720
J. Opt. Soc. Am. B/Vol. 4, No. 5/May 1987
S.-I Chu and R. Y. Yin
W
0
Classicaland quantal nonperturbativetreatments of
U,
multiphoton and above-threshold ionization
0
0C.)
Shih-I Chu and R. Y. Yin
Department of Chemistry, University of Kansas,Lawrence, Kansas
66045
k
44
Received November 24, 1986; accepted January 13, 1987
L2 non-Hermitian Floquet studies of intensity- and frequency-dependent
threshold shifts of atomic hydrogen in
intense laser fields are presented. We also performed a Monte
Carlo classical trajectory study of the motion of an
electron under the influence of both the Coulomb and the laser fields.
We found that classicalresults are consistent
with quantum-mechanical predictions with regard to continuum
threshold shift and above-threshold ionization
(ATI).
In addition, our classical treatment reveals detailed mechanisms
responsible for ATI processes.
1.
INTRODUCTION
The phenomena of resonance-enhanced multiphoton ionization (MPI) and above-threshold ionization (ATI) that occur
when atoms are irradiated with intense laser fields have
recently received much attention both experimentally' and
theoretically.2 Quantum-mechanical nonperturbative
study of MPI/ATI of atomic hydrogen has been pursued
recently by Chu and Cooper,3 using the L non-Hermitian
Floquet matrix method.4 In this paper, we extend this work
and present more detailed results regarding the nature of
threshold shifts induced by laser fields. To gain insights
about the actual mechanisms occurring in MPI/ATI processes, we performed
a Monte Carlo classical trajectory
study of the electron motion under the influence of both the
Coulomb and oscillating electric fields. In addition, to confirm the quantum-mechanical prediction of the existence of
the continuum threshold shift due to the ponderomotive
potential, we found that the ATI phenomenon can be induced by two different mechanisms: (1) direct excitation of
the atom from the ground state to the individual continuum
by nonresonant multiphoton absorption and (2) sequential
excitation, in which an electron is first multiphoton excited
to a Rydberg orbit and then followed by absorption of additional photon(s) to the continuum.
In Section 2 we review briefly the L2 non-Hermitian
Flo-
quet theory and present the results for intensity-dependent
threshold shifts. The classical trajectory studies of ATI
processes are presented in Section 3 along with a detailed
discussion of electron dynamics in intense laser fields.
2. L2 NON-HERMITIAN
METHOD
FLOQUET MATRIX
The L non-Hermitian Floquet formulation4 ,5 was recently
extended to the study of intensity-dependent threshold
shifts and above-threshold MPI of atomic hydrogen in intense laser fields.3 The method permits nonperturbative
and self-consistent treatment of intense field effects (in that
all atomic levels are simultaneously shifted and broadened
by the periodic external field) and straightforward inclusion
of free-free transitions
R(r, t) = -(h2 /2m)v 2 - e2 /r + eFz cos t,
(1)
describing the interaction of atomic hydrogen with a monochromatic, linearly polarized, coherent field of frequency co
and peak field strength F, an equivalent time-independent
Hamiltonian FIF(r) may be obtained by an extension of the
semiclassicalFloquet Hamiltonian method of Shirley.6 The
resulting block structure is shown in Fig. 1, where Vl,l's
are
dipole coupling elements and the angular-momentum blocks
S, P D, . . represent the projection of the atomic electronic
Hamiltonian onto states of total L = 0, 1, 2,. . ., etc. Thus,
in the case of atomic hydrogen, the S block consists of the s,
2s, 3s,. . ., ns,. . . bound states and the entire ks Coulomb
continuum. The Hamiltonian of Fig. 1 has no discrete spectrum, and the time evolution is dominated by poles of the
resolvent (E - F)-' near the real axis but on higher Riemann sheets. These complex poles, which correspond to
decaying complex quasi-energy states (QES's), may be
found directly from the analytically continued Floquet
Hamiltonian, 1qF(a), obtained by the dilatation transformation7 r - reia. This transformation effects an analytical
continuation of (E - 1F)-l into the lower half-plane on an
appropriate higher Riemann sheet, allowing the complex
QES to be determined by solution of a non-Hermitian
eigen-
problem. The real parts of the complex eigenvalues of
FIF(a) provide the ac Stark shifts, whereas the imaginary
parts determine directly the total MPI widths (rates). In
practice, the atomic blocks are made discrete by use of a
finite subset of the complete Laguerre basis
2
electronic continua.
