Dressed bound states for attosecond dynamics in strong laser fields

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by  Vladislav S. Yakovlev

Dressed bound states for attosecond dynamics in strong laser fields
V. S. Yakovlev,1,2,∗ M. Korbman,2 and A. Scrinzi3
1Ludwig-Maximilians-Universit¨at, Am Coulombwall 1, 85748 Garching, Germany
2Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
3Ludwig-Maximilians-Universit¨at, Theresienstraße 37, 80333 Munich, Germany
Wepropose atheoretical approach for theinterpretation of pump-probemeasurementswherean
attosecond pulse is absorbed in the presence of an intense laser pulse. This approach is based on
abstractly defined dressed bound states, which capture the essential aspects of the interaction with
the laser pulse and facilitate a perturbative description of transitions induced by the attosecond
pulse. Necessary properties of dressed bound states are defined and various choices are discussed
and compared to accurate numerical solutions of thetime-dependentSchr¨odinger equation.
2
1
0
INTRODUCTION we search for an analytical description of resonant tran-
2
sitions that an attosecond pulse of ultraviolet (UV) or

n extreme-ultraviolet (XUV) radiation drives in the pres-
a The ionization of atoms and molecules in the field
ence of a strong near-infrared field.
J of an intense laser pulse is one of the most fundamen-

7 tal phenomena in the interaction of light with matter.
2
Despite decades of intensive experimental and theoret-
DRESSED BOUND STATES
ical research, there are still open conceptual questions,
]
h highlighted by recent experiments [1, 2]. A key ques-
p tion is whether one can define the instantaneous ioniza- Letusconsideraquantumsystem(atom,ion,molecule
t- tion probability in the presence of an ionizing field. A etc.) described by a Hamiltonian Hˆ0. In the absence
n of any external field, this system has a set of orthonor-
closely related question is to which extent the detach-
a mal bound stationary states that satisfy the stationary
ment of an electron by multiphoton absorption or quan-
u
Schr¨odinger equation:
q tum tunneling can be described by an ionization rate
[ (see, for example, [3, 4]). Such a description is com- Hˆ |ni=ǫ(0)|ni. (1)
mon, but it can be made rigorous only in stationary 0 n
3
v or quasi-stationary situations. Two such cases are the Atypicalattosecondmeasurementconsistsinletting the
3 static and time-periodic externalelectric fields. In static
systeminteractwiththe electricfields ofanintenselaser
8 fields, the system decays exponentially at a well-defined pulse E (t), which may be regarded as a pump pulse,
7 L
rate. Intime-periodicfields,Floquettheoryallowsdefin-
6 and a relatively weak probe pulse (or a train of pulses)
. ing cycle-averaged decay rates (see e.g. [5] or [6] for a E (t). AnXUVprobepulsemaybeasshortasafew
0 recent application). In both situations, real and imagi- probe
tens of attoseconds [13, 14]. Varying the delay between
1
nary parts ofcomplex energiesgive energypositions and
1 the two pulses with an attosecond accuracy and observ-
decay widths. Non-Hilbert space resonance states can
1 ing the outcomes of their interaction with a quantum
v: be associated with the complex energies, either by us- system provides valuable insights into phenomena trig-
ing complex scaling [7, 8] or by iteratively adapting the
i geredandsteeredbythesetwopulses[15]. Ifasignificant
X boundary conditions (Siegert boundary conditions) [6].
fraction of atoms or molecules is ionized within a single
r None of these approaches, however, provides a fully sat-
half-cycleofthelaserpulse,theinteractionwiththepulse
a
isfactory framework for investigating the interaction of
is highly non-perturbative, so that an accurate descrip-
few- to single-cycle ionizing pulses with matter. At the
tion of this interaction calls for a numerical solution of
same time, the establishment of attosecond absorption
the time-dependent Schr¨odinger equation (TDSE). We
spectroscopy [9, 10] and the recent advances in the gen-
assume that such a solution |ψ (t)i is available, and one
L
eration of optical waveforms [1, 11] call for a deeper un-
ofourgoalsistointerpret strong-fielddynamicsinterms
derstanding of dynamics unfolding within a laser cycle.
