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1-1-2009
Quantum Simulation of High-Order Harmonic
Spectra of the Hydrogen Atom
A. D. Bandrauk
Département de Chimie, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Quebéc, Canada
S. Chelkowski
Département de Chimie, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Quebéc, Canada
Dennis J. Diestler
University of Nebraska - Lincoln, ddiestler1@unl.edu
J. Manz
Institut für Chemie und Biochemie, Freie Universität Berlin, 14195 Berlin, Germany
K.-J. Yuan
Département de Chimie, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Quebéc, Canada
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Bandrauk, A. D.; Chelkowski, S.; Diestler, Dennis J.; Manz, J.; and Yuan, K.-J., "Quantum Simulation of High-Order Harmonic Spectra
of the Hydrogen Atom" (2009). Agronomy & Horticulture -- Faculty Publications. Paper 352.
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PHYSICAL REVIEW A 79, 023403 共2009兲
Quantum simulation of high-order harmonic spectra of the hydrogen atom
1
A. D. Bandrauk,1 S. Chelkowski,1 D. J. Diestler,2,3 J. Manz,3 and K.-J. Yuan1,3,*
Département de Chimie, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Quebéc, Canada J1K 2R1
2
Department of Agronomy and Horticulture, University of Nebraska-Lincoln, Lincoln, Nebraska 68583, USA
3
Institut für Chemie und Biochemie, Freie Universität Berlin, 14195 Berlin, Germany
共Received 22 May 2008; published 3 February 2009兲
Three alternative forms of harmonic spectra, based on the dipole moment, dipole velocity, and dipole
acceleration, are compared by a numerical solution of the Schrödinger equation for a hydrogen atom interacting with a linearly polarized laser pulse, whose electric field is given by E共t兲 = E0 f共t兲cos共␻0t + ␩兲 with Gaussian
carrier envelope f共t兲 = exp共−t2 / ␦2兲. The carrier frequency ␻0 is fixed to correspond to a wavelength of 800 nm.
Spectra for a selection of pulses, for which the intensity I0 = c␧0E20, duration T ⬀ ␦, and carrier-envelope phase
␩ are systematically varied, show that, depending on ␩, all three forms are in good agreement for “weak”
pulses with I0 ⬍ Ib, the over-barrier ionization threshold, but that marked differences among the three appear as
the pulse becomes shorter and stronger 共I0 ⬎ Ib兲. Except for scalings by powers of the harmonic frequency, the
three forms differ from one another only by “limit contributions” proportional to the expectation values of the
dipole moment 具z共t f 兲典 or dipole velocity 具ż共t f 兲典 at the end 共t f 兲 of the pulse. For long, weak pulses the limit
contributions are negligible, whereas for short, strong ones they are not. In the short, strong limit, where
具ż共t f 兲典 ⫽ 0 and therefore 具z共t兲典 may increase without bound 共i.e., the atom may ionize兲, depending on ␩, an
“infinite-time” spectrum based on the acceleration form provides a convenient computational pathway to the
corresponding infinite-time dipole-velocity spectrum, which is related directly to the experimentally measured
“harmonic photon number spectrum” 共HPNS兲. For short, intense pulses the HPNS is quite sensitive to ␩ and
exhibits not only the usual odd harmonics but also even ones. The analysis also reveals that most of the
harmonic photons are emitted during the passage of the pulse. Because of the divergence of 具z共t兲典 the dipolemoment form does not provide a numerically reliable route to the harmonic spectrum for very short 共fewcycle兲, very intense laser pulses.
DOI: 10.1103/PhysRevA.79.023403
PACS number共s兲: 42.65.Ky, 32.80.Wr, 42.50.Hz
I. INTRODUCTION
High-harmonic generation 共HHG兲—the phenomenon in
which low-frequency 共infrared兲 radiation is converted, by the
interaction of a laser pulse with a low-density gas, for example, to higher frequencies that are integer multiples 共harmonics兲 of the low frequency—holds promise as a source of
intense, coherent, short-wavelength radiation for such new
practical applications as time-dependent x-ray scattering 关1兴
and attosecond pulse generation 关2兴. Most theoretical descriptions of HHG consider a single atom 共or molecule兲,
which is usually assumed to have a single “active” valence
electron driven by an external electric field representing the
laser pulse 共see, however, Ref. 关3兴兲. The frequency distribution of the harmonic radiation, to which we shall henceforth
loosely refer in the present context as the “harmonic spectrum,” is variously described for single atoms as being proportional to the squared magnitude of the Fourier transform
共FT兲 of the expectation value of either the dipole moment
关4–12兴, the dipole acceleration 关1,8,9,11–19兴, or the dipole
velocity 关12,20兴. We are aware of just a single previous comparison of dipole, dipole-velocity, and dipole-acceleration
forms based on accurate solutions of the time-dependent
Schrödinger equation, but this was restricted to rates of harmonic generation by periodic, or continuous-wave fields at
*Corresponding
author.
kaijun.yuan@usherbrooke.ca
1050-2947/2009/79共2兲/023403共14兲
Email
address:
low intensity and infinite duration 关21兴. Expressions for the
harmonic spectrum that are proportional to the 共double兲 FT
of the time-autocorrelation function of the dipole moment
关4兴 or the dipole velocity 关20兴 scale as the square of the
density of the gas, reflecting the cooperativity of HHG, as
observed, for example, by Lorin et al. 关22兴, who employed
coupled Maxwell-Schrödinger equations to simulate HHG in
a one-dimensional model of H2+ gas.
An important work in the present context is that of Burnett et al. 关13兴. Alluding to a remark by Sundaram and
Milonni 关4兴 that the total power radiated by an atomic dipole
is proportional to the expectation value of the squared acceleration, they questioned the use of the dipole form of the
harmonic spectrum, implying that the correct form is proportional to the FT of the squared magnitude of the expectation
value of the dipole acceleration. By means of integration by
parts, they derived a relationship between the FTs of the
dipole and dipole acceleration, which involve limit contributions depending on the expectation values of the dipole moment and dipole velocity at the end of the laser pulse. They
compared “power spectra” for a one-dimensional model
atom, finding good agreement between dipole and dipoleacceleration spectra at low intensity, but marked differences
between them at high intensity. Stating that it is impractical
to compute the dipole-acceleration spectrum by correcting
the dipole-moment spectrum with the limit contributions,
Burnett et al. recommended calculating the acceleration
spectrum directly. Some researchers have subsequently followed their advice, while others have adhered to the dipole
023403-1
©2009 The American Physical Society
PHYSICAL REVIEW A 79, 023403 共2009兲
BANDRAUK et al.
form, or in some cases have used both. For example, numerically accurate simulations of HHG for the three-dimensional
共3D兲 hydrogen-molecular ion show that the acceleration
form is preferable for the strong-field approximation to angular distributions of the harmonic radiation 关9兴. However,
other simulations of HHG for a one-electron 3D diatomic
molecule, based on the strong-field approximation 关7兴, indicated that the dipole-velocity form of the strong-field limit
“gives the closest agreement with exact results” 关12兴.
The purpose of this paper is to explore the connections
among the three forms 共i.e., dipole-moment, dipole-velocity,
and dipole-acceleration兲 of the harmonic spectrum that have
been considered previously 关1,3–19兴. We present and compare the three forms computed from accurate numerical solutions of the Schrödinger equation for the interaction of a
hydrogen atom 共H兲 with a linearly polarized laser pulse.
Note that for the present purpose we invoke no approximations 共e.g., the strong-field approximation 关7,12兴兲, because
they may affect the results for the three forms differently. We
fix the carrier frequency of the pulse. We vary the intensity
from “low” 共I0 ⬍ Ib, the over-barrier ionization threshold兲 to
“high” 共I0 ⬎ Ib兲. Motivated by recent experimental advances
in the production of single-cycle pulses 关23兴, we vary the
duration from 16 down to 3 optical cycles. Likewise, results
of a “double-slit” experiment in the attosecond time domain
关24兴, which are quite sensitive to the carrier-envelope phase
共CEP兲 of the laser pulse, have induced us to examine the
influence of the CEP on the harmonic spectra. Special attention is paid to the dipole-velocity form, which is directly
related to the “harmonic photon number spectrum” 共HPNS兲
关20兴 measured experimentally for dilute atomic gases.
the H atom occupies the ground state 共1s兲 initially 共i.e., at
t = ti兲. Since the potential energy to which the electron is
subject is cylindrically symmetric, as is the initial state of the
atom, the wave packet must remain cylindrically symmetric
as it evolves. The z component of L therefore vanishes; that
is, the magnetic quantum number m is restricted to zero. As
a consequence, the wave function can be represented in the
共restricted兲 basis of normalized eigenfunctions of L2 as
⌿共r,t兲 = ⌿共r, ␪, ␾,t兲 = 兺 Rl共r,t兲Y l0共␪, ␾兲,
共2.3兲
Y l0共␪, ␾兲 = 冑共2l + 1兲/4␲ Pl共cos ␪兲.