Corresponding to the time-dependent Hamiltonian
and the effects of coupling among
rI+le-xrLn2l+2(Xr),
where Xis an adjustable parameter and n
= 0, 1, 2, ....
This yields a Pollaczeck quadrature represen-
tation of the bound and continuum contributions to the
spectral resolution of the hydrogenic Hamiltonian. In general, the convergence of MPI calculation may be achieved to
arbitrary precision by systematically increasing the basis
size and the number of angular momentum blocks.
A.
Intensity-Dependent Threshold Shifts
The ionization potential in intense fields may be defined as
th(F =
0740-3224/87/050720-06$02.00
©1987 Optical Society of America
osc
+
IER(F)I,
(2)
Vol. 4, No. 5/May 1987/J. Opt. Soc. Am. B
S.-I Chu and R. Y. Yin
721
CD
CD
a
HF =
I-
X
O.n
40
V)
S
VIP
0
Vpd
0
0
D
Vdf
0
Mf
0
0
0g
pqP-W
WHERE
o
A:
VdP
o
Vtd
0
o0
F-wI Vfg
0
0-
0
o
0
0
0
0
VP5
0
Vpd
0
0
o
0
0
0
0
o0
Vfd
0
V g
0
0
0
F.
AND
o
Fig. 1.
0
(a.u.)
threshold shift AEth(F) = AEc + AER
Fig. 2. Intensity-dependent
- ER(F) is the ac Stark shift
for w = 0.5 a.u. (Nm = 1). AER = ER(O)
2
2
2 2
is the
of the ground state, while AEc = e F /4mW = e2Frms/2mW
1.0
Frms
of
strength
field
rms
The
continuum threshold upshift.
2
a.u. corresponds to a rms intensity of 7.0 X 1016 W/cm .
Structure of the Floquet Hamiltonian for atomic MPI/ATI.
5.0
where ER(F) (<O) is the field-dependent perturbed groundstate energy obtained from the complex QES calculation and
f0sc
8
2
= e20 /4mw
(0=0.20 a.u.
4.0
(3)
is the average quiver kinetic energy (also known as the ponpicked up by an electron of mass m
potential)
deromotive
and charge e driven sinusoidally by the field. Since in the
V
/
3.
limit of high quantum numbers a Rydberg electron becomes
a free electron, the continuum threshold is shifted up by the
0..h
Electrons traversing a laser beam
amount equal to f
scatter elastically from regions of high light intensity by the
ponderomotive potential.
Thus an electron with energy
2.02.
Cel(F)less than Zos,cannot escape from the Coulomb poten-
~~~~~~~~AE,
Q
tial and is trapped. From Eq. (2), we can define the threshold shift as
AEth(F)= th(F)-th(F
(4)
= ),
0.~
The total energy of the emitted electron in the field can be
written as
2
2 2
Eel(F) = Nhw + ER(F) = e F /4mo +
0
An
PT2/
2
m,
(5)
where N = (Nm + S) is the total number of photons absorbed
by the electron near the atom and Nm is the minimum number of photon energy required to ionize the electron.
Since a
........
/~~~~~~~~~~~~~~,E
1.
where Eth(F = 0), the field-free ionization threshold, is equal
to 0.5 a.u.
n
nr
0.10
F.. (a.u.)
Fig. 3.
3).
Intensity-dependent
threshold shifts for w = 0.2 a.u. (Nm =
722
0
0
U,
¢
0
*^
J. Opt. Soc. Am. B/Vol. 4, No. 5/May 1987
S.-I Chu and R. Y. Yin
free electron cannot absorb or emit photons after leaving the
Coulomb field, the electron has an energy el that is the same
in the laser field as it is at the detector. Thus the ponderomotive potential acts to alter the kinetic energy (PT2/2m)of
the electron from its value outside the laser field to a lower
value inside the laser.
0X
.).
