PhysRevA 49 2117
PhysRevA 49 2117
PhysRevA 49 2117
ate kinetic energy to produce a given harmonic at the motivation of the present work was to find an interme-
time of return. This classical interpretation shows that diate approximate solution to the problem of harmonic
in order to control harmonic-generation processes, one generation valid in a low-&equency high-intensity regime
should try to control the motion of free electrons in the that would provide a link between the methods of Refs.
laser field. Shaping appropriately electron trajectories [6] and [8,9], on one hand, and would allow one to study
might allow for various fascinating applications, includ- the effects of multicolor or elliptically polarized light on
ing, for example, the generation of subfemtosecond high- harmonic generation, on the other hand.
frequency pulses [12]. Recently, several authors [20—22] have emphasized the
In a recent paper [7], we have discussed the harmonic- importance of bound states in harmonic generation on
generation cutoff in a low-&equency high-intensity the formation of the plateau (see also [23,24]). Other
regime, presenting experimental data and calculations in- cutoff laws, difFerent from Eq. (1), e.g. , involving Rabi
volving the response of a single-atom and of the macro- frequencies, have been derived [21,22]. The present the-
scopic medium. Systematic measurements of the har- ory applies to a regime of parameters where bound states
monic generation yields in neon have been performed us- should be relatively unimportant: it is valid in a low-
ing a short-pulse low-&equency laser. The experimental frequency, high intensity limit (U„) I„), and for high
cutofF energy was found to be approximately Ip + 2Up, harmonics, with energy higher than, say, the ionization
therefore lower than that predicted in single-atom the- energy. It is well adapted to a discussion of the cut-
ories [9,6, 13]. A simple, analytic, and fully quantum- off location, but it obviously cannot describe well those
mechanical theory of harmonic generation valid in the low-order harmonics below or just above the ionization
tunneling limit has been formulated. It agrees well with limit. On the other hand, it includes important quantum-
the predictions of other single-atom theories and in par- mechanical effects such as quantum diffusion, quantum
ticular with the cutoff law. It recovers the semiclas- interferences, depletion of the ground state which are not
sical interpretation of harmonic generation in this low- included in the classical approaches [8,9,25] and it allows
&equency regime. The difference between the prediction us to get a better physical understanding of harmonic
for the single-atom cutoff and the experimental results generation in the tunneling limit.
have been explained by accounting for the inQuence of The plan of the paper is the following. Section II
propagation effects [14] in a tight focusing geometry [7]. contains a general presentation of our theory and a dis-
A good agreement between theory and experiment has cussion of its quasiclassical interpretation. We present
been obtained. general expressions for harmonic spectra and harmonic
In the present paper, we give a full account of the the- strengths. Section III is divided into several subsections
oretical part of Ref. [7] which deals with the response of in which we describe applications of our theory to vari-
a single atom. Propagation effects will be discussed in ous model potentials. In particular, we study the case of
a future paper. We give a detailed formulation of the a transition &om a 1s state to a continuum for a Gaus-
theory with, in particular, a discussion of its range of va- sian model and also for hydrogenlike atoms. We address
lidity. We study different potentials and we investigate the question of electron rescattering, i.e. , the process in
the inBuence of the various relevant parameters. One which the electron returning to the nucleus is scattered
could argue that the fully quantum and exact theory of away from it instead of being recombined. In Sec. IV, we
HG has been already formulated in terms of the solu- reexamine our theory using the saddle-point technique.
tion of the time-dependent Schrodinger equation (TDSE) The most important result of this section is the deriva-
[6,15—18] or, equivalently, in terms of the solution of the tion of the quantum-mechanical cutoff law. We find that
time-independent Floquet equations [19]. On the other it actually differs &om the phenomenological expression
hand, the simple quasiclassical model of Refs. [8,9] al- I„+3.17U„and has a form 3 17U~+I„F(I„/. U„) where the
lows for a physical interpretation of the results of Ref. factor F(I~/U„) is equal to 1.3 for I~ (( U~ and decreases
[6]. Therefore, at 6rst sight, there is no need for any slowly towards 1 as I~ grows. In Sec. V, we discuss the
approximate theory that would link the two mentioned efFects of saturation and depletion of the atomic ground
approaches. However, the TDSE method requires a lot state. Finally, Sec. VI contains the conclusions, whereas
of computer time. In particular, although it works well the Appendixes A and B are devoted to some more tech-
for linearly polarized laser fields, it cannot be easily ex- nically involved derivations of the formulas used in the
tended, for the time being, to elliptically polarized fields main body of the paper.
or nonmonochromatic fields with a time-dependent po-
larization. In contrast, the present theory can easily de-
scribe the interaction with laser light of arbitrary po- II. THEORY OF HARMONIC GENERATION
larization and frequency content, allowing one to study
strong-field coherent control of high-harmonic emission,
for example in the field of two lasers with related fre- The theoretical problem that we attempt to solve is the
quencies. On the other hand, the semiclassical approach same as in the case of the TDSE method. We consider
used in Refs. [8,9] mixes classical and quantum argu- an atom (or an ion) in a single-electron approximation
ments (first quantum tunneling, then classical motion, under the influence of the laser field E cos (t) of linear
then quantum recombination). It does not account for polarization in the x direction (we use atomic units, but
many important quantum effects, such as quantum dif- express all energies in terms of the photon energy). In the
fusion of wave packets, quantum interferences, etc. The length gauge, the Schrodinger equation takes the form
THEORY OF HIGH-HARMONIC GENERATION SY I.O%-. . .
