Bohmian-trajectory analysis of high-order harmonic generation: Probing the local dynamics

1.8 MB·English

by  J. Wu

Bohmian-trajectory analysis of high-order harmonic generation:
Probing the local dynamics
J. Wu,1 A. S. Sanz,1,2 B. B. Augstein,1 and C. Figueira de Morisson Faria1
1Department of Physics and Astronomy, University College London,
Gower Street, London WC1E 6BT, United Kingdom
2Instituto de F´ısica Fundamental (IFF–CSIC), Serrano 123, 28006 Madrid, Spain
(Dated: January 10, 2013)
In this article, we perform detailed studies of high-order harmonic generation (HHG) employing
3 Bohmiantrajectories,withparticularemphasisonwhichspatialregionsarethemostrelevantforthis
1 phenomenon. Thesetrajectories, directlyextractedfrom thetime-dependentSchr¨odingerequation,
0 playtheroleof“tracerparticles”,thusbeingveryhelpfulformappingtheflowofthetime-dependent
2 wavefunction. In particular, they allow us to revisit several issues, such as the use of the dipole
lengthorthedipoleacceleration inthecomputationofharmonicspectra,theinfluenceofnumerical
n
filterson theflowof thewavefunction inreal time, ortheroleof theinnermost andoutermost core
a
regions and of ionization in the spectra. Our results show that the Bohmian trajectory starting
J
at the innermost part of the core is the most relevant in order to reproduce typical HHG spectra,
9
regardlessofthepulseshapeandofthepotentialrangeemployed. Thisprovidesadditionalinsighton

thesuccessofstrong-fieldmodelsthatconsidertheatomtobeasourceterm,suchasthestrong-field
]
h approximation. A quantitative agreement between the spectra obtained from Bohmian-trajectory
p computationsand thetime-dependentSchr¨odingerequation is also obtained. Furthermore,we also
- establish a parallel between the use of the dipole acceleration and the Hanning filter, by analyzing
m
how, in both cases, Bohmian trajectories away from thecore are removed.
o
t PACSnumbers: 32.80.Rm,42.50.Hz
a
.
s
c I. INTRODUCTION sufficestoassumethatthecoreisstaticandtotreatitas
si a source term, located at the origin r=0 of the coordi-
y natesystem. ThisisthekeyassumptionbehindtheSFA,
h High-order harmonic generation (HHG) is a well-
whichapproximatesthecontinuumbyfield-dressedplane
p known phenomenon that occurs when matter interacts
waves. Ontheonehand,thissimplifiesthe problemcon-
[ with a strong laser field, of intensities of the order of
siderablyandallowsforintuitive,semi-analyticalmodels
or higher than I 1013 W/cm2. Since the late 1980s,
1 ∼ to be developed. On the other hand, it does not allow
thisphenomenonhasbeeninvestigatedboththeoretically
v for quantitative predictions of harmonic spectra and un-
6 and experimentally. It has attracted a great deal of at-
derplays the role of the core. In fact, even for a simple,
1 tention due to the myriad of possible applications, such
one-electron system, such as the hydrogen atom, pulling
9 astable-top,extremeultraviolet(XUV)sources[1],high-
out one electron will alter the charge distribution in the
1 frequencylightpulsesofattosecondduration[2,3](for a
. core considerably. On a more general note, it has been
1 review,seeRef.[4]),and,inrecentyears,ithasalsobeen
recently shown that core polarization and excitation [8],
0 identified as a powerful tool for the attosecond imaging
aswellasthe influence ofthe Coulombpotential,areex-
3 ofdynamic processesinmatter [5–9]. Theseapplications
1 pected to play animportantrolein strong-fieldphenom-
have been made possible due to a very intuitive phys-
: ena [16, 17]. Specific examples are the so-called low en-
v ical interpretation of HHG in terms of electron orbits,
ergy structure [18] and interference patterns [19] in ATI.
i in which an electron, under the influence of the external
X
laser field, reaches the continuum by tunnel ionization, Abinitiomethods,suchastheTDSE,though,provide
r
a is accelerated by the field, and, at a subsequent time, the full time-dependent wavefunction, but do not give a
recollides with its parent ion [10]. Upon recollision, the clearspace-timepictureintermsofelectronorbits. Typi-
electron may recombine with a bound state, thus emit- cally,thispictureisextractedbyusingwindowedFourier
ting high-frequency radiation. This physical picture has transforms and constructing time-frequency maps. Oth-
been put across in the context of the three-step model erwise,the mainfocus is onhow to minimize the numer-
(TSM). It has been explored both in classical models ical background and spurious reflections in the spectra.