of some dressed states. These states, if introduced prop-
Such understanding implies the development of approxi-
erly,mayserveasaconvenientstartingpointforbuilding
mate analytical theories,and the aim of this article is to
approximate models describing the interaction with the
make a step towards creating a useful theoretical frame-
probe pulse. Before we discuss different approaches to
work.
definingsuchdressedstates,letuslistafewrequirements
Our work is closely related to the development of adi- that they should fulfill.
abatic or quasistatic approximations to strong-field ion- Let |ϕ (t)i be a bound dressed state associated with
n
ization, such as described in [12] and the papers cited a stationary state |ni. This means that |ϕ (t )i = |ni
n 0
therein. Incontrasttothatresearch,wemakenoattempt at some moment t before the interaction with external
0
to analytically describe strong-field ionization. Instead, fields. As a first requirement, we demand that dressed
2
statesshouldformanorthonormalsetofwavevectorsat Assuming that time-dependent wave vectors |ϕ (t)i
n
all times: satisfying all the above requirements exist, let us use
them to develop a general perturbative description of
hϕm(t)|ϕn(t)i=δmn. (2) the interaction of a laser-dressed system with a weak
high-frequency probe pulse. For simplicity, we assume
Orthonormalityisnotonly convenientforcomputations,
that the pump (laser) and probe (UV or XUV) pulses
but it is also needed for the physical interpretation of
are polarized along the z-axis. In the dipole approx-
the states: only for orthonormal states does finding a
imation, the length-gauge Hamiltonian takes the form
systeminstate|ϕn(t)iwithprobability|hϕn(t)|ψL(t)i|= Hˆ(t)=Hˆ +Vˆ (t)+Vˆ (t) with
0 L probe
1 imply that all other probabilities are 0.
A strong laser field can significantly ionize a quan- Vˆ (t)=eE (t)Zˆ, (7)
L L
tum system. As a free electron cannot absorb a photon,
theprobepulsepredominantlyinteractswithboundelec- Vˆprobe(t)=eEprobe(t)Zˆ, (8)
trons. Therefore, it would be desirable to have dressed
statesthatserveasabasisfor describingthe boundpart wheree>0istheelectronchargeandZˆ isthedipoleop-
of an electron wavefunction even in the presence of a erator. The interaction with the laser pulse is described
strongexternalelectricfield. Whenthispropertyissatis- by
fied,strong-fieldionizationshoulddepletebounddressed
d
states without changing their shape in a fashion similar i~ |ψ (t)i= Hˆ (t)+Vˆ (t) |ψ (t)i (9)
L 0 L L
to the decay of quasi-stationary states. dt
Except for losses due to ionization, each dressed state (cid:0) (cid:1)
or, in the operator form,
should fully account for distortions caused by the laser
field. Therefore,we assumethat the pump pulse induces
∂
no transitions between dressed states. This is the most i~ UˆL(t,t0)= Hˆ0+VˆL(t) UˆL(t,t0) (10)
∂t
important requirement that we place on dressed states,
(cid:0) (cid:1)
and it means that ϕ (t ) Uˆ (t ,t ) ϕ (t ) must be with|ψ (t)i=Uˆ (t,t )|iiforaninitialstate|ii. Astan-
m 2 L 2 1 n 1 L L 0
negligibly small if mD6= n, w(cid:12)here Uˆ (t(cid:12),t ) is Ethe propa- dard approachto developing a perturbation theory with
(cid:12) L 2(cid:12) 1 respect to the probe pulse consists in writing the com-
gatordescribing the interact(cid:12)ionwith th(cid:12)e laser pulse (the
plete TDSE
exact definition of the propagator is given below). In
particular, we expect that the interaction with the laser
d
pulse brings the quantum system from its initial state i~ |ψ(t)i=Hˆ(t)|ψ(t)i (11)
dt
|ni to the corresponding dressedstate |ϕ (t)i multiplied
n
withaprobabilityamplitudean(t),themodulusofwhich in the following integral form:
accounts for the depletion by the laser pulse:
i t
a (t)= ϕ (t) Uˆ (t,t ) n , (3) Uˆ(t,t0)=UˆL(t,t0)− ~ dt′Uˆ(t,t′)Vˆprobe(t′)UˆL(t′,t0).