&ee particle moving in the electric field with no effect of where a(t) 1 is the ground-state amplitude, and
V(x). b(v, t) are the amplitudes of the corresponding contin-
Assumption (b) can be used only for intensities smaller uum states. We have factored out here &ee oscillations
then saturation intensity. Otherwise, the depletion of the of the ground-state amplitude with the bare frequency
ground state has to be taken into account, as discussed I„. The Schrodinger equation for b(v, t) reads as
in Sec. V. Assumption (c) is non-questionable for short-
range potentials, but is also valid for hydrogenlike atoms, b(v, t) = .
i /
—+Ip
fv'
E2 /
b(v, t)—
provided U„ is large enough. It is important, however,
to be aware of what is the regime of validity of the as- Bb(v,' t)—
sumptions (a) and (c). Generally speaking, they hold
Ecos(t) +iEcos(t) d (v). (4)
Bv~
when there are no intermediate resonances and when the Here d(v) =
(v[x~0) denotes the atomic dipole matrix
Keldysh parameter p = QIp/2Up is smaller then one, i.e. , element for the bound-free transition and d (v) is the
in the tunnebng or over-the-barrier ionization regimes. component parallel to the polarization axis. In writing
The latter condition requires Ip & 2Up and implies that Eq. (4) we have neglected the depletion of the ground
(i) when the electron appears in the continuum it is under state, setting a(t) = 1 on the right-hand side. The whole
the inHuence of a very strong laser field, and (ii) when information about the atom is thus reduced to the form
it comes back to the nucleus it has a large kinetic en- of d(v), and its complex conjugate d'(v).
ergy, so that the atomic potential force can be neglected. The Schrodinger Eq. (4) can be solved exactly and
Obviously, the latter implication concerns only highly en- b(v, t) can be written in the closed form,
ergetic electrons, responsible for the production of har- t
monics of order 2M + 1 & I„. b(v, t) = i dt'E cos(t') d (v + A(t) —A(t') )
There are several theoretical approaches that incorpo- t
rate assumption (c) in solving Eq. (2). Ammosov et aL xexp —i dt" ~+ A t —A t" 2+ Ip
study the "classical" dynamics of the electron with time gl
where A(t) = ( —Esin(t), 0, 0) is the vector potential of (8) is to interpret it as a Landau-Dyhne formula for tran-
the laser field. sition probabilities applied to the evaluation of the ob-
In order to calculate the x component of the time- servable z [33]. Finally, note that Eq. (8) is evidently
dependent dipole moment, we have to evaluate z(t) gauge invariant.
(@(t)~z~tI)'(t)). Using Eqs. (3) and (5) we obtain The expression (8) can be easily generalized to the case
of laser fields E(t) of arbitrary polarization and temporal
shape. If we want to evaluate the component of the time-
z(t) =
f d vd (v')b(vt)+, cc. (6)
dependent dipole moment along the direction n, where n
is an unit vector, the result is
In writing the above formula, we have neglected the con-
tribution from the C-C part [32], i.e. , we have considered
p n d*(p —A(t))
only the transitions back to the ground state. Introduc-
z„(t) = i
0 f
dt' d
ing a new variable which is a canonical momentum x E(t') d(p —A(t')) exp [—iS(p, t, t')] + c.c.,
p = v+ A(t) (10)
we get the final expression In the present work, we shall restrict ourselves to the
simple case of the linearly polarized monochromatic field,
t with the time-dependent dipole moment given by Eq.
z(t) = i dt' d pEcos(t')d (p —A(t')) (8). The dipole matrix elements that enter Eq. (8)
0
x d* (p —A(t) ) exp [—iS(p, t, t')] + c.c., change typically on a scale of the order of p On I„.
(8) the other hand, the quasiclassical action (9) changes on
where a characteristic scale p2 I/(t —t'), due to quantum
diffusion effects. For t —t' of the order of one period of
the laser field the quasiclassical action varies thus much
S( t tt) dttt
(
[P
2
( )]
+I (9) faster than the other factors entering Eq. (8). Therefore,
the major contribution to the integral over p in Eq. (8)
comes &om the stationary points of the classical action,
Equation (8) has a nice physical interpretation [33]
as a sum of probability amplitudes corresponding to V, S(p, t, t') = 0.
the following processes: The first term in the integral,
Ecos(t')d (p —A(t')), is the probability amplitude for On the other hand, V'~S(p, t, t') is nothing else but the
an electron to make the transition to the continuum at difference between the position of the free electron at
time t' with the canonical momentum p. The electronic time t and time t',
wave function is then propagated until the time t and
acquires a phase factor equal to exp[ — iS(p, t, t')], where (12)
S(p, t, t') is the quasiclassical action. The effects of the Therefore we conclude that the stationary points of the
atomic potential are assumed to be small between t' and classical action correspond to those momenta p for which
t, so that S(p, t, t') actually describes the motion of an the electron born at time t' returns to the the same po-
electron &eely moving in the laser field with a constant sition at time t. It is also evident that x(t) must be
momentum p. Note, however, that S(p, t, t') does in- close to the origin, because it is the only position where
corporate some effects of the binding potential through the transitions to the ground state (and from the ground
its dependence on Iz. The electron recombines at time state) can possibly occur. Mathematically, this state-
t with an amplitude equal to d*(p —A(t)), which gives ment follows &om the fact that the Fourier transforms
the last factor entering Eq. (8). of d (p —A(t')) and d*(p —A(t)) are localized around
Strictly speaking, Eq. (7) defines the canonical mo- the nucleus on a scale comparable to ao, where ao is the
mentum at time t, which does not have to be the same Bohr radius.