[10] and in quantum-mechanicalapproaches,such as the From this perspective, it is well known that using the
strong-field approximation (SFA) [11]. Furthermore, the expectation value of the dipole acceleration instead of
recollisionpicture has also been inferredfrompurely nu- that of the dipole length leads to higher-quality spec-
merical models, such as the numerical solution of the tra [20, 21]. Physically, it has been argued that, since
time-dependent Schr¨odinger equation (TDSE), by time- the light emitted by a moving charge, i.e., the electron,
frequency analysis (see, e.g., [12–15]). is proportional to its dipole acceleration, it would make
The above-stated picture is consistent with the fact more sense to use this quantity as the observable from
that, in the modeling of HHG, to first approximation, it which HHG spectra are calculated. On a more techni-
2
cal level, the dipole acceleration probes regions near the HHG phenomenon. To that end, we focus the discussion
core, while the dipole length emphasizes large distances. onthedipolelengthvsdipoleaccelerationissue,andpro-
This implies that the latter observable is far more sen- vide an alternative insight on the interplay between the
sitive with regard to reflections at the edges of the box electronandthecoredynamicsinHHG.Accordingly,we
employed in the numerical integration. This can be re- have organized this work as follows. In Sec. II we pro-
duced using absorbing boundaries and a Hanning filter, vide the necessarytheoreticalbackgroundto understand
but obviously one alsoexpects that these devices will al- our subsequent results, which includes the model con-
ter the system dynamics. sidered, some remarks about Bohmian mechanics, the
Apartfromtheaforementionedapproaches,thereexist choice of initial conditions, and the propagation meth-
other theoreticalmethods, which allow a space-time pic- ods used here. The results obtained are presented in
tureofHHGintermsofelectronorbitsandstillinvolvea Sec. III. More specifically, in Sec. IIIB we commence by
lesserdegreeofapproximationthantheSFA.Oneofsuch making a qualitative assessment of the different contri-
methods is Bohmian mechanics [22, 23]. In strong-field butions to the spectra for specific trajectories, near the
physics, so far, Bohmian mechanics has only been em- core and near the boundary. Thereafter, in Sec. IIIC
ployedveryrecentlyin ahandful ofpublications [24–30]. we present ensemble computations, in which we identify
It is, however, a powerful tool in order to assess the in- which regions of the core lead to a quantitative agree-
terplaybetweentheresidualpotentialandthelaserfield. ment between the Bohmian-trajectory computation and
Withinananalyticworkingscheme[31],Bohmiantrajec- the TDSE, for both length and acceleration. Finally, in
tories are directly obtained from the full solution of the Sec.IIID,weassesstheinfluenceofHanningfiltersonthe
TDSE. Hence they do not come from any sort of phys- electrondynamics,farawayandinthe coreregion. Here
ical approximation, but they contain all the quantum weshowthatBohmianmechanicsallowsareconstruction
phase information necessary for the full description of of the equivalent dynamics to having a filter, something
the problem. Because of this reason, Bohmian trajecto- which is not possible at all with the wavefunction. Note
riesareemployedinmanyareasofphysicsandchemistry that once the expectation value (either the length or the
as “tracerparticles”[32, 33], whichprovide aninsightful acceleration) is computed to obtain the corresponding
hydrodynamic-likepictureoftheprobability-densityflow spectrum, all the information about the local configura-
associated with the quantum mechanical wavefunction tion space dynamics is irreversibly lost (in other words,
(indeed,thispictureisprettymuchrelatedtoMadelung’s one cannot obtain the wavefunction or the probability
former hydrodynamical reformulation of quantum me- density from such an expectation value). Closing this
chanics [34]). work, the main conclusions and results are summarized
in Sec. IV.
Recently, we have computed HHG spectra employing
Bohmian trajectories [30], and compared such results
with those obtained from the numerical solution of the
II. THEORY
one-dimensionalTDSE.Wefoundthatthetrajectorylo-
cated at the innermost part of the core [i.e., starting at
A. The model
x(0) = 0] best reproduces the HHG spectra obtained
from the dipole acceleration. Both spectra were quali-
tatively very similar, apart from an overall difference in For the sake of clarity,we have developedour analysis
intensity. This provided additional insight on the role in one dimension, which suffices to illustrate the time-
played by specific spatial regions in HHG. First, since dependent electron dynamics, and yet contains the es-
theinnermostregioncontributesthemost,itislegitimate sential physical elements for linear polarization. Making
to assume that, to first approximation, the core may be use of the dipole approximation and the length gauge,
representedbyasourcetermattheorigin,x=0,ascon- the electron dynamics is then described by the time-
sidered in the SFA. Second, using time-frequency maps, dependent Hamiltonian (in atomic units)
wehaveshownthataspatiallyconfinedBohmiantrajec-
1
tory may contain both bound and continuum dynamics. H = 2+V(x) xE F(t), (1)
0
In fact, the specific time-frequency profile obtained from −2 ∇ −
the above-mentionedBohmian trajectorywas associated
where E F(t) denotes the driving field and V(x) the
with that of an ensemble of classical trajectories under- 0
binding potential (note that the minus sign in the last
going an unbound motion. In contrast, the motion of
term arises from the electron charge: in atomic units,
bound classical trajectories may be associated with spe-
e= 1).
cific, confined regions in space. A still open question, −
The quantities that we are interested in here are the
though,is whatspecific regionsinspacearenecessaryto
dipole length,
obtain a quantitative agreement between the Bohmian-
trajectory and the full TDSE computations.
x¯(t)= ΨxΨ , (2)
Inthe presentworkwediscussanumber ofissuesthat h | | i
will give a robust idea about why Bohmian mechanics which, in atomic units, is equal to the dipole moment,
is an excellent tool to probe the local dynamics in the butwithoppositesign,sinceµ(t)=ex¯(t),andthedipole
3
acceleration, isregardedasdescribinga(quantum)transportproblem
[32], with a local velocity field being defined as v=J/ρ.
′
a¯(t)= ΨV (x)Ψ , (3) In either case one obtains the equation of motion
−h | | i
where V′(x) = dV(x)/dx is defined as the acceleration S J ~ Ψ∗ Ψ Ψ Ψ∗
operator. In these expressions,Ψ(x,t) denotes the wave- x˙ = ∇ = = ∇ − ∇ , (10)
m ρ 2im Ψ2
function solution of the TDSE, (cid:18) | | (cid:19)
which, after integration, renders the corresponding
∂Ψ(x,t)
i~ =HΨ(x,t), (4) Bohmian trajectories, i.e., the paths along which the
∂t probability flows.
Once the Bohmian trajectories are synthesized, two
with H being the Hamiltonian operator (1). Following
types of calculations have been carried out with them.
the standard procedure, from these quantities one ob-
The first type consists in the direct evaluation of their
tains the corresponding power spectra,
individual spectrum in order to determine how each tra-
2 jectorycontributestotheexpectationvalues(2)and(3).
I(ω)= g(t)e−iωtdt , (5) That is, we have studied the power spectrum (5) for a
Z  series of individual trajectories, substituting g(t) by the
 
  time-dependenttrajectoryx(t)obtainedafterintegrating
where g(t) genericallydenotes eitherx¯(t) or a¯(t) [or a
Eq. (10) with the corresponding initial condition, x(0).
Bohmian trajectory, x(t), as seen below].