n n L 0 Zt0
D (cid:12) (cid:12) E (12)
while (cid:12)(cid:12) (cid:12)(cid:12) Here, Uˆ(t,t0) is the propagator associated with the full
Hamiltonian Hˆ(t):
ϕ (t) Uˆ (t,t ) n ≈0 if m6=n. (4)
m L 0
As long asDthe abo(cid:12)(cid:12)(cid:12)ve assum(cid:12)(cid:12)(cid:12)ptiEons hold, the effect of the i~∂∂tUˆ(t,t0)=Hˆ(t)Uˆ(t,t0) (13)
laser field on any dressed bound state can be expressed
via the probability amplitudes: and|ψ(t)i=Uˆ(t,t )|iifortheinitialcondition|ψ(t )i=
0 0
|ii. Furthermore, if only single-photon processes play a
a (t )
ϕ (t ) Uˆ (t ,t ) ϕ (t ) ≈ n 2 , (5) roleinthe interactionwiththe probepulse,thenthe un-
n 2 L 2 1 n 1
an(t1) knownoperatorUˆ(t,t′)ontheright-handsideofEq.(12)
D (cid:12) (cid:12) E
(cid:12) (cid:12) may be approximated with Uˆ (t,t′), which we consider
and the time evo(cid:12)lution of t(cid:12)he system can be approxi- L
to be known. This yields
mated as
a (t ) i t
Uˆ (t ,t )≈ |ϕ (t )i n 2 hϕ (t )|. (6) |ψ(t)i≈|ψ (t)i− dt′Uˆ (t,t′)Vˆ (t′)|ψ (t′)i.
L 2 1 n 2 a (t ) n 1 L ~ L probe L
n n 1 Zt0
X (14)
Note that the approximate time evolution is no longer The probe pulse may cause bound-bound, as well as
unitary as it accounts for ionization by losses from the bound-continuum transitions. In this paper, we focus
dressed-state space. on transitions between bound states, leaving the direct
3
photoionization by the probe pulse aside. In this case, to dressed bound states and use our general theory to
Eq. (14) can be written as investigate their performance in comparison with exact
solutions of the TDSE. We will consider four different
i t definitions of dressed bound states.
|ψ(t)i≈|ψL(t)i− ~ dt′ UˆL(t,t′)|ϕl(t′)i× Asaverycrudeapproximation,onecanattempttouse
Xl,mZt0 ( the unperturbed stationary states:
ϕ (t′) Vˆ (t′) ϕ (t′) hϕ (t′)|ψ (t′)i . (15) ϕ(u)(t) =|ni. (20)
l probe m m L n
)
D (cid:12) (cid:12) E (cid:12) E
(cid:12) (cid:12) Another option, freq(cid:12)uently encountered in literature, is
Let us eval(cid:12)uate the p(cid:12)robability amplitude that a sys- (cid:12)
to use time-dependent eigenstates of the Hamiltonian
tem,initiallypreparedinstate|ψ(t0)i=|ii,willbefound Hˆ(t) evaluated in the subspace of unperturbed bound
instate|ϕ (t)iatalatertimet. Multiplyingbothsides
n6=i states and normalized to satisfy Eq. (2):
of Eq. (15) with hϕ (t)| from the left, keeping in mind
n
the fact that |ψL(t)i ≈ ai(t)|ϕi(t)i, and using Eq. (5), |mi m Hˆ0+VˆL(t) ϕ(na)(t) =ǫ(na)(t) ϕ(na)(t) .
we obtain
Xm D (cid:12) (cid:12) E (cid:12) E
(cid:12) (cid:12) (cid:12) (21)
(cid:12) (cid:12) (cid:12)
α (t)= ϕ (t) Uˆ(t,t ) i =hϕ (t)|ψ(t)i≈ In the context of strong-field ionization, these states are
ni n 0 n
known under many different names, such as adiabatic
−D~iean((cid:12)(cid:12)(cid:12)t)Zt0tdt′(cid:12)(cid:12)(cid:12)EEprobe(t′)aani((tt′′))Zni(t′) (16) ssttaatteess[[1129,],1fi6e],ldq-uaadsaispttaetdicssttaatteess[[2107],,1a8n]d,pahdaiaseb-aatdiciafibealtdic-
following dressed states [21, 22]. We will refer to them
for n6=i, where
as “adiabatic states”.