共2.4兲
l
where
Substituting Eq. 共2.3兲 into Eq. 共2.2兲 and projecting both
members of the resulting equation onto Y l0 共i.e., multiplying
* . . ., where d⍀
both sides by the operator 兰d⍀Y l0
= sin ␪d␪d␾ represents the element of solid angle兲, we obtain
the following set of coupled second-order partial differential
equations for the radial wave functions
i
再 冉 冊
+
lrE共t兲Rl−1共r,t兲
冑共2l − 1兲共2l + 1兲
H共t兲 =
p2 1
− + zE共t兲 = Ha + W共t兲,
2 r
共2.1兲
where Ha ⬅ p2 / 2 − 1 / r refers to the isolated atom. The form
of the matter-radiation interaction indicates that the description is cast in the length gauge in the long-wavelength approximation. Here and below, except where it is stated otherwise, we employ atomic units: a0, ប / mea0, and ប2 / m2e a30
for length, velocity, and acceleration, respectively; Eh / ea0
for the electric field; and 共ea0兲2 for harmonic spectra. 共me
stands for the mass of the electron, e for the magnitude of its
charge, a0 for the Bohr radius, and ប for Planck’s 共modified兲
constant; Eh ⬅ ប2 / mea20.兲
The wave packet describing the relative motion in spherical polar coordinates satisfies the Schrödinger equation
i
再
冉 冊
⳵⌿共r, ␪, ␾,t兲
1 ⳵ 2⳵
L2 1
= − 2
r
+ 2−
⳵t
2r ⳵r
⳵r
2r
r
冎
+ r cos ␪ E共t兲 ⌿共r, ␪, ␾,t兲,
共2.2兲
where L is the orbital angular momentum. We assume that
+
共l + 1兲rE共t兲Rl+1共r,t兲
冑共2l + 1兲共2l + 3兲
,
共2.5兲
l = 0,1,2 . . . ,
where we have used Eq. 共A2兲 of Appendix A. The 兵Rl共r , t兲其
are subject to the initial conditions
II. THEORY
Consider a linearly 共z-兲 polarized laser pulse with z component of the electric field E共t兲 interacting with an H atom.
The Hamiltonian governing the motion of the electron relative to the nucleus can be expressed as
冎
⳵Rl共r,t兲
1 ⳵ 2⳵
l共l + 1兲 1
= − 2
r
+
−
Rl共r,t兲
⳵t
2r ⳵r
⳵r
2r2
r
Rl共r,ti兲 =
再
2e−r ,
l=0
0,
l 艌 1,
冎
共2.6兲
where 2e−r is the normalized radial factor of the 1s wave
function.
We describe the harmonic spectrum D共␻兲 in three alternative forms: D共␻兲 is equal to the squared magnitude of the
FT of the expectation value of either the dipole moment 共−z兲,
the dipole velocity 共−ż兲, or the dipole acceleration 共−z̈兲.
Thus, we set
D␨共␻兲 = 兩␨共␻兲兩2 ,
共2.7兲
冕
共2.8兲
where
␨共␻兲 ⬅
tf
dt exp共− i␻t兲具␨共t兲典,
ti
and
具␨共t兲典 = 具⌿共t兲兩␨兩⌿共t兲典,
␨ = − z,− ż,− z̈.
共2.9兲
As indicated by Eq. 共2.9兲, in atomic units the dipole moment
␨ = −z is the negative of the z coordinate of the electron relative to the nucleus. Since the harmonic spectra defined by
Eqs. 共2.7兲 and 共2.8兲 do not depend on the sign of ␨, for
notational economy we suppress the minus sign, although we
continue to refer to the coordinate z loosely as the dipole
moment, bearing in mind the absent minus sign.
023403-2
PHYSICAL REVIEW A 79, 023403 共2009兲
QUANTUM SIMULATION OF HIGH-ORDER…
Using Eqs. 共2.3兲, 共2.4兲, and 共2.9兲, we can write the expectation value of the dipole moment itself as
具z共t兲典 = 兺 兺
k
l
= 2兺
l
冕
⬁
drr Rl*共r,t兲Rk共r,t兲
3
0
冋冕
共l + 1兲
冑共2l + 1兲共2l + 3兲 Re
冕
⬁
* cos ␪Y
d⍀Y l0
k0
=−兺兺
册
⫻
where the second line of Eq. 共2.10兲 follows from the properties of the Legendre polynomials 关see Appendix A, Eq.
共A2兲兴.
The dipole velocity is given by
共2.11兲
where pz is the z component of the relative momentum and
the last member of Eq. 共2.11兲 depends on Eq. 共2.1兲. Hence
the expectation value of the dipole velocity can be written
具ż共t兲典 = 具⌿共t兲兩pz兩⌿共t兲典 = − i
冕
共2.13兲
we can rewrite Eq. 共2.12兲 as
冋
l
k
0
册
共2.14兲
再 冋冕
冋冕
⬁
0
drr2
⳵Rl共r,t兲 *
Rl+1共r,t兲
dr
⬁
− l Im
0
drrR R*
l l+1
册冎
.
册
共2.18兲
where we utilize Eq. 共A2兲 to obtain the third line of Eq.
共2.18兲.
The three forms of harmonic spectra we are considering
can be related to one another as follows. Using the definitions in Eqs. 共2.8兲 and 共2.9兲, we write the FT of ż explicitly
as
冕
tf
dte−i␻t具ż共t兲典.
共2.19兲
Integrating by parts, we obtain
ż共␻兲 = ei␻t f 具z共t f 兲典 + i␻z共␻兲,
共2.20兲
where the condition 具z共ti兲典 = 0, which follows from the
spherical symmetry of the initial state, is implicit. From Eqs.
共2.7兲 and 共2.20兲 we deduce the relationship between the dipole and dipole-velocity forms of the harmonic spectrum
z̈ = 共i兲−1关ż,H兴 = 共i兲−1关pz,H兴 = − ⳵H/⳵z.
共2.16兲
From the last member of Eq. 共2.16兲 and Eq. 共2.1兲 we deduce
⳵r−1
− E共t兲 = − zr−3 − E共t兲.
⳵z
Combining Eqs. 共2.3兲 and 共2.17兲, we get
where the limit contribution of the dipole moment 具z共t f 兲典 is
manifest. The alternative integration by parts of ż共␻兲 yields
z̈共␻兲 = i␻ż共␻兲 − ei␻t f 具ż共t f 兲典,
共2.22兲
Dz̈共␻兲 = − 具ż共t f 兲典2 + 2具ż共t f 兲典Re关ei␻t f z̈共␻兲兴 + ␻2Dż共␻兲.
共2.15兲
In analogy with Eq. 共2.11兲, we can express the dipole
acceleration as
z̈ =
0
drRl*共r,t兲Rl+1共r,t兲兴 − E共t兲,
where we have invoked the symmetry-dictated condition
具ż共ti兲典 = 0. 关Note that this condition differs from that imposed
in the tunneling-ionization model 关25兴, which assumes the
instantaneous initial condition ż共ti兲 = 0; see the discussion
just above Eq. 共4.1兲.兴 Using Eqs 共2.7兲 and 共2.22兲, we obtain
the following relationship between the dipole-velocity and
dipole-acceleration forms of the harmonic spectrum:
共l + 1兲
冑共2l + 1兲共2l + 3兲
⫻ Im
冕
⬁
共2.21兲
*
d⍀Rl*共r,t兲Y l0
A lengthy analysis 共see Appendix B兲 yields
l
共l + 1兲
Dż共␻兲 = 具z共t f 兲典2 − 2␻具z共t f 兲典Im关ei␻t f z共␻兲兴 + ␻2Dz共␻兲,
⳵ sin ␪ ⳵
⫻ cos ␪ −
Rk共r,t兲Y k0 .
⳵r
r ⳵␪
具ż共t兲典 = 2 兺
⫻Re关
⬁
drr2
drRl*共r,t兲Rk共r,t兲
0
ti
Using Eq. 共2.3兲 and the relation
冕 冕
⬁
冑共2l + 1兲共2l + 3兲
ż共␻兲 =
共2.12兲
具ż共t兲典 = − i 兺 兺
冕
* cos ␪Y − E共t兲
d⍀Y l0
k0
l
⳵
dr⌿*共r,t兲 ⌿共r,t兲.
