Floquet analysis of real-time wavefunctions without solving the Floquet equation

1.3 MB·English

by  V. Kapoor

Floquet analysis of real-time wavefunctions without solving the Floquet equation
V. Kapoor and D. Bauer
Institut fu¨r Physik, Universita¨t Rostock, 18051 Rostock, Germany
(Dated: February 1, 2012)
We propose a method to obtain Floquet states—also known as light-induced states—and their
quasi-energiesfromreal-timewavefunctionswithoutsolvingtheFloquetequation. Thisisusefulfor
the analysis of various phenomena in time-dependent quantum dynamics if the Hamiltonian is not
strictly periodic, as in short laser pulses, for instance. There, the population of the Floquet states
dependson thepulse form and is automatically contained in thereal-time wavefunction butnot in
2
the standard Floquet approach. Several examples in the area of intense laser-atom interaction are
1
exemplarily discussed: (i) the observation of even harmonics for an inversion-symmetric potential
0
2 withasingleboundstate;(ii) thedependenceofthepopulation ofFloquetstateson(gauge) trans-
formations and the emergence of an invariant, observable photoelectron spectrum; (iii) the driving
n of resonant transitions between dressed states, i.e., the dressing of dressed states, and (iv) spec-
a
tralenhancementsat channelclosings duetotheponderomotiveshift of above-thresholdionization
J
peaks.
1
3 PACSnumbers: 32.80.Rm,02.70.Hm,32.80.Wr

]
h
I. INTRODUCTION so that information about LIS are not directly available.
p
- AsmanyinterestingphenomenasuchastheACStarkef-
m fect, Rabi oscillations, or stabilization against ionization
The time-dependent Schr¨odinger equation (TDSE)
o [9, 10] is most conveniently analyzed in terms of LIS, it
withatime-periodicHamiltonianhassolutionswhichcan
t is desirable to extract the “Floquet information” from
a be expressed in a time-periodic basis. This basis is re-
the real-time wavefunction “on-the-fly” while propagat-
s. ferred to as the Floquet basis, and eigenstates in this ing(orbypost-processing)it,withouthavingtosolvethe
c basis are the Floquet states [1–4]. Time-periodic poten-
i Floquet equation as well. We present such a method to
s tialsnaturallyarisewhenmatterisexposedtolaserfields.
analyze non-perturbative, laser-driven quantum dynam-
y In this context, Floquet states are also known as “light-
h icsviathe(time-resolved)Floquetinformationcontained
inducedstates”(LIS)[5],astheyarethenewstatesofthe
p in the corresponding real-time wavefunction.
combined system “target + laser field”. In fact, Floquet
[
theory has been used to determine, e.g., very accurate

2 ionization rates [6, 7]. Using so-called R-matrix Floquet
The paper is organized as follows: in Sec. II we re-
v theory, the method has been extended to multi-electron
view the basics of Floquet theory. In Sec. III we briefly
8
systems [8]. Strict periodicity of the Hamiltonian with
1 summarizethegeneralderivationofharmonicgeneration
the laser period implies physically that the laser pulse
5 selection rules before we present the (at first sight sur-
was always on and will be on forever. Then the prob-
1 prising)presence of peaks at even harmonics of the laser
. lem ariseshow the field-free system under study, e.g., an frequency in the case of an inversion-symmetric poten-
0
atom, gets into the laser field in the first place and how
1 tial with only one bound state. In Sec. IV we intro-
the field-free observablesemerge. In fact, the population
1 duceourmethodtoobtaintheFloquetinformationfrom
1 of the Floquet states depends on the laser pulse form. If the real-time wavefunction, e.g., the populated states
: the (up anddown) rampingof the laserfield is adiabatic and their energies, and use them to explain the pres-
v
andthelaserfrequencyisnon-resonantweexpectthesys-
i enceofhyper-Ramanlinesatevenharmonicfrequencies.
X tem to followjust a single Floquet state, namely the one
In Sec. V we investigate how the population of Floquet
r which is adiabatically connected to the field-free initial states changes under (gauge) transformations while the
a state. However, for non-adiabatic ramping or resonant
Floquet energies and the observable photoelectron spec-
interactions a superposition of Floquet states is created.
tra remain invariant. In Sec. VI time-resolved Floquet
An example for non-adiabatic population of several Flo-
spectra of real-time wavefunctions in the so-called ve-
quet states, leading to an apparent generation of even
locity gauge and in the Kramers-Henneberger frame-of-
harmonics in inversion-symmetric potentials, is given in
reference are compared. In Sec. VII the channel-closing
Sec. III of this paper.
phenomenon and related spectral enhancements are in-
Instead of converting the TDSE into the time- terpreted in terms of Floquet state-crossings. A conclu-
independent Floquet equation [eq. (13) below] one may sion is given in Sec. VIII. In this work we restrict our-
alternatively solve it directly in real-time. In the latter selves to spatially one-dimensional (1D) model Hamil-
case there are no assumptions about periodicity or adia- tonians. It is straightforward to extend the method to
batic rampingand, e.g., the effect of different laserpulse higher dimensions, as indicated in Appendix A. Atomic
formscanbestudied. However,thedirectsolutionofthe units (a.u.) |e| = m = ~ = 4πǫ = 1 are used unless
e 0
TDSE in real-time does not involve the Floquet basis noted otherwise.