In a second step, we have stressed the ensemble feature
of Bohmian mechanics, analyzing the emergence of the
B. Bohmian trajectories usual dipole length and acceleration as the number of
Bohmiantrajectorieswasincreased[andbearinginmind
that their initial positions distributed randomly accord-
In order to determine the Bohmian trajectories, first
ing to ρ(x,0)]. In this case, we have taken into account
the above wavefunction is recast in polar form,
thefactthattheBohmianversionoftheexpectationval-
Ψ(x,t)=ρ1/2(x,t)eiS(x,t)/~, (6) ues (2) and (3) are given by
whereρistheprobabilitydensityandS isthereal-valued 1 N
x¯ (t) = x (t), (11)
phase. After substituting this form into the TDSE (4) B N i
and rearranging terms, one obtains a set of two real- Xi=0
valued differential equations, which account for the evo- 1 N dV(x)
a¯ (t) = , (12)
lutionofthesetwoquantities. Oneisthecontinuityequa- B
N dx
tion, Xi=0 x=xi(t)

respectively,whichfollowausualstatisticalapproach. In
∂ρ
+ J=0, (7) these expressions, x (t) denotes the ith Bohmian trajec-
∂t ∇· i
toryfromanensembleandN isthetotalnumberofthese
whereJ=ρ S/misthe usualquantumprobabilitycur- trajectories in each numerical experiment performed.
∇
rentdensity. TheotherequationisaquantumHamilton-
Jacobi equation,
C. Initial conditions and dynamics
∂S ( S)2
+ ∇ +V +Q=0, (8)
∂t 2m
As the initial condition for the wavefunction, we have
considered the ground state of the bare or field-free
where
Hamiltonian,
~2 2ρ1/2 ~2 2ρ 1 ρ 2
Q(x,t)= ∇ = ∇ ∇ 1
−2m ρ1/2 −4m" ρ − 2(cid:18) ρ (cid:19) # H0 =−2 ∇2+V(x). (13)
(9)
istheso-calledquantumpotentialandS istheequivalent This eigenstate has been numerically obtained by means
of the classical action of the imaginary time propagation method [35]. Then,
As happens with the classical analog of Eq. (8), one the exact time-propagation of the wavefunction accord-
can postulate [22, 23] the existence of trajectories (char- ing to the TDSE (4) has been carried out by combining
acteristics)that are solutions to this equation. As in the the split-operator technique [36] with the fast Fourier
classicalcase,thesetrajectoriesareperpendiculartosur- transform (FFT) technique [37]. More specifically, the
faces of constant phase and therefore the corresponding latter has been used to achieve a high performance in
equationof motionis proportionalto S. Alternatively, the evaluation of the action of the kinetic operator on
∇
the same can also be straightforwardly stated if Eq. (7) the wavefunction. We have also taken advantage of this
4
decomposition of the wavefunction to recast purposely withx =4.102a.u. Thisensuresthatthetime-evolution
c
the wavefunction as ofmostoftheprobabilitydensitywillbewellmonitored,
Ψ(x,t)= cn(t)eipnx/~. (14) aosf tthheetionttaeglrpartoebdapbriloitbya.bility for |x| ≥ xc is only 0.151%
n
X
Here, the c (t) coefficients denote the Fourier weights
n
(in general,complex valued) associated with each one of III. RESULTS AND DISCUSSION
the plane-wave components with momentum p of the
n
decomposition at time t. A. Driving fields and binding potentials
To ensure that all the relevant dynamics are incorpo-
rated for the parameter range of interest, we have set
In the results that follow, we will mainly assume that
the box boundaries located far enough from the core re-
the atomic potential, V(x), is given by the soft-core in-
gion (at x = 150 a.u.). Furthermore, special care has
| | teraction
been taken in order to avoid reflections and spurious ef-
fects near the box edges. First, we have employed a 1
Vsc(x)= , (16)
mask function in the form of a cos1/8–function, which at −√x2+a2
becomes active near the boundaries. This function is
smoothenough,butstillcapabletoabsorbingwithahigh where a is the impact parameter. Far away from the
efficiency,thusavoidingnonphysicalreflections. Wehave core, this potential decays as x−1, behaving similarly to
tested this fact by considering a wide range of box sizes. a Coulomb-type, long-range interaction. Equation (16)
Second, the maximum/minimum value of the total po- gives dV(x)/dx=x/(x2+a2)3/2 for the dipole accelera-
tentialfunctionhasbeentruncatedinordertoavoidalso tion operator in Eq. (3), and the acceleration (12) along
nonphysical accelerations towards the box edges. In this the Bohmian trajectories reads as
regard,thesizeoftheboxwaschosensuchthatthistrun-
cationtakesplace close to the regionwhere the absorber 1 N x (t)
i
a¯ (t)= . (17)
becomes active. B N [1+x2(t)]3/2
Additionally, to avoid effects associated with the loss Xi=0 i
ofprobabilitydue to the presence ofthe absorber,which
Here we have chosen the impact parameter to be a = 1,
becomes more prominent for higher values of the exter-
which yields a ground-state energy ǫ = 0.66995 a.u.
0
nal field intensity, in the calculations special care was −
for the field-free Hamiltonian (13).
taken in properly renormalizing the wavefunction. Fur- In Sec. IIIB2, the long-range of the potential Vsc(x)
at
thermore, unless otherwise stated (see Sec. IIID), no fil-
is eliminated by re-writing it as
ter functions (e.g.,Hanning windows)wereusedinorder
to avoid influencing the long-range dynamics and there- 1
V (x)= f(x), (18)
fore the topology of the Bohmian trajectories. tr −√x2+a
RegardingtheBohmianequationofmotion(10),ithas
beenintegrated“onthefly”,substitutingthevalueofthe where the truncating function is described as
wavefunction Ψ(x,t) at each time into the last term of
thisequation. Now,giventhegeneralform(14),Eq.(10) 1, x <a0
| |
can be recast as f(x)=cos7 π|x|−a0 , a x L . (19)
0
x˙ = 1 m,npn|cm(t)||cn(t)|cos[∆pnmx/~+∆nm(t)], 0, (cid:18)2 L−a0 (cid:19) ≤x|>|L≤
m c (t) c (t) cos[∆p x/~+∆ (t)] | |
P m,n| m || n | nm nm (15) Theexplicitvaluesofa andLarechosenintwodifferent
P 0
where the cn(t) coefficients have been expressed in po- ways,whichwillbelateronspecifiedinthatSection(see
lar form, i.e., cn(t)= cn(t)eiδn(t), with cn(t) and δn(t) Table I for details). A similar truncation was formerly
| | | |
denoting its (time-dependent) modulus and phase, re- used in Ref. [38] to assess the influence of the potential
spectively. Moreover, we have defined ∆pnm pn pm tail on resonant HHG enhancements.