as p at t'. Note, however, that between t' and t, p is The physical meaning of the mathematical result ex-
a conserved quantity, due to neglection of the effects of pressed by Eq. (12) is clear: the dominant contribution
V(x). For this reason, our interpretation is correct, since to the harmonic emission comes from the electrons which
we can identify p equally well as a canonical momen- tunnel away &om the nucleus but then reencounter it
tum at t' or t It is worth st. ressing that Eq. (8) allows while oscillating in the laser Geld. Thus, our quantum
also for an alternative interpretation in which the elec- theory justifies one of the basic assumptions of the semi-
tron appears in the continuum at time t with the kinetic classical model of Refs. [8,9].
momentum p —A(t), is then propagated back until t', According to the above discussion, the integral over p
and recombines back to the ground state ~0) with the might be performed using a saddle-point method. The
amplitude E cos(t')d (p —A(t')). This interpretation is result is
not as intuitive as the previous one, but, owing to the
invariance of the problem with respect to time reversal, tzct 3/2
d'(p, q(t, 7) —A (t))
p q
is equally correct. This is an example of a situation in z(t) = i d7 ~
) e+z7 2) ~
p
which approximate versions of the ordinary S matrix and
time-reversed S matrix approaches give the same results x d (p.~ (t, r) —A (t —7)) E cos (t — T).
(compare with Ref. [29]). Another way of looking at Eq. x exp[ —iS,), (t, r)] + c.c. (13)
49 THEORY OF HIGH-HARMONIC GENERATION BY LOW-. . . 2121
In writing Eq. (13), we have introduced a new variable IV. However, it is worth mentioning here that Eq. (13)
(the return time) r = t t— ' and extended integration over shows a further relation between our quantum theory and
it to oo. The stationary value of the xth component of the semiclassical model of Refs. [8,9). Indeed, in the limit
the momentum, p, q(t, r), allows the electron trajectory Ip Up the saddle point of the integral over w in Eq.
starting near the origin at t —w to return to the same (13) tends to the stationary point r = r, t of the classical
position at t. It is equal to action S,t(t, r) = f, dt" (p, t —A(t")) /2 [Eq. (15)].
One can easily see that this point corresponds to the zero
p, t,. (t, r) = E[cos(t) —cos(t —r)]/r. (14) value of the initial velocity, v(t —r) = p, (t, r) —A(t —r) =
q
Owing to the properties of the dipole matrix elements A(t —r)) /2. The dashed line in Fig. 1 represents this
due to the central symmetry of the atomic potential, all gain in kinetic energy, calculated by using the classical
even Fourier components on the right-hand side of the Newton equation for the case when the electron appears
above expression vanish. Typically bM(7) decreases quite in the continuum with zero initial energy at the moment
rapidly with ~M~ (for instance, in the case of the broad t —7 and revisits the nucleus at t. The two curves, classi-
Gaussian model discussed in Sec. III, bM's are nonzero cal b, Eg;o/U„and quantum 2~C(r) (resulting from inte-
= 0, +1, —2 only).
~
~
).
oo 3/2 3, 5
*21c+i = i
( y 3.17
e+ir 2) 3.0
0
which is shown in Fig. 1 (solid line). 2C(r) determines 0 10 15 20 25 30
the variation of S,t(t, r) as a function of t. Since the f
action is the integral of the kinetic energy Eg;„(t") plus FIG. 1. The function 2~C(r) (solid line) and the ki-
I~ over t", the maxima of 2C(r) correspond therefore to
~
As we see &om Fig. 1, both functions have several turning to the nucleus is scattered away instead of being
maxima. They correspond to trajectories of the electron recombined. Section IIIF discusses the efBciency of har-
that contain one, two, or more returns to the vicinity of monic emission for various laser frequencies.
the nucleus (with the last one at t) T. he first maximum
appears at r, „z 4.08 and 2~C(r, „z) = 3.17. The follow-
~
ing maxima are between 2 and 2.4. Since the Bessel func- A. Gaussian model
tion Jic(z) becomes exponentially small when K
from Eq. (18) we conclude that the absolute cutoff of HG
)
z,
We assume that the ground-state s-wave function can
for U„)) I„ is at JZ/ „= 2K + 1 2~C(r, „z)~U„ be written in the form
3.17Up. In the range of 2 4Up & 2K & 3 17Up only the
contributions from the trajectories that contain one re- (19)
turn are relevant. As we shaB see in Sec. IV, there are
typically two such trajectories that correspond to two dif- where o, is a parameter of the order of I„.