Yet another reasonable approach is to obtain dressed
Z (t)= ϕ (t) Zˆ ϕ (t) (17)
ni n i bound states by solving the TDSE in the basis ofunper-
D (cid:12) (cid:12) E turbed bound states:
is the transition matrix element(cid:12) b(cid:12)etween dressed states.
(cid:12) (cid:12)
Eq. (16) is the desired expression, where the interaction i~ d ϕ(d)(t) = |ni n Hˆ +Vˆ (t) ϕ(d)(t) (22)
with the probe pulse is accounted for in first-order per- dt n 0 L n
turbationtheory,whiletheeffectofthestronglaserpulse (cid:12)(cid:12) E Xn D (cid:12)(cid:12) (cid:12)(cid:12) E
is represented by quantities related to dressed states. with th(cid:12)e initial condition ϕ(cid:12)(d)(t ) = (cid:12)|ni. With this
n 0
BasedonEq.(16),wecanevaluatequantitiesobservable choice, we assume that virt(cid:12)ual excitEations into field-free
in measurements. For example, the probability that, at continuum states do not in(cid:12)(cid:12)fluence the dynamics. How-
some final time tf after the interaction, the system will ever, contrary to ϕ(a)(t) , these states do not follow
n
be found in a bound stationary state |fi is given by
the laser field adia(cid:12)baticalEly, but rather allow for non-
(cid:12)
2 instantaneous resp(cid:12)onse to the laser field. We will call
p =|hf|ψ(t )i|2 = α (t )hf|ϕ (t )i . (18) them “dynamically dressed states”.
fi f ni f n f
(cid:12)(cid:12)Xn (cid:12)(cid:12) The above three alternativesallow one to evaluate the
(cid:12) (cid:12) dressed bound states without solving the TDSE in the
As an example that is(cid:12)more relevant to tra(cid:12)nsient ab-
(cid:12) (cid:12) fullHilbertspace,althoughsuchsolutions|ψ (t)iarere-
sorptionspectroscopy,thedipoleresponsedue tobound- L
quired later to obtain the probability amplitudes a (t)
boundtransitionsinducedbytheprobepulsecanbeeval- n
from Eq. (3). The last option that we are going to con-
uated as
sider is different: we will evaluate dressed bound states
d(t)= ψ(t) Zˆ ψ(t) ≈ α∗ (t)Z (t)α (t). (19) by projecting solutions of the TDSE obtained in the
mi mn ni
wholeHilbertspace(9)ontothesubspaceofunperturbed
D (cid:12)(cid:12) (cid:12)(cid:12) E Xm,n boundstationarystatesandorthonormalizingthemwith
(cid:12) (cid:12)
So far, the dressed bound states were treated as an the aid of the Gram-Schmidt procedure:
abstract mathematical concept. It is worth mentioning
that without the requirement that dressed states should ϕ˜(np)(t) = 1−Pˆl(t) |mi m UˆL(t,t0) n ,
describe bound electrons, the wavevectors |ϕn(t)i = (cid:12) E lY<n(cid:16) (cid:17)Xm D (cid:12) (cid:12) E
UˆL(t,t0)|ni wouldexactlysatisfy the restofour require- (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (23)
ments, but such dressed states would not serve their
main purpose, which is to facilitate the interpretation ϕ(np)(t) = ϕ˜(np)(t) / ϕ˜(np)(t)|ϕ˜(np)(t) , (24)
