Harmonic-generation control
A. Sanpera,
1
J. B. Watson,
1
M. Lewenstein,
2
and K. Burnett
1
1
Clarendon Laboratory, Physics Department, University of Oxford, Oxford OX1 3PU, United Kingdom
2
Comissariat a ` l’Energie Atomique, DSM/DRECAM/SPAM, Centre d’Etudes de Saclay, 91191 Gif-sur-Yvette, France
Received 1 May 1996
In this paper we examine some effects of quantum interference on high harmonic generation. We demon-
strate in particular that preparing the initial state in a coherent superposition of bound states leads to a harmonic
spectrum with distinct plateaus with different conversion efficiencies. We show how this scheme may provide
a way of controlling the coherent output that is produced in an experiment. S1050-29479603611-6
PACS numbers: 32.80.Rm, 42.50.Hz
I. INTRODUCTION
While some aspects of harmonic generation are both theo-
retical and experimentally quite well understood, the way in
which the coherent radiation output can be controlled for
practical purposes is still a matter of debate. By controlling
harmonic generation we mean the possibility of modifying
the spectrum to our convenience, for example, by enhancing
only a narrow range of harmonics around a desired energy in
the total spectrum or extending the harmonic spectrum to
higher orders than the ones predicted by the well-known cut-
off law U
i
+3 U
p
1where U
p
=E
2
/4
2
is the ponderomo-
tive energy expressed in atomic units, is the frequency of
the driving laser field, and U
i
refers to the ionization poten-
tial of the atomic state.
The semiclassical model of harmonic generation 1,2 is,
in spite of its simplicity, correct in showing the importance
of the electron’s classical returning trajectories in the
harmonic-generation process. In particular, it suggests that it
should be possible to change the spectrum by modifying
these trajectories. Early work 3 using two different lasers
with commensurate frequencies interacting simultaneously
with an atom reinforces this idea. For that combination of
fields E
1
sin(t)+E
2
sin(3t+), the plateau structure ex-
tends up to U
i
+kU
p
with k 3 here U
p
still refers to the
ponderomotive potential of the fundamental field. The value
of k is determined by the intensity of the fields as well as
their relative phase. The two-color field modifies the return-
ing trajectories and a classical analysis of those trajectories
once again gives an accurate prediction of the position of the
cutoff. Furthermore, when a second color is superimposed it
not only extends the spectrum to higher orders but also pro-
duces a clear enhancement up to 2–3 orders of magnitude
in the low-energy region of the spectrum with energies be-
low U
i
+U
p
) 4. To our knowledge, however, it is not pos-
sible to enhance the higher-energy harmonics by modifying
the relative phase or the intensity ratio between both fields.
Achieving such a goal would require the selective injection
of electrons into the continuum only or mainly at those
times that correspond to returning trajectories with maximum
kinetic energy. At present this does not look as though it can
be straightforwardly achieved 5.
It appears at first sight easier and more beneficial to
modify the classical aspects of the process, i.e., the recollid-
ing trajectories, than to modify the steps that deal with the
intrinsically quantum aspects of it. In this paper, however,
we shall demonstrate that more profound effects are pro-
duced when the ionization and rescattering events are modi-
fied. This can be achieved, for example, by tailoring the ini-
tial state to allow different paths in the recombination event.
If the initial state is prepared in a coherent superposition of
different bound states 6,7, in which only the more loosely
bound state becomes ionized by the action of the field, the
harmonic spectrum contains two distinct set of harmonic pla-
teaus 8. Our aim is to show how one may take advantage of
this feature each plateau presents a distinct conversion effi-
ciency to manipulate the harmonic spectrum.
The paper is organized as follows. In the next section we
shall introduce the model we use and discuss the main fea-
tures of the recombination event in terms of the values and
phases of the dipole matrices. In Sec. III we analyze the
dependence of the conversion efficiency on the initial states.
This aspect is further discussed in the Appendix. In Sec. IV
we shall demonstrate how to use these features to control the
harmonic generation. Finally, in Sec. V we discuss the fea-
sibility of the practical realization of the scheme we propose.
II. COHERENT SUPERPOSITIONS
Our results are based on quantum interference effects in
recombination via different states. We prepare the initial
state in a superposition of the ground state | g and some
excited state denoted by | e with a fixed though arbitrary
phase difference between both states
r , t =
| g + e
-i
| e 1
( | |
2
+| |
2
=1). We shall take =0 to simplify the nota-
tion. The laser parameters ( I and ) are chosen such that
only the excited state is depleted by ionization. We do this
because it is sufficient, and requires much lower intensities,
to promote the electron into the continuum from the excited
state. Since we aim to describe a rather general way of pos-
sible control over the harmonic emission, we perform our
calculations for a simple hydrogenic ion He
+
. At this point it
is worth pointing out that the results we present are not di-
rectly related to the structure of the atomic potential and
therefore can be straightforwardly extended to any type of
ion or atom.
PHYSICAL REVIEW A NOVEMBER 1996 VOLUME 54, NUMBER 5
54 1050-2947/96/545/43207/$10.00 4320 © 1996 The American Physical Society