2
II. BASIC THEORY Fora monochromaticlaserfield the interactionHamilto-
nian Wˆ(x,t) can be written as
Consider a linearly polarized laser field E(t) of fre- Wˆ(x,t)=Wˆ+(x)exp(iω t)+Wˆ−(x)exp(−iω t), (12)
quency ω in dipole approximation, polarized along the 1 1
1
x-directionandinteractingwithanelectroninsomebind- leading to the time-independent Floquet equation
ing potential V. The Hamiltonian in length gauge reads
(ǫ+n~ω −Hˆ )ϕ (x) (13)
1 0 n
Hˆ(t)=Hˆ +Wˆ(x,t), Wˆ(x,t)=xE(t) (1) = Wˆ+(x)ϕ (x)+Wˆ−(x)ϕ (x).
0 n+1 n−1
The index n of the Floquet state is known as the “block
with
index,” which may be interpreted as the number of pho-
1 ∂2 tons involved in the process under study. Hence, the
Hˆ =− +V(x). (2)
0 2∂x2 Floquet equation (13) couples any Floquet block n with
itsneighboringblocksn±1viaabsorptionoremissionof
a photon.
A. Floquet theory In principle, (13) is an infinite-dimensional set of dif-
ferential equations. In practice, it is truncated so that
For sufficiently long laser pulses nmin ≤ n ≤ nmax. In obtaining the eigenvalue equation
(13),weassumedstricttime-periodicity,whichphysically
2π means that the laser pulse is always on.
E(t+T)=E(t), T = (3)
ω
1
holds to high accuracy, and thus also Wˆ(t+T) = Wˆ(t) B. Non-Hermitian Floquet Theory
so that
We are interested in systems which, in the field-free
Hˆ(t+T)=Hˆ(t). (4)
situation,possessbesidesboundstatesalsoacontinuum.
In the presence of a laser field such a system may ion-
The Floquet theorem [1–4] states that in this case the
ize, i.e., the field-free stationary states are turned into
TDSE
field-dressed,quasi-stationarystates. The simplest cases
∂ of only a few (field-free) bound states (allowing for res-
i Ψ(x,t)=Hˆ(t)Ψ(x,t) (5)
onances) plus a continuum dressed by laser fields have
∂t
been discussed in the literature since long ago (see, e.g.,
has solutions of the form [11]and[12]forareview). Inanactualimplementationof
Floquettheory,thedecayofquasi-stationarystatesneeds
Ψ(x,t)=e−iǫtΦ(x,t), (6) to be taken into account when solving (13) by applying
Siegert boundary conditions for the outgoing waves [7],
Φ(x,t) being periodic itself,
leading to complex Floquet energies
Φ(x,t)=Φ(x,t+T). (7) Γ
ǫ=Reǫ−i (14)
2
ǫ is called the quasienergyor Floquet energy. The wave-
where Γ is the ionization rate. The difference between
functions Φ(x,t) fulfill the Schr¨odinger equation
Reǫ and the field-free ǫ(0) is the AC Stark shift.
Hˆ(t)Φ(x,t)=ǫΦ(x,t) (8)
C. Finite-grid, finite-pulse TDSE solution
with
∂ We solve
Hˆ(t)=Hˆ(t)−i . (9)
∂t ∂
i Ψ (x,t)=Hˆ(t)Ψ (x,t) (15)
# #
∂t
If ǫ is an eigenvalue and Φ(x,t) the correspondingeigen-
state, also on a numerical grid of size L, −L < x < L for times
2 2
0 < t < t with t the total simulation time. The
sim sim
ǫ′ =ǫ+mω1, Φ′(x,t)=eimω1tΦ(x,t), m∈Z (10) binding potential V(x) is centered at x=0. In all cases
discussedinthisworkwestartfromthefield-freeground
are solutions of (8). Owing to the time periodicity of state on the grid Ψ (x,0) = Ψ(0)(x). Probability den-
Φ(x,t) we can expand # #
sity approaching the grid boundary is absorbed by an
imaginary potential.
∞
Φ(x,t)= ϕn(x)e−inω1t. (11) Ouraiminthefollowingwillbeto analyzeΨ#(x,t) in
n=X−∞ terms of Floquet energies and states.
3
III. HARMONIC GENERATION
Inthefirstexampleweapplyourmethodtoinvestigate
the origin of apparently even harmonics in an inversion-
symmetric potential with only one bound state.
There are many ways to derive selection rules for har-
monic generation (HG). Most elegant, rigorous, and ap-
propriate for our purpose is the approachemploying dy-
namical symmetries [13, 14]. Consider the stationary
Schr¨odinger equation
Hˆ Ψ(x)=EΨ(x), (16)
0
with Hˆ given by (2). If the potential V is inversion-
0
symmetric, V(x) = V(−x), the Hamiltonian Hˆ is in-
0
variant under spatial inversion as well,
FIG. 1: (color online). Logarithmically scaled HG strength
Pˆ f(x)=f(−x), Pˆ2 =1, Pˆ−1 =Pˆ , (17) ω4|d(ω)|2 vs harmonic order and excursion amplitude αˆ =
p p p p Aˆ/ω (ω =1,vectorpotentialA(t)rampedupanddownover
1 1
4 cycles and held constant with amplitude Aˆ for 30 cycles).