ofboundelectrondynamicsinthe presenceofanintense (cid:12) E (cid:12) E rD E
laserpulse. We donotknowifperfectdressedstatessat- wh(cid:12)(cid:12)ere the pro(cid:12)(cid:12)jector Pˆl(t) is defined as
isfying all our requirements including their bound char-
acter exist, but we can consider several approximations Pˆl(t)= ϕ(lp)(t) ϕ(lp)(t) . (25)
(cid:12) ED (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4
We will call ϕ(p)(t) “projected dynamical states”, as 0.05
n 1
a)
tdhyenyamreiscusltinfrcol(cid:12)(cid:12)(cid:12)umdicnaglcsuEtlraotniogn-sfietlhdatioanciczoautinotnf,obruatlltehleectoruotn- 0.98 (cid:12)(cid:12)(cid:12)a(1u)(t)(cid:12)(cid:12)(cid:12)2 0.04 nits)
cfjreoecmeteebdsooudfnydtnh-asemtsaeitcecaalclocsnutatlaetetnisto,.ntNshoaetreeprtpohrajoetjc,etceetvdeadlduoraentstisnoegdththgeerfiopuerlndod-- 0.96 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)aa(1(1ad))((tt))(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)22 0.03 tomicu
state is only normalized without any orthogonalization, 0.94 (cid:12)(cid:12)(cid:12)a(1p)(t)(cid:12)(cid:12)(cid:12)2 0.02 )|(a
the first excited dressed state is forced to be orthogonal (t
to the dressed ground state and so on. 0.92 0.01 EL
|
0.9 0
-6 -4 -2 0 2 4
NUMERICAL RESULTS t (fs)
1
inUthnilsesssecsttiaotne:d~ot=heerw=isme,ew=e u1s,ewahteormeitcheunuintsits(aotf.eun.)- 0.8 (cid:12)(cid:12)(cid:12)a(1ub)()t)(cid:12)(cid:12)(cid:12)2 00..018 nits)
eorarfgdtyihueasne(ld5e.cl2te9rnigc×tfih1e0lad−re1i1s15mH.4)a,1r2rter×sepe1e0=c1t1i2vV7e.l/2ym1. .eOVTnoeacnaodtmotmphaeicrBeuotnhhietr 0.6 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)aa(1(1ad))((tt))(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)22 0.06 tomicu
differentapproachestodefiningdressedboundstatesand 0.4 (cid:12)(cid:12)(cid:12)a(1p)(t)(cid:12)(cid:12)(cid:12)2 0.04 )|(a
to illustrate the power of our analytical theory, we solve (t
the TDSE numerically for one electronin one spatialdi- 0.2 0.02 EL
|
mension with a soft-core model potential:
0 0
-6 -4 -2 0 2 4
1 t (fs)
V(z)=− . (26)
z2+a2
SC
FIG.1. Theprobabilitiesofbeinginvariouslydefineddressed
With the soft-core parampeter being equal to aSC = states |ϕ1(t)i, provided that the quantum system is initially
0.3, the deepest two energy levels have energies ǫ(0) = prepared in state |1i. The modulus of the laser field is
0 shown with black dots. The two panels correspond to dif-
(0)
−1.75=−47.49 eV and ǫ1 =−0.41=−11.11 eV. The ferent peak intensities Imax of the laser pulse: a) Emax =
TDSE was solved on a grid with a step of ∆z = 0.1 0.02atomic units I =1.4×1013 W/cm2 andb)E =
max max
(cid:0) (cid:1)
atomic units in a box as large as 819.2atomic units. We 0.04 atomic units I =5.6×1013 W/cm2 .