There are sev-
ferent values of 7. When 2K becomes smaller than 2Up eral models for which Eq. (19) applies. They all, how-
more and more maxima of 2~C(r) contribute, giving rise
~
ever, correspond to short-range potentials that describe
to a more complex interference structure. negative ions. First of aB, we can consider a truncated
Physically, the quantum interference results from the harmonic-oscillator potential
existence of more than one possible trajectory that leads
to the same 6nal state. In particular, it becomes more X —P,
V{x) = (20)
and more complex as the number of trajectories returning
to the nucleus several times grows. Obvioulsy, when I„ is
comparable to Up, the eKects of the atomic potential on for ~x~ & /2P/n2 and V(x) = 0 otherwise. Here P
these secondary returns can be very signi6cant. Electron is another parameter of the order of I~. If P is large
rescattering will dramatically afFect the phases accumu- enough, the ground-state wave function for a potential
lated along those trajectories and will destroy the inter- (20) takes approximately the form (19). The ground-
ference patterns due to multiple returns. Note, however, state energy is then I„=
—P—+3n/2. It is worth stress-
that even if multiple returns are not possible because of ing, however, that Gaussian wave functions approximate
scattering on the atomic potential, due to the existence well ground states of other short-range potentials, such
of two trajectories with a single return the quantum in- as the screened Coulomb potential discussed in Ref. [34].
terference between them is unavoidable. Since we are interested in transitions to and from
highly energetic states in the continuum, it is safe to as-
sume that electronic wave functions in the continuum can
III. HARMONIC SPECTRA FOR VARIOUS be described as plane waves. The dipole matrix element
ATOMIC POTENTIALS takes also a Gaussian form,
)s/ 2()R+i ~ /
z2K+i —/U dr
/
.
~
exp[ —iF~(r)]
(
a2
(zrnj [
o
]
(1 n+zr 2) )
where the functions B(r), C(7), D(r), and F~(r) are B. Gaussian model and saddle-point technique
given in Appendix A. Note that Eq. (22) describes z2Ic+i When n is large enough (which, in fact, is usually
in the units in which the laser frequency is one. In order the case) we expect that the saddle-point integration
to express harmonic strengths in atomic units one has to over p should give a reasonable approximation to the
multiply z2a+i by the factor QI„/IFo~, where Eo is the expressions (22) and (Al) —(A4). In fact, the harmonic
ground-state energy in a.u. strengths when calculated with this method are given by
The above results are now compared with approxi- the same expression as Eq. {22) except that this time the
mated formulas that make use of the saddle-point in- functions B(r), C(r), D(r), and F~(r) take a different
tegration over p. form (see Appendix A).
49 THEORY OF HIGH-HARMONIC GENERATION BY LOW-. .. 2123
11
Finally, it is interesting to note that in the limit of large 10
terms. The result for the harmonic strength again has the
form (22) and the functions B(7.) and C(7 ) are the same
10 13
as the ones defined in Eqs. (A6) and (A7), respectively.
The other functions can be found in Appendix A.
10-14
The expression (22) in the broad Gaussian limit have
been used by us in Ref. [7]. It is interesting to compare
the results from the three approaches discussed above 10
0 20 40 60 80 100
which we denote by GEX (for the exact Gaussian model), Harmonic order 2M+1
GSP (for the Gaussian model with saddle-point approx-
imation), and GBR (for the broad Gaussian limit). In FIG. 2. Comparison of harmonic spectra obtained with
Fig. 2, we present spectra obtained &om the three corre- GEX (open squares), GSP (stars), and GBR (black squares)
sponding sets of expressions. These models give similiar methods; I~ = 13.6, U„= 20, a = 2I„.
results. For moderate harmonic orders (in the middle
of the plateau), GBR, GSP, and GEX are practically
indistinguishable. For high harmonic orders, GSP and D. Hydrogenlike atoms
GEX are indistinguishable. Both of them give results
smaller than GBR, since they account for the energy de- It is a little more difficult to calculate harmonic spectra
pendence of the dipole matrix element (21) that falls ofF for hydrogenlike atoms. In this case, the ground state s-
slowly as the energy increases. GBR, however, produces wave function takes the form
the same kind of spectra as the other two, except for a
global shift towards higher harmonic strengths by a fac- (23)
tor slowly changing with the harmonic order.
This is a very general property of our theory. In any
of its realizations (GBR, GSP, or GEX), it produces the where this time a = 2I„. Again, since we consider transi-
same spectral shape except for a slowly varying energy tions to and from the highly energetic continuum states,
dependent factor. The same is also true for individual it is legimate to treat continuum states as plane waves,
harmonic strengths. They follow the same intensity de- even though the Coulomb potential is a long-range one.
pendences in the three cases, except for a constant factor, The dipole matrix element takes in this case the form [36]
which is almost independent of the laser intensity. Mack-
(27/2~5/4 )
lin [35] compared the results from our theory with the
exact results obtained for the zero-range potential [13].
d(p) = i I
)I (, +p ), (24)
+2K+1
s= —~
x (i) e ' iBs(r) Jr+i s(U&C(7)) + e' Bs(r) J~ s(U~C(w)) (25)
In Appendix B, we explain how analytic formulas for the 35th harmonic enters the plateau region. It determines
coefficients Bs(7 ) can be derived by performing an inte- the location of the cutofF at this particular intensity (for
gration in the complex plane. The functions C(v) and discussion see [7]). Note that the cutoff location for hy-
Fa (v) are given in Appendix A by Eqs. (A9) and (A10), drogen is shifted downwards in comparison to the Gauss-
respectively. %e compare the results for the hydrogen- ian model, since the 35th harmonic reaches it at a higher
like atoms with those for the Gaussian model in Fig. 3. intensity. An explanation for this eKect is presented in
The 35th harmonic strengths calculated from GBR and Sec. IV.