Q
Q
.,4
4-
.0
Figures 2 and 3 show typical examples of intensity- and
frequency-dependent threshold shifts of atomic hydrogen in
intense laser fields, where AER = ER(F = 0) - ER(F) is the
ac
Stark shift, AEc = e 2 F 2 /4mW2 is the continuum threshold
upshift, and Ath(F) is the net threshold shift defined by Eq.
(4) and is equivalent to the sum of AER and AEc. Figure
2
shows the threshold shifts typical of the one-photon (N =
0
1) dominant process (w
0.5 a.u.), while Fig. 3 shows the
typical phenomena for the multiphoton (Nm = 3 in this case)
(bP~
dominant process ( < 0.5 a.u.). Note the marked differ-
ence between the two cases. For N
= 1 (Fig. 2), both the
ground state [ER(F)> ER(O)]and the continua are upshifted,
with the ac Stark shift AERI being greater than AEc. The
resulting net threshold shift Aeth(F)becomes more negative
as the field strength increases. Hence the ionization potential decreases with increasing F. On the other hand, for Nm
> 2, such as the case co= 0.2 a.u. shown in Fig. 3, the
groundstate energy is shifted downward [ER(F) < ER(O)], while AEc
shifts the continuum threshold upward. The result is a
large positive net threshold shift, and the ionization potential increases rapidly with increasing field strength F. As a
general rule, the ponderomotive
potential
F=. (a.u.)
Fig. 4.
tions.
AEc becomes
Generalized one-photon
dominant
ionization
cross sec-
more and more important than the ac Stark shift IAERIas Nm
increases or
decreases.
The consequence is that the ion-
ization potential increases rather rapidly with both F and
Nm. The disappearance of the lowest-energy electrons
in
the MPI/ATI experiment of xenon8 (Nm = 11), for example,
may be attributed to this threshold-shift effect. As will be
shown below, similar phenomena are also observed in our
classical dynamical studies.
B. Intensity-Dependent Generalized Ionization Cross
Sections
The generalized ionization cross section can be defined as
ON = Wi_
/IN_,
(6)
where I is the laser intensity, N is the minimum number of
photon energy required to ionize the electron, and W is the
total ionization rate from the ground state, which is proportional to the total width r1 8 [= -2 Im(Els)] of the perturbed
ground state. In the frequency region w 2 0.5 a.u., the direct
one-photon ionization channel is open for ground-state
atomic hydrogen:
H(ls) + h
H+ + l-.
(7)
In Figure 4 we show the one-photon dominant generalized
cross sections &i as a function of frequency and field
strength. At lower fields (Frms S5 0.01 a.u.), W , cc , and
1
the cross sections &1are intensity independent identically to
the usual photoelectric cross sections calculated using the
exact hydrogenic ls ground and p-wave continuum states.
At higher fields, ATI processes occur, and one might expect
that the cross section will increase.
It is evident from Fig. 4
that, for frequencies near threshold, there is some substantial high field enhancement
of the cross sections.
hancement dies off rapidly at higher frequencies.
This en-
For Nm ' 2 or w < 0.5 a.u.,
Nm near multiphoton
reso-
nances are strongly flux dependent. For two-photon dominant ionization processes, the intensity-dependent &2'shave
been studied in Ref. 4 for the case of linearly polarized light
and in Ref. 9 for the case of circularly polarized light.
C. On the Continuity of Photoionization Cross Section
Across the Field-Free Threshold
Subsection 2.Ashows that the ionization potential of atomic
hydrogen decreases with increasing field strength F. This
raises the followinginteresting question: can atomic hydrogen be ionized by a laser beam with frequency w just below
0.5 a.u. by a one-photon dominant process? Figure 2 indi-
cates that the ionization potential decreases by more than 1
eV at w = 0.5 a.u. and Frms = 0.20 a.u. This is a considerable
effect. Under such circumstances, one might expect to see
&1(w) vary smoothly across the field-free continuum thresh-
old down to the subthreshold
region (w < 0.5 a.u.). Further,
the finite lifetimes (widths) of both ground and excited
states could lead to a smearing of the threshold. More study
is needed to understand this interesting subthreshold behavior.
3.