max
(cid:0) (cid:1)
usedthefirstfiveboundstatestoinvestigatevariousdef-
initions of dressed states. The most excited of these un-
perturbedstateshasabindingenergyofǫ(0) =−0.077=
4
−2.10 eV. In Fig. 1, we compare the probabilities |an(t)|2 =
We define the light pulses via their vector potentials: |hϕn(t)|ψL(t)i|2 evaluated using the four different def-
A(t) = −A cos2(t/T)sin(ωt) for |t| < πT/2 and initions of dressed bound states introduced in the pre-
max
A(t) = 0 for |t| ≥ πT/2. Here, A = E /ω is the vious section. In these simulations, we set n = 1,
max max
peak value reached by the envelope of the vector poten- that is, each simulation begins with the electron be-
tial, E is the peak value reached by the envelope of ing in first excited state |1i. For the peak electric field
max
the electric field, ω is the central frequency, and T is Emax = 0.02 = 1010 V/m, the ionization probability is
related to the full width at half maximum (FWHM) of as little as 0.3%. If we knew the perfect dressed bound
the pulse intensity as T = FWHM/ 2arccos 2−1/4 . states, then, for such a small ionization probability, our
Given a vector potential A(t), the electric field is evalu- model system would largely remain in |ϕ1(t)i, so that
atedasF(t)=−A′(t). Inoursimulatio(cid:2)ns,weus(cid:0)eda3(cid:1).5(cid:3)- |a1(t)|2 wouldonlytakevaluesbetween|a1(t0)|2 =1and
femtosecond laser pulse (TL =126.78=3 fs) and a 300- |a1(tf)|2 = 0.997, decreasing more or less monotonously
attosecond XUV probe pulse (T = 10.84= 0.26 fs). as the initial state is slightly depleted. Fig. 1 illustrates
probe
The central wavelength of the laser pulse was set to 760 the well-known fact that the unperturbed bound states
nanometers (ω = 0.06 = 2.48 fs−1). The central fre- are very far from this ideal: |a(u)(t)|2 reaches values be-
L 1
quencyoftheXUVpulsewaschosentoberesonantwith low 0.95. This happens mainly because the initial state
the transition between the first two stationary states: isdistortedbytheexternalfield,whichreducesthevalue
ω = ǫ(0) − ǫ(0) = 1.34 = 36.38 eV. Although we of |hn|ψ (t)i|2. So, even though |a(u)(t)|2 can be inter-
probe 1 0 L 1
make no attempt to model a realistic atom, the excita- pretedastheprobabilityofbeingintheunperturbedsta-
tion from |0i to |1i may be viewed as analogous to a tionarystate,itshouldnotbegiventhephysical meaning
dipole-allowed transition from an inner-shell orbital of a ofbeinginsomedistortedbutboundstatethat,afterthe
multi-electron atom to an unoccupied orbital. interaction,turns into the initialunperturbed stationary
5
state.
1
In this respect, dressed state ϕ(a)(t) , defined by 0] a)
1 ≡
p(u)
Eq. (21), performs much better, (cid:12)but it Eis not perfect L 0.95 01
(cid:12) E p(a)
either. Even though ϕ(1a)(t) acc(cid:12)ounts for the distor- SE[ p0(1d)
itnionFigo.f 1thaestiinllitihaalssptarto(cid:12)(cid:12)(cid:12)enoiunntcheEed emxitneirmnaalafiteltdim, e|as(1aw)(hte)r|2e TD]/pL01 0.9 pT0p1(00Dp11S)E
EL(t) has its zero crossings,and |a(1a)(t)|2 reaches values [E01 0.85
that are significantly smaller than the final probability p
of being in state |1i. The main reason for this is that 0.8
-4 -3 -2 -1 0 1 2 3
ϕ(a)(t) is a stationary solution, while the average ve- τ (fs)
1 probe
l(cid:12)ocity oEf a bound electron can become relatively big in 1
(cid:12)
a(cid:12)n intense laser pulse. As this velocity has maxima at 0] 0.9 b)
zero-crossingsof the electric field, ϕ(a)(t)|ψ (t) 2 has ≡ 0.8 p(u)
1 L L 0.7 01
mbrienaikmdaowatntohfestehetimadeisa.bTathicis(cqaunasail(cid:12)(cid:12)(cid:12)ssDtoatbiec)inatpeprprorxetiEmed(cid:12)(cid:12)(cid:12)aatisona DSE[E 00..56 pp(0(0a1d1))
[a2n3d].tAhelsmoitnhiemsamoafll|ad(eul)a(yt)s|2b,evtwiseibenletihneFeixgt.r1ema,apoofinEtLo(utt) T]/pL01 00..34 pT0p1(0Dp1S)E
