-dependent electric field.
Fig. 3. The instantaneous photodetachment cross section for the H− ion in the static electric field plus different oscillating electric field. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The photon energy Ep = 1.0 eV. The oscillating electric field strength is as follows: (a) F1 = 0.05F0 ; (b) F1 = 0.1F0 ; (c) F1 = 0.5F0 ; (d) F1 = 1.0F0 .
D.-h. Wang / Journal of Electron Spectroscopy and Related Phenomena 214 (2017) 20–28
25
Fig. 4. The average photodetachment cross section of H− ion in a static electric filed plus an oscillating electric filed. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The oscillating electric field strengths are given in each plot. The black line denotes the photodetachment cross section in a static electric filed plus an oscillating electric filed, while the dotted line shows the cross section in a static electric field only.
under this condition, the oscillatory structure in the cross section is relatively simple. With the increase of the oscillating electric field strength, the oscillatory structures become much more complex. The reason can be interpreted as follows: with the increase of the electric field strength, the period of the closed orbit becomes shorter. The interference effect of the returning electron wave with the initial outgoing electron wave becomes significant, which makes the oscillatory structure in the time-dependent cross section becomes complex. In Fig. 4, we calculate the average photodetachment cross section of H− ion in a static electric filed plus an oscillating electric filed under the condition F1 ≤ F0 . In order to see the influence of the oscillating electric filed on the photodetachment cross section clearly, we also plot the photodetachment cross section of H− ion in a static electric field[10], which is denoted by the dotted line. In Fig. 4(a), we calculate the photodetachment cross section with the oscillating electric field strength F1 is far less than F0 , F1 = 0.001F0 . Under this condition, the cross section is nearly the same as only the static electric field exists[10], and the influence of the oscillating electric field can be neglected. The correspondence of our method with the result given by Du for the static electric field case suggests the correctness of our time-dependent theoretical calculation. As we increase the strength of the oscillating electric field, its influence becomes apparent. Fig. 4(b) shows the photodetachment cross section with the oscillating electric field strength F1 = 0.03F0 . We find as the photon energy is less than 1.18ev, the influence of the oscillating electric field on the photodetachment cross section is very
small, but with the increase of the photon energy, the oscillating electric field begins to weaken the photodetachment cross section and the oscillating amplitude in the cross section becomes reduced. As we further increase the oscillating electric field strength, the region of the photon energy that can weaken the cross section becomes enlarged and the oscillating amplitude becomes smaller. For example, in Fig. 4(d), F1 = 0.1F0 , the cross section is weakened as Ep > 0.96ev. As F1 ≥ 0.5F0 , the oscillation in the cross section is nearly disappeared, as we show in Fig. 4(e) and (f). Next, we study the influence of the oscillating electric field on the photodetachment cross section with the oscillating electric field strength F1 > F0 . Under this condition, the whole electric field oscillates with the time, and there are four types closed orbit for the detached electron as we demonstrated in Fig. 1. Fig. 5 shows the returning time plots for these four types of closed orbit with F1 = 3F0 , the photon energy Ep = 1.0 eV. From this figure, we can get the initial, outgoing time ti for each possible closed orbit returning back to the atom center at time t. For the up closed orbit, the initial outgoing time ti can be positive or negative, but for the other three types of closed orbits, the initial outgoing time ti should be negative. Considering the contribution of these four types of closed orbits,we calculate the average photodetachment cross section of H− ion in the time-dependent electric field with F1 = 3F0 . The result is given in Fig. 6. Fig. 6(a) shows the total average photodetachment cross section. Compared to the average photodetachment cross section with the oscillating electric field strength F1 is less than
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Fig. 5. Returning time plot for the four types closed orbit of the detached electron in a static electric filed plus an oscillating electric filed. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The oscillating electric field strength F1 = 3F0 . The photon energy Ep = 1.0 eV.