≡ −
and ∆nm(t)≡δn(t)−δm(t). In this way, since the cn(t) Regardingtheexternallaserfield,F(t)=f(t)sin(ω0t),
are known at each time through the FFT operation, the two types of envelopes, f(t), have been considered,
Bohmiantrajectoriesx(t)areobtainedstraightforwardly namely a flat-top trapezoidal pulse,
by integratingthe analyticalform(15), whichis anordi-
narydifferentialequation. Inparticular,inthis workthe (t/τ ), 0 t<τ
0 on
≤
integration has been performed by means of the Euler f(t)= 1, τ t<τ , (20)
on off
 ≤
method, which has been proven to be accurate enough. 1 (t τ )/τ , τ t τ
 − − off on off ≤ ≤ f
Unless otherwise stated, the initial conditions x(0)
considered for the Bohmian trajectories have been ran- and a sin2–function,
domly chosen to mimic the initial probability distribu-
tion, ρ(x,0) = Ψ(x,0)2, within the interval [ x ,x ], f (t)=sin2(πt/τ ). (21)
c c s2 f
| | −
5
In both cases, ω = 0.057 a.u. and τ = 2π/ω denote
0 0 0
the frequency and period of the field, respectively. More 0.12
1.2 (a) (b)
specifically,in(20)thetimerequiredfortheexternalfield 0.08
twτooifftrhe=amc2ha.x2ii5tmsτu0mm+axaNimmτpu0lmisttuaadnmed,pslaifntoudrdtτehfies=dτuτoonrffa=t+io2n2..22o55fττt0h0,erwefihfeeilrldes x(t) (a.u.)00..48 a(t) (a.u.) 00..0004
to the time at which the field is switched off (and our 0.0 -0.04
quantum simulations are completed); to ensure a high
-0.08
monochromaticity in the flat-top pulse, we have chosen -0.4
-0.12
N = 10 (τ = 14.5τ ). The propagation described in 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Sec. IIC h1.afsBbS.oefetn-Sciotnaregk0lepeno-tteurnpatjiteaolctaton=rdyflτafan.t-atolypspisulse log Harmonic spectra (arb.units)---106420246 (c) t (cycles) log Harmonic spectra (arb.units)10 ---642024 (d) t (cyc les)
We will first revisit the issue of the dipole length vs -8 -8
0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80
acceleration for the soft-core potential (16) employing Harmonic order ( Harmonic o rder (
Bohmian dynamics. The dipole length over-emphasizes FIG. 1: (Color online) Dipole length (a) and dipole acceler-
regions near the box edges due to its linearity with the ation (b) computed from the TDSE for a model consisting
distance. Hence, even if the probability density far from of asoft-core atomic potential andaflat-top laser pulsewith
frequency ω = 0.057 a.u., and intensities E = 0.05 a.u.
the core is negligible, x can reach relatively high values, 0 0
(dashed red line) and E =0.075 a.u. (black solid line). The
thus compensating this fact and enhancing the associ- 0
corresponding power spectra are displayed in panels (c) and
ated features in the corresponding spectra. In contrast,
(d), respectively. The cutoff frequencies in the cases consid-
thedipoleaccelerationfocusesonthecoreregion. Specifi- ered are ω = 22.5ω for E = 0.05 a.u. and ω = 35.8ω for
0 0 0
callyforthepotential(16),itincreasesuptox= 1/√2, E =0.75 a.u..
and then decreases to zero again very rapidly, li±ke x−2. 0
Hence, any contributionfromportions of the probability
density outside this region is strongly suppressed [20].
In Fig. 1, we show the time-dependent dipole length E2/4ω2 is the ponderomotive energy. These features,
0 0
[panel (a)] and acceleration [panel (b)] obtained from shown in Fig. 1(d), are in full agreement with the pre-
Eqs. (2) and (3), respectively, and computed using the dictions of the TSM. In the harmonic spectra computed
TDSE. Moreover, the corresponding harmonic spectra fromthedipolelength,however,thesefeaturesarenotso
are provided in the lower panels. As seen in Fig. 1(a), easily identifiable. In particular, for the higher driving-
as the field strength E increases, the amplitude of the field intensity, the harmonic peaks are obfuscated by a
0
oscillations for the dipole length becomes comparatively large background, with no visible plateau. Overall, the
large. Furthermore,atthe endofthe pulse,the expecta- HHG spectra in Fig. 1(c) are monotonically decreasing.
tion value of the dipole operatoris non-vanishing for the Nonetheless,formoderateintensitiesoftheexternalfield
larger driving-field intensity. This can be traced to irre- (see curve for E0 = 0.05 a.u.) the cutoff could be as-
versibleionizationandthecontributionsfarfromthecore signedtothe dramaticdecreaseintheintensityobserved
region,whichareover-estimatedinthis case. The dipole around ω 30ω0.
≈
acceleration[see Fig. 1(b)], however,behaves in a rather We will now employ Bohmian trajectories in order
different way. First, there is a much less pronounced to analyze in detail why the spectra obtained from the
increase in the amplitude of a(t) with the driving-field dipole accelerationandfromthe dipole lengtharesodif-
strength. Comparing the two graphs, we note that their ferent (in particular, why the cutoff frequency does not
maindifferenceliesinthesubstructureobservedfora(t). appear clearly in the spectra obtained from the dipole
Second,theaccelerationvanishesattheendofthepulse. length). In Figs. 2(a) and (b) two sets of Bohmian tra-
These differences stem from the fact that a(t) only de- jectories are displayed for the same driving-field intensi-
pendsonthepartofthewavefunctionwheretheelectron tiesasinFig.1. As E increasessomeofthe trajectories
0
undergoesthestrongestinteractionwiththe core,asdis- startto oscillatewith increasinglylargeramplitudes and
cussed above. eventuallyleavethecoreregion[seepanel(b)]. Actually,
When comparing the spectra for the dipole length itisthisbehaviorwithE whatleadstotheeventualde-
0
and the acceleration for such intensities, however [see pendence on the boundaries of the problem. That is, as
Figs. 1(c) and (d)], one readily notices that the spec- E increases, the trajectories undergo larger excursions
0
tra obtained from the dipole accelerationexhibit a large out of the core and therefore, when their average (11) is
plateau consisting of high-order harmonics extending up computed, one will obtain results highly dependent on
to a cutoff energy of ω ǫ + 3.17U , where U = theboundaryconditionschosentosolvetheTDSE.Note
c 0 p p
≈ | |
6
if the trajectories are launched from x(0) = 3 a.u., or
−
120 even x(0)=1.8 a.u., the corresponding spectra will only
4 (a) (b)
80 consistofthe fundamentalandofauniformbackground.