⳵z
⳵ sin ␪ ⳵
⳵
,
= cos ␪ −
⳵r
⳵z
r ⳵␪
冕
= − 2兺
共2.10兲
ż = 共i兲−1关z,H兴 = ⳵H/⳵ pz = pz ,
k
l
drr Rl*共r,t兲Rl+1共r,t兲 ,
3
0
具z̈共t兲典 = − 具⌿共t兲兩zr−3兩⌿共t兲典 − E共t兲
共2.17兲
共2.23兲
In Eq. 共2.23兲 the limit contribution of the dipole velocity
appears, in analogy with that of the dipole itself in Eq.
共2.21兲.
From Eqs. 共2.21兲 and 共2.23兲 it is clear that the three forms
of harmonic spectra 共with proper scalings by powers of ␻兲
may be identical, or nearly so, or may differ radically, depending on the values of 具z共t f 兲典 and 具ż共t f 兲典 共i.e., whether
these values are zero, or nearly so, or are quite different from
zero兲. We show that 具z共t f 兲典 and 具ż共t f 兲典 depend significantly on
the parameters that specify the laser pulse 共e.g., the duration,
intensity, and CEP兲. Suppose, for example, that 具z共t f 兲典
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PHYSICAL REVIEW A 79, 023403 共2009兲
BANDRAUK et al.
⬇ 具ż共t f 兲典 ⬇ 0. Then the three forms are simply related by the
expression
for ␩ = 0. The full width at half maximum of the carrier envelope of the intensity 共i.e., 关f共t兲兴2兲 is
Dz̈共␻兲 ⬇ ␻2Dż共␻兲 ⬇ ␻4Dz共␻兲.
FWHM = ␦共2 ln 2兲1/2 .
共2.24兲
The expression Dz̈共␻兲 ⬇ ␻4Dz共␻兲 is implicit in relations
given by Burnett et al. 关13兴. We note that all members of Eq.
共2.24兲 have dimensions Q2L2T−2 共where Q stands for charge,
L for length, and T for time兲. Following a previously established convention 关19兴, we define the various harmonic spectra by
Pz共␻兲 ⬅ Dz共␻兲/T ,
共2.25a兲
Pż共␻兲 ⬅ Dż共␻兲/T2␻2 ,
共2.25b兲
Pz̈共␻兲 ⬅ Dz̈共␻兲/T2␻4 ,
共2.25c兲
2
where T ⬅ t f − ti is the pulse duration. All of the harmonic
spectra now have dimensions Q2L2. Note that Pż共␻兲 corresponds to the HPNS 关20兴, whereas Pz共␻兲 and Pz̈共␻兲 refer to
complementary, albeit related, dynamic properties based on
具z共t兲典 and 具z̈共t兲典, respectively 关26兴.
We take the electric field of the laser pulse to be
E共t兲 =
再
E0 f共t兲cos共␻0t + ␩兲,
ti ⬍ t ⬍ t f
0,
elsewhere,
冎
f共t兲 = exp共− t2/␦2兲.
III. NUMERICAL METHODS
For convenience we rewrite Eq. 共2.5兲 in terms of auxiliary
radial wave functions defined by
Rl共r,t兲 ⬅ r−1␺l共r,t兲.
i
再
冕
dtE共t兲 = 0
共2.28兲
ti
holds 关27兴. The duration of one optical cycle is ␶ = 2␲ / ␻0;
the wavelength is ␭ = ␶c, where c is the speed of light.
We note that the maximum electric field given by Eq.
共2.26兲 and the corresponding maximum intensity 共I = c␧0E2,
where ␧0 is the electric permittivity of vacuum兲 关28兴 depend
on ␩. 共Here and for the remainder of this Section we use SI
units.兲 Likewise, the ponderomotive energy
U p = e2E2/4me␻20 ,
⳵␺共r,t兲
= H共r,t兲␺共r,t兲 = 关Ha共r兲 + W共r,t兲兴␺共r,t兲.
⳵t
Here ␺ is the column vector whose lth element is ␺l共r , t兲; Ha
is the diagonal matrix with elements
关Ha兴ll = −
1 ⳵2 l共l + 1兲 1
+
− .
2 ⳵r2
2r2
r
共3.4兲
The elements of the tridiagonal matrix W, which represents
the interaction of the electric field with the dipole, are given
by
关W共r,t兲兴kl =
冦
lrE共t兲/冑共2l − 1兲共2l + 1兲,
k=l−1
共l + 1兲rE共t兲/冑共2l + 1兲共2l + 3兲,
k=l+1
0
otherwise.
共2.32兲
冧
共3.5兲
The solution of Eq. 共3.3兲 can be written formally as
␺共r,t + ⌬t兲 = U共t + ⌬t,t兲␺共r,t兲.
再
共2.30兲
关i.e., the number of the harmonic beyond which Pz̈共␻兲 rapidly falls off兴 depend on ␩. For reference, we characterize the
intensity of a given pulse in terms of the maximum electric
field 共E0兲 by
I0 = c␧0E20
共3.2兲
The time-evolution operator U is given by
共2.31兲
,
共3.3兲
冉冕
U共t + ⌬t,t兲 = P exp − i
where I p is the ionization potential 共=0.5Eh for the ground
state of the H atom兲, as well as the cutoff number 关25兴
Nm = 共I p + 3.17U p兲/ប␻0
共l + 1兲rE共t兲␺l+1共r,t兲
Equation 共3.2兲 can be recast compactly in matrix form as
共2.29兲
and hence the so-called Keldysh limit for ionization 关29兴
␥ = 冑I p/2U p ,
lrE共t兲␺l−1共r,t兲
冑共2l − 1兲共2l + 1兲 + 冑共2l + 1兲共2l + 3兲
l = 0,1,2 . . . .
Note that f is symmetric about t = 0. We set t f ⯝ −ti and
choose the CEP ␩ so that the condition
tf
冎
⳵␺l共r,t兲
1 ⳵2 l共l + 1兲 1
= −
+
−
␺l共r,t兲
2 ⳵r2
⳵t
2r2
r
+
共2.26兲
共2.27兲
共3.1兲
Substitution of Eq. 共3.1兲 into Eq. 共2.5兲 yields
i
where the carrier envelope is of Gaussian form
共2.33兲
t+⌬t
dt⬘关Ha + W共t⬘兲兴
t
共3.6兲
冊冎
,
共3.7兲
where P signifies the Dyson chronological operator 关30兴. To
second order in ⌬t, U can be approximated by 关31兴
U共t + ⌬t,t兲 ⬵ e−iHa⌬t/2e−iW共t兲⌬te−iHa⌬t/2 .
共3.8兲
The vector ␺共r , t兲 is propagated one step in time from t to
t + ⌬t in three stages corresponding to the successive action
of the three exponential operators in Eq. 共3.8兲 关32,33兴. In the
first stage ␺共r , t兲 evolves to ␺共1兲共r兲 according to the relation
023403-4
PHYSICAL REVIEW A 79, 023403 共2009兲
QUANTUM SIMULATION OF HIGH-ORDER…
共1 + iHa⌬t/4兲␺共1兲共r兲 = 共1 − iHa⌬t/4兲␺共r,t兲.
共3.9兲
Note that since Ha is diagonal, Eqs. 共3.9兲 decouple. We then
solve each implicit differential equation for ␺共1兲
l 共r兲 by the
finite-difference method 关35兴. The second derivative in Ha is
approximated by the three-point central-difference formula.
The 兵␺共1兲
l 共r兲其 are required to vanish at r = 0 and r = rmax.
In the second stage ␺共1兲共r兲 evolves to ␺共2兲共r兲 through the
electric-field-dipole interaction according to
␺共2兲共r兲 = e−iW共r,t兲⌬t␺共1兲共r兲.
with the corresponding numerical first and second derivatives of 具z共t兲典. We have also verified that the numerical solutions satisfy the relations in Eqs. 共2.21兲 and 共2.23兲.
IV. RESULTS
We have computed harmonic spectra for a selection of
laser pulses at fixed wavelength ␭ = 800 nm 共optical cycle
time ␶ = 2.67 fs兲. We examine the influence of variations of
1.0
␺共1兲共r兲 = e−iHa⌬t/2␺共r , t兲, which can be recast 共approximately兲
in the Crank-Nicholson form 关34兴
and substitute the “inverse” of Eq. 共3.11兲, W共r , t兲
= S共r , t兲wdiag共r , t兲S†共r , t兲, into Eq. 共3.10兲 to get
-0.5
␺共2兲共r兲 = S共r,t兲e−iwdiag共r,t兲⌬tS†共r,t兲␺共1兲共r兲.
-1.0
E
共3.11兲
0.0
We diagonalize W共r , t兲 by the unitary transformation 关36兴
S†共r,t兲W共r,t兲S共r,t兲 = wdiag共r,t兲,
共3.12兲
Thus, we obtain ␺共2兲共r兲 by multiplying ␺共1兲共r兲 successively
by the matrices S†, e−iwdiag共r,t兲⌬t, and S. In the third and final
stage we propagate ␺共2兲共r兲 to ␺共r , t + ⌬t兲 by a second application of the Crank-Nicholson finite-difference technique.
