Quantum beats in the polarization response of a dielectric to intense few-cycle laser pulses

0.82 MB·English

by  Michael Korbman

Quantum beats in the polarization response of a dielectric to intense few-cycle laser
pulses
M. Korbman,1 S. Yu. Kruchinin,1 and V. S. Yakovlev2,1,∗
1Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
2Ludwig-Maximilians-Universit¨at, Am Coulombwall 1, 85748 Garching, Germany
Wehaveinvestigatedthepolarizationresponseofadielectrictointensefew-cyclelaserpulseswith
a focus on interband tunnelling. Once charge carriers are created in an initially empty conduction
2 band,theymakeasignificant contributiontothepolarization response. Inparticular, thecoherent
1 superposition of conduction- and valence-band states results in quantum beats. This quantum-
0 beat part of the polarization response is affected by the excitation dynamics and attosecond-scale
2 motion ofcharge carriers in anintenselaser field. Ouranalysis shows that,with theonset of Bloch
oscillations or tunnelling, the non-linear polarization response becomes sensitive to the carrier-
v
o envelopephase of the laser pulse.
N
2 I. INTRODUCTION [12]. The theoretical interpretation of that experiment
2 wasgivenin terms of Wannier-Starklocalization[13, 14]
Multiphoton strong-field excitation of valence-band that can be viewed as a frequency-domaindescription of
h] electrons has been intensively investigated over several Bloch oscillations. These developments created a new
p decades. Until recently, the main motivation for exper- perspective for studying electron dynamics in solids ex-
- imental research has been laser damage of optical ma- posed to strong fields, where the emphasis is placed on
m
terials [1–3]. Detailed time-resolved measurements of attosecond-scale dynamics.
o electron dynamics used to be out of reach, as the char- The foundations for the theoretical description of ex-
at acteristic times of the most essential processes belong tremely non-linear interaction with strong fields, where
. to the attosecond domain. The situation has recently conventionalperturbationtheorybreaksdown,werelaid
s
c changed due to the progress in the generation of few- by L. V. Keldysh in his seminal papers [15, 16] where
i cycle laserpulses andthe developmentofattosecondsci- multiphoton and tunnelling excitations [17], which are
s
y ence [4, 5]. In this context, several recent papers should two extreme regimes of interband transitions, were dis-
h be pointed out. Gertsvolf et al. observed that an el- tinguished by the Keldysh parameter
p
liptically polarized40-fs pulse of near-infrared(NIR) ra-
[ ω mE
diation changes its ellipticity as it propagates through γ = L g. (1)
2 a few micrometres of fused silica [6]. Based on their peF0
v numerical calculations, the authors concluded that the
This parameter was introduced for a monochromatic
8 change of ellipticity is due to sub-cycle, attosecond-scale
3 laser field in the approximation of parabolic bands. In
22 ecxoncictlautsiioonnsdwynearemiccosnfiorfmveadlenbcye-Mbaintrdofealneocvtroents.al.,Twhehior Eisqt.h(e1)r,edωuLciesdtmhealsassoerf afrneqeuleecntcryo,nman=d(a1/hmolee,+E1/m=h~)ω−1
used a non-collinear pump-probe technique to observe g g
. is the bandgap, e is the absolute value of the electron
0
an attosecond-scale modulation of the electron density
1 created by a 5-fs NIR pulse in a SiO sample [7]. A dif- charge, and F0 is the amplitude of the laser field in the
2 2 medium. In the limit γ 1, where the field is weak
ferent optical strong-field phenomenon was observed by ≫
1 and/or its frequency is large, an electron is excited from
: Ghimire et al. in ZnO exposed to high-power few-cycle thevalencebandbyabsorbinganintegernumberofpho-
v
mid-infraredlaserpulses: theyobservedthegenerationof
i tons in a way that is well described by conventionalper-
X highharmonicswithpropertiesthatcannotbeexplained
turbation theory. In the opposite extreme γ 1, where
by the conventional non-linear optics [8]. The origin of ≪
r the field is strong and its frequency is sufficiently small,
a these harmonics was linked to a combination of anhar-
it is quantum tunnelling that promotes electrons to the
monic electron motion and multiple Bragg reflections at
conductionband. Obviously,this is the caseif the exter-
the boundaries of the first Brillouin zone, also known as
nal field is constant (ω = 0), and interband tunnelling
“Bloch oscillations” [9]. Another manifestation of Bloch L
as a concept was first introduced by Zener [18] for time-
oscillations observed by the same team was a redshift in
independent fields. In the tunnelling regime, the exci-
the optical absorptionin ZnO [10]. Most recently,it was
tation rate is often assumed to be determined by the
demonstrated that an intense few-cycle laser pulse not
instantaneous value of the electric field F (t) [7].