[Hˆ ,Pˆ ]=0, (18) Thenumericalfast-Fouriertransformwasperformedoverthe
0 p
pulseduration,i.e.,t =38cycles,usingaHanningwindow.
sim
sothatfornon-degenerateenergiesE theeigenstateΨ(x)
is alsoan eigenstate ofthe spatial-inversionoperatorPˆ .
p
Because of Pˆp2 =1 the eigenvalues can only be ±1 (par- Assuming that the numerically determined, exact wave-
ity): function on the grid is well described by just a single
Floquet state, using (6), (11), and (14) yields
Pˆ Ψ(x)=±Ψ(x). (19)
p
L
2
ThefullHamiltonian(1)andtheFloquet-Hamiltonian d(ω)∼ ϕ∗ (x)xϕ (x)dx (27)
(9)arenotinvariantunderspatialinversionbutunderthe XnmZ−L2 m n
dynamical symmetry operation “spatial inversion com-
bined with a translation in time by half a period”,
tsim
× et{i[ω−ω1(n−m)]−Γ}dt.
[Hˆ(t),Pˆpt]=[Hˆ(t),Pˆpt]=0, (20) Z0
The spatial integral is non-vanishing only if ϕ has the
n
Pˆ f(x,t)=f(−x,t+π/ω ), Pˆ2 =1, (21) opposite parity of ϕ , i.e.,
pt 1 pt m
n−m=2k+1, k ∈Z. (28)
Pˆ =Pˆ Pˆ =PˆPˆ , Pˆf(x,t)=f(x,t+π/ω ). (22)
pt p t t p t 1
The temporal integral thus leads to peaks centered at
For non-degenerate ǫ frequencies
PˆptΦ(x,t)=±Φ(x,t). (23) ω =(2k+1)ω1 (29)
Because of (11) we observe that with widths determined by t (frequency-time uncer-
sim
tainty) and Γ (decay). The selection rule (29) is the
PˆptΦ(x,t)= (−1)ne−inω1tPˆpϕn(x), (24) well-known result that an inversion-symmetric target in
Xn a linearly polarized laser field generates odd harmonics
only. Notethattheabovederivationalsoholdsformulti-
and with (23) follows that
electrontargetsbecausethe electron-electroninteraction
Pˆ ϕ (x)=±(−1)nϕ (x), (25) is also invariant under the symmetry operations Pˆp and
p n n
Pˆ .
pt
i.e., the ϕ (x) haveanalternatingparity withrespect to
n
the Floquet block index n.
Numerically, the HG spectrum ∼ ω4|d(ω)|2 is calcu- A. Hyper-Raman lines at even harmonics of the
lated via the Fourier-transformeddipole moment laser frequency
d(ω)∼ tsimdt L2dxΨ∗ (x,t)xΨ (x,t)eiωt. (26) It is known that HG peaks at positions different from
Z0 Z−L # # odd multiples of the fundamental laser frequency ω1 are
2
4
to be expected for an inversion-symmetricpotential if at
leasttwoFloquetstates ofopposite parityarepopulated
[15, 16]. Physically, the superposition of two Floquet
statesmayamountto,e.g.,theabsorptionofnphotonsof
energyω butemissionofonephotonofenergynω −∆ǫ,
1 1
with ∆ǫ being the energy difference between initial and
finalstate. Thisshouldleadtohyper-Ramanlinesinthe
spectrawhich,however,aretypicallyweak[16,17]. Nev-
ertheless,ifobservable,theyappearatevenharmonicsof
the laser frequency in the case of degeneracy, ∆ǫ=0.
We consider an electron in the Po¨schl-Teller potential
1
V(x)=− (30)
cosh2x
and subject to a laser field. The potential (30) supports
only a single bound state Ψ (x) of energy E = −0.5. FIG.2: (coloronline). LogarithmicplotofR=|Q+|2+|Q−|2
Hence,superpositionsoffield-0freeboundstates0areruled- vsenergy E and excursion amplitude αˆ =Aˆ/ω1, showing the
quasi-energies of the (populated) field-dressed states. The
out. As a consequence,perturbationtheoryinthe exter-
laser frequency was ω =4. The pulseshapewas trapezoidal
nal field can certainly not predict hyper-Raman lines or 1
(4,1200,4) inthevectorpotentialofamplitudeAˆ. Foreachαˆ
even harmonics. However, Fig. 1 shows the logarithmi-
themaximum in R was renormalized to unity.
cally scaled HG strength ω4|d(ω)|2 as obtained from the
numericalsolutionoftheTDSE.TheHGstrengthisplot-
ted vs harmonic order ω/ω and the amplitude αˆ of the
1 ǫ. The Fourier-transformeddipole will be
excursion
L
t d(ω)∼ 2 ϕ∗ (x)xϕ (x)dx (33)
α(t)=Z A(t)dt (31) βXγnmZ−L2 γm βn
with A(t) the vector potential of the laser field. The
veleeccttorricpfioeteldntiiaslgaivmepnlitbuydeE(Aˆt), t=he−ex∂ctAur(sti)o.n Gamivpelnitutdhee ×Z tsimet{i[ω−ω1(n−m)−(Reǫβ−Reǫγ)]−(Γβ+Γγ)/2}dt.