The radial finite-difference grid contains nr points, separated
by distance ⌬r, so that rmax = nr⌬r; ⌬r is fixed at 0.125a0.
The choice of nr is dictated essentially by the parameters of
the laser pulse. The time step is fixed at ⌬t = 0.025ប / Eh
= 0.025⫻ 24.2 attoseconds.
To avoid spurious effects due to the reflection of the wave
packet from the boundary at r = rmax, we multiply ⌿共r , t兲 by
a “mask function” 关33兴
兵cos关␲共r − r0兲/2共rmax − r0兲兴其
1/8
, r0 ⬍ r ⬍ rmax .
0
drr2
冕
d⍀兩⌿共r, ␪, ␾,t兲兩2 = 兺
l
冕
r0
8.0
0.0
0.5
1.0
6.0
-1.0
1.0
-2.0
-1.0
-0.0
1.0
2.0
3.0
-1.0
-0.5
-0.0
0.5
1.0
1.5
c)
-0.5
-1.0
r0
4.0
-0.5
E
冎
共3.13兲
冕
2.0
b)
-3.0
For all results reported here we set rmax − r0 = 32a0. The consequence of the wave packet’s entering the “absorber” domain 关r0 , rmax兴 is that the probability of presence of the system in the complementary domain 关0 , r0兴, which is given by
Pr0共t兲 =
-4.0 -2.0 -0.0
0.5
r ⬍ r0
1,
-6.0
0.0
再
-8.0
E
g共r兲 =
a)
0.5
共3.10兲
drr2兩Rl共r,t兲兩2 ,
-1.5
0
t
共3.14兲
falls below unity. We set the maximum value of l to 80 and
monitor Pr0共t兲 to assure that it remains equal to one.
Once the wave packet has been propagated to time t, the
expectation values 具z共t兲典, 具ż共t兲典, and 具z̈共t兲典 are computed by
Eqs. 共2.10兲, 共2.15兲, and 共2.18兲. 关We note that for 具z共t兲典 and
具z̈共t兲典 the angular integrals are evaluated by means of 81point Gauss-Legendre quadrature 关37兴, whereas 具ż共t兲典 is calculated directly from the expression given in Eq. 共2.15兲.兴 The
radial integrations, as well as the time integrations in the FTs
关Eqs. 共2.8兲兴, are done by the trapezoidal rule.
As additional checks on the reliability of the numerical
methods, we have verified that 具ż共t兲典 and 具z̈共t兲典 computed
using the expressions given in Eqs. 共2.15兲 and 共2.18兲 agree
FIG. 1. 共Color online兲 Electric field 关in units of maximum amplitude E0, defined in terms of the intensity by Eq. 共2.32兲兴 versus
time t 共in units of optical-cycle time ␶兲 for selection of laser pulses
specified by Eqs. 共2.26兲–共2.33兲. All pulses have wavelength ␭
= 800 nm, corresponding to optical-cycle time ␶ = 2.67 fs. Durations
of pulses shown in panels 共a兲–共c兲 are, respectively, T = 16␶, 6␶, and
3␶, corresponding to FWHM= 9.33, 3.99, and 2.05 fs, respectively.
Continuous and dashed lines correspond to CEPs ␩ = 0 and −␲ / 2,
respectively. Electric-field amplitudes E0 = 0.0377Eh / a0e for weak
共see Figs. 2–4兲 and E0 = 0.119Eh / a0e for strong 共see Figs. 5–8兲
pulses correspond, respectively, to maximum intensities I0
= 1014 W cm−2 共⬍Ib = 1.4⫻ 1014 W cm−2, the threshold for overbarrier ionization兲 and I0 = 1015 W cm−2 共⬎Ib兲. Respective Keldysh
parameters for weak and strong pulses are ␥ = 1.5 and ␥ = 0.48.
023403-5
PHYSICAL REVIEW A 79, 023403 共2009兲
-8
-12
0.0
log10[Pz ]
.
FIG. 2. 共Color online兲 Expectation values 共in atomic units兲 of
dipole-acceleration 具z̈共t兲典 共eប2 /
a30m2e 兲 共a兲, dipole-velocity 具ż共t兲典
共eប / a0me兲 共b兲, and dipole moment
具z共t兲典 共ea0兲 共c兲 versus time t 共in
units of optical cycles ␶兲, along
with respective harmonic spectra
Pz̈共␻兲 共d兲, Pż共␻兲 共e兲, and Pz共␻兲 共f兲
关in atomic units of 共ea0兲2兴 as a
function of frequency 共in units of
carrier frequency ␻0兲, for longest
weak pulse 共I0 = 1014 W cm−2, E0
= 0.0377Eh / a0e兲.
Solid
and
dashed lines refer to CEPs ␩ = 0
and ␩ = −␲ / 2, respectively 关see
Fig. 1共a兲兴. Initial time ti of pulse is
set to −8␶; final time t f is chosen
close to, but slightly larger than
−ti such that 具ż共t f 兲典 = 0. Actual duration of pulse is therefore slightly
longer than T = 16␶.
e)
-12
-16
0.0
log10[Pz ]
.
-8
-4
b)
f)
-4
c)
-8
-20
-0.2
-16
-12
log10[Pz ]
0
-0.1
<z>
0.1
0.2
-20
-0.01
.
<z>
0.01
-20
-0.02
-16
-0.01
..
<z>
d)
-4
a)
0.01
0.02
BANDRAUK et al.
-8
-6
-4
-2
0
2
4
6
8
0
10
20
30
40
ω
t
intensity I0, duration T = t f − ti, and CEP ␩. Plots of the electric field versus t for the considered pulses are displayed in
Fig. 1. The results, namely 具z共t兲典, 具ż共t兲典, and 具z̈共t兲典 and the
corresponding harmonic spectra Pz共␻兲, Pż共␻兲, and Pz̈共␻兲 are
presented in Figs. 2–7. We consider two sets of results corresponding to 共1兲 “weak” pulses 共I0 = 1014 W cm−2, below the
over-barrier ionization threshold Ib = 1.4⫻ 1014 W cm−2兲 with
durations that correspond to 16, 6 and 3 optical cycles ␶; 共2兲
“strong” pulses 共I0 = 1015 W cm−2 ⬎ Ib兲 of the same durations. For each of these six pulses we look at two CEPs, ␩
= 0 and ␩ = −␲ / 2.
Figure 2 shows the results for the longest weak pulses
共T = 16␶, FWHM= 9.33 fs, ␩ = 0, −␲ / 2兲. These pulses cause
transient oscillations of rather low amplitude in 具z共t兲典, 具ż共t兲典,
and 具z̈共t兲典, as indicated in panels 2共c兲, 2共b兲, and 2共a兲, respectively. We note that the maxima in 具z共t兲典 occur at the minima
of the electric field, in accordance with adiabatic following at
low intensity. After the passage of the pulse, 具z共t兲典 remains
oscillating with small amplitude about nearly constant values
depending on ␩, which are close to, but not necessarily identical to, the initial value 具z共ti兲典 = 0. Consequently, both 具ż共t兲典
and 具z̈共t兲典 oscillate about zero with small amplitude. This
allows us to define the final time t f as the time after −ti when
具ż共t f 兲典 first vanishes 共i.e., the pulse duration T = t f − ti is taken
to be just slightly larger than −2ti, which is 16␶ for the
present case兲.
The resulting harmonic spectra for the three forms are
nearly identical, except for some significant deviations of
Pz共␻兲 for ␩ = −␲ / 2. Moreover, they are nearly insensitive to
␩, again with the exception of Pz共␻兲. The near identity of
Pz̈共␻兲 and Pż共␻兲 may be understood as a consequence of the
vanishing of the dipole-velocity limit contribution 共i.e.,
具ż共t f 兲典 = 0兲 in Eq. 共2.23兲. Likewise, relation 共2.21兲 suggests
that Pż共␻兲 and Pz共␻兲 would be identical if 具z共t f 兲典 were equal
to zero. Figure 2共c兲 shows that this condition is satisfied for
␩ = 0, in contrast with small negative deviations of 具z共t f 兲典
from zero for ␩ = −␲ / 2. As a consequence, Pz共␻兲 for ␩
= −␲ / 2 no longer exhibits the pronounced “falloff” 关i.e., the
precipitous drop in the magnitude of the spectrum for harmonics beyond Nm defined by Eq. 共2.31兲兴 beginning at Nm
= 15 that is evident in the other two spectra, regardless of ␩.
The results for the next shorter weak pulse 共T = 6␶,
FWHM= 3.99 fs兲 are illustrated in Fig. 3. The main findings
are similar to those for the weak T = 16␶ pulse, except for a
significant influence of ␩, which is due to the fact that as ␩
departs from zero, the maximum in the electric field 关see Fig.