only excites charge carriers in SiO , but it also drives L
2 Electron dynamics in the intermediate regime γ 1
measurable currents that can be steered by controlling ∼
areparticularly complex—alaser fieldcannot be treated
the carrier-envelope phase (CEP) [11] of the laser pulse
as a small perturbation in this regime, and, at the same
time, interband transitions cannot be described by pure
tunnelling. A general expression for the cycle-averaged
∗ [email protected] excitation rate was given by L. V. Keldysh in [16], but
2
a completely satisfactory description of electron dynam- andexpandthispseudopotentialintheplane-wavebasis:
ics within an optical cycle of an ultrashort pulse is still
missing. The dynamics can be modelled by solving the U(r)=NmaxU eiKjr, (5)
time-dependent Schr¨odinger equation (TDSE) in a suit- j
able approximation,but there isno unambiguouswayto Xj=1
1
distinguish between valence- and conduction-band elec- U = e−iKjrU(r)dr, (6)
j
trons as long as the external field is present. This is Ω
ZΩ
analogous to the problem of determining the ionization
where Ω is the volume of a unit cell. Writing the unper-
rateofatomsandmoleculesinstrongfields[19],forwhich
turbed Hamiltonian as
manypracticalsolutionsexist,butthereisnomathemat-
ically rigorous solution in the case of a few-cycle pulse. ~2
Hˆ = 2+U(r) (7)
This motivated us to study the polarization response of 0 −2m ∇
0
a dielectric in the regime γ 1, as interband transi-
∼ and substituting ansatz (3) into Eq. (2), we obtain the
tions directly affect polarization and, at the same time,
following eigensystem for the expansion coefficients Cn
polarizationisaquantum-mechanicalobservablethatde- k,i
and energies En:
termines all optical properties of a solid. k
Nmax ~2
II. THEORY 2m (k+Kj)2δij +UKi−Kj Ckn,j =EknCkn,i.
j=1 (cid:18) 0 (cid:19)
X
(8)
We are mainly interested in the strong-field regime Here, m is the electron rest mass, and the eigensystem
0
where perturbation theory shows its shortcomings. In can be solved independently for each quasimomentum
particular, the polarization response of a solid to a k. The solutions of this eigensystem are normalized to
laser pulse is no longer described by a set of lin- satisfy
ear and non-linear susceptibilities; therefore, quantum-
mechanical simulations are necessary to model it. We φm φn = 1 [φm(r)]∗φn(r)d3r=δ , (9)
k k k k mn
solve the TDSE numerically in the velocity gauge, using h | i ΩZΩ
themethoddescribedin[20]. Foreachvalueofthequasi-
which implies Cn 2 =1. Here and in the following,
momentum k, the electron wave function is represented j| k,j|
a scalar product implies integration over one unit
in the basis of Bloch states φnk , which are evaluated by P h·|·i
| i cell.
solving the single-electron stationary Schr¨odinger equa-
tion with an unperturbed Hamiltonian Hˆ0: baHsisavtiongsoelvvaeluthaeteTdDBSloEchansdtattheuss|φmnkoi,dweletuhseeinthteermacatisona
Hˆ φn =En φn . (2) of electrons with a laser pulse:
0 k k k
| i | i
∂
Here, n is a band index. This equation is solved in the i~ ψk(t) = Hˆ0+Hˆint(t) ψk(t) , (10)
∂t| i | i
basis of Nmax plane waves: (cid:16) (cid:17)
where Hˆ (t) is the interaction Hamiltonian. We per-
int
Nmax form our simulations in the velocity gauge, where an
φnk(r)= Ckn,jexp[i(k+Kj)r], (3) external electric field is described by the vector poten-
j=1 tial A(r,t). We also employ the dipole approximation:
X
A(r,t) A(t). InSIunits,thiscorrespondstousingthe
whereK denotesreciprocallatticevectors,eachofwhich ≡
j following form of the interaction Hamiltonian:
can be expressed as a combination of the primitive vec-
tors b1,b2,b3 : Hˆ (t)= e A(t)pˆ. (11)
{ } int
m
0
3
K= m b , m Z. (4) Note that in CGS units the right hand side of Eq. (11)
α α α
∈ must be divided by the vacuum speed of light c. The
α=1
X substitution of the ansatz
Ansatz (3) ensures that all Bloch states satisfy φn(r+
k Nmax
R) = φnk(r)exp[ikR], where R is a vector in the direct ψk(t) = αnk(t) φnk (12)
Bravais lattice. This property immediately follows from | i | i
n=1
exp[iKR]=1. X
Thecoefficientsinexpansion(3),aswellastheenergies into Eq. (10) leads to the following system of coupled
En,aredeterminedbydiagonalizingtheHamiltonianHˆ differential equations:
k 0
inthebasisofplanewavesexp[iK r]. Weassumethatthe
j ∂αq(t) e Nmax
interaction of a particular electron with lattice ions and i~ k =Eqαq(t)+ A(t) pqlαl(t). (13)
k k k k
other electrons is described by a pseudopotential U(r) ∂t m0
l=1
X
3
Here, current density averagedover a unit cell:
Nmax ∗ e 1
pqkl = φqk|pˆ|φlk =~ (k+Kj) Ckq,j Ckl,j (14) jk,n(t)=−m V d3r Re ψk∗,n(r,t)pˆψk,n(r,t) +
(cid:10) (cid:11) Xj=1 (cid:16) (cid:17) e ZV (cid:18) (cid:2) (cid:3)
are the matrix elements of the momentum operator pˆ. +eA(t)ψk,n(r,t)2 . (17)
| |
An importantadvantageof the velocitygaugeis that, as (cid:19)
longasthedipoleapproximationholdsandelectronscat- Here and in the following, we add index n to quanti-
teringisneglected,thereisnocouplingbetweendifferent ties related to single-electron wave functions in order to
values of k. Mathematically, this is a consequence of indicate the initial band. Thus, jk,n(t) represents the
contribution from all the bands to j(t) in a calculation
(φq (r))∗pˆφl(r)d3r= where the electron initially occupied band n. The term
k′ k
R3 containingthevectorpotentialisrequiredinthevelocity
Z
= φqk′ pˆ φlk ei(k−k′)Rj δkk′, gauge.