&om expression (25) are plotted as a function of intensity
in a logarithmic scale. As we see, apart from a constant E. Electron rescattering
absolute factor, both curves follow a similar intensity de-
pendence and interference pattern. The change of slope One of the important problems connected with the
in both curves corresponds to the intensity at which the present theory deals with rescattering of the electrons
2124 M. LEWENSTEIN et al. 49
10
10
'2
10 10
10 10
10 10
10-18 10 "
0-20 20
1 10 I I
0.6 0' 8 1.0 1.2 1.4 0.6 0.8 1.0 1.2 1.4 1.6
log10 U log10 U
that. First, the number of harmonic we have to produce technique is quite eKcient in calculating the integral over
to get 00 is lower for higher laser &equency. Second, for p, we apply the same technique to evaluate the remaining
higher laser &equency the laser period is shorter. There- integrals over 7 and t. This method is asymptotically
fore, the spreading of the wave packet before the moment exact provided Uz, Iz, and K are large enough. The
of re-encounter with the nucleus is smaller. As we have saddle-point equations that arise &om the derivatives of
seen in the previous subsection, this is a very dramatic ef- the classical action (9) take the form
fect which strongly reduces the intensity of emission due
to electron-parent ion collision. Indeed, the wave packet S(p, t, r) = x(t) —«(t —~) = 0, (27)
spreads as w /, and for the harmonic intensity this factor
should be squared. BS(p, t, 7.) —A(t —~)]2
We performed calculations for I& —24 eV and laser fre- 87
[p
2
+Ip —0, (28)
quencies 1 and 2 eV, at intensities around 10is W/cm2
and harmonic &equeneies around 50 eV. In all calcula- —A(t)l' —A(t — )]'
tions the harmonic intensities were 1—2 orders of magni- ~S(» t ) [ [ = 2K+1.
tude higher for 2-eV laser light than for 1-eV light. For 2 2
these laser frequencies the efFect of spreading alone will (29)
give only a factor of 2 = 8 advantage to the shorter
wavelength laser. Therefore, the fact that the harmonic The first of these equations indicates, as we already men-
number we need to obtain the given radiation &equency tioned, that the only relevant electron trajectories are
is lower for shorter wavelength is also quite important. those where the electron leaves the nucleus at time t —w
Summarizing, shorter wavelength radiation can be sig- and returns at t. Equation (28) has a somewhat more
nificantly more eKcient for generating harmonic radia- complicated interpretation. If I„were zero, it would sim-
tion at a given, not very high, frequency. This conclusion ply state that the electron leaving the nucleus at t —~
is consistent with experimental data [2]. should have a velocity equal to zero. In reality, 0 I„g
and in order to tunnel through the Coulomb barrier the
electron must have a negative kinetic energy at t —~.
IV. SADDLE-POINT ANALYSIS AND THE This condition cannot be fulfilled for real ~'s, but can
EXACT CUTOFF LAW easily be fulfilled for complex 7's. The imaginary part of
r can then be interpreted as a tunneling time, just as it
The main aim of this section is the derivation of the ex- has been done in the seminal papers of Ammosov et al.
act quantum cutoff law in the limit U„~ oo, with I„/2U„ [11]. Finally, we can rewrite the last expression (29) as
constant, but smaller than one. In the semiclassical pic- —A(t)l
[
ture of Ref. [9] the cutoff law results from the energy con-
2
+I, = Ek;„(t)+I, =2Ky1.
servation principle. One calculates the maximal kinetic
energy of the electrons born at the origin with zero ve- This is simply the energy conservation law, which gives
locity and returning to the origin. This maximal kinetic the final kinetic energy of the recombining electron that
energy turns out to be 3.17U„. One argues then that the generates the (2K + 1)-th harmonic.
maximal energy of the photons emitted due to recombi- These equations can be used to derive the cutofF law.
nation is I„+ 3.17U„. Obviously, in quantum case, this Indeed, Eq. (30) clearly says that the maximum emitted
picture cannot be exactly valid because of quantum tun- harmonic &equency is given by the maximum possible
neling and diffusion effects. First of all, in the quantum kinetic energy the electron has at the moment t of col-
theory, the tunneling electrons are not born at the origin, lision with the nucleus. Qualitatively, this conclusion is
but rather at zo such that I~ = Exocos(t —7'). When fully consistent with the classical model of Refs. [8,9].
they return to xo with Eg;„— —3.17U„, they may thus Quantitatively, there is a difFerence because Eqs. (27)
acquire an additional kinetic energy as they move &om and (28), which have to be considered together with Eq.