1共b兲兴 declines, causing U p and hence Nm to decrease. The
influence of ␩ becomes even more apparent for the shortest,
weak pulse 关T = 3␶, FWHM= 2.05 fs, see Fig. 1共c兲兴. Equa-
023403-6
PHYSICAL REVIEW A 79, 023403 共2009兲
0.02
QUANTUM SIMULATION OF HIGH-ORDER…
d)
-8
-16
0.012 -0.02
0.006
e)
-4
b)
FIG. 3. 共Color online兲 Same as
Fig. 2, except for the pulse of
nominal duration 6␶; ti = −3␶ and
t f determined as described in Fig.
2, such that actual duration is
slightly longer than 6␶.
-16
-12
-8
0.0
log10[Pz ]
.
-0.012 -0.006
-4
f)
-12
-0.14
0
log10[Pz ]
0.14
c)
-8
.
<z>
<z>
-12
0.0
log10[Pz ]
.
-0.01
..
<z>
0.01
-4
a)
-3
-2
-1
0
1
2
3
4
0
10
20
30
40
ω
t
tions 共2.29兲 and 共2.31兲 imply that Nm = 15 for ␩ = 0 compared
with Nm = 13 for ␩ = −␲ / 2. The harmonic spectra plotted in
Fig. 4 bear this out. For the purpose of this paper, however,
the most important conclusion is that for the weak pulse
共I0 ⬍ Ib, the domain of tunneling ionization兲 the harmonic
spectra for all three forms are in agreement, and this is
mainly a consequence of the nearly vanishing mean dipole
velocity and also rather small shifts of the mean dipole moment at the end of the laser pulse.
We now present results for the series of strong pulses
共I0 = 1015 W cm−2 ⬎ Ib兲, starting with the longest 关T = 16␶,
FWHM= 9.33 fs, ␩ = 0 , −␲ / 2 共see Fig. 5兲兴. We note first that
the influence of ␩ is not very pronounced, for the same reasons as in the case of the longest weak pulse 共see Fig. 2兲.
Second, we observe that the amplitudes of the oscillations in
具z共t兲典, 具ż共t兲典, and 具z̈共t兲典 are much greater than for the corresponding weak pulses. The high electric field associated with
the strong pulse induces sudden large-amplitude movement
in 具z共t兲典 关see Fig. 5共c兲兴, not from the very beginning of the
pulse, but only rather close to its maximum intensity, when
the
effective
potential-energy
共Coulombic+ electricdipole interactions兲 is suppressed below the energy of the
initial 共1s兲 state, so that the electron can escape over the
barrier. The 具ż共t兲典 and 具z̈共t兲典 exhibit corresponding sudden
large-amplitude oscillations, compared with those induced
by the weak pulse. The much larger amplitudes of 具z共t兲典,
具ż共t兲典, and 具z̈共t兲典 naturally correlate with large increases in
the magnitudes of the harmonic spectra. Another consequence of the higher intensity of the strong pulse is a dramatic increase from Nm = 15 for the weak pulse to Nm = 70 for
the strong pulse.
Regardless of the substantial influence of intensity evident
in Fig. 5, the most important conclusion, for the present purpose, is that Pż共␻兲 and Pz̈共␻兲 are again very similar, at least
for the domain of the harmonics below Nm. This can again be
explained by Eq. 共2.23兲 共i.e., the two spectra should agree for
the ideal case 具ż共t f 兲典 = 0兲. The apparent discrepancy between
Pż共␻兲 and Pz̈共␻兲 in the falloff domain may be a real effect, or
a numerical artifact, for the following reason: close analysis
reveals that it is almost impossible to realize numerically the
constraint 具ż共t f 兲典 = 0. In the present case, we could achieve
具ż共t f 兲典 = 8 ⫻ 10−6共eប / a0me兲, which is very small, yet still sufficiently large that the two 具ż共t f 兲典-dependent terms in Eq.
共2.23兲 appear to be dominant, compared to Dz̈共␻兲, as ␻ enters the falloff domain. For numerical reasons it is therefore
essentially impossible to decide whether Pż共␻兲 should also
exhibit a falloff 关as implied by Eq. 共2.23兲 for the ideal case
具ż共t f 兲典 = 0兴 for the case where 具ż共t f 兲典 is very small but still not
zero. We demonstrate below for the case of the shortest
strong pulse that the apparent discrepancy between Pz̈共␻兲
023403-7
PHYSICAL REVIEW A 79, 023403 共2009兲
d)
-16
-12
0.0
log10[Pz ]
.
-8
-4
a)
e)
-4
b)
FIG. 4. 共Color online兲 Same as
Fig. 2, except for pulse of nominal
duration 3␶; ti = −1.5␶ and t f determined as described in Fig. 2, such
that actual duration is slightly
longer than 3␶.
-16
-12
-8
0.0
log10[Pz ]
.
0.2 -0.012 -0.006
.
<z>
0.006 0.012 -0.002 -0.001
..
<z>
0.001
0.002
BANDRAUK et al.
f)
-12
-16
-0.2
-1.5
-8
log10[Pz ]
0
-0.1
<z>
0.1
-4
c)
-1.0
-0.0
-0.5
0.5
1.0
1.5
0
20
40
60
80
ω
t
and Pż共␻兲 can be resolved. We note, however, that Pz共␻兲 is
definitely different from either Pż共␻兲 or Pz̈共␻兲 in almost all
domains. This is a consequence of the rather large deviations
of the limit contribution 具z共t f 兲典 from zero 关see Eq. 共2.21兲兴.
Thus, the plots of Figs. 1共a兲 and 5 suggest an important
hypothesis: the HPNS 关20兴 should be derivable from either
the dipole-velocity or dipole-acceleration forms in the domains n ⬍ Nm, but not from the dipole form.
Results for the next shorter strong pulse 共T = 6␶, FWHM
= 3.99 fs兲 are documented in Fig. 6, where it can be seen that
␩ has a heavy impact on 具z共t f 兲典. This is also reflected in
具ż共t f 兲典, which oscillates with small amplitude about zero for
␩ = −␲ / 2, but about a negative value for ␩ = 0. Therefore,
Pż共␻兲 depends markedly on ␩. A prominent qualitative difference is that for ␩ = −␲ / 2, Pż共␻兲 should fall off for n
⬎ Nm, whereas for ␩ = 0 it should not. At the same time the
magnitude of Pż共␻兲 for the higher harmonics is greater for
␩ = 0 than for ␩ = −␲ / 2. This observation supports a working
hypothesis: the previous finding of no falloff for the longer
pulse 关see Fig.5共e兲兴 may also be a real effect. Moreover, the
application of Eq. 共2.23兲 for ␩ = −␲ / 2, with 具ż共t f 兲典 ⬇ 0, implies that Pż共␻兲 and Pz̈共␻兲 should be similar, whereas for ␩
= 0, with 具ż共t f 兲典 ⫽ 0, Pż共␻兲 and Pz̈共␻兲 should differ 关compare
Figs. 6共d兲 and 6共e兲兴. We note again that Pz共␻兲 differs drastically from both Pż共␻兲 and Pz̈共␻兲.
Results for the shortest strong pulse 共T = 3␶, FWHM
= 2.05 fs兲 are displayed in Fig. 7, where it is clear that the
radically different asymptotic behavior of 具z共t兲典 for ␩ = 0 and
␩ = −␲ / 2 results in 具ż共t f 兲典 ⫽ 0. It follows that Pż共␻兲 and
Pz̈共␻兲 differ substantially for the two phases 关see Figs. 7共d兲
and 7共e兲兴. Most prominent is that Pż共␻兲 Ⰷ Pz̈共␻兲. Moreover,
Pz̈共␻兲 exhibits the expected falloff for both CEPs, with different cutoffs close to Nm = 70 for ␩ = 0 versus Nm = 52 for
␩ = −␲ / 2, in accord with Eq. 共2.31兲. In sharp contrast, Pż共␻兲
exhibits no such falloff.
It is instructive to analyze the results in Fig. 7 in terms of
the two-step quasistatic tunneling model 关38兴. In the first step
the H atom is ionized, predominantly during a short time
about the maximum of the electric-field amplitude, where the
carrier envelope remains approximately constant. It is assumed that the electron is liberated at t = t0 with dipole moment r共t0兲 and dipole velocity ṙ共t0兲 = 0 关25兴. In the second
step the electron moves classically under the action of only
the field of the pulse. Its Coulombic attraction to the proton
is ignored. Newton’s second law then yields
where
023403-8
z共t兲 = z共t0兲 + zosc + żd共t0兲关t − t0兴,
共4.1a兲
ż共t兲 = żd共t0兲 − 共E0/␻0兲sin共␻0t + ␩兲,
共4.1b兲
PHYSICAL REVIEW A 79, 023403 共2009兲
0.12
QUANTUM SIMULATION OF HIGH-ORDER…
0
-4
-8
-16
-12
log10[Pz ]
0.0
-20
-0.12
FIG. 5. 共Color online兲 Same as
Fig. 2, except for strong pulse 共I0
= 1015 W cm−2, E0 = 0.119Eh / a0e兲
of nominal duration 16␶. ti = −8␶
and t f determined as described in
Fig. 2, such that actual duration is
slightly longer than 16␶.