| | ∝ With the aid of Eq. (12), the single-electron cur-
j
(cid:10) (cid:11)X rentdensity canbe expressedvia probability amplitudes
wherethesummationisperformedoverallthevectorsof αqk(t):
theBravaislatticeR ,andweassumethatbothkandk′
j
belong to the first Brillouin zone. Due to this property,
e
Eqs. (13) can be solved independently for each quasi- jk,n(t)= eA(t)+
momentum k. According to our experience, the down- −me
side of the velocity gauge is a larger number of bands ∗
required for convergence, as compared to length-gauge +Re αqk,n(t) αlk,n(t)pqkl . (18)
simulations, where different values of k are coupled to (cid:20)Xq,l (cid:16) (cid:17) (cid:21)!
each other [21, 22], but even the two-band approxima-
tion may be sufficient [23]. Three different physical effects determine the induced
Atthe beginningofasimulation,allelectronsaresup- currents: polarizationduetothedisplacementofvalence-
posed to reside in the valence bands (VB), while all the band electrons, the light-driven motion of conduction-
conduction bands (CB) are empty. We solve Eqs. (13) band electrons, and quantum beats due to coherent su-
for each of the electrons independently and then add perpositions of valence- and conduction-band states. In
single-electron contributions together to evaluate physi- the presenceofastrongexternalelectricfield,these con-
cal observables such as current density or polarization. tributionscannotbefullyseparatedfromeachother,but
Obviously, our approach is only applicable if (i) lat- the overallinduced current density j(t) is defined unam-
tice vibrations have no significant effect on the inves- biguously. It is this induced current that determines the
tigated dynamics, (ii) effects related to electron corre- change of the refractive index and the generation of new
lation and electron-hole interaction are not important, frequencycomponents. However,itisconventionaltode-
and (iii) distortions of the pseudopotential U(r) upon scribeopticalresponseintermsofpolarization,whichwe
electronicexcitationsarenegligible. While theseapprox- define as
imation obviously have their limitations, single-electron
t
models proved to be useful for studying basic physics P(t)= j(t′)dt′. (19)
related to phenomena occurring on attosecond and few- −∞
Z
femtosecond time scales [20, 24, 25].
Notethatthepolarizationdefinedthiswayisduetoboth
The initial conditionfor solving Eqs.(13) for a partic-
bound and free electrons. Ordinarily, P(t) would be as-
ular k is
signed to bound electrons, while j(t) would describe the
αq(t )=δ . (15) motion of free electrons. We make no attempt to make
k min qn
this distinction because, to the best of our knowledge,
Here, n is a band where the electron was before the in- there is no rigorous way to distinguish between bound
teraction with a laser pulse. and free electrons as long as the external field is present
In this paper, we are most interested in the macro- [19]. It is only after the laser pulse that projecting a
scopic electric current density j(t) induced by the laser wave function onto conduction-band Bloch states yields
pulse physically relevant excitation probabilities.
For simplicity, we perform our numerical simulations
j(t)= jk,n(t)d3k, (16) in one spatial dimension x, which we assume to be the
polarizationdirectionofthelaserfield. Mostoftheequa-
n∈VBZ
X tions in this section can be transformed to their one-
where the integral is taken over the first Brillouin zone dimensional forms by replacing all vector quantities and
andjk,n(t)representssingle-electroncontributionstothe three-dimensional integrals with scalar quantities and
4
one-dimensional integrals: r x, k k. The only ex- of electrons—this field is assumed to be a part of A(t)
→ →
ception is Eq. (16), where this procedure would result in and F(t).
j(t)beingmeasuredinwrongunits. Reducingthedimen-
sionality of our original problem, we essentially assume
that physical observables do not depend on coordinates 15 E 1
k
y and z, so that the integralover the first Brillouin zone E 2
k
in Eq. (16) reduces to 10 E 3
k
V)
e 5
j(t)=η jk,n(t)dk, (20) y (
g
nX∈VBZ ner 0
e
where the factor η, measured in m−2, accounts for the
E = 8.95 eV
integration over k and k . In practice, the value of -5 g
y z
η should be chosen to approximate some known opti-
cal properties of the solid, such as its refractive index or -10
-1 -0.5 0 0.5 1
absorption.