CLASSICAL TREATMENT
OF
MULTIPHOTON AND ABOVE-THRESHOLD
IONIZATION
In this section we consider the solution of the classical equations of the motion of an electron under the influence of both
the Coulomb and an intense oscillating electric field. In the
absence of a field, the electron moves in an elliptical orbit
with characteristic binding energy and frequency. In the
absence of the Coulomb field, the electron moves uniformly
on average, but superimposed upon this uniform motion is a
723
Vol. 4, No. 5/May 1987/J. Opt. Soc. Am. B
S.-I Chu and R. Y. Yin
-0.40
sinusoidal oscillation with characteristic mean kinetic energy and frequency. In the presence of both the Coulomb and
,_Z.
a
CD
the oscillating electric fields, the electron can gain or lose
energy and undergoes multiphoton absorption and ATI, as
we show below.
The classical Hamiltonian function is derived from the
quantum-mechanical Hamiltonian given in Eq. (1). The
effect of the manner in which fields are turning on and off is
not considered here but will be addressed in a subsequent
publication. Hamilton's equations of motion for the elecusing the variable-order varitron are solved numerically,
0 The initial conditions for the
able-step Adams method.'
motion of the electron at t = 0 were chosen by standard
Monte Carlo methods" from a microcanonical distribution
with energy E = -0.5 a.u. to simulate the statistical distribution characteristic of the initial physical conditions of the
classical hydrogen-atom ground state.
To classify the behavior of classical trajectories, it is convenient to introduce the compensated energy Ec advocated
by Leopold and Percival.11 This compensated energy allows
for the oscillation produced by the field. In the absence of
the Coulomb field, the velocity of a free electron in an oscil-
'11i
as
-0.50
I
1::
ItIt ii III:::
:11:
iI I:
IL
LI
L
0-1
II ii
is defined as
2
E = (m/2)1v 8 + vy2 + [v. - (eF/)sin
2
wt]2 } - e /r.
(9)
C.
L;
n
-0.60
2.5,
.
i
2
3
4
.4
.
7
5
9
710
L..
10
(8)
In the presence of both the Coulomb and the laser fields, Ec
I
Iii
lating field eFR cos wt is
v = vo + (eF/mw)z sin wt.
LL
Time (optical cycle)
Fig. 6. a, Etot(t), and b, r(t), for a bounded (unionized)
Parameters and notations same as in Fig. 5.
trajectory.
When the electron is ionized, the Coulomb potential is weak,
and EC is positive and nearly constant in time.
Figure 5b shows a typical electron trajectory r(t), which
ionizes rapidly within a few optical cycles. The compensated energy EC(t) (solid line) and the total energy of the electron Etot(t) (dashed line) are shown in Fig. 5a. Notice that
EC(t) rapidly approaches a positive constant value Ec once
the electron is detached as postulated. The total energy of
the electron Etot(t) oscillates periodically in time on ionization (r - a). The total mean energy of the electron (averaged over one oscillation) becomes
2
2
Etot = Ec + e2 F /4mw ,
(10)
which is the sum of the compensated energy Ec and the
ponderomotive potential. From Eqs. (5) and (10) we see Ec
=PT-/2m. Thus Ec is equivalent to the mean kinetic energy
of the electron over one oscillation.
Following Eq. (10), we
see that classically an electron cannot be ionized unless
Etot '
a
Eth(C)
2
e2 F 2 /4mW .
(11)
The quantity Eth(C)is the classical threshold shift due to the
presence of the external field. It is interesting to see that
this classical threshold shift is identical to the quantummechanical continuum threshold shift discussed in Section
2. As will be seen below, the classical threshold shift has a
Time (optical cycle)
Fig. 5.
a, Compensated
energy Ec(t) and total electronic energy
t
Etot(t), and b, electron radius trajectory r(t), as a function of time
(measured in number of field optical cycles) for a direct ionization
classical trajectory. The physical parameters used are w = 0.2 a.u.
and F = 0.2 a.u.
dramatic effect on the ATI processes.