the static electric field strength F0 , F1 ≤ F0 , we find the oscillating amplitude is increased and the oscillatory structure gets complex. In order to show the influence the oscillating electric field on the photodetachment cross section clearly, we calculate the oscillating part in the total photodetachment cross section, which is shown in Fig. 6(b). From this figure, we find that the oscillating photodetachment cross section exhibits a multi-periodic oscillatory structure, in contrast to the single sinusoidal curve for the case in the static electric field. Fig. 6(c–f) shows the contribution of each kind of closed orbit to the oscillating cross section, respectively. Fig. 6(c) corresponds to the contribution of the up closed orbit to the cross section, it is clearly seen that the oscillating amplitude is very large but the oscillating frequency is relatively small. Fig. 6(d) shows the contribution of the down closed orbit to the cross section, the oscillating amplitude is decreased a little but its oscillating frequency gets increased. Fig. 6(e) is the contribution of the up-down closed orbit to the cross section, we find the oscillating amplitude is further decreased. Fig. 6(f) shows the contribution of the down-up closed orbit to the cross section, we find the oscillating amplitude is very small compared to the contribution of the other three types of closed orbits. From this figure we find that the contribution of the up and down closed orbit to the cross section is significant and play the dominate role, however, the contribution of the downup closed orbit to the cross section is relatively small and can be omitted. Finally, we calculate the oscillating photodetachment cross section for different oscillating electric field strength under the condition F1 > F0 . Fig. 7(a) shows the oscillating cross section with
F1 = 2F0 . Under this condition, due to the influence of the four types of closed orbit, the oscillatory structure in the cross section gets complicated, but the oscillating amplitude is relatively small. With the increase of the oscillating electric field strength, the amplitude in the oscillating cross section gets increased. As we can see from Fig. 7(b–c) clearly. 5. Conclusion In summary, according to the time-dependent closed orbit theory, we have studied the photodetachment of H− ion in a static electric field plus an oscillating electric field. Firstly, by solving the time-dependent Hamiltonian canonical equations of motion, we find out all the closed orbit of the detached electron. It is found that the number of the closed orbit is dependent on the oscillating electric field strength sensitively. When the strength of the oscillating electric field is less than the static electric field, there is only one closed orbit of the detached electron, which is similar to the photodetachment in the static electric field. However, as the strength of the oscillating electric field is larger than the static electric field, there are four different types of closed orbit for the detached electron. Next, we derive an analytical formula for calculating the photodetachement cross section of this system, which can be written as a smooth background term plus many oscillating terms. This formula is universal no matter the oscillating electric field strength is less or larger than the static electric field. Finally, we calculate the photodetachment cross section of this system for different oscillating electric field strength. Our results suggest that
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(a)
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(b)
(c)
(d)
(e)
(f)
Fig. 6. (a) The average photodetachment cross section of H− ion in a static electric filed plus an oscillating electric filed. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The oscillating electric field strength F1 = 3F0 . (b) The oscillating part in the total photodetachment cross section of this system. The black line denotes the oscillating cross section in a static electric filed plus an oscillating electric filed, while the dotted line shows the oscillating cross section in a static electric field only. (c) The oscillating cross section induced by the up closed orbit. (d) The oscillating cross section induced by the down closed orbit. (e) The oscillating cross section induced by the up-down closed orbit. (f) The oscillating cross section induced by the down-up closed orbit.
due to the interference effect between the returning electron wave traveling along the closed orbit with the initially outgoing electron wave, oscillatory structures appear in the photodetachment cross section. Compared with the case in the static electric field, the number of the closed orbit for the detached electron in the
time-dependent oscillating electric field is increased, which makes the oscillatory structures in the photodetachment cross section becomes much more complex. It is found when the strength of the oscillating electric field is less than the static electric field, the oscillating electric field can weaken the photodetachment cross section
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Fig. 7. The oscillating photodetachment cross section of H− ion in a static electric filed plus different oscillating electric filed. The static electric field strength F0 = 500 kV/cm. The frequency of the oscillating electric field is ω = 2.42 × 10-5 a.u. The oscillating electric field strength is as follows: (a)F1 = 2F0 ; (b)F1 = 5F0 ; (c)F1 = 8F0 .
greatly. However, as the strength of the oscillating electric field is larger than the static electric field, the oscillating amplitude in the photodetachment cross section gets increased again. In this work, in order to compare our result with the photodetachment of H− ion in the static electric field, we only consider the influence of the z-polarized laser field on the photodetached dynamics of the electrons. As to the influence of the laser light field polarized along other directions, we will discuss in our future work. This study provides a clear and intuitive picture for the photodetachment processes of negative ion in the presence of a time-dependent electric field. We hope that our work will be useful in guiding the future experimental research for the photodetachment dynamics in the time-dependent electric field. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 11374133, and Taishan scholars project of Shandong province (ts2015110055). References [1] H.C. Bryant, A. Mohagheghi, J.E. Stewart, J.B. Donahue, et al., Phys. Rev. Lett. 58 (1987) 2412.
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