x(t) (a.u.) 02 x(t) (a.u.)400 Filnoosgrt.mthoetsieonse(tasrooufnindittihael ccoonrrdeistpioonnsd,itnhgeincoitniafilnpedosoitsicoinll)atis-
-2 -40 Apart from that, the further from x(0)=0 the initial
position of a trajectory is, the higher the overall inten-
-80
-4 sity in the spectra will be. Thus, even if the number of
-120
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 trajectories that leave the core regionis relatively small,
log Harmonic spectra (arb.units)10 ----86420246 (c) t (cyc les) log Harmonic spectra (arb.units)10 1--04202468 (d) t(cyc les) ttmciwtRnhuoohegoednfiess.terrtfrraae[iem2brtcva1ertuoea]trr,ibtjniesehoawicbancaithnktlsgoeetgrriterreniihenooeeessuntmtiehnirhgzeeditanissvhdtptieianbieoswpncantoticrhtrtlkiahehasegt-eolrtHaestohtnrheHueoegenoGtosadebhbentsvssmwadpeepiraernaeovacssecfatlkdtrattrioainthuofrtgremdnwroiespabm.orsuiufsTnltatstohehRtehfdet,eeersmaieftib.lonnaafiu[dngt2nttten0hedierin]der---,
0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80
Harmonic order ( ) Harmonic order ( ) probabilitydensityneartheedgesoftheintegrationbox.
FIG. 2: (Color online) Top: Bohmian trajectories obtained
with a flat-top pulse with intensity: (a) E = 0.05 a.u. and
0
(b) E0 = 0.075 a.u. Bottom: Power spectra computed from 2. Short-range potentials and flat-top pulse
individual trajectories xi(t) selected from among the sets at
thetoppanels. (c)Trajectoriesstartingatx(0)=0a.u.(black
Inordertoinvestigatetheinfluenceofconfinementfur-
solid line) and x(0) = −3 a.u. (red dashed line), for E =
0 ther and whether the above-discussed picture remains
0.05 a.u. (d) Trajectories starting at x(0) = 0 a.u. (black
valid across all model potentials, we will consider the
solid line), x(0) = 1.8 a.u. (blue dotted line) and x(0) =
−3 a.u. (red dashed line), for E = 0.075 a.u. Colors/line- truncated short-range potentials given by Eq. (18) [38].
0
styles are in correspondence with the trajectories displayed In this section, the parameters a0 and L in Eq. (18) are
at therespective top panels. chosen in two distinct ways. For the truncated poten-
(1)
tial V (x), the long-range tail is eliminated, but the
tr
core region is kept practically unaltered, i.e., the field-
freeeigenenergiesobtainedforthe untruncatedpotential
thatthisaverageformallycorrespondstotheexpectation Vsc and V(1)(x) are very close, and many weakly bound
value ofthe dipole moment. Onthe contrary,if one only at tr
states are present. In contrast, the truncated potential
focussesonthosetrajectoriesthatalwayskeeposcillating
V(2)(x) is so short ranged that it can only support only
aroundthe core, the averagebehavior will resemble that tr
two strongly bound states. Throughout, we keep the
of the dipole acceleration.
ground-state energy roughly the same. A list with the
The interpretationprovidedaboveissupportedbythe
eigenvaluespectrumforthese three potentialsis givenin
harmonicspectraobtainedfromsingleBohmiantrajecto-
Table I.
ries x (t), depicted in the lower panels of the figure. As
i These potentials, together with the soft-core model
an overall feature, a large plateau followed by a sharp
(16), are illustrated in Fig. 3. Panels (a) and (b) show
cutoff is always obtained if the trajectory starting at
the field-free potential V(x) and the effective potential
x(0) = 0 is taken, regardless of the driving-field inten-
barrierV (x)=V(x) xE when the driving laserfield
sity. This trajectory corresponds to the innermost part eff − 0
reaches its peak value E = 0.075 a.u.. This higher in-
of the wavefunction, which is strongly localized at the 0
tensity will be employed throughout this section, as it
core. This spatial region resembles that spanned by the
causes more irreversible ionization. For V(1)(x), the po-
dipole acceleration. In contrast, the spectra from tra- tr
tentialwell(coreregion)isessentiallythesame,whilethe
jectories whose initial positions lie at a few atomic units
long-rangetailis significantlydecreased. Inthe presence
of distance, i.e., at the outermost regions of the wave-
of the field, however, V (x) is practically not affected
function, will be strongly influenced by the driving-field eff
by the truncation. For the second truncated potential
intensity. For moderate driving-field intensity [Fig. 2(c)]
the spectra from both innermost and outer trajectories Vt(r2)(x), thewidthofthe potentialwellhasdecreasedre-
arequitesimilarinshape,apartfromanoverallintensity markably as well as the potential range. This leads to a
difference. This is caused by the fact that both trajec- much higher ionization barrier Veff(x).
tories perform an oscillatory motion near the core. If, InFig.4, wedisplaythe dipole lengthandthe acceler-
however,theintensityincreases,thespectraoftheinner- ationcomputed with the TDSE for V(1)(x) and V(2)(x),
tr tr
mostandoftheoutermosttrajectoriesdiffermarkedly,as together with the corresponding power spectra [upper
exemplifiedinFig.2(d). Inthefigure,oneobservesthat, and lower panels, respectively]. In Fig. 4(a) we observe
7

TABLEI:Eigenvaluesfortheuntruncatedsoft-core(16)and 0.2
twotruncatedsoft-core(18)potentials. Weshallrefertothese (a) 1.5 (b)
0.0
sLLmuop==dpe5o7l0r.s5t,e5asd.secfiNborynosdtttehttrteruhuunannctc,atartituneedndp,c,raooitrnrecVdViptt(r(pr1l2e)o),((txxet)nh),,teiffaoonlrruamwwphbhpiierccrohhaocaafh00ee=i=sgei51nn..fis00tnaaaittnneydds; ---V(x) (a.u.)000...642 - V(x) (a.u.)0010eff....0505
inourcalculations,though,weobtainafinitenumberofthem
-0.8 -1.0
becauseoftheboundariesofthegridweareusingtosolvethe
TDSE. All quantitiesare given in a.u. -1.0 -1.5
-40 -20 0 20 40 -20 -10 0 10 20
n untruncated truncated 1 truncated 2 x (a.u.) x (a.u.)