Figures 6a and 6b show the behavior of Etot(t) and r(t) for
classical trajectory on an invariant torus that does
typical
a
not ionize. Notice that Etot(t) oscillates quasi-periodically
around the unperturbed ground-state energy and r(t) oscillates around one Bohr radius.
e
04
O
0
724
J. Opt. Soc. Am. B/Vol. 4, No. 5/May 1987
S.-I Chu and R. Y. Yin
0
Figures 7a and 7b show an interesting
U,
0
0
bJ
0
typical case in
which the electron is first pumped by multiphoton absorption to an excited state where EC(t) becomes nearly constant
though negative. This corresponds to an electron's moving
in a large Kepler elliptical orbit with wobbles due to
the
external field. After moving around the orbit for a
few
optical cycles, the electron reaches the perihelion point close
to the nucleus, absorbs additional photon(s), and is suddenly
.).
C.a
ionized.
1_,
4-a
~.F)
Besides the direct (Fig. 5) and the two-step sequential
(Fig. 7) ionization mechanisms, the electrons can also
be
ionized through a multistep sequential mechanism such
as
a
Cd
that shown in Fig. 8. Here an electron is first pumped
by
external fields to an excited orbit, and then it wobbles
around the elliptical orbit for a few optical cycles, similar
to
LL
the case shown in Fig. 7. When the electron comes
close to
U
1
2
3
4
5
6
7
8
the nucleus, it absorbs additional photon energies and jumps
to a highly excited but bounded orbit. There the electron
moves rather slowly around an extended Rydberg orbit for
a
few tens of optical cycles until it reaches again near
the
perihelion point. At that time the electron can absorb a
few
more photons and becomes detached from the nucleus.
Figures 9a-9d depict the classical branching-ratio histogram for ATI corresponding to F(peak field strength) = 0.07,
10
9
0.10, 0.15, and 0.20 a.u., respectively.
Time (optical cycle)
About 500 to 1,000
trajectories were run for each case. The field frequency
is
set to be at co= 0.20 a.u., so that the minimum number
of
photons (Nm)required to ionize the electron is three. P(N)
Fig. 7. a, Ec(t) and Etot(t), and b, r(t), for a two-step
sequentialexcitation-type trajectory, where the electron is first
excited to some
stable Rydberg states and followed by absorption of
additional
photon(s) to the continuum. Parameters and notation
same as in
Fig. 5.
2
3.0
a-
a
2.5
,,
2.0
1.5
Id
ii,,
1911
1.0
-0
2
U
a.
0.0
-
0.5
0
.
5
10
15
20
25
30
35
40
45
0
0
2
aL7
2
a(.
Time (optical cycle)
Fig. 8.
a, EC(t) (solid line) and Etot(t) (dashed line), and b,
r(t), for
a three-step sequential-excitation-type trajectory.
same as in Fig. 5.
Parameters
N number of photons absorbed)
Fig. 9. Classical branching ratio histogram for atomic
hydrogen
ATI. See text for an explanation of notation and symbols.
Vol. 4, No. 5/May 1987/J. Opt. Soc. Am. B
S.-I Chu and R. Y. Yin
is the relative population of the continuum box corresponding to the absorption of N photons from the ground state.
Each continuum box has a width equal to the photon frequency a, with the center of the box indicated by N beneath.
The arrow under each figure denotes the field-free threshold
position. The width of the blackened area measures the ac
Stark shift of the ground state (from quantal calculations),
whereas the width of the hatched area is equal to the classical continuum threshold shift due to the pondermotive potential. Thus an electron with energy less than that at the
right-hand edge (where the compensated energy Ec = 0) of
the hatched area is trapped in fields and cannot escape.
(The choice of the center of the continuum box is somewhat
nonunique as the ac Stark shift concept is not well defined in
classical dynamics.
However, for frequency w well below 0.5
a.u., the pondermotive frequency shift is much larger than
the ac Stark shift of the ground state. Thus, to a good
approximation, the center of the box may be taken simply at
the Nhw position.)