is αˆ = Aˆ/ω , the field amplitude Eˆ = Aˆω . The laser 0
1 1
pulse parametersarespecifiedin the figurecaption. One
Again, in order for the spatial integral to not vanish
sees that for sufficiently strong excursion amplitude αˆ
the parity of ϕ and ϕ must be different. However,
peaks at even harmonics of the laser frequency appear βn γm
nowthiscanbethecasenotonlyforn−m=2k+1,but
too. Picking an even harmonic at αˆ > 15 (e.g., the 6th)
alsoforn−m=2kiftheparityof,e.g.,ϕ isoppositeto
and tracing it back to low αˆ reveals that the peak splits β0
the one of ϕ . Hence, one expects the above-mentioned
and rapidly drops in magnitude (e.g., around αˆ ≃ 2 for γ0
hyper-Raman peaks at
the 6th harmonic). In the next Section we will use our
real-time Floquet method to show that the appearance
ω =kω +∆ǫ, k ∈Z (34)
ofevenharmonicsisduetothepopulationofseveralLIS 1
that become quasi-degenerate as αˆ increases.
where ∆ǫ = Reǫ − Reǫ is the difference between
β γ
the real parts of the two Floquet quasienergies involved.
Thus, in order to observe even harmonics at exactly
B. Superposition of Floquet states ω = 2kω a degeneracy Reǫ = Reǫ is required. Such
1 β γ
adegeneracybetweenthe(field-dressed)initialstateand
In the case of a non-adiabatic transfer of the field-free another one of opposite parity is also likely to populate
state to field-dressed states one has to allow for a super- the latter one.
positionofFloquetstatesinordertorepresenttheexact,
numerically determined wave function on the grid,
IV. FLOQUET STATE ANALYSIS OF
Ψ (x,t)≃ e−iǫβtΦ (x,t)= e−it(ǫβ+nω1)ϕ (x). REAL-TIME WAVEFUNCTIONS
# β βn
Xβ Xβn
(32) TheextractionofFloquetinformationcontainedinthe
Here we assume that the expansion coefficients are in- real-timewavefunctionisusefultoanalyzeanyfeatureof
cludedinΦ (x,t)andϕ (x). Forcontinuousquasiener- interest in HG spectra. We startwith the determination
β βn
giesthesumoverβshouldbereplacedbyanintegralover of the (real part of the) quasi-energy of the populated
5
finite-grid TDSE simulations the width of the peaks in
|Q (E)|2 alsodepend onthe integrationtime t −t and
± 2 1
the grid-size because of the absorbing grid boundaries.
Onlyforaflat-toplaserpulseanda verylongsimulation
time a stationary absorption rate at the grid boundaries
would be established, and Γ could be determined from
β
the peak-width. This, however, is exactly the regime
where the standardFloquet approachshouldbe applied.
We focus here on aspects of our method complementary
tothe conventionalFloquetmethod, inparticularits ap-
plicability to finite pulses and time-resolved studies.
Ifwemultiplythe wavefunction(32)byexp(itE)(with
E real)andintegrateovertime, mainlytheFloquetstate
ϕ forwhichthephaseisstationary,i.e.,E =Reǫ +nω
E β 1
“survives,”
t2
ϕ (x)∼ eitEΨ(x,t)dt. (37)
E
Z
t1
The integration time t −t has to be sufficiently long
2 1
inordertocovermanytemporaloscillationsofthewave-
function.
Startingfromthegroundstateinthepotential(30),we
solved the TDSE for a high-frequency laser field of vec-
tor potential A(t)=−Aˆ(t)sinω t for ω =4 and Aˆ(t) a
1 1
trapezoidal pulse shape with linear up- and down-ramps
over 4 cycles and 1200 cycles constant amplitude Aˆ (de-
noted in the form (4,1200,4) in the following). Figure 2
shows
FIG.3: Field-dressed groundstatewavefunctionϕ˜0n forαˆ= R=|Q |2+|Q |2 (38)
4. (a) Floquet block n=0, (b) n=−1. + −
(with the time-integral in (35) performed over the en-
tire pulse) as a contour plot vs the excursion amplitude
Floquetstates. Oncetheseenergiesareknownthecorre-
αˆ = Aˆ/ω and energy E for an energy interval within
spondingFloquetstatescanbe obtained. Themethod is 1
the zeroth Floquet block n = 0. Plotting |Q |2 and
similartotheoneproposedin[18]forfield-freedynamics. +
|Q |2 individually allowsto distinguish the parity of the
The numerical solution of the time-dependent −
states (labeled ’even’ or ’odd’ in Fig. 2). For αˆ → 0
Schr¨odinger equation in real time yields Ψ (x,t). Upon
# only the field-free state at E = −0.5 remains. How-
multiplication of (32) by an even or odd test function
ever,withincreasingexcursionamplitudeαˆlight-induced
q (x), spatial integration, and Fourier transformation
± quasi-bound states emerge, which are populated due to
from the time to the energy domain,
the finite rise-time of the laser field. From the popu-
t2 L2 lations (see color-coding) one infers that around αˆ = 6
Q±(E)= e−it(ǫβ+nω1−E)dt q±(x)ϕβn(x)dx, besides the field-dressedground state the second excited
Xβn Zt1 Z−L2 field-dressed state is more populated than the first ex-
(35) cited. For increasing αˆ the field-dressed ground state
0 ≤ t1 < t2 ≤ tsim one can extract from the peak posi- and the field-dressed first excited state become almost
tions in |Q±(E)|2 the real part of the Floquet energies degenerate so that ∆ǫ→0 in (34), explaining the peaks
at even harmonics of the laser frequency due to hyper-
ReE =Reǫ +nω (36)
βn β 1
Raman scattering.