a k / π
From this point on, we will write our equations in 0
atomic units (at.u.), unless stated otherwise. In atomic
units, ~ = e = m0 = 1, a unit of energy is equal 1.56 model
to 27.21 eV, and a unit of electric field is equal to fused silica
5.142 1011V/m. Because~=1,wewillinterchangeably 1.54
×
use the words “energy” and “frequency”. ex 1.52
We adjusted our model potential U(x) to obtain a nd
bandgapcloseto thatofSiO : 8.95eV[26]. Tothis end, e i 1.5
2 v
we set the lattice constant to a0 = 0.5 nm = 9.45 at.u., acti 1.48
whichisoneofthe lattice constantsinα-quartz,anduse efr
r 1.46
the following expression for the central unit cell:
1.44
U(x)= 0.7(1+tanh[x+0.8])
− × 1.42
(1+tanh[ x+0.8]) (21) 400 600 800 1000 1200 1400
× −
wavelength (nm)
for x a /2. Outsidethecentralunitcell,thepotential
0
| |≤
is continued periodically: U(x+a ) = U(x). The band
0 FIG. 1. (a) The uppermost valence band and the two lowest
structure that corresponds to this potential is shown in conductionbandsofthemodelpotential(21). (b)Acompar-
Fig.1(a). Atk =0theenergiesEnofthefirstfourbands ison of the refractive index evaluated according to Eq. (24)
k
are 35.48, 8.33,0.62,and13.13eV.Weregardthefirst with therefractive index of fused silica [27].
− −
two bands with negative values of En as valence bands
k
(the lowest valence band is not shown Fig. 1(a)), while For a weak pulse, the polarization response is linear,
all the bands with En > 0 are regarded as conduction so that, in the frequency domain, P˜(ω) = χ(ω)F˜(ω),
k
bands. andtherefractiveindexcanbeevaluatedfromthelinear
We usedthe following expressionfor the vectorpoten- susceptibility χ(ω):
tial of the (linearly polarized) laser field acting on elec-
trons in the solid: n(ω)= 1+χ(ω). (24)
A(t)= θ(τ t)F0 cos4 πt sin(ω t+ϕ ). (22) In Fig. 1(b), we comparepthe refractive index of fused
L L CEP
− −| | ωL (cid:18)2τL(cid:19) silica with n(ω) evaluated using our model. The re-
fractive index is plotted against the laser wavelength
Here, F is the amplitude of the electric field, ω is the
0 L λ=2πc/ω. Forthesesimulations,weusedseveralpulses
central frequency of the laser pulse, ϕ is its carrier-
CEP with F =10−5 at.u.=5 106 V/m, FWHM=4π/ω ,
envelope phase, θ(x) is the Heaviside step function, and 0 × L
and values of ω that allowed us to cover the spectral
τ isrelatedtothefullwidthathalfmaximum(FWHM) L
L range presented in Fig. 1(b). Normalizing the polariza-
of A(t)2 byFWHM=4arccos 2−1/8 τ /π 0.5224τ .
| | L ≈ L tion response,we set η =0.111at.u. in Eq.(20) anduse
The FWHM of A(t)2 is very close, although not pre-
| | (cid:0) (cid:1) this valuehenceforth. Giventhe simplicityofourmodel,
ciselyequal,tothe FWHM of F(t)2. The externalelec-
| | wefindtheagreementwiththemeasuredrefractiveindex
tric field acting on electrons is, by definition,
very satisfactory.
′ The next section reports on simulations with much
F(t)= A(t). (23)
− moreintensefields,wheremultiphotonexcitationspopu-
Note that, in this work, we do not make any attempt to late conduction bands. For those simulations, we had to
evaluate the screening field created by the displacement use15bandsinordertoachievenumericalconvergencein
5
solving the TDSE. A smaller number of bands results in 1 x2
W(x)= exp ix . (26)
discrepancies that are most visible in the spectral range √τ − 2τ2
W (cid:20) W(cid:21)
occupied by low-order harmonics, while increasing the
number of bands to 20 has a negligible effect on the po- The parameter τ determines the size of the temporal
W
larization response even for the most intense pulses that window, and it must be chosen to provide an optimal
we used in our modelling. compromise between the temporal and spectral resolu-
tion. Our choice was τ =3ω /ω .
W g L
The outcome of this analysis, S(t,ω), is shown as a
false-colour diagram in Fig. 2(b), the bandgap energy
III. RESULTS AND DISCUSSION beingindicatedbyadashedwhiteline. Fortheseparam-
eters,thequantum-beatsignalcompletelydominatesthe
third-order harmonic, which is barely visible in the plot.
For all the simulations in this section, we used pulses
In this time-frequency analysis, the third-harmonic sig-
with the FWHM equal to three optical oscillations:
nal is centred at 6.7 eV, and it is temporally confined to
FWHM = 3 2π/ω . The electric field F(t) of such
a pulse is sho×wn in LFig. 2(a) by the dotted blue line the FWHM of the light pulse: ωL|t|/(2π).1.5.
Oneofthe moststrikingfeaturesobservedinFig.2(b)
plotted against the number of optical cycles ω t/(2π).