0 -0.66995 -0.66995 -0.65651 FIG. 3: (Color online) (a) Comparison between the soft-
1 -0.27508 -0.27503 -0.17864 coremodel(16)(blacksolid line) andthetruncatedsoft-core
model (18) for two different values of the parameters a and
2 -0.15158 -0.15059 0
L. Thebluedashedlinedenotesthetruncatedpotentialwith
3 -0.09276 -0.08714
a = 5.0 and L = 50, while the red dotted line stands for
0
4 -0.06358 -0.05013 a = 1.0 and L = 7.55. In panel (a) and (b) we display the
0
5 -0.04552 -0.02390 samepotentials,butacted bythelaser fieldat themaximum
6 -0.03462 -0.00754 of its amplitude, which is taken as E0=0.075 a.u..
. .
. .
. .
14 -0.00826

15 -0.00707 1.8 0.12
(a) (b)
16 -0.00670
1.2 0.08
thatx¯(t)undergoeslongerexcursionswiththefirsttrun- x(t) (a.u.)00..06 a(t) (a.u.) 00..0004
cated potential V(1)(x) than even with the untruncated -0.6 -0.04
tr
case. Furthermore, there is a marked increase in the -1.2 -0.08
dipole-length amplitude with regard to the time, which
-1.8 -0.12
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
boiiaVetsnonfset(rcai1ttnfoe)bohh(noslxeeoilezao)fncawtetsotttlsuerhinio.acpentntciprntfiFFofoiuoveriniurgrletidsm.cVtml,o,1t.(wrnat(1bhang)aFoe()y-vux.srteeahlr)potn.BahosgtTotcseheakterhehtlemtiyetasuseoibblnfftrwooeteperuw,hciuulntaallfnslovdribcrndisoasegsutVrtgaesttct(gdrththa1eree)nesos(p.tnxiboocgI)atenonlenyritenzehtxetihcenipdreaorcleeluaprpgpieaairwnaolreneeisntsldddes-l. log Harmonic spectra (arb.units)---106420246 (c) t (cycles) log Harmonic spectra (arb.units)10 ---642024 (d) t (cyc les)
-8 -8
This enhanced ionization leads to a large background in 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80
Harmonic order ( Harmonic o rder (
the corresponding harmonic spectra [see dashed line in
Fig. 4(c)]. For the truncated potential V(2)(x), in con- FIG. 4: (Color online) Dipole length (a) and dipole accel-
tr eration (b) for the truncated soft-core model (18) with two
trast, x¯(t) essentially oscillates within the core region.
sets of parameters: a = 5.0 and L = 50 (red dashed line),
0
This happens because this potential only contains two and a =1.0 and L=7.55 (black solid line), computed from
0
bound states, the excited one being also quite far from the TDSE. The pulse considered was a flat-top one, as given
thecontinuum. Hence,theionizationbarrierremainsrel- above,withintensityE =0.075a.u. andthesamefrequency
0
ativelyhighand, inthe presenceofthe field, thereis less asinthepreviousfrequencies. Thecorrespondingpowerspec-
couplingwiththecontinuum. Thisoutweighstheabsence tra are displayed in panels (c) and (d), respectively. For the
of a long-range tail mentioned above. This suppression potentials Vt(r1)(x) and Vt(r2)(x), the predicted cutoff energy
of ionizationconsiderablyreduces the backgroundin the lies at ω=35.8ω0 and ω=35.6ω0, respectively.
harmonicspectra and the overallharmonicintensity [see
solid line in Fig. 4(c)].
In this regard, one should notice that the expectation
values ofdipole accelerationobtainedforboth truncated samereason,theharmonicspectraexhibitamuchclearer
potentials, displayed in Fig. 4(b), are very similar. This cutoff, near ω =35ω0 and similar shapes [see Fig. 4(d)].
isexpected,astheaccelerationprobesthespatialregions Furthermore,thediscrepancyinthe overallharmonicin-
close to the core and the main differences are due either tensities is not as extreme as in Fig. 4(c). Still, due to
to the high-lying excited states, whose spatial extension thesuppressionofionization,thespectrumobtainedwith
islarge,ortothedifferentpotentialbarriers. Forthevery V(2)(x) is a few orders of magnitude weaker.
tr
8
More detailed information about the ionization dy-

namics is provided by the outcome of the Bohmian- 120 120
(a) (b)
trajectory computation, exhibited in Figs. 5(a) and (b). 80 100
ooFffortiohtnehiepzasuhtlisooern,t-arissanseegsesenepniotntiaeplnlatyniaepllrV(oatd(r)1u,)c(wxehd)i,lietnhinethltaehregceelnsottnragam-lraopnuagnrett x(t) (a.u.)400 x(t) (a.u.)6800
40
soft-core model (16) we found that ionization was pro- -40
20
duced in a more homogeneous way along the pulse du- -80
ration. This is related to Vt(r1)(x) being short ranged: -1200 2 4 6 8 10 12 14 00 2 4 6 8 10 12 14
stpntontohhounteooheepacrapetprellnaraoarttnectshhhiksteenaeesnrcniortcrotfiodfioernale,aalrelolefsj.odoeotef,rcarseitCeiientoglolvhoyniraeoteniitnbehnzstsstaeo.ereutquaacNoaiucnroouleteednlnityvnndoweetdgusrxalsuoytthrcth,semdihoitedsmrteeeltoeemdp-otsrtrrasssahoatt,tnobehajgewtgatiicrcgevebaoathseirporjlnedepeirortcislttyeratebhesioyntflaegrs.etrioiioiserawnaTcsintelam.,heroVpnitOossteah(srcr2nnieelilt)elerdy(laieataxmhddtto)ioeess--f, log Harmonic spectra (arb.units)10 1---6420246800(c1)0 20 t3 (0cyc4 0les5)0 60 70 80 log Harmonic spectra (arb.units)10 ----864202460 (d1)0 20 t3 0(cy4c 0les5)0 60 70 80
ing to irreversible ionization. For the same reason, i.e., Harmonic order ( ) Harmonic order ( )
theshorterrangeofthepotential,theywillremaininthe FIG. 5: (Color online) Top: Bohmian trajectories for the
continuum in spite of the laser field amplitude becoming truncated soft-core model (18) with two sets of parameters:
(a) a = 5.0 and L = 50, and (b) a = 1.0 and L = 7.55, in
verysmallnearthe pulse turnoff,as showninpanel(b). 0 0
a flat-top pulse with intensity E =0.075 a.u. and the same
Similarly to what has been observed for the long-range 0
frequency as in the previous figures. Bottom: Power spectra
potential, the central trajectory, starting at x(0) = 0, is
corresponding to some selected trajectories from among the
the one that best reproduces the behavior displayed by
sets. (c) Trajectories starting at x(0) = 0 a.u. (black solid
the dipole acceleration, while marginal trajectories ren- line) and x(0) = −3 a.u. (red dashed line). (d) Trajecto-
der spectra which decrease monotonically, except for a ries starting at x(0) = 0 a.u. (black solid line), x(0) = +3
single peak at the field frequency, ω =ω0 [see panels (c) a.u. (dotted blue line) and x(0)=−3 a.u. (red dashed line).