belonging to even or odd Floquet states ϕ , respec- Using (37) we extracted field-dressed states. Fig-
βn
tively. Theeventestfunctionis,e.g.,simplyunityforall ure3showsthefield-dressedgroundstatefortheFloquet
−L <x< L, the odd test function may be chosen 1 for blocks n=0 (a) and n=−1 (b) for αˆ =4. The integra-
2 2
x>0 and −1 for x<0. The purpose of these test func- tion time was again the pulse duration. Equation (37)
tionsistoextracttheevenandoddparityFloquetstates in general yields a complex wavefunction ϕ = ϕ˜ eiθ.
E E
separately. Of course,only the energiesof the populated The plots in Fig. 3 show the real wavefunction ϕ˜ . It is
E
(and thus relevant) Floquet states ϕ are obtained in seen that the parity indeed changes as one decreases n
βn
this way. by one. For n = 0 and αˆ = 0 the ground state must be
TheimaginarypartΓ /2ofǫ contributestothewidth even. Hence, for n = −1 it is odd, in accordance with
β β
of the peaks in |Q (E)|2. However, in our finite-time, (25).
±
6
V. TRANSFORMATIONS
We consider transformations Gˆ(t) which are periodic
intimeandreducetounityasthelaserfieldgoestozero,
Gˆ(t+T)=Gˆ(t), Gˆ(t)| =ˆ1. (39)
α,E,A=0
Now, since each Floquet state Φ fulfills (8),
β
Gˆ(t)Hˆ(t)Gˆ−1(t)Gˆ(t)|Φ (t)i=Hˆ′(t)|Φ′ (t)i=ǫ |Φ′(t)i
β β β β
(40)
where Hˆ′(t) = Gˆ(t)Hˆ(t)Gˆ−1(t) is the transformed
Floquet-Hamiltonian and |Φ′ (t)i = Gˆ(t)|Φ (t)i the
β β
transformed Floquet state. The quasi-energy ǫ is not
β
affected by the transformation, and |Φ′ (t)i is also peri-
β
odic because of (39), so that with (11)
FIG. 4: (color online). R vs energy E and second-laser
frequencyω˜ for first-laserexcursion αˆ=2.5 and second-laser e−inω1t|ϕ′ i= e−i(n+m)ω1tGˆ |ϕ i, (41)
field strength amplitude E˜ˆ =0.01. βn m βn
Xn Xnm
where Gˆ(t)= me−imω1tGˆm, and thus
P
|ϕ′ i= Gˆ |ϕ i. (42)
βℓ ℓ−n βn
It is knownthat if the laserfrequency is tuned around Xn
resonances field-dressed states originating from different
Floquet blocks (and corresponding to the coupled field- We now specialize on transformations Gˆ that commute
free states) display avoided crossings. These crossings with the dynamical symmetry operation Pˆ ,
pt
have been shown to be related to localization, and to
chaos in the corresponding classical system [19]. The [Gˆ(t),Pˆ ]=0. (43)
pt
separationofthetwodressedstatesinvolvedcorresponds
to the Rabi frequency and is proportional to the field Examples are gauge transformations, e.g., for the trans-
strength of the driving laser. We will now show that the formation from velocity gauge, where
same is observedfor transitions between already dressed
1
states, i.e., we use the laser of frequency ω1 to dress Wˆ(t)=pˆA(t)+ A2(t), (44)
the system and a second, weaker laser of frequency ω˜ 2
toinducetransitionsbetweendressedstates. Thesecond
to the length gauge one has
laser will dress the already dressed system [20], and the
“dressed2”states(ortwocolor-dressedstates)shoulddis-
G (t)=exp[ixA(t)]. (45)
LG
playavoidedcrossingsasthefrequencyω˜ istunedaround
the energy gap of two dressed states. Another example is the Pauli-Fierz or Kramers-
Henneberger (KH) transformation, which is not a gauge
From Fig. 2 one infers that for an excursion ampli- transformation (although one frequently finds the term
tude, αˆ = 2.5 the energy difference between the field- “KHgauge”intheliterature). Ifwestartfromtheveloc-
dressed ground state and the field-dressed first excited ity gauge interaction (44) the KH transformation reads
state is Reǫ −Reǫ ≃ 0.155. Hence, we tune the fre-
1 0
quency ω˜ of the second laser around this energy differ- i t
Gˆ (t)=exp A2(t′)dt′+iα(t)pˆ . (46)
ence. The pulse envelope was the same for both lasers, KH (cid:20)2Z (cid:21)
∞
and the electric field amplitude of the second laser was