L isthe factthatthequantum-beatsignalinitiallyappears
The solid red line in this figure shows the polarization
at frequencies exceeding ω by roughly 2 eV/~. The in-
response P(t) evaluated according to Eq. (19) for the g
stantaneous frequency of quantum beats then gradually
case where the central frequency is equal to a quarter
decreases,approachingω atthe tailing edgeofthe laser
of the bandgap (ω = ω /4, λ = 2πc/ω = 557 nm), g
L g L L pulse. In other words, the quantum-beat signal is neg-
and the peak intensity is equal to I =3 1013 W/cm2,
L × atively chirped—its lower-frequency components are de-
whichcorrespondstoanamplitudeofthe electricfieldof
layed with respect to the higher-frequency ones. This
F = 0.029 at.u. = 1.5 V/˚A. The scales chosen for the
0 cannot be explained by the presence of the fifth-order
figure emphasize that, at the leading edge of the laser
harmonic—even though the fifth harmonic occupies the
pulse,P(t)isproportionaltoF(t). Atlatertimes,thepo-
relevantspectralrange(5~ω =11.2eV),itmustbecon-
larization response becomes increasingly non-linear and L
fined to an even smaller temporal range than the third
high-frequencyoscillationsappear,whichpersistevenaf-
harmonic, so that S(t,ω) can be considered free from
ter the laser pulse ends. The frequency of these oscilla-
tionsisclosetothebandgapfrequencyω =E /~. Thus, its contribution for ωLt/(2π) & 1, where most of the
g g quantum-beat signal is observed.
these are quantum beats appearing due to the presence
We seek an explanation for the chirp of the quantum-
of coherent superpositions of valence- and conduction-
beat signal in the laser-drivenmotion of charge carriers.
band states. This part of the polarization response will
Theaveragekineticenergyofanelectron-holepairisthe
be the main topic of our further discussion. In order to
distinguish it from conventional harmonics due to χ(3), ponderomotive energy. For a monochromatic field with
χ(5), χ(7),... nonlinearities, we chose the central laser an amplitude F0, the ponderomotive energy is given, in
the approximationof parabolic bands, by
frequency to be an even divisor of the band gap, tak-
ing advantage of the fact that harmonics of even orders
F2
are absent if the potential U(x) possesses the inversion U = 0 . (27)
symmetry U( x)=U(x), like our model potential (21). p 4mω2
L
−
Note that, due to the same symmetry, the four-photon
transition from the uppermost valence band to the low- Here, m is the reduced mass of an electron and a hole.
est conduction band at k = 0 is forbidden, while inter- We obtain the effective masses of electrons and holes by
bandtunnellingcanefficientlypopulateconduction-band fittingparabolastothelowestconductionanduppermost
states in the middle of the Brillouin zone. Indirectly, valence bands in the region k < π/(2a0), which yields
| |
this is confirmed by the fact that the fast oscillations me = 0.34 at.u. and mh = 3.61 at.u. These values are
in Fig. 2(a) only appear at very high intensities. At a in good agreement with those that can be found in lit-
peakintensityofI =1013 W/cm2,whichisjust3times erature [29, 30]. Thus, the reduced effective mass m in
L
smaller than the one used for Fig. 2, the amplitude of Eq. (27) is equal to m = (1/me+1/mh)−1 = 0.35 at.u.
quantum-beat oscillations is smaller by approximately a The cycle-averaged total energy of an electron-hole pair
factorof400,whichcorrespondsto adropoftheirinten- is equalto Eg+Up(t), where Up(t) is to be evaluatedby
sity by five orders of magnitude. replacing F0 in Eq. (27) with the pulse envelope. This
quantity is plotted as a red line in Fig. 2(c). It is fairly
We investigate the quantum-beatpart of the polariza-
closetotheenergieswhereS(t,ω)haslocalmaximawith
tion response by means of the waveletanalysis using the
respect to ω. We observed this kind of negative chirp in
Morlet wavelet [28]:
numerous simulations with different parameters of the
laser pulse, some of which are shown in Fig. 3. There-
∞ 2
′ ′ ′ fore, we conclude that the frequency of quantum-beats
S(t,ω)=ω P(t)W ω(t t) dt , (25)
(cid:12)Z−∞ − (cid:12) exceeds ωg by approximately the ponderomotive energy.
(cid:12) (cid:0) (cid:1) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6
a)
3 P(t)
3
F(t)
s) 2 2 s)
nit nit
c u 1 1 c u
mi mi
o 0 0 o
at at
P(t) ( -1 -1 F(t) (
310 -2 -2 210
-3
-3
-4 -2 0 2 4 6 8
time (optical cycles)
c) 12 d) 10
E +U (t)
11.5 g p 8 F2(t)
QB peak
6
11 4
V) s 2
y (e 10.5 unit 0
nerg 10 arb. -2
e
9.5 -4
-6
9
-8
8.5 -10
0 1 2 3 4 5 6 7 8 -4 -2 0 2 4 6 8
time (optical cycles) time (optical cycles)
FIG.2. (a)Thepolarization P(t)inducedbyalaserpulsewithωL=ωg/4(λL =557nm),apeakintensityof3×1013 W/cm2,
and ϕCEP =0. Theelectric field F(t)isshown bythedottedblueline; theFWHMofthepulsecorresponds tothetimerange
from −1.5 to 1.5 optical cycles. (b) S(t,ω), which is the result of applying the wavelet transform to P(t). The dashed white
line marks the energy Eg = ~ωg. (c) The solid red line is the sum of the bandgap energy Eg and the ponderomotive energy
Up(t). For each time t, the dashed blue line depicts the above-bandgap frequency where S(t,ω) has a maximum. (d) The
quantum-beat part of the polarization response F−1[w˜(ω)P˜(ω)] (grey) and its envelope (red). The spectral window used for
theanalysis is given by Eq.(28).