zo to the origin. Moreover, the electrons are not local- (30), naturally account for the tunneling process and its
ized in the quantum case, due to the finite size of the infIuence on the electron kinetic energy at the moment it
ground-state wave function and to quantum difFusion ef- encounters the nucleus again.
fects. The "additional" kinetic-energy gain therefore has In order to obtain the cutofF law, we must use the first
to be averaged over all electron trajectories. As we shall two of saddle-point equations to express any two of the
see below, the exact quantum cutofF law actually difFers variables p, t, v via the remaining third. Then we have
slightly from I~ + 3.17U„. to substitute the results into Eq. (30) and find the maxi-
In order to derive the quantum cutofF law, we shall mum of the left-hand side expression as a function of the
examine the asymptotic behavior of Eqs. (8) and (18) remaining variable. In practice, it is very convenient to
in the limit when U„, I„, and K are large. The Fourier use the return time v and solve Eqs. (27) and (28) for p
components of x(t) are defined as and t as functions of v. Thus, the cutofF law reads as
tp+2m
(t) (2K+1)it (2K+ 1) „=maxRe C [p(r) —A(t(~))] )+
2K
(26) I„(31)
)
/
~
to
Since we know already from Sec. III that the saddle-poiat under the constrat~t that the imaginary part of the right-
2126 M. LEWENSTEIN et al. 49
hand side expression is equal to zero. is C(T) = 2s(T)a(T). From Eq. (32), we obtain
Inserting the solution of Eq. (27), i.e. the expression
for p, (, (t, T), into Eq. (28) it is easy to solve the latter i)/s~ a(s)ks(~)/s (~)+a (~)+
with respect to sin(t —T/2) and cos(t —7/2). Namely, ("(-)+-'(-)I
~
Eq. (28) reduces to the form
cos(t —T/2)
' '
= 'E~s (,( ),
(T)+a(T) ps(7 )+as (7 )+
( )I
(36)
2'",
—T/2)a(T) —cos(t —T/2)s(T) = i Ip
sin(t ' (32) There exists also a pair of complex-conjugated solutions.
The choice of the appropriate pair is dictated by the re-
where quirement that the resulting imaginary part of the classi-
cal action must be negative, so that it causes an exponen-
a(T) = cos(T/2)— 2sin T 2
(33) tial decrease of the corresponding transition amplitudes.
The results do not depend on the choice of the sign in
front of the square root in Eqs. (35) and (36), provided
8(T) = sin(T/2). it is the same in both of them.
Inserting these solutions into Eq. (29), we obtain a
Note that the function C(T), as expressed by Eq. (16), closed form equation for 7. ,
I
I etus first consider a limiting case I„= 0. Eq. (37) solutions of Eq. (38) then acquire a significant imag-
takes then a simple form inary part which introduces an exponentially decreas-
ing factor to exp[ —iS(p, t, T)] at the saddle point and
a (T)8 (T) 2K +1 causes a sharp (exponential) cutoff in the harmonic spec-
8
[8 (T) + a (7.)[ U~
(38) trum. %e conclude that for I„=
0, the cutoff occurs at
2E + 1 3.17'.
It is much more difficult to study the case of finite I„.
The function on the left-hand side of Eq. (38) is noth- One way to do it is to perform a systematic expansion in
ing else but the classical kinetic-energy gain, AEi„„(T),
Iz/Uz It is easy .to see, however, that the zeroth order (in
as plotted in Fig. 1 (dashed line). Equation (38) allows
I„/U~) solution of Eq. (28) is doubly degenerated and the
for real solutions (tunneling time equal to zero) provided corresponding saddle point is not Gaussian but rather of
K is not too large. In order to find out how large it third order. Integration around such a saddle point gives
can be, we have to find the maxima of EEk, „(T) . As rise to Airy functions [39] and will be discussed elsewhere.
we already learned in Sec. II, these maxima occur ex- In the present paper, we shall use a simple approximation
actly at the same points as the maxima of the function valid in the limit Iz && Uq and a numerical evaluation in
C(T). There are two families of maxima, corresponding the general case, where I„ is of the order of U„or slightly
to a(T) = s(T) and to a(— T) = s(T). The first family are smaller.
the solutions of the equation I et us denote
1
tan(T/2) = (39) G (T)8 (T) 8 (T) +G (T) + ~&
f(T)= 8 (41)
["(T) + a'(T)l'
and contains m & wi
tion vi
(
2a, w3 4', etc. The first solu-
4.08 is the same as 7,„& discussed in Sec. II. For G(T)8(T)[a (T) —8 (T)] 8 (T) +G (T) +
this solution, the kinetic-energy gain attains the absolute g(T)= 8
maximum equal to 3.17. Further solutions give maxima
["(T)+ '( )]'
in the range 2 —2.4. The second family [a(T) = s(T)] ful-
(42)
fills Equation (37) then takes a simple form
tan(T/2) =
2/T
1
+ 1' (40)
f(T) + i
I„2E+
=
g(T)
1
and contains v2 3', v4 47t etc. , with values of max- cutofF law [see Eq.
ima also in the range 2 —2.4. Strictly speaking the exact quantum
For 2K + 1 )
3.17U~ there are no real solutions to (31)] is thus
Eq. (38). The reason is not because of the impossi-
bility of tunneling, but rather because of the impossi-
(2K+ 1) „=U~maxRe f(T) +i " g(T), (44)
bility of gaining sufficiently large kinetic energy. The 2 Up
)
49 THEORY OF HIGH-HARMONIC GENERATION BY LOW-. .. 2127
( = 0.