E˜ˆ = 0.01 = A˜ˆω˜ = α˜ˆω˜2 for all ω˜. Figure 4 shows results This amounts to a translation in position space by the
free electron excursion α(t) (31) and a purely time-
for the Floquet energy spectrum R vs energy and ω˜ for
dependent contact transformation. The KH Floquet-
αˆ =2.5. If the two frequencies ω and ω˜ are incommen-
1
Hamiltonian is
surate the Hamiltonian is not periodic at all. However,
ourapproachdoesnotrequireperiodicity,andweexpect 1 ∂
a Floquet analysis to be meaningful as long as the two- Hˆ′(t)=HˆKH(t)= pˆ2+V[x+α(t)]−i . (47)
2 ∂t
color Hamiltonian is approximately periodic, namely in
T˜ = 2π/ω˜ because ω ≫ ω˜. In fact, the avoided cross- As a consequence of (43),
1
ings of Reǫ with Reǫ −ω˜ and of Reǫ +ω˜ with Reǫ
0 1 0 1
around ω˜ =0.155 are clearly visible in Fig. 4. Pˆ |Φ′ (t)i=Gˆ(t)Pˆ |Φ (t)i=±|Φ′ (t)i (48)
pt β pt β β
7
with the eigenvalue ±1 the same as for Pˆ |Φ (t)i =
pt β
±|Φ (t)i. One also finds Gˆ = (−1)mPˆ Gˆ Pˆ and
β m p m p
Pˆ |ϕ′ i = ±(−1)ℓ|ϕ′ i, i.e., the transformed (primed)
p βℓ βℓ
states have the same symmetry as the original states.
Figure 5 shows the KH and the velocity gauge proba-
bility density for the excursion amplitude αˆ = 10. The
target energy was E = −0.08 where in Fig. 2 the al-
most degenerate ground and first excited state energies
for αˆ = 10 are. The KH probability density fits to the
KH potential
1 2π
V (x)= V[x+αˆsinτ]dτ, (49)
KH
2π Z
0
shown in the lower panel. The actual calculation was
performed for ω = 4 and a trapezoidal (10,1180,10)-
1
pulse. The target energy E in (37) is scanned through
the energy region of interest, and the Floquet energy is
hitwhenthevalueoftheintegralismaximum. Ifoneuses
the same integration time for different E the integral
L
2
N = |ϕ (x)|2dx (50)
E E
Z−L
2
isarelativemeasureforthepopulationoftherespective
Floquet state in the actual pulse.
The Floquet energies are invariantunder the transfor-
mations Gˆ(t) while both the Floquet states |ϕ i and
βn
their populations are not. In particular, in the high-
frequency limit one expects that only the eigenstates in
FIG.5: (a)KHandthevelocitygaugeprobabilitydensityfor
the KH potential (49) matter [9]. These states corre- theexcursionamplitudeαˆ=10andtargetenergyE =−0.08.
spondtotheFloquetenergiesintheFloquetblockn=0. (b) Corresponding KH potential.
Hence,the energyspectrumintheKHframeisexpected
to be muchmorelocalizedaroundn=0than invelocity
gauge. This is confirmed by Fig. 6. Instead of using the Teller potential have energies E >0. With laser field all
evenoroddtestfunctions in(35)andspatialintegration continuum and bound states are contained in each Flo-
we analyzed the wavefunction Ψ (x,t) at x = 2, i.e., quet block so that overlapsof dressedbound states from
# test
we calculated one block with continua from other blocks with lower
n are possible. However, we expect the dressed bound
t2
Q′(E)= e−it(ǫβ+nω1−E)dt ϕ (x ). (51) states of the n=0 block to dominate since they are the
βn test
Z
Xβn t1 mainonesbeingpopulatedduringtheswitching-onofthe
laser. Let us firstdiscuss the case where ω >minReǫ ,
This avoids the transformation of the entire wavefunc- 1 β
i.e., a single photon is sufficient for ionization. Then
tion to the KH frame and yields similar results as long
the dressed bound state in Floquet block n with energy
as one chooses x in a region where the wavefunction
test Reǫ +nω overlapswithcontinuumstatesofalltheFlo-
is sizable and both odd and even parity wavefunctions β 1
quet blocks m < n. In particular, Reǫ +nω overlaps
contribute (for x = 0 only contributions from even β 1
test with the continuum state of energy ǫ of the zeroth Flo-
Floquet states would be visible). Figure 6 confirms that p
quetblock,wherep indicatesthe asymptoticmomentum
for transformations of the type (39) the populations of
of this continuum state.