Fig. 2(d) presents a different kind of analysis that as the quantum-beat signal is due to interband excita-
highlights the quantum-beat part of the polarization re- tions, and the rate of the interband excitations is ex-
sponse. In order to suppress the third- and fifth-order pected to reach its peak at the peak of the laser pulse.
harmonics, we multiplied the Fourier transform of P(t) Another prominent feature apparent from Fig. 2(d) is
with a soft spectral window that cuts all frequency com- the decay of quantum beats that begins at the trailing
ponents that are further than ω /2 from the bandgap edge of the laser pulse. As our model does not account
L
frequency: for any relaxation phenomena (such as electron-phonon
interaction or spontaneous emission of radiation), the
π ω ω origin of this decay lies in the dephasing of dipole os-
ω | |− g
w˜(ω)=θ(cid:18) 2L −(cid:12)|ω|−ωg(cid:12)(cid:19)cos2 (cid:12)(cid:12)(cid:12) ωL (cid:12)(cid:12)(cid:12). (28) clailtleadtovrsaleansscoec-iaatneddcwointdhuecaticohn-pbaainrdofstcaothese.renEtalychposupcuh-
(cid:12) (cid:12)  (cid:12) (cid:12) coherent superposition creates a current j (t) described
(cid:12) (cid:12)   k
(cid:12) (cid:12)   by Eq. (18). If the laser pulse ends at a time t , then
ThegreylineinFig.2(d)showstheinverseFouriertrans- 1
formofw˜(ω)P˜(ω). Inotherwords,itrepresentsthe part αqk(t)=αqk(t1)exp[−i(t−t1)Ekq], so that
oclfotsheetpootlahreizbaatniodngarepspfroenqsueetnhcayt.oTschilelaetnesvealtopfreeqoufetnhceisees jk(t)=−Re (αqk(t1))∗αlk(t1)pqklei(t−t1)(Ekq−Ekl). (29)
q,l
fast oscillationsis shownas a solid redline, andthe dot- X
ted blue line depicts the square of the laser field. From For brevity, we have omitted the index specifying the
this analysis, it is obvious that the quantum-beat part initial band of the electron. For a given pair of bands
of the polarization response experiences its most rapid q = l, Eq. (29) describes a current oscillating with a
increase at a time that lies about two optical cycles af- k-d6 ependent frequency Eq El. Because of this depen-
k− k
ter the peak of the laser pulse. This is counter-intuitive, dence, the net current density (20) attenuates as most
7
FIG.3. Thewaveletanalysisofthepolarizationinducedby3-cyclepulseswithapeakintensityof3×1013W/cm2,ϕCEP =0,and
thefollowing central frequencies: (a) ωL =ωg/5 (λL=697nm);(b) ωL =ωg/6 (λL=836nm);(c) ωL =ωg/8 (λL =1108nm);
(d) ωL =ωg/10 (λL =1385nm). The unitsin colour coding are arbitrary, but theyare thesame as those used for Fig. 2(b).
currents j (t) get out of phase with each other. plot the sum of all conduction-band populations at the
k
Fig. 3 presents the wavelet analysis of polarization end of our simulations in Fig. 4. These are the probabil-
response functions for several other values of the cen- ities to find an electron with a given quasimomentum k
tral laser frequency ω . The dashed white lines depict in one of the conduction bands at a time t > τ where
L L
E +U (t). InFigs.3(a)and3(b), theselinescloselyfol- the laser field is absent (since we do not account for re-
g p
lowlocalmaximaofS(t,ω). Forsmallerlaserfrequencies, laxation phenomena, populations do not change in the
the ponderomotivepotentialU significantlyexceedsthe absence of external fields). The distribution of excited
p
widthofthelowestconductionband(seeFig.1(a)). This electronsdramaticallychangesasω decreasesfromω /4
L g
is the regime where Bloch oscillations [31] must play an to ω /10: the pronouncedpeak atk =0 disappears,and
g
important role [8]. many irregular peaks appear across the Brillouin zone.
In Fig. 3(a), where valence-band electrons can be ex- At the same time, the distributions become increasingly
cited by five-photon absorption, we made an exception sensitive to the carrier-envelopephase of the laser pulse.
from our choice to use those values of ω that are even We consider two physical phenomena that may be re-
L
divisors of ω . Still, the time-frequency analysis looks sponsible for this trend: interband tunnelling and Bloch
g
very similar to that for ω = ω /4, shown in Fig. 2(b), oscillations.
