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IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 215003 (5pp) doi:10.1088/0953-4075/41/21/215003
Detachment from negative ions by an
electric pulse: from symmetric to fully
asymmetric momentum distribution
D Dimitrovski1and E A Solov’ev2
1Lundbeck Foundation Theoretical Center for Quantum Systems Research, Department of Physics
and Astronomy, University of Aarhus, 8000 Aarhus C, Denmark
2Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna,
Russia
E-mail: esolovev@theor.jinr.ru
Received 13 August 2008, in final form 2 September 2008
Published 27 October 2008
Online at stacks.iop.org/JPhysB/41/215003
Abstract
The momentum distribution of ejected electrons from a negative ion induced by an electric
pulse is calculated. The transformation of the momentum distribution from symmetric to fully
asymmetric form with the increase of the pulse duration τfrom zero to infinity is studied. In
the adiabatic limit (τ →∞), the ionization is determined by the tunneling mechanism. In this
case, an analytic expression for the momentum distribution is derived. A comparison of
numerical calculations with perturbation theory, adiabatic approximation and strong-field
approximation shows good agreement in the regions of their validity.
1. Intriduction
In a previous paper [1], we have studied the ionization of
negative ions and atoms by electric pulses. We have revealed
an effect of zigzag behaviour of the ionization probability
as a function of pulse duration at weak fields. This effect
appears as a result of competition between two mechanisms:
perturbation and tunneling. The typical curve of ionization
probability as a function of pulse duration is shown in figure 1.
For short pulses the probability of ionization rises as a function
of pulse duration, exhibits a maximum and then decreases
to ‘lock’ to the tunneling curve. The zigzag shape is more
strongly pronounced the smaller F0is. However, momentum
distributions of ejected electrons were not considered. In this
paper, we study momentum distributions in the whole range
of pulse duration, from sudden to the adiabatic limit.
We consider detachment from negative ions subject to
unidirectional (half-cycle) electric pulses. For the sake of
definiteness, we choose an electric pulse with the Gaussian
time-dependence
F(t) =F0e−(t/τ )2,(1)
directed along the z-axis. The time-dependent Schrodinger
equation describing this problem is
i∂(r,t)
∂t =−1
2r+V(r)−F(t)z
(r,t), (2)
where the interaction of active electron with neutral core is
described by the zero range potential (ZRP) [2]
V(r)=2π
αδ(r)∂
∂r r. (3)
This potential supports only one bound state having binding
energy α2/2. The potential (3) enables one to scale the
problem (2) by using the transformation t→α2t,r→αr.
Then F
0=F0/α3,τ=α2τand the energy of the bound state
is fixed: E
0=−1/2. In what follows, results are presented
for the case α=1 and for an arbitrary value of αthey can
be obtained using these scaling rules. Equation (2)issolved
using the method presented elsewhere [1]. The momentum
distribution of the ejected electron W(p)is obtained by
the projection of the wave packet of ejected electron onto
plane waves φp(r)=(2π)−3/2eiprat t→∞. Since the
problem has an axial symmetry the momentum distribution in
cylindric coordinates pz,p
ρand pϕdoes not depend on the pϕ
component of momentum: W(p)=W(p
z,p
ρ).
0953-4075/08/215003+05$30.00 1© 2008 IOP Publishing Ltd Printed in the UK
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 215003 D Dimitrovski and E A Solov’ev
10
0
10
1
10
2
τ α
2
[au]
10
-4
10
-3
Ionisation probability
Figure 1. Typical dependence of ionization probability as a function
of pulse duration at (scaled) pulse amplitude F
0=0.1. Full line:
numerical result; dashed line: first-order time-dependent
perturbation theory; and dotted line: tunnelling curve.
2. Perturbation theory
With short and weak pulses, the ionization can be
described within the first-order time-dependent perturbation
approximation. In this approximation, the shape of
the momentum distribution is determined by the dipole
matrix element p|z|0between the bound state φ0(r)=
(α/2π)1/2e−αr /r and the plane wave φp(r)and the amplitude
of the momentum distribution is
aper(p)=−ip|z|0∞
−∞
F(t)exp i(p2+α2)t
2dt. (4)
-2 -1 0 1 2 0
0.5
1
1.5
2
0
2
4
6
W(pz,pρ)×105
pz
pρ
-2 -1 0 1 2 0
0.5
1
1.5
2
0
2
4
6
Wper(pz,pρ)×105
pz
pρ
-2 -1 0 1 2 0 0.5
1
1.5
2
0
1
2
3
W(pz,pρ)×103
pz
pρ
-2 -1 0 1 2 0
0.5
1
1.5
2
0
1
2
3
Wper(pz,pρ)×103
pz
pρ
Figure 2. Momentum distributions of the ejected electron for pulse amplitude F0=0.1 au with pulse durations τ=0.2 (first row) and
τ=2 au (second row); left column: numerical calculation; right column: perturbation theory equation (6).