and (d)]. Colors/line-styles are in correspondencewith thetrajectories
displayed at therespective top panels.
3. Soft-core potential and sinusoidal pulse
figure exhibit the Bohmian trajectories obtained for this
kind of pulse [panels (b) and (c)], with the correspond-
Another interestingquestionthatariseshere isthatof
ing spectra [panels (e) and (f)]. For sinusoidal driving
the role played by the overall shape of the laser pulse.
pulses,wefindthesamepatterndescribedabove,namely
Up to now we have considered a flat-top pulse, which to
that there is a good qualitative agreement between the
some extent may be viewed as a reasonable approxima-
spectrumobtainedfromthedipoleaccelerationwiththat
tion to a monochromatic wave. One could argue, how-
computed only counting the centraltrajectoryofthe en-
ever, that the pulses employed in experiments are far
semble. Nonetheless, one could now ask the question of
from monochromatic,and the pulse shape influences the
whether a quantitative agreement between the outcome
dynamics. For that reason, we will perform a Bohmian
oftheTDSEandtheresultsobtainedfromtheBohmian-
trajectory analysis for the long-range potential model
trajectory computations can be reached. This issue will
(16), but assuming now that the driving field is a pulse
be addressed in the next section.
whose envelope is given by Eq. (21).
The outcome of our computations is shown in Fig. 6,
for the two peak strengths employed in the remaining
sections of this work. In the left panels we display the C. Ensemble analysis
dipole acceleration and the corresponding power spec-
tra obtained from the TDSE, which will be employed In this Section we focus on what distributions of
as benchmarks. In general, the HHG spectra follow the Bohmiantrajectoriesleadaquantitativeagreementwith
cutoff law ǫ +3.17U . However, the resolution of the the TDSE solution and the Bohmian-trajectory compu-
0 p
| |
harmonicpeaksismuchworsethanthatobtainedforthe tation. For such a purpose, we have considered a set
flat-top pulse. Indeed, only in the cutoff regionit is pos- of initial conditions obtained from uniformly generated
sibletooidentifytheharmonicpeaks. Thisisexpectedas randomnumberswithinacertaininterval[ x ,x ],sym-
c c
−
sharp harmonic peaks stem from the field periodicity: a metricwithregardtotheorigin,x=0. Thisintervalhas
trapezoidalpulsecontainsmanyidenticalcycles,whereas been chosen such that the probability density related to
a sinusoidalpulse does not. The remaining panels of the the initial wavefunction, Ψ(x,0), is recovered up to over
9

0.08 120
(a) (b) (c)
4
0.06
80
0.04
2
u.)0.02 u.) u.) 40
a(t) (a.0.00 x(t) (a. 0 x(t) (a. 0
-0.02
-40
-2
-0.04
-80
-0.06
-4
-0.08 -120
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
log Harmonic spectra (arb.units)10 ----0248642 (d) t (cyc les) log Harmonic spectra (arb.units)10 --012321 (e) t (cyc les) log Harmonic spectra (arb.units)10 --01234521 (f) t(cyc les)
-10 -3 -3
0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80
Harmonic o rder ( ) Harmonic o rder ( ) Harmonic o rder ( )
FIG. 6: (Color online) Left panels: Dipole acceleration [panel (a)] and its power spectra [panel (d)]computed from theTDSE
for a model consisting of a soft-core atomic potential and a sin2-shaped laser-field pulse with frequency ω = 0.057 a.u., and
0
intensities E = 0.05 a.u. (dashed red line) and E = 0.075 a.u. (black solid line). Panels (b) and (c): Bohmian trajectories
0 0
obtained for the same field and atomic parameters as in the left panels, for peak intensities E = 0.05 a.u. and E = 0.075
0 0
a.u., respectively. Panels (e)and(f): Powerspectrafrom someselected trajectories from amongthesetsinpanels(b)and(c).
Panel (e) shows the spectra from the trajectories starting at x(0) = 0 a.u.(black solid line) and x(0) = −3 a.u. (red dashed
line), for E =0.05 a.u., while panel (f) exhibits spectra computed from the trajectories starting at x(0)=0 a.u. (black solid
0
line), x(0) = 1.8 a.u. (blue dotted line) and x(0) = −3 a.u. (red dashed line), for E = 0.075 a.u. Colors/line-styles are in
0
correspondence with thetrajectories displayed at therespective top panels.