L g
and the contribution from the fifth harmonic generation The first phenomenon, interband tunnelling, repre-
is so weak that it is practically invisible in our colour sents an extreme regime of interband excitations where
scheme. For all the cases shown in Fig. 3, we observe the Keldyshparameterγ is muchsmaller than1. In this
the same qualitative features as those in Fig. 2(b): the regime, interband transitions should be viewed not as a
frequency components of the polarization response that resultofabsorbingacertainnumberofphotons,butasa
lieabovethebandgapappeardelayedwithrespecttothe resultofquantumtunnelling[18]. Also,inthetunnelling
peak of the laser pulse, and the instantaneous frequency regime, the carrier-envelope phase is an important pa-
of these oscillations decreases with time, eventually ap- rameter because it is the field of the laser pulse, rather
proaching ω . We also observe a rapid decrease in the than its envelope, that controls the tunnelling rate. Ac-
g
magnitude of S(t,ω). cording to Fig. 4 and all the other similar simulations
To further understand the origin of these trends, we that we have performed, the onset of the CEP depen-
8
a) 0.08 b) 0.08
ϕ = 0 ω = ω /4 ϕ = 0 ω = ω /6
CEP L g CEP L g
0.07 ϕ = π/2 γ = 0.93 0.07 ϕ = π/2 γ = 0.62
CEP CEP
0.06 ωB/ωL = 1.1 π 0.06 ωB/ωL = 1.6 π
0.05 0.05
0.04 0.04
0.03 0.03
0.02 0.02
0.01 0.01
0 0
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
a k / π a k / π
0 0
c) 0.08 d) 0.08
ϕ = 0 ω = ω /8 ϕ = 0 ω = ω /10
CEP L g CEP L g
0.07 ϕ = π/2 γ = 0.50 0.07 ϕ = π/2 γ = 0.40
CEP CEP
0.06 ωB/ωL = 2.1 π 0.06 ωB/ωL = 2.7 π
0.05 0.05
0.04 0.04
0.03 0.03
0.02 0.02
0.01 0.01
0 0
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
a k / π a k / π
0 0
FIG.4. Theprobabilityforanelectronthatinitiallyoccupiesastateintheuppermostvalencebandwithacertainquasimomen-
tumk tobeexcitedtooneoftheconductionbandsbytheendofa3-cyclelaserpulsewithapeakintensityof3×1013 W/cm2.
Resultsobtainedforcosinepulses(ϕCEP =0,solidredlines)arecomparedwiththoseforsinepulses(ϕCEP =π/2,dottedblue
lines). ThecentrallaserfrequencyωL,theKeldyshparameterγ,andtheratiooftheBlochfrequencyωB tothelaserfrequency
are specified in theplots.
dence coincides with γ decreasing below a value approx- is comparable to π/a . Introducing the Bloch frequency
0
imately equal to 0.7. Furthermore, the tunnelling pic-
ture provides an intuitive explanation for the multitude ea0F0
ω = (SI units), (30)
ofpeaksinFigs.4(c)and4(d): theymaybeattributedto B ~
theinterferenceamongelectronwavepacketslaunchedto
this condition can be expressed as ω /ω = π. The la-
conductionbands within differenthalf-cycles of the laser B L
bels in Figs. 4(a-d) provide values of ω /ω in units of
pulse, which is analogous to the above-threshold ioniza- B L
π for all the four simulations. The appearance of the
tionspectraofatomsexposedtointenselaserpulses[32].
CEP-dependence and the complex structures in the dis-
tribution of conduction-band electrons coincide not only
withγ becomingsmallerthan1,butalsowithω /ω be-
B L
What the tunnelling picture alone cannot explain is coming larger than π. In fact, this is not a coincidence,
the disappearance of the peak at k = 0. Indeed, all as
analytical models for the tunnelling rate predict that it
shouldrapidlydecreaseasthe bandgapincreases[16,18, ωB a0 mEg
= (SI units), (31)
33], and our model potential has the smallest bandgap ω ~γ
L p
at k = 0. Therefore, the motion of excited electrons in
conductionbands mustplaya significantrole[23], Bloch as follows from Eqs. (1) and (30). For our parameters,
oscillations being the most prominent manifestation of ω /ω 1.02π/γ, so that the onset of tunnelling coin-
B L
≈
thismotion. Anelectronexcitedtothebottomofaband cides with the onset of Bloch oscillations, irrespectively
at a zero-crossing of the vector potential (A(t ) = 0) of the laser pulse parameters.
0
may reach the boundary of the first Brillouin zone if, at Without interband excitations, Bloch oscillations ob-
some later moment t , the condition A(t ) = π/a is viously cannot occur, but is it the nature of interband
1 1 0
| |
satisfied. Thus, Bloch oscillations should be considered transitions or the Bloch oscillations themselves that are
importantiftheamplitude ofthevectorpotentialF /ω mainly responsible for the observed CEP dependencies?
0 L
9
a) 100 b) 100
ϕCEP = 0 ωL = ωg/4 = 2.2 eV ϕCEP = 0 ωL = ωg/6 = 1.5 eV
10-1 ϕCEP = π/2 γ = 0.93 10-1 ϕCEP = π/2 γ = 0.62
s) ωB/ωL = 1.1 π s) ωB/ωL = 1.6 π
nit 10-2 nit 10-2
u u
b. b.
ar 10-3 ar 10-3
y ( y (
nsit 10-4 nsit 10-4
e e
nt nt
i i
10-5 10-5
10-6 10-6
4 6 8 10 12 14 4 6 8 10 12 14
photon energy (eV) photon energy (eV)
c) 100 d) 100
ϕCEP = 0 ωL = ωg/8 = 1.1 eV ϕCEP = 0 ωL = ωg/10 = 0.9 eV
10-1 ϕCEP = π/2 γ = 0.50 10-1 ϕCEP = π/2 γ = 0.40
s) ωB/ωL = 2.1 π s) ωB/ωL = 2.7 π
nit 10-2 nit 10-2
u u
b. b.
ar 10-3 ar 10-3
y ( y (
nsit 10-4 nsit 10-4
e e
nt nt
i i
10-5 10-5
10-6 10-6
4 6 8 10 12 14 4 6 8 10 12 14
photon energy (eV) photon energy (eV)
FIG.5. Thepowerspectraofthepolarization responseforϕCEP =0(solidredlines)andϕCEP =π/2(dottedbluelines). The
parameters of the laser pulse are the same as those used for Figs. 4(a-d), the values of the laser frequency ωL are specified in
plot labels. Thedouble-dashed black vertical lines mark thebandgap energy. Note that a part of thequantum-beatspectrum
appears below thebandgap.