Since the bound state in ZRP is a s-state, the dipole matrix
element is nonzero for final states of p-type
p|z|0=2√αp cos θ
π(α2+p2)2,(5)
where θis the angle between pand F. After substituting (1)
and (5) into equation (4) the momentum distribution in the
first-order sudden-perturbation theory obtains its final form
Wper(p)=|aper (p)|2=4α(F0τp)2cos2θ
π(α2+p2)4
×exp −(α2+p2)2
8τ2.(6)
The typical shape of the momentum distribution in this region
is given in figure 2: it is a symmetric double-peaked structure
with respect to the origin, i.e. the electron has no preferred
direction, although the electric pulse is unidirectional. This
symmetry is due to the shake mechanism of ionization. The
first row in figure 2demonstrates the perfect agreement
between numerical calculations for the momentum distribution
W(p)and the analytic expression (6). With the increase
of the duration of electric pulse the agreement becomes less
satisfactory and at τ=2 (the second row in figure 2) one can
already observe some departure of the results of the numerical
calculations from the double-peaked structure predicted by the
sudden-perturbation theory.
3. Adiabatic approach
In the adiabatic limit τ→∞, the momentum distribution has
an asymmetric form. The asymmetric momentum distribution
is a consequence of the tunneling mechanism. Namely, the
2
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 215003 D Dimitrovski and E A Solov’ev
electron penetrates through the potential barrier and goes out
as a compact wavepacket in the direction opposite to the
electric field. The adiabatic approximation for the tunneling
momentum distribution can be obtained for the momentum
distribution integrated over the (pρ,p
ϕ)plane normal to the
direction of the electric field,
W(p
z)=2π∞
0
W(p
z,p
ρ)pρdpρ,(7)
as follows. At the moment tthe ejected electron appears at the
external turning point having pz=0 and after that accelerates
by an electric field up to the final value
pz(t) =∞
t
F(t)dt.(8)
It is a monotonically decreasing dependence—with increasing
time tthe component pz(t) decreases. At the moment t,the
probability of tunneling is
Ptun(t ) =1−exp −t
−∞
(F(t)) dt,(9)
where the width of the state (F) has to be calculated
numerically from a transcendental equation as detailed in [1].
Taking into account equations (8) and (9), the momentum
distribution over pzbecomes
Wtun(pz)=dPtun (t)
dtt=t(p
z)
dt(p
z)
dpz=(F(t(pz)))
F(t(p
z))
×exp −t(p
z)
−∞
(F(t)) dt,(10)
where t(p
z)is an inverse function to the function pz(t) defined
by (8). When the field is weak, the width (F) is exponentially
small with respect to 1/F and has a form [3]
(F) =F
2αexp −2α3
3F.(11)
In this case, we can neglect the width in the exponent of
expression (10) and obtain the momentum distribution in the
form
Wtun(pz)=1
2αexp −2α3
3F0
exp t(p
z)
τ2.(12)
The above expression attains the maximum at t(p
z)=0or,
from (8), at
pmax
z=√π
2F0τ. (13)
The comparison between the analytic momentum
distribution (10) and the numerical calculation (7)isgiven
in figure 3. The position of the peak in the momentum
distribution is correctly represented by equation (13). For
longer pulse durations, (see the τ=30 au case in
figure 3), where the tunneling mechanism clearly dominates,
the agreement of the analytic momentum distribution with the
numerical calculation is almost perfect.
4. Strong-field approximation
Beside the sudden-perturbation (τ →0)and adiabatic
approximation (τ →∞)the strong-field approximation
-1 0 1 2 3 4 5
p
z
[au]
0
1
2
3
4
5
6
7
8
9
W (p
z
) × 10
4
τ=30 au
τ=10 au
Figure 3. Momentum distributions over pzfor pulse amplitude
F0=0.1 au with pulse durations τ=10 and τ=30 au; full
line—numerical calculation (7) and dashed line—adiabatic
approximation (10). Positions of the maximum (13) are indicated by
arrows.