-8
0.0
log10[Pz ]
.
-4
0
e)
30
2 -20
-16
-12
0.12
b)
-0.12
f)
0
-2
-8
-30
-20
-6
-10
0
log10[Pz ]
10
20
c)
-4
.
<z>
<z>
d)
.
-0.06
..
<z>
0.06
a)
-8
-6
-4
-2
0
2
4
6
8
0
20
40
t
zosc ⬅ 共E0/␻20兲关cos共␻0t + ␩兲 − cos共␻0t0 + ␩兲兴, 共4.2a兲
żd共t0兲 ⬅ 共E0/␻0兲sin共␻0t0 + ␩兲.
共4.2b兲
At long times the last term on the right side of Eq. 共4.1a兲
dominates. The electron drifts away from the proton with
approximately constant velocity żd共t0兲. From Eq. 共4.2b兲 it
follows that żd共−t0兲 = −żd共t0兲 when ␩ = 0 and żd共−t0兲 = żd共t0兲
when ␩ = −␲ / 2. Hence, since the ionization rate is approximately symmetric in time about the maximum, the average
drift 共dipole兲 velocity of the electron should vanish when ␩
= 0, whereas it should be either negative or positive when
␩ = −␲ / 2. In other words, when ␩ = 0, electrons are emitted
in the positive and negative z directions with equal likelihood, whereas when ␩ = −␲ / 2, they are emitted preferentially in either the negative or the positive z direction. The
two CEPs ␩ = 0 and ␩ = −␲ / 2, respectively, correspond to
symmetric and asymmetric photoelectron kinetic-energy
spectra 关38兴. The plots in Figs. 7共b兲 and 7共c兲 are roughly in
accord with the model. Although 具ż共t f 兲典 ⫽ 0 when ␩ = 0, it is
much smaller than 具ż共t f 兲典 for ␩ = −␲ / 2. We note that since
the 共mean兲 drift, or postpulse, velocity is constant, the
共mean兲 postpulse acceleration vanishes 关see Fig. 7共a兲兴.
ω
60
80
The predicted harmonic spectra depend on the length of
the period over which photons are collected. Until now we
have regarded this period to be the same as the duration T of
the pulse. We now consider the extreme case of collecting
photons forever 共i.e., from ti to ⬁兲. For this purpose we employ a numerical trick: we define t f not by 具ż共t f 兲典 = 0, as for
the results shown in Figs. 2–5, but rather by choosing t f
slightly greater than −ti so that 具z̈共t f 兲典 = 0. Furthermore, we
set 具z̈共t兲典 = 0 for times t ⬎ t f , assuming that there are no significant effects of accelerations after the passage of the pulse.
Thus, in the FT 关see Eq. 共2.8兲兴 the upper limit can be extended to ⬁ for the acceleration form, simply because there is
no contribution beyond t = t f . We refer to the acceleration
form of the spectrum thus calculated as the infinite-time
form, denoted by Pz̈inf共␻兲. It is numerically advantageous
since it can be calculated rigorously from the numerically
generated 具z̈共t兲典 and then used in Eq. 共2.23兲 to compute a
corresponding infinite-time dipole-velocity spectrum Pżinf共␻兲.
A comparison of Pż共␻兲 and Pżinf共␻兲 is made in Fig. 8 for ␩
= 0 关see Fig. 7共b兲兴. The agreement is nearly quantitative.
Similar results are obtained for the case represented by Fig.
6. We conclude that most of the photons are produced during
T and that collecting photons for longer times will not significantly alter the harmonic spectra.
023403-9
PHYSICAL REVIEW A 79, 023403 共2009兲
BANDRAUK et al.
-4
0
d)
1.2
-16
-0.06
-12
-8
log10[Pz ]
.
0.0
..
<z>
0.06
a)
-4
0
e)
FIG. 6. 共Color online兲 Same as
Fig. 5, except nominal duration is
6␶. ti = −3␶; t f chosen close to, but
slightly larger than −ti, such that
具z̈共t f 兲典 = 0. Actual duration is
slightly longer than 6␶.
-8
0.0
log10[Pz ]
.
2
40
-1.2
-12
-0.6
.
<z>
0.6
b)
f)
0
-6
-40
-4
-2
log10[Pz ]
0
-20
<z>
20
c)
-3
-2
-1
0
1
2
3
4
0
20
40
60
80
ω
t
Finally, we note that harmonic spectra corresponding to
the long pulses 共see Figs. 2 and 5兲 manifest the “rule” that
only odd harmonics appear in the domain n ⬍ Nm 关33兴, in
stark contrast to the spectra produced by the shortest pulses
共Figs. 4 and 7兲, which disobey the rule. The reason for this
breakdown is that the single carrier frequency ␻0 dominates
the long pulses, whereas many frequencies effectively contribute to the short ones. In the extreme limit of ultrashort
共few-attosecond兲 pulses, the distinction between odd and
even harmonics disappears and the HHG spectrum becomes
continuous 关24,39兴. This is also reflected in the strong asymmetry seen in simulations of photoelectron kinetic-energy
spectra 关40兴.
V. CONCLUSION
In this paper we have presented comparisons of three
forms of the harmonic spectrum, based on dipole, dipole
velocity, and dipole acceleration, by solving the Schrödinger
equation numerically for the hydrogen atom interacting with
a laser pulse whose electric field is specified by E共t兲
= E0 f共t兲cos共␻0t + ␩兲, where f共t兲 = exp共−t2 / ␦2兲. We have examined the influence of intensity I0 共proportional to E20兲, duration T 共proportional to ␦兲, and the phase ␩ of the carrier
envelope. We have considered two sets of pulses: 共1兲 “weak”
共I0 = 1014 W cm−2 ⬍ Ib = 1.4⫻ 1014 W cm−2, the threshold intensity for over-barrier ionization兲 with durations T between
3 and 16 optical cycles; 共2兲 “strong” 共I0 = 1015 W cm−2 ⬎ Ib兲
with the same durations. The carrier frequency ␻0 corresponds to the wavelength 800 nm for all pulses.
The principal finding is that for long, weak pulses the
three forms of the harmonic spectrum are in good agreement.
This is so because the limit contributions by which the forms
differ 共except for scalings by powers of frequency兲 in general, according to Eqs. 共2.21兲 and 共2.23兲, are negligible when
the pulse is long and weak. This confirms previous findings
by Telnov and Chu 关21兴 for HHG rates based on weak, periodic, or continuous-wave fields. On the other hand, as the
pulse becomes stronger and shorter, the limit contributions
increase gradually and the three spectral forms tend to diverge.
Following the passage of a short 共few-cycle兲, intense
pulse that ionizes the H atom, 具z共t兲典 increases without bound
共corresponding to unidirectional ionization 关41,42兴兲 while
具ż共t兲典 oscillates about a nonzero value, whereas 具z̈共t兲典 decays
to zero. By a mathematical artifice we can exploit this behavior to reconcile the divergent forms. In particular, for the
strongest, shortest pulse studied here, we demonstrate explicitly how the infinite-time dipole-velocity spectrum Pżinf共␻兲
023403-10
PHYSICAL REVIEW A 79, 023403 共2009兲
0
a)
-8
-4
d)
-20
0.5 -0.06
-16
-12
0.0
log10[Pz ]
.
-0.03
e)
0
b)
FIG. 7. 共Color online兲 Same as
Fig. 5, except nominal duration is
3␶. ti = −1.5␶ and t f determined as
described in Fig. 6, such that actual duration is slightly longer
than 3␶. The inset in 共c兲 displays
具z共t兲典 on fine scale.
2 -12
20 -1.0
-8
-0.5
.
<z>
log10[Pz ]
0.0
.
-4
..
<z>
0.03
0.06
QUANTUM SIMULATION OF HIGH-ORDER…
f)
-1.0
-0.0
-0.5
0
-2
log10[Pz ]
0.5
1.0
1.5
0
20
40
80
60
ω
t
-2
-4
0
-2
2
since the postpulse 共mean兲 dipole velocity is constant and the
共mean兲 dipole acceleration vanishes, the acceleration form of
the harmonic spectrum provides a superior computational
pathway to the number of harmonic photons. Finally, we
note that our results are also relevant to the analysis of
log10[Pz ]
.