While we cannot answer this question with certainty, we ities, the CEP-dependence is very weak for ω = ω /4,
L g
canarguethatinterbandtunnellingaloneisnotsufficient but it becomes very pronounced already at ω = ω /6,
L g
for a polarization response to be CEP-dependent. In- and it affects not only the quantum-beat part of the po-
deed, tunnelling must be the dominant excitation mech- larization response, but also the conventional harmonics
anism in the parameter regime of Fig. 4(a) because, in of third, fifth etc orders.
spite of the forbidden four-photon transitions at k = 0, Consistently with the results shown in Fig. 3, the
thebottomofthe conductionbandisstronglypopulated quantum-beat signal rapidly decreases with decreasing
after the laser pulse. Yet the polarization response is al- ω . Atthesametime,accordingtoFig.4,theexcitation
L
mostCEP-independent. Therefore,our simulationswith probabilitiesarecomparableinallthefourcases. Conse-
ω = ω /4 provide an example where the dominance of quently, the decrease in the polarization response is due
L g
interbandtunnellingover(perturbative)multiphotonex- to dephasing. Most importantly, we used longer pulses
citations does not lead to a significant CEP dependence. forlongerwavelengthsinorderto havethe samenumber
of optical cycles in all the pulses. As a result, dephas-
Thequantum-mechanicalobservablesthatwehavedis-
ingplayedamoresignificantroleforthe 14-fspulse with
cussedsofarcannotbedirectlymeasuredinarealisticex-
ω =ω /10 than for the 5.5-fs pulse with ω =ω /4. In
periment. The most accessible quantity for experiments L g L g
addition to that, the dephasing time is inversely propor-
is the spectrum of emitted radiation. In order to eval-
tionaltothespectralwidthoftheexcitation,andexcited
uate such spectra, one needs to account for absorption
electronsoccupyalargerpartoftheBrillouinzoneinthe
and phase-matching [9], which is beyond of scope of this
tunnelling regime (see Fig. 4).
work. Nevertheless, important conclusions about these
spectra can be made by analyzing the Fourier transform
of the polarization response P(t) that we evaluate with
IV. CONCLUSIONS AND OUTLOOK
ourmodel. InFig.5,weplotpolarizationspectraforthe
same simulations as those presented in Fig. 4. We see
thatthe spectraarealsosensitiveto the carrier-envelope Wehaveanalyzedthepolarizationresponseofamodel
phaseofthelaserpulse. Similarlytoexcitationprobabil- dielectric resembling SiO to few-cycle laser pulses that
2
10
are strong enough to efficiently excite electrons from va- nelling (γ 1) coincides with the onset of Bloch os-
∼
lence to conduction bands. These interband transitions cillations (ω πω ). In the tunnelling regime, we
B L
∼
create coherent superpositions of states that manifest have observed that the non-linear polarization and the
themselves as quantum beats in the polarization. Ob- final distribution of charge carriers are sensitive to the
viously, the dynamics of interband excitations and de- carrier-envelope phase of a laser pulse. In particular,
phasing determine the properties of the quantum-beat the spectrum of the emitted high-frequency radiation
part of the polarization response. It is less obvious that depends on the CEP, which should be possible to ob-
theinstantaneousfrequencyofquantum-beatoscillations serve in experiments. Such a measurement would have
changes with time, approaching the bandgap frequency to overcome obstacles related to the strong absorption
as the laser field attenuates. We find that, during the above the bandgap and the phase mismatch below the
laser pulse, the quantum-beat frequency (averaged over bandgap [9]. Also, it must be mentioned that the coher-
a laser cycle) is larger than the bandgap frequency by ent few-femtosecond pulse radiated by quantum beats
U /~, unless the ponderomotive energy U exceeds the spectrally overlaps with the incoherent radiation due to
p p
width of the lowest conduction band. Thus, the emitted fluorescence,whichourmodeldoesnotaccountfor. Some
burst of high-frequency radiation is negatively chirped. ultrashorttemporalgatingmaythereforebenecessaryto
Wehavealsoobservedthatquantumbeatsappearinthe isolatetheradiationduetoquantumbeats. Still,inspite
polarization response significantly delayed with respect of all these obstacles, such measurements should be fea-
to the peak of a laser pulse, even though most electron- sible at the current stage of technology.
hole pairs must appear when the laser field is strongest.
Apparently, the effect of a strong field on charge carri-
erssuppressessingle-photontransitionsfromconduction- ACKNOWLEDGEMENTS
tovalence-bandstates. Asatisfactoryexplanationofthis
phenomenonprobablyrequiresthedevelopmentofanan- Theauthorsgratefullyacknowledgefruitfuldiscussions
alytical theory, which may be triggered by this work. with Prof. F. Krausz and Prof. K. Yabana. Supported
by the DFG Cluster of Excellence: Munich-Centre for
For a dielectric like SiO , the onset of interband tun- Advanced Photonics.
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