1.4—
Im f(r)+i P
2Up
g(~) (45) 1.32
) CL
1.3—
Ip &'(&~)
2Up f"(rg) '
(46) saddle-point equations for I„= 0, we recover
the result
(2K+ 1) = 3.17Up + 1.32I„. (ii)
The electron under-
and & ~ = 0. The primes denote here derivatives. After goes diffusion which tends to average and decrease the
complicated, but otherwise elementary calculations, we efFect of the additional kinetic-energy gain for larger Ip.
obtain the explicit form of the cutofF law for Iz && U&, The exact cutoff law is asymptotically valid when both
Up and Ip tend to infinity, so that the ratio I„/2U„
remains constant (and not too large). Strictly speak-
(2K+1) „=3.17Up+Ip 1+ 8 l
U„m oo, but rather slowly. These corrections counter- where the complex rate p(t) is defined by,
act the cut-off shift discussed in this section and move
the cutoff location to a lower energy. They are responsi- =
ble for the difference in the position of the cutoff between
p(t)
f d p
0
dvEcos(t)d (p A(—
t))
hydrogen and the GBR model observed in Fig. 3. These xEcos(t —w)d (p —A(t —w))e ' i~" ). (52)
points will be discussed in detail in a future publication.
V. GROUND-STATE DEPLETION p(t) is in general time dependent and becomes periodic
as t grows, having maxima at the peak values of the elec-
It is possible to generalize our theory to include the tric 6eld, when the electron has a much larger chance
effect of the ground-state depletion. To this aim, we con- of tunneling. Equation (52) can be analyzed using the
sider the Schrodinger equations, saddle point method and in the limit of U„)& I„&& 1,
one recovers the tunneling rate obtained by Ammosov et
al. [11]. Note, however, that our theory is more general,
o(t) =tEcos(t)
f d v d (v)b(vt), , (49)
since it takes into account quantum interference effects
and the returns of the electron to the nucleus, which are
completely absent in the approach of Ref. [11]. We leave
b(v, t) = i. fv2 ~
—
— + I„~ b(v, t) —Ecos(t) (9b(v, t)
Bv
the detailed discussion of the time-dependent rate to a
future publication and we limit our discussion here to the
+iEcos(t) d (v)a(t). (50) case where p(t) can be substituted by its time average
OO
We solve Eq. (50) and insert the solution into (49). As- = — dt' p(t').
p lim
suming that a(t) changes slowly we can set a(t') a(t). —
t +oo g
The differential equation for a(t) takes the form
For the Gaussian models (GEX, GSP, and GBR) the ex-
a(t) = -~(t)a(t) (51) pression for p takes this form,
p = 4((U„)
x
—"
q n
—
[,
'
) sn' o
d7
(1/n+ i~/2) ] . )
exp[ i+o(&)]
( B(r) J2(U„C(r)) + [B(w) + D(7 ) cos(w)] Jp(U„C(v)) + 2i[B(w) cos(r) + D(r)/2] Ji(UpC(~)))t (54)
In Fig. 7, we present the intensity dependence of p as obtained from the GBR model. As we see, both the real and
the imaginary parts of p [see Figs. 7(a) and 7(b)] increase (in absolute value) with intensity. The real part of p, p~
describes the rate of depletion of the ground state. For the laser pulse duration tD the saturation intensity is achieved
at p, ttDE 1, i.e. , when the atom is fully ionized during the interaction with the laser pulse. The imaginary part
of p, pl describes a dynamical shift of the ground-state energy (not to be confused with the ponderomotive shift). It
is proportional to the laser intensity and becomes quite large as pR
Taking into account the depletion, the expression for the time-dependent dipole moment takes the form
t
s(t) = d p dcd (p —A(t))Fco's(t —v)d, (p —A(t —v)) ~ (55)
sf 0
The Fourier transform of x(t), or rather its modulus squared, now consists of a sequence of Lorentzian peaks with
width 2pR. The harmonic strengths can be calculated as the total area covered by each of these peaks (equal to the
total energy radiated at the corresponding harmonic frequency).
Using Eq. (18) we obtain
tz)
z(A) =
27(
) p
dT bR M(7) JM(U~C(w))(i) e* h(r, 0 —2M —1), (56)
where
—2p~ a+i(O —2M —1)7 —2p~ t ~+i(O —2M —1)t~
h(v', 0 —2M —1) = e~ (57)
2pR+i 0 —2M —1
The peaks are centered at 0 = 2M+1. The harmonic strengths ~x&M+i are thus approximately equal to ~x(2M+1)
times a factor that accounts for the area of the peak, 2vrpR/[1 —exp( —
~ ~
/ 2(.)M+ ~ ( & q /
l»M+il
2 = — — exp[ —iFM(r)]h(w, 0)
I exp( 2p~tD) gvra) a2 o q 1/a + i7/2)
x ( B—
(7.) JM+2 (UpC(w) ) + i[B(~)e' + D(w)] JM+z (UpC(7. ) )
A similar expression can be derived for hydrogenlike cycle, the electron still has a chance to come back at least
atoms. once to the origin.