For peak field strength F < 0.07 a.u., classical dynamics
(Fig. 9a) predicts the dominance of the first ionization peak
(box) (N
=
3).12 AtF
=
0.10 a.u. (see Fig. 9b), nearly half of
the first peak is hindered by the threshold upshift, and the
second peak (N = 4) now becomes dominant. At F = 0.15
a.u. (Fig. 9c), the first electron peak is nearly completely
hindered, and the second (N = 4) and the third (N = 5) peaks
are the dominant ones. At further higher field strength, F =
0.20 a.u. (Fig. 9d), the first peak (N = 3) completely disappears, the second peak (N = 4) is partially hindered, and the
third (N = 5), fourth (N = 6), and fifth (N = 7) peaks are all
substantially populated, and so on. These classical predictions are in semiquantitative agreement with the quan3
tum-mechanical results and are consistent with experimenphenomena.l
ATI
tal observation of
In conclusion, classical dynamical treatment provides a
powerful complementary tool to quantum-mechanical
methods for the exploration of detailed physical mechanisms responsible for MPI/ATI processes in intense laser
fields. Extension of the method to complex atoms is in
725
ACKNOWLEDGMENTS
This research was supported in part by the U.S. Department
of Energy (Division of Chemical Sciences).
Acknowledge-
ment is also made of the Donors of the Petroleum Research
Fund, administered by the American Chemical Society, for
partial support of this work. S.-I Chu is grateful to J. Cooper for stimulating discussions on threshold shift and to K.
Wang for the preparation of some of the graphs.
theoretical
chemical physics from Har-
vard University in 1974. After spending
two years as a research associate at Joint
Institute for Laboratory Astrophysics
(JILA) (1974-1976) and two years as a J.
Willard Gibbs Lecturer at the Department
of Physics
of Yale University
(1976-1978), he joined the Department
of Chemistry of the University of Kansas
in 1978. He is currently a professor of
chemistry. In 1985, he was a visiting
fellow at JILA. His research interests
are molecular astrophysics, many-body resonances, atomic and molecular collisions, intense-field multiphoton and nonlinear optical
processes, and classical and quantum chaos. He is a member of
American Physical Society and the American Chemical Society.
O
a
e
0
0
REFERENCES AND NOTES
1. See for example, P. Agostini, F. Fabre, G. Mainfray, G. Petite,
and N. K. Rahman, Phys. Rev. Lett. 42,1127 (1979);P. Kruit, J.
Kimman, and M. J. van der Wiel, J. Phys. B 14, L597 (1981); L.
A. Lompr6, A. L'Huillier, G. Mainfray, and C. Manus, J. Opt.
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man, T. J. McIlrath, and L. F. Dimauro (preprint).
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Phys. B 18, 1927 (1985); Z. Deng and J. H. Eberly, Phys. Rev.
Lett. 53, 1810 (1984); M. H. Mittleman, J. Phys. B 17, L351
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Phys. B 18, L427 (1985); H. G. Muller and A. Tip, Phys. Rev. A
30, 3039 (1984).
3. S.-I Chu and J. Cooper, Phys. Rev. A 32, 2769 (1985).
4. S.-I Chu and W. P. Reinhardt, Phys. Rev. Lett. 39, 1195 (1977);
A. Maquet, S.-I Chu, and W. P. Reinhardt, Phys. Rev. A 27,2946
(1983).
5. S.-I Chu, Adv. At. Mol. Phys. 21, 197 (1985).
6. J. H. Shirley, Phys. Rev. 138, B 979 (1965).
7. B. R. Junker, Adv. At. Mol. Phys. 18, 207 (1982); W. P. Reinhardt, Annu. Rev. Phys. Chem. 33, 223 (1982).
8. P. Kruit, J. Kimman, H. G. Muller, and M. J. van der Wiel,
Phys. Rev. A 28, 248 (1983).
9. S.-I Chu, Chem. Phys. Lett. 54, 367 (1978).
10. G. Hall and J. M. Watt, eds., Modern Numerical Methods for
Ordinary Differential Equations (Clarendon, Oxford, 1976).
11. J. G. Leopold and I. C. Percival, J. Phys. B 12, 709 (1979).
12. For weaker fields, the fraction of classical trajectories leading to
ionization is very small (-1% for F = 0.07 a.u.). Thus a larger
swarm of trajectories may be needed to achieve the convergence
of ATI branching ratio.
R. Y. Yin
Shih-I Chu received the Ph.D. degree in
0o
01
progress.
Shih-I Chu
F
R. Y. Yin graduated from Peking University, Beijing, China, in 1965. He received the M.S. and Ph.D. degrees from
the University of Pittsburgh in 1982and
1986, respectively.
He is currently a re-
search associate at the University of
Kansas.
Dr. Yin is a member of Ameri-
can Physical Society.
M
,nX