Floquet states in different frames (or gauges) are differ-
We will now turn to the question of how the manifold
ent while the Floquet energies are the same. The latter,
ofmixtures ofbound andcontinuum Floquet states con-
dressed levels could be probed with a second laser [21].
verts to an observable photoelectron spectrum when the
Ofcourse,anygauge-orframe-dependenceshouldvanish
pulseisswitchedoff. Figure7showsatime-resolvedFlo-
whenfield-freeobservables,suchasphotoelectronspectra
quet spectrum in velocity gauge for a N = 100-cycle
are considered. cyc
sin2-pulse
ω t
VI. PHOTOELECTRON SPECTRA A(t)=Aˆsin2 1 sinω1t (52)
(cid:18)2N (cid:19)
cyc
WithoutlaserfieldthecontinuumstatesofthePo¨schl- for 0 < t < N 2π/ω and zero otherwise. The other
cyc 1
8
FIG. 7: (color online). Time-resolved Floquet spectra for a
100-cycle sin2-pulse of amplitude αˆ = Aˆ/ω = 10, ω = 4,
1 1
x =2 (i.e., “inside” the potential), and a time-window of
FIG. 6: Floquet spectra for αˆ = 10, ω = 4, and a test
1 widtht =t −t =50. Theverticallineindicatestheendof
(10,1180,10)-pulse in (a) velocity gauge (with the A2(t)/2- w 2 1
thepulse. Panel(b)isaclose-upoftheenergyregionaround
term transformed away) and (b) in the KH frame. In the
ǫ(0) =−0.5in(a). Thecalculation was performed invelocity
KH frame the n = 0-Floquet block dominates while in ve- 0
gauge (with theA2(t)/2-term transformed away).
locity gauge thepopulation is broadly distributed overmany
Floquet blocks.
pulse parameters are given in the figure caption, and
x =2(i.e.,“inside”thepotential)andatime-window
test
of width t = t −t = 50 were chosen for (51). The
w 2 1
time onthe horizontalaxisis t sothat the spectrumfor
1
times t >100T =157.1 (indicated by the verticalblack
1
line) shows field-free states, i.e.,
t1+tw
Q(0)(E,t ) = eiEtΨ (x ,t)dt (53)
1 # test
Z
t1
= ϕ(β0)(xtest)Z t1+twe−it(ǫ(β0)−E)dt.
Xβ t1
Figure7ashowsthatwhilethepulseisonthepopulation
is distributed over many Floquet blocks. As the pulse is
switched off, all the Floquet populations for n 6= 0 dis-
appear, and only the ground state population inside the FIG.8: (coloronline). SameasinFig.7butforxtest =471.3.
(0)
potential with energy ǫ remains. This is because we
0
analyzed the spectrum at the position x =2. Contri-
test
9
butions to the wavefunction corresponding to electrons
in the continuum, traveling with an asymptotic momen-
tum p, decay at x = 2. Figure 7b shows a close-up
test
of the region around ǫ(0). With increasing amplitude of
0
the laser pulse the dominant Floquet population shifts
adiabatically from the field-free value ǫ(0) =−0.5 to the
0
ground state energy of the KH potential ǫ(KH) ≃ −0.09
0
(see Fig. 2 for αˆ = 10) and back. Note that although
the calculation was performed in velocity gauge the KH
groundstate energyis relevanthere because the Floquet
quasi-energies are frame- and gauge-independent.
Figure8showsthesameanalysisforx =471.3,i.e.,
test
“faraway”fromtheatomsothatittakessometimeuntil
probabilitydensityarrivesthere,namelyaroundt=100.
Itisinterestingtoobservethatinvelocitygaugethis“ar-
FIG.9: (color online). SameasFig. 8butin theKHframe.
rivaltime”duringthepulseisindependentoftheenergy.
As the laser pulse is switched off at t=157.1 many Flo-
quet channels close. However, because electrons are still
on their way from the atom to the “virtual detector” The TDSE was solved for a trapezoidal pulse of fre-
at xtest = 471.3 we are able to “measure” the field-free quency ω1 = 0.08. On the energy scale of the ioniza-
photoelectron spectrum of the electrons emitted in that tionpotentialthe Floquetblocksare packedmuchcloser
direction. The time these free electrons need to pass the in this case, meaning that many photons are necessary
virtual detector decreases with increasing energy, as is for ionization. In Fig. 10 we plot the Floquet energy
seeninFig.8 where the width ofthe tracesfor t>157.1 spectrum R in a certain range of excursion amplitude
decrease with increasing energy. The five traces visible αˆ = Eˆ/ω2 and energy E around the field-free contin-
1
are separated by ω1 and correspond to above-threshold uum threshold (other relevant parameters given in the
ionization (ATI) peaks (see, e.g., the review[22] or [23]). figure caption). The calculation was performed in ve-
They are quite broad in energy because of the change locity gauge using again the potential (30). There is a
of the ionization potential (from field-free value to KH clear down-shift of all the populated Floquet levels with
value and back). Their figure-eight shape in the contour increasing laser amplitude. This AC Stark shift is also
plot of Fig. 8 is a peculiarity of the sin2-pulse shape. referred to as the “ponderomotive shift” because the ef-
Figure 9 shows the corresponding result obtained in fective ionization potential is increased by U . In fact,
p
the KH frame. We see that in the KH frame only those the energy in the photoelectron spectrum is given by
statesarepopulatedinthelaserfieldwhichactuallycon-
tribute to the final field-free spectrum. This is because p2
E = =n~ω −(|E |+U ), (54)
the KH potential at xtest = 471.3 is almost identical to 2 1 0 p
the field-free one so that outgoing electrons are not af-