In order to study the effects of depletion, we have first In order to study the full dynamical infiuence of the
treated p as a &ee parameter, [i.e. , not determined &om depletion on harmonics, we have evaluated p &om Eq.
Eq. (53)] and we have calculated harmonic spectra for (54) (see Fig. 7) and the harmonic strengths &om Eq. (58)
U„= 20 and several values of p. In particular, we have for the GBR model. An example for the 35th harmonic is
performed calculations for pl —0 and pR —0.01 and 1. shown in Fig. 8. Again, the main effect of the depletion
The duration time of the laser pulse was chosen to be 20 consists of the decrease of the harmonic strength. For
optical cycles. The spectrum for pR —0.01 differs only high enough intensities, the harmonic strength becomes
slightly &om the result obtained for pR —0, even though a slowly decreasing function of the laser intensity.
the system is already close to saturation. A further in- It is now clear that in order to get higher conversion
crease of pR decreases the harmonic strengths quite sig- eQciency of harmonic emission one should use shorter
nificantly, and smoothes out the spectrum, i.e. , reduces laser pulses. Indeed, according to the results of this sec-
the effects of quantum interferences. The depletion does tion and Sec.III E, the main contribution to the harmonic
not change the cutofF location for the single-atom spec- emission is given by the first re-encounter of the contin-
tra. That can be understood since even for pR —1, when u»m electron with the nucleus, and the effects of sec-
the ionization takes place more or less within one optical ondary collisions are not very strong. Therefore, one can
—
use very short a few cycles long laser pulses of high —
intensity. Even though the ground state of the atom will
be depleted very fast, the electron will still be able to
10
return to the nucleus and collide with it at least once.
As a result, the absolute harmonic intensity will not de-
crease significantly. On the other hand, the energy of
the shorter incident laser pulse will be reduced. In other
words, at high intensities, when the depletion of the atom
is very fast and harmonic emission occurs only at the be-
ginning of the pulse, most of the pulse energy is wasted
C
IQ and using short pulses will solve this problem.
10 I
10 20
Ponderomotive potential U
10
P
10 '—
0, 5
10
10
CJ
C7
6
CV
0
C7
0.2
10
10
0. 1
10
10
10
0.6 0.8 1.0 1.2 1.4 1.6 1.8
I lag10 U
10 20
Ponderomotive potential U
P
FIG. 8. Comparison of intensity dependences of the 35th
harmonic with (open squares), and without (black squares)
FIG. 7. Intensity dependence of the depletion rate pR for depletion. Both curves are obtained for the GBR model with
the GBR model; I~ = 5, a = 2I~; (b) Same as (a) for the I„= 5, o. = 2I„. The duration of the pulse t~ is taken to be
ac-Stark shift of the ground-state energy —7I. 50 optical cycles.
2130 M. LEWENSTEIN et al. 49
( ra)
[13] but it is not restricted to a zero range potential. We
can describe hydrogenlike atoms and account for electron 2 sin (r/2) sin(r) 1
rescattering processes (as we did in Sec. III). The effects 2 2' (A6)
of the Coulomb potential, introducing corrections to the
classical action could be included as in Ref. [11]. We can D(r) = —2B(r) —1+ cos(r), (A7)
also investigate the influence of the ground-state deple-
tion on the high harmonic production and on the cutoff
formula (see Sec. V).
FJc(r) = (I~ + U~ —K)r
Finally, we should emphasize that the approach de-
veloped in the present paper for a linearly polarized 2iU„4U„sin (7/2) ( 2i 5
r 1+ a7)
monochromatic laser field can easily be generalized to \
cLz
Bs(r) = 2
. H(z), (B2)
c
where the integral runs around the unit circle along the contour C. The function H(z) is given by
z'+'W. (z)
'
[Wg(z) W2(z)]s
and the quadradic polynomials W;(z), i = I, 2, 3, take the form
Wq(z) = U~[s(r) —ia(r)] e ' z + (U~[s (r) + a (r)] + n) z+ Uz[s(r) + ia(r)) e', (B4)
W2(z) = U„(s(r) + ia(r)) e ' z + (U~[s (r) + a (r)] + o.') z+ U„[s(r) —ia(r)] e',
Ws(z) = U„[s(r) + a (7.)] (e ' z + 2z+ e* ) . (B6)
The function H(z) has two singularities inside the contour C. These are
2(s +a ) +u'/Up
—Q(a'/Up) +4(s + a )a'/Up, .
z2- e' . (B8)
2(s —ia) 2
At these points, the function H(z) has poles of third or- for the values of r when either s(r) or a(r) vanish that
der at least. Using Cauchy's theorem, we obtain analytic is not the case. The singularity is then of the sixth order
expressions for the coefBcients Bs(r) as and the calculations become very tedious. In numerical
calculations, it is very convenient to solve this problem
Bs(r) = — )
i=1,2
lim
d2
(z —z;) H(z) (B9)
zq
f
by using o. cr', at least in the vicinity of r's for which
—z2. For o;, o," of the order of 20, it is enough to
Strictly speaking expression (B9) is valid only if the two set the difference between them to about one to assure a
poles are not degenerated, i.e. , zq P zq. Unfortunately good convergence of the results.
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