3
4
5
-8
-6
-4
2
-10
can be computed from the infinite-time acceleration spectrum Pz̈inf共␻兲. Pżinf共␻兲 is shown to agree almost quantitatively
with the spectrum Pż共␻兲 computed for the finite time T = t f
− ti, namely, the pulse duration. Apparently most of the photons that contribute to the spectrum are produced during the
passage of the pulse. Hence, even though the 共idealized兲 experimental spectrum 共HPNS兲 关20兴 is expressed in terms of
the dipole-velocity form, the acceleration form affords a convenient numerical route to the HPNS.
The relations in Eqs. 共2.21兲 and 共2.23兲 are especially useful for calculating and interpreting harmonic spectra currently being generated with intense, few—to single-cycle laser pulses 关23,24兴. We emphasize the extreme sensitivity of
spectra to the CEP ␩ for intense, ultrashort pulses, which was
also observed in previous simulations 关43兴. One needs the
precise formulation represented by Eqs. 共2.21兲 and 共2.23兲 in
order to handle such pulses, which give rise to nonvanishing
final 共mean兲 dipole moment 具z共t f 兲典 and 共mean兲 dipole velocity 具ż共t f 兲典. These postpulse “residues” have a critical impact
on the symmetry of photoelectron kinetic-energy spectra 共a
signature of the CEP 关27,38,40,41,44兴兲, as well as on the
“parity” of the harmonics 共i.e., both odd and even harmonics
appear in the spectra 关39兴兲. It is particularly noteworthy that,
-6
-1.5
-4
-60
0
0.8
-20
η=0
-40
<z>
0
c)
0
10
20
ω
30
40
FIG. 8. 共Color online兲 Comparison of infinite-time spectrum
Pżinf共␻兲 关in atomic units of 共ea0兲2, as a function of frequency in units
of the carrier frequency ␻0兴 共dashed line, computed as described in
text兲 with finite-time spectrum Pż共␻兲 关solid line, displayed in Fig.
7共e兲兴 for shortest, strong pulse with ␩ = 0.
023403-11
PHYSICAL REVIEW A 79, 023403 共2009兲
BANDRAUK et al.
orientation-dependent molecular high-harmonic spectra in
orbital tomography 关17兴, which leads to confirmation of the
three-step recollision model 关25兴 for molecules 关7兴.
Substituting Eq. 共2.4兲 into Eq. 共B1兲 and making the change
of variables x = cos ␪, we get
ACKNOWLEDGMENTS
The authors are grateful to Ingo Barth 共Berlin兲 for diligently checking various formulas and to Robert Gordesch
共Berlin兲 for checking selected numerical results. They are
also thankful to Professor Paul Corkum 共Ottawa兲 for clarifying the HHG experiment. A.D.B. and J.M. thank the Alexander von Humboldt Foundation for financial support of
A.D.B. J.M. also gratefully acknowledges the support of the
Deutsche Forschungsgemeinschaft 共through Project No. Sfb
450-TP C1兲 and of the Fonds der Chemischen Industrie.
D.J.D. thanks the Freie Universität Berlin for financial support for several visits.
APPENDIX A
For the convenience of the reader in following the developments in Sec. II, we summarize here relevant properties of
the 共spherical兲 Legendre polynomials Pl共cos ␪兲 关45兴.
冕
␲
d␪ sin ␪ Pl共cos ␪兲Pk共cos ␪兲
0
=
冕
⫻
冕
=
共A1兲
⫻
+
再冕
冕
冋
+ 2k␦k,l+1/关共2k + 1兲共2k − 1兲兴,
共A2兲
dPk共x兲
= − kxPk共x兲 + kPk−1共x兲.
dx
共A3兲
APPENDIX B
⫻
冕
冕
冕
0
⫻
drrRl*共r,t兲
drr2Rl*共r,t兲
⳵Rk共r,t兲
⳵r
冕
冎
1
dxPl共x兲xPk共x兲
−1
drrRl*共r,t兲Rk共r,t兲
冕
1
dxPl共x兲xPk共x兲 + k
−1
冕
1
dxPl共x兲Pk−1共x兲
−1
i
兺 兺 关共2l + 1兲共2k + 1兲兴1/2
2 l k
+
冕
⬁
0
Rk共r,t兲
r
* 共␪, ␾兲sin ␪
d⍀Y l0
−1
dPk共x兲
,
dx
册冎
.
再冕
⬁
0
drr2Rl*
⳵Rk
⳵r
关共2l + 1兲共2l − 1兲兴 + 2k␦k,l+1/关共2k + 1兲共2k − 1兲兴兲
⳵Rk共r,t兲
drr Rl*共r,t兲
⳵r
2
* 共␪, ␾兲cos ␪Y 共␪, ␾兲
d⍀Y l0
k0
⬁
−
0
dxPl共x兲共1 − x2兲
⫻ 共2l␦l,k+1/
Equation 共2.14兲 can be rewritten as
k
1
Inserting the relations in Eqs. 共A1兲 and 共A2兲 into Eq. 共B3兲
yields
具ż共t兲典 = −
l
冕
dxPl共x兲xPk共x兲
−1
共B3兲
= 2l␦l,k+1/关共2l + 1兲共2l − 1兲兴
再冕
⬁
⫻ −k
dxPl共x兲xPk共x兲
具ż共t兲典 = − i 兺 兺
⬁
0
0
1
⬁
drrRl*共r,t兲Rk共r,t兲
1
i
兺 兺 关共2l + 1兲共2k + 1兲兴1/2
2 l k
−1
共1 − x2兲
冕
⳵Rk共r,t兲
⳵r
where we have performed the integration on the azimuthal
angle. Utilizing the differential relation 共A3兲, we can rewrite
Eq. 共B2兲 as
1
冕
⬁
drr2Rl*共r,t兲
共B2兲
d␪ sin ␪ Pl共cos ␪兲cos ␪ Pk共cos ␪兲
0
⬁
0
0
具ż共t兲典 = −
dxPl共x兲Pk共x兲 = 2␦lk/共2l + 1兲,
再冕
冕
+
−1
␲
i
兺 兺 关共2l + 1兲共2k + 1兲兴1/2
2 l k
具ż共t兲典 = −
drrRl*Rk关− k共2l␦l,k+1/关共2l + 1兲共2l − 1兲兴
冎
+ 2k␦k,l+1/关共2k + 1兲共2k − 1兲兴兲 + k2␦l,k−1/共2l + 1兲兴 ,
冎
⳵Y k0共␪, ␾兲
.
⳵␪
共B4兲
共B1兲
which can be expanded in detail as
023403-12
PHYSICAL REVIEW A 79, 023403 共2009兲
QUANTUM SIMULATION OF HIGH-ORDER…
具ż共t兲典 = −
⫻
i
2
再兺 兺
冕
关共2l + 1兲共2k + 1兲兴1/2
k
l
⬁
0
drr2Rl*
⬁
0
冉
drr2Rl*
冊
冉
冊
2l␦l,k+1
⳵Rk
+ 兺 兺 关共2l + 1兲共2k + 1兲兴1/2
⳵r 关共2l + 1兲共2l − 1兲兴
k
l
2k␦k,l+1
⳵Rk
− 兺 兺 关共2l + 1兲共2k + 1兲兴1/2
⳵r 关共2k + 1兲共2k − 1兲兴
k
l
− 兺 兺 关共2l + 1兲共2k + 1兲兴1/2
l
冕
k
冕
⬁
0
drrRl*Rk
冉
冊
冕
⬁
drrRl*Rk
0
冉
2kl␦l,k+1
关共2l + 1兲共2l − 1兲兴
2k2␦k,l+1
+ 兺 兺 关共2l + 1兲共2k + 1兲兴1/2
关共2k + 1兲共2k − 1兲兴
k
l
冕
冊
⬁
drrRl*Rk
0
冉
2k␦l,k−1
共2l + 1兲
冊冎
.
共B5兲
Carrying out the summations on l in the first and third terms and on k in the second, fourth, and fifth terms, respectively, in Eq.
共B5兲, we obtain
i
2
具ż共t兲典 = −
⫻
再
冕
冕
关共2k + 1兲共2k + 3兲兴1/2
2共k + 1兲
共2k + 1兲共2k + 3兲
⬁
0
⫻
兺k
⬁
0
drr2Rl*
冕
⬁
0
*
drr2Rk+1
关共2l + 1兲共2l + 3兲兴1/2
⳵Rk
2共l + 1兲
+兺
共2l + 1兲共2l + 3兲
⳵r
l
关共2k + 1兲共2k + 3兲兴1/2
⳵Rl+1
2k共k + 1兲
−兺
共2k + 1兲共2k + 3兲
⳵r
k
drrRl*Rl+1 + 兺
l
冕
⬁
0
关共2l + 1兲共2l + 3兲兴
2共l + 1兲共2l + 3兲
共2l + 1兲共2l + 3兲
1/2
关共2l + 1兲共2l + 3兲兴1/2
2共l + 1兲2
共2l + 1兲共2l + 3兲
* R −
drrRk+1
k 兺
冕
冎
l
⬁
0
共B6兲
drrRl*Rl+1 .