fected anymore by the oscillating KH binding potential. (providedtheACStarkshiftoftheinitialstateisnegligi-
It is also seen in Fig. 9 that the most energetic electrons ble, which for atomic ground states at long wavelengths
arrive earlier at xtest, unlike the velocity gauge-result in often is the case). E0 is the initial electron energy and n
Fig. 8. isthenumberofphotonsabsorbed. Inordertoreachthe
continuum at all n > (|E |+U )/~ω photons have to
0 p 1
be absorbed. As the intensity, and thus U , is increased,
p
moreandmorephotons areneededforionization. When
VII. CHANNEL-CLOSINGS
n photons are no longer sufficient but n+1 photons are
needed the n-photon channel closing occurs. In the plot
So far we studied mainly high-frequency phenomena shown in Fig. 10 a channel closing manifests itself as a
where the Floquet blocks are well separated on the crossing of a Floquet quasi-energy and the continuum
atomic energy scale because the laser frequency exceeds threshold. Now, the interesting feature in Fig. 10 is the
thegroundstateionizationpotential. However,thereare zero-energyLIS.SuchLISwerealsoobservedinRef.[26],
plentyofinteresting,non-perturbativephenomenaoccur- wheretheir connectionwith experimentallyobserveden-
ring at low frequencies where the ponderomotive energy hancementsinthephotoelectronspectraathighenergies
Up = Eˆ2/4ω12 can be large at nowadays available laser [27] was established. The parity of both states involved
intensities Eˆ2. Examples are tunneling ionization and in the crossing in Fig. 10 is even, and it is known that
high-order ATI due to rescattering of electrons [23, 24]. depending on the parity of the states, channel closings
In this Section we choose the so-called“channel-closing” affect the photoelectron spectrum differently [26, 28].
(see [25] and references therein) as a low-frequency phe- In our model, for the first even channel closing eight
nomenon to illustrate our method. photons are needed. According (54) it is expected at
10
FIG. 10: (color online). Logarithmic plot of R = |Q |2 +
+
|Q−|2 vs energy E and excursion amplitude αˆ, showing the
(populated) field-dressed states for ω = 0.08 and a trape-
1
zoidal (4,40,4)-pulse.
αˆ = 9.354, which indeed is close to where the crossing
is observed in Fig. 10. The small discrepancy is because
of the AC Stark shift of the initial state, neglected in
(54). One wouldexpect that channel closings only affect
low-energy electrons because the kinetic energy of the
electrons whose channel is about to close is low. Hence,
as the intensity is increased the yield of ATI peaks at
energies, say, > 5Up should increase monotonously as FIG. 11: (color online). Photoelectron spectra around 5Up,
well. However, near even photon channel closing there parameters as in Fig. 10. (a) Non-monotonic behavior of the
is a marked increase in the photoelectron yield at high yield(opensquaresαˆ=13.0,solidsquares13.3,circles13.55,
triangles13.8). (b)Sameforanoddchannelclosing,showing
energies [25, 26, 28]. Instead, when in odd photon chan-
a monotonic behavior of the yield with increasing intensity
nel closings the odd-parity LIS crosses the zero-energy
(triangles α=11.8, circles 11.55, open squares α=11.3).
LIS, such enhancements are absent. The first odd pho-
ton channel closing occurs around αˆ = 11.55, the next
even photon channel closing occurs around αˆ = 13.55.
The photoelectron spectra obtained using our Floquet ond laser, the properties of Floquet states under time-
method confirm the presence and absence of enhance- periodictransformations,theemergenceofinvariant,ob-
ments at even and odd channel closings, respectively, as servable photoelectron spectra after the laser pulse, and
shown in Fig 11. photoelectronenhancementsatchannelclosingsweredis-
cussed. The method is straightforwardly extendable to
three dimensions. We think the method is most useful
VIII. CONCLUSIONS forresearchersrunningcodestosolvethetime-dependent
Schr¨odinger equation in real time. By saving the wave-
functionatselectedspatialpositionsasafunctionoftime
We described a method for obtaining Floquet infor-
during the interactionwith the laser field the analysis in
mation from real-time wavefunctions. In this approach,
terms of light-induced states can be easily performed a
it is not necessary to assume strict periodicity. In fact,
posteriori. The application of the method to correlated
it is possible to follow the time-resolved Floquet quasi-
multi-electron systems may be very fruitful, as the un-
energies as they shift during a laser pulse. Moreover,
derstandingoffield-dressed,multiply-excitedorautoion-
the populations of the Floquet states can be determined
izing states is still poor.
so that especially cases where superpositions of Floquet
states play a role can be identified. The usefulness of
the method was illustrated by several examples employ-
ingtheone-dimensionalPo¨schl-Tellerpotentialwithonly
Acknowledgment
a single field-free bound state. In particular, the ori-
gin of peaks at even harmonics of the laser frequency
in an inversion-symmetric potential, avoided crossings This work was supported by the SFB 652 of the Ger-
of dressed already field-dressed states induced by a sec- man Science Foundation (DFG).