99.8%. For the parameters employed in this work, we panels (c) and (d).
have verified that all the relevant probability density is
incorporated if we take x = 4.102 a.u. These random After having established the good agreement between
c
numbers are multiplied by weights, which are chosen in bothcomputations,alegitimatequestioniswhatregions
such a way that the probability density related to the of the core are the most important for HHG. To investi-
ground state of the soft-core potential, ψ (x)2, is mim- gatethis issue,wehavegraduallyvariedthe rangeofthe
0
icked. For our parameters, this amount|s to fo|urteen in- aforementionedrandom-numberdistributionbychanging
tervals at each side of the origin, x = 0, and around the boundary xc. These results are displayed in Fig. 8
N 1,000 random numbers in total. Each of these for the time-dependent dipole accelerationandusing the
num≈bers thus corresponds to the initial condition for a same intensities as above. As seen, overall, the time
Bohmian trajectory. dependence of all the distributions considered is very
similar. What is affected is the amplitude of the time-
This statistical prescriptionhas been used to compute dependent acceleration. In fact, if only the acceleration
thedipoleaccelerationaccordingtoEq.(12)forthesoft- along the central trajectoryis taken (red lines in the fig-
core potential and the trapezoidal pulse. The same field ure),weseethatthis amplitude isuptofivetimes larger
strengths as in Fig. 1 have been used. In Figs. 7(a) thanthatdeterminedbytheaccelerationcomputedwith
and (b), we compare a¯ (t) with the dipole acceleration theTDSE.Themaineffectofincreasingtherangeofthe
B
computed from the TDSE. The outcomes of both ap- integration around this central trajectory is to decrease
proachesexhibitanexcellentagreement,bothqualitative this amplitude. Indeed, for both intensities a quantita-
andquantitative. Thisagreementisalsoobservedforthe tive agreement is found for x = 2.344 a.u., which cor-
c
corresponding high-order harmonic spectra, displayed in responds to about 96% of the total probability density.
10

0.06 0.3 0.4
(a) 0.08 (b) (a) (b)
0.04 0.2
a(t) (a.u.)00..0002 a(t) (a.u.)00..0004 a(t) (a.u.)00..01 a(t) (a.u.)00..02
-0.02 -0.04 -0.1
-0.04 -0.2
-0.08 -0.2
-0.06
log Harmonic spectra (arb.units)-101----8642002400 (c1)02 20H4arm30to 6(nciycc4 o l0ers8d)er5 0(106)01270 1480 log Harmonic spectra (arb.units)10---02464200 (d1)20 20H4arm30to 6(nciycc4 o l0erds8e)r5 (0106)01270 1480 a(t) (a.u.)---0000000.......21301230(c)2 4Tim6e (c yc8les) 1T xxxxC0ccccD e====Sn Et0112r1....a62532l (((((14789)332597951))))4 a(t) (a.u.)----000000000.........3210123440(d)2 4 Tim6e (c y8cles) 1T xxxxC0ccccD e====Sn Et0112r1....a62532l (((((14789)332597951))))4
-0.3 -0.4
5.00 5.25 5.50 5.75 6.00 5.00 5.25 5.50 5.75 6.00
FIG. 7: (Color online) Dipole acceleration for a model con- Time (cy cles) Time ( cycles)
sisting of a soft-core atomic potential Vastc(x) and a flat-top
trapezoidal pulse with frequency ω0 = 0.057 a.u and field FIG. 8: (Color online) Dipole acceleration obtained from a
strengths E0 = 0.05 a.u. [panel (a)] and E0 = 0.075 a.u. series of sets of Bohmian trajectories randomly distributed
[panel (b)], computed from the TDSE (black solid line) and within the interval [−xc,xc] for two values of the laser field:
from a Bohmian trajectory simulation employing 987 trajec- (a) E = 0.05 a.u. and (b) E = 0.075 a.u. Enlargements
0 0
tories(reddashedline). Thespectracorrespondingto(a)and of thebehavior of this quantity within one cycle of the pulse
(b) are represented respectively in panels (c) and (d). are given respectively in panels (c) and (d). The different
valuesofxc usedinoursimulationsarelabeledwithdifferent
types of colors/line-styles. For details, see the legend in the
lower panels (in parenthesis, we provide the total number of
However, if one is interested in a qualitative comparison Bohmian trajectories used in each case). In all cases, the
it is fully legitimate to assume, to first approximation, blacksold linerepresentstheresult obtained from theTDSE
that the core is a source term located around x(0) = 0. and the red dashed line the result from the central Bohmian
trajectory.
In fact, this is the key assumption of the existing ana-
lytic strong-field approaches (e.g., the SFA), which is in
agreement with our previous results in [30].
ning) filter,
Finally, in Fig. 9 we investigate how this influences
the high-harmonic spectra. Due to the above-mentioned 2πt 2πt
w(t)=0.5 1 cos =sin2 . (22)
difference in amplitude, the harmonics in the spectrum
− τ τ
from the central trajectory are several orders of magni- (cid:20) (cid:18) f (cid:19)(cid:21) (cid:18) f (cid:19)
tudehigherthanthosefromtheTDSE[seeFigs.9(a)and This filter is specially useful to force (artificially) the
(b)]. As the spatialrangeis increased,we observethat a time-dependent function g(t) and its derivative to start
reasonablequantitativeagreementisreachedforthehar- at zero and decay to zero at the end of the simulation in
monics in the cutoff region, already if 75% of the overall a smooth fashion. With it, Eq. (5) reads as
probability density is considered (x = 1.172 a.u.). For
c
2
the below-threshold and low-plateau harmonics, a good
I (ω)= g(t)w(t)e−iωtdt . (23)
agreementis reachedonly if 90%of the totalprobability w
density is taken (xc = 1.8 a.u.). This is still, however, Z 
less than the requirements in the time-dependent case. A legitimate questionrelated to this kind of spectra is
therefore if they can be associated with a certain dy-
namical behavior of the system. If g(t) refers to the
expectation values x¯(t) or a¯(t) of the dipole length or
D. Effective dynamics induced by numerical filters
acceleration, then the effective values x¯ (t) = w(t)x¯(t)
eff
and a¯ (t) = w(t)a¯(t) say very little about the behavior
eff
Finally, we will address the type of effective dynam- of an also effective wave function that suits such val-
ics induced by filters or window functions, which are ues. In contrast, if g(t) refers to the Bohmian trajecto-
often used to stress some particular aspects from the ries, one can reconstruct an effective dynamics, since (i)
corresponding spectra. For example, a widespread way x (t)=w(t)x(t) isatrajectory(actually,justthe prod-
eff
of eliminating the background introduced by the dipole uct of the actual Bohmian trajectory and the window
length inthe harmonic spectra is to apply a Hann(Han- function) and, as seen above, (ii) an ensemble of such