By redefining dummy indices and combining terms we can simplify Eq. 共B6兲 to
具ż共t兲典 = − i
再兺
l
−兺
l
共l + 1兲
关共2l + 1兲共2l + 3兲兴1/2
l共l + 1兲
关共2l + 1兲共2l + 3兲兴1/2
冕
冋冕
⬁
0
⬁
0
drr2Rl*
冕
⳵Rl+1
+
⳵r
* +
drrRlRl+1
兺
l
⬁
*
drr2Rl+1
0
⳵Rl
⳵r
共l + 1兲共l + 2兲
关共2l + 1兲共2l + 3兲兴1/2
册
冕
冎
⬁
共B7兲
drrRl*Rl+1 .
0
Now consider the integral
I⬅
冕
⬁
0
drr2Rl*
dRl+1
,
dr
共B8兲
t fixed,
which appears in the first term on the right side of Eq. 共B7兲. Integration by parts yields
I = 兩r2Rl*Rl+1兩⬁0 −
冕 冋
⬁
dr 2rRl* + r2
0
dRl*
dr
册
共B9兲
Rl+1 .
The first term on the right side of Eq. 共B9兲 vanishes by virtue of the boundary conditions. Substitution of Eq. 共B9兲 into Eq.
共B7兲 gives
具ż共t兲典 = − i
再兺
l
−兺
l
冋
共l + 1兲
−2
关共2l + 1兲共2l + 3兲兴1/2
l共l + 1兲
关共2l + 1兲共2l + 3兲兴1/2
冕
⬁
0
冕
0
再兺
l
−兺
l
drrRl*Rl+1 −
* +
drrRlRl+1
兺
l
Combining like terms in Eq. 共B10兲, we obtain
具ż共t兲典 = − i
⬁
冋冕
冋冕
共l + 1兲
关共2l + 1兲共2l + 3兲兴1/2
l共l + 1兲
关共2l + 1兲共2l + 3兲兴1/2
drr2
drr2Rl+1
⳵Rl*
0
⳵Rl *
R −
⳵r l+1
⬁
0
⬁
l共l + 1兲 + 2共l + 1兲
关共2l + 1兲共2l + 3兲兴1/2
⬁
0
冕
* −
drrRlRl+1
冕
冕
⳵r
冕
drr2
⬁
*
drr2Rl+1
冎
⬁
drrRl*Rl+1 .
⳵Rl*
0
⬁
0
冕
0
0
⬁
+
drrRl*Rl+1
⳵r
册冎
Rl+1
⳵Rl
⳵r
册
共B10兲
册
.
Now observing that the pairs of terms in brackets are complex conjugates of each other, we can recast Eq. 共B11兲 as
023403-13
共B11兲
BANDRAUK et al.
具ż共t兲典 = − i
再兺
l
= 2兺
l
PHYSICAL REVIEW A 79, 023403 共2009兲
冋冕
共l + 1兲
2i Im
关共2l + 1兲共2l + 3兲兴1/2
再
冋冕
共l + 1兲
Im
关共2l + 1兲共2l + 3兲兴1/2
⬁
drr2
0
⬁
0
drr2
册
冋冕
l共l + 1兲
⳵Rl *
−兺
2i Im
R
关共2l
+
1兲共2l + 3兲兴1/2
⳵r l+1
l
册
冋冕
⳵Rl *
− l Im
R
⳵r l+1
关1兴 T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 共2000兲.
关2兴 P. B. Corkum and F. Krausz, Nat. Phys. 3, 381 共2007兲.
关3兴 S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve,
Phys. Rev. Lett. 97, 123003 共2006兲.
关4兴 B. Sundaram and P. W. Milonni, Phys. Rev. A 41, 6571
共1990兲.
关5兴 J. L. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev.
Lett. 68, 3535 共1992兲.
关6兴 H. Xu, X. Tang, and P. Lambropoulos, Phys. Rev. A 46,
R2225 共1992兲.
关7兴 M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and
P. B. Corkum, Phys. Rev. A 49, 2117 共1994兲.
关8兴 X.-M. Tong and S.-I. Chu, Chem. Phys. 217, 119 共1997兲.
关9兴 G. Lagmago Kamta and A. D. Bandrauk, Phys. Rev. A 71,
053407 共2005兲.
关10兴 J. Chen and S. G. Chen, Phys. Rev. A 75, 041402共R兲 共2007兲.
关11兴 A. D. Bandrauk, S. Barmaki, and G. Lagmago Kamta, in
Progress in Ultrafast Intense Laser Science, edited by K. Yamanouchi 共Springer, New York, 2007兲, Vol. 3.
关12兴 C. C. Chirilă and M. Lein, J. Mod. Opt. 54, 1039 共2007兲.
关13兴 K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys. Rev.
A 45, 3347 共1992兲.
关14兴 A. Sanpera, P. Jönsson, J. B. Watson, and K. Burnett, Phys.
Rev. A 51, 3148 共1995兲.
关15兴 J. Prager, S. X. Hu, and C. H. Keitel, Phys. Rev. A 64, 045402
共2001兲.
关16兴 S. X. Hu and L. A. Collins, Phys. Rev. A 69, 033405 共2004兲.
关17兴 J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C.
Kieffer, P. B. Corkum, and D. M. Villeneuve, Nature 共London兲
432, 867 共2004兲.
关18兴 M. Lein, J. Phys. B 40, R135 共2007兲.
关19兴 J. J. Carrera, X. M. Tong, and Shih-I. Chu, Phys. Rev. A 74,
023404 共2006兲.
关20兴 D. J. Diestler, Phys. Rev. A 78, 033814 共2008兲.
关21兴 D. A. Telnov and Shih-I. Chu, Phys. Rev. A 71, 013408
共2005兲.
关22兴 E. Lorin, S. Chelkowski, and A. D. Bandrauk, New J. Phys.
10, 025033 共2008兲.
关23兴 E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J.
Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T.
Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, Science 320, 1614 共2008兲.
关24兴 F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goul-
关25兴
关26兴
关27兴
关28兴
关29兴
关30兴
关31兴
关32兴
关33兴
关34兴
关35兴
关36兴
关37兴
关38兴
关39兴
关40兴
关41兴
关42兴
关43兴
关44兴
关45兴
023403-14
⬁
0
*
drrRlRl+1
册冎
.
⬁
0
*
drrRlRl+1
册冎
共B12兲
ielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and
G. G. Paulus, Phys. Rev. Lett. 95, 040401 共2005兲.
P. B. Corkum, Phys. Rev. Lett. 71, 1994 共1993兲.
See, for example, Y.-S. Huang and K.-H. Lu, Found. Phys. 38,
151 共2008兲.
D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, J.
Phys. B 39, R203 共2006兲.
W. H. Flygare, Molecular Structure and Dynamics 共PrenticeHall, Englewood Cliffs, New Jersey, 1978兲, Chap. 1.
L. V. Keldysh, Sov. Phys. JETP 20, 1307 共1965兲.
See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation 共Wiley, New York, 1973兲, p. 64ff.
M. D. Feit and J. A. Fleck, Jr., J. Chem. Phys. 78, 301 共1983兲.
P. L. DeVries, J. Opt. Soc. Am. B 7, 517 共1990兲.
J. L. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. A
45, 4998 共1992兲.
See, for example, S. Chelkowski and A. D. Bandrauk, Phys.
Rev. A 41, 6480 共1990兲.
See, for example, W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77
共Cambridge University Press, Cambridge, 1986兲, Chap. 19.
Ref. 关35兴, Chap. 11.
Ref. 关35兴, p. 140.
S. Chelkowski, M. Zamojski, and A. D. Bandrauk, Phys. Rev.
A 63, 023409 共2001兲.
A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E.
Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A.
Scrinzi, T. W. Hänsch, and F. Krausz, Nature 共London兲 421,
611 共2003兲.
S. Chelkowski, A. D. Bandrauk, and A. Apolonski, Phys. Rev.
A 70, 013815 共2004兲.
S. X. Hu and A. F. Starace, Phys. Rev. A 68, 043407 共2003兲.
A. D. Bandrauk, S. Chelkowski, D. J. Diestler, J. Manz, and
K.-J. Yuan, Int. J. Mass. Spectrom. 277, 189 共2008兲.
A. deBohan, P. Antoine, D. B. Milošević, and B. Piraux, Phys.
Rev. Lett. 81, 1837 共1998兲.
G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 91, 253004
共2003兲.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions 共Dover Publications, Inc., New York, 1972兲, Chap.
22.