(SFA) can be applied which treats, in some sense, the atomic
potential as a perturbation. The transition amplitude from the
ground state to the continuum state of momentum pin the SFA
formalism [4–6]is
Tp0=−i∞
−∞p+A(t ) −A(∞)|z|0F(t)exp[iS(t)]dt,
(14)
where
A(t) =−t
−∞
dtF(t)(15)
is the vector potential and the dipole matrix element that
appears in (14) is equal to the dipole matrix element of
equation (5) with preplaced with p+A(t)−A(∞).The
phase S(t) that appears in equation (14)isgivenby
S(t) =−t
−∞
dt[Ei−Ef(t)].(16)
In the above definition, Eiand Efare instantaneous energies
of the initial and final states, respectively, given by
Ei=−α2
2(17)
and
Ef(t)=1
2(p+A(t)−A(∞))2.(18)
The appearance of the value of the vector potential A(∞)in
equation (14) enables one to include in the analysis the cases
of nonzero value of the vector potential at the end of the pulse
[7]. Namely, with the correction A(∞), the Volkov state goes
over into a usual plane wave when the interaction with the
field is over (in t→∞), as it should be. The total detachment
probability obtained in SFA is compared in figure 4. The SFA
results are obtained by taking modulus square of the transition
amplitude of equation (14) and integrating over all momenta p.
For weak fields the SFA result is in excellent agreement with
the numerical results for the whole interval of pulse durations,
3
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 215003 D Dimitrovski and E A Solov’ev
1 10 100
τ [a.u.]
10
-4
10
-3
10
-2
10
-1
10
0
probability
SFA
numerics
tunnelling
F0=0.08 au
F0=0.1 au
F0=0.2 au
F0=0.5 au
Figure 4. Total detachment probability as a function of the pulse
duration τat constant pulse amplitude F0. Full line with open
circles—numerical result; dotted line—tunneling probability
(adiabatic result) of equation (9) in the limit t→∞; dashed line
with open squares—SFA.
-2 -1 0 1 2 0
0.5
1
1.5
2
0
0.5
1
1.5
W(p
z
,p
ρ
)×10
3
p
z
p
ρ
-2 -1 0 1 2 0
0.5
1
1.5
2
0
0.5
1
1.5
W
SFA
(p
z
,
p
ρ
)×10
3
p
z
p
ρ
-2 -1 0 1 2 0
0.5
1
1.5
2
0
1
2
3
W(p
z
,p
ρ
)×10
3
p
z
p
ρ
-2 -1 0 1 2 0
0.5
1
1.5
2
0
1
2
3
W
SFA
(p
z
,
p
ρ
)×10
3
p
z
p
ρ
-2 -1 0 1 2 0
0.5
1
1.5
2
0
1
2
3
W(p
z
,p
ρ
)×10
3
p
z
p
ρ
-2 -1 0 1 2 0
0.5
1
1.5
2
0
1
2
3
W
SFA
(p
z
,
p
ρ
)×10
3
p
z
p
ρ
Figure 5. Momentum distributions in cylindric coordinates for field amplitude F0=0.1 au and pulse durations τ=1(firstrow),τ=3
(second row) and τ=10 au (third row). First column: numerical results; second column: results within the strong field approximation.
from sudden to the adiabatic limit. However, this accordance
is due to the specific features of the ZRP model. For large pulse
amplitudes F0, the SFA ceases to be valid, overestimating the
detachment probability.
5. Transient region
Finally, we consider momentum distributions in the region of
the pulse duration τwhere there is no clear dominance of the
perturbative or tunneling mechanism. This region begins right
after the local peak in the ionization probability (see figure 1).
The local peak can be almost perfectly reproduced by first-
order time-dependent perturbation theory [1]. In figures 5,
the momentum distributions are shown for three values of
pulse durations τ. With the increase of the pulse duration τ
the momentum distribution changes from almost symmetric
(τ =1)to fully asymmetric distribution (τ =10)through the
transient region (τ =3), where the peak at negative momenta
is eventually suppressed and the minimum between two peaks
disappears. From this figure one can see that the strong-field
4
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 215003 D Dimitrovski and E A Solov’ev
approximation does not reproduce the momentum distribution
in the transient region; instead of the suppression of the peak
at negative momenta this approximation predicts symmetric
merging of two peaks.
Acknowledgments
We are grateful to John Briggs for valuable comments.
This work was partially supported by Heisenber–Landau
Programme.
References
[1] Dimitrovski D and Solov’ev E A 2006 J. Phys. B: At. Mol. Opt.
Phys. 39 895
[2] Demkov Yu N and Ostrovskii V N 1988 Zero Range Potentials
and their Application in Atomic Physics (New York: Plenum)
[3] Demkov Yu N and Drukarev G F 1964 Sov. Phys.–JETP
20 614
[4] Keldysh L V 1965 Sov. Phys.–JETP 20 1307
[5] Faisal F H M 1973 J. Phys. B: At. Mol. Phys. 6L89
[6] Reiss H R 1980 Phys. Rev. A22 1786
[7] Hu S X and Starace A F 2003 Phys. Rev. A68 043407
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