1 E
some non-adiabaticity of the electron response. [1 0.2
0
p 0.1
The intuitively expected step-like decrease of |a1(t)|2, 0
-4 -3 -2 -1 0 1 2 3
as the laser field depopulates the initial state, is best
τ (fs)
probe
reproduced by the dynamically dressed states ϕ(d)(t)
1
FIG. 2. The effect of the laser field on the probability to
and the projected dynamical states ϕ(1p)(t) ,(cid:12)(cid:12)(cid:12) defineEd findourmodelquantumsysteminstate|0iatafinaltimetf,
by Eqs. (22) and (24), respectively. (cid:12) HowevEer, unlike the initial state being |1i. The XUV pulse, which is mainly
(cid:12)
ϕ(u)(t) and ϕ(a)(t) , these dressed s(cid:12)tates do not nec- responsible for the transition, arrives at a time τprobe. The
n n probabilities are normalized to the transition probability in
e(cid:12)ssarilyEturn i(cid:12)nto |niEonce the laser pulse is gone. In the absence of the laser field. The approximate probabilities
(cid:12) (cid:12)
o(cid:12)ther words, |a(cid:12)(d,p)(t )|2 6= |h1|ψ (t )i|2 at a final time p(u), p(a), p(d), and p(p) are evaluated with Eq. (27), using
1 f L f 01 01 01 01
t . As longasthe discrepancyis small,it is legitimate to different definitionsof dressed boundstates. Theexact tran-
infterpret |a (t)|2 as the probability of being in the first sition probabilities pT01DSE, obtained by numerically solving
1 theTDSE,areshown with filled circles. Thepeakintensities
excited state dressed by the laser field. In these simula-
ofthelaserpulsearethesameasthoseinFig.1: E =0.02
tions, this discrepancy is much smaller for |a(p)(t)|2, as for panel a) and E =0.04 for panel b). max
1 max
compared to |a(d)(t)|2, which is especially easy to see in
1
Fig. 1b.
number of states used in Eq. (18):
The mere fact that, for a certain choice of dressed
states, |a (t)|2 does not show the intuitively expected 2
n max{i,f}
step-like behavior, does not necessarily mean that these
p = α (t )hf|ϕ (t )i . (27)
dressedstatesshouldnotbe usedinouranalyticalresult fi (cid:12) ni f n f (cid:12)
(cid:12) n=0 (cid:12)
(16) for the interaction with a probe pulse. In Fig. 2, (cid:12) X (cid:12)
(cid:12) (cid:12)
weplottheresultsofsimulationswhereourmodelatom, Thefilledcircle(cid:12)(cid:12)sinFigs.2aand2breprese(cid:12)(cid:12)nttheresults
initially prepared in state |1i, interacts with the near- of solving the TDSE on a grid. Comparing these “ex-
infraredlaserpulseEL(t)andadelayedattosecondXUV act” transition probabilities with the predictions of our
pulse Eprobe(t−τprobe). We plot the probability to find analytical model, we clearly see that projected dressed
the atom in state |0i at the end of the simulation, nor- states ϕ(p)(t) , defined by Eq.(24), givethe mostaccu-
n
malized to the probability of the same transition in the
rate re(cid:12)sults. AEt the same time, the unperturbed states
absenceofthelaserpulse. Applyingourmodel,wefound (cid:12)
that care should be taken implementing Eq. (18). If a ϕn(u) (cid:12)=|ni,whichareourmostprimitivedressedstates,
highly excited bound state is strongly depleted by the a(cid:12)lso pEerform surprisingly well. For the calculations with
(cid:12)
laser pulse, an(t′) in the denominator of Eq. (16) may t(cid:12)he higher intensity, presented in Fig. 2b, these states
become very small. As a result, even small inaccuracies even outperform both the adiabatic states ϕ(a)(t) and
n
in the approximations that we made deriving Eq. (16)
may result in big errors. To avoid this, we restrict the the dynamically dressed states ϕn(d)(t) in(cid:12)(cid:12)terms Eof the
(cid:12)
(cid:12) E
(cid:12)
(cid:12)
6
agreementwiththe TDSE calculations. Forthis highin-
0.26
tensity, the biggest discrepancy between the exact and
)
s
analytical results is found for the dynamically dressed nit 0.24
states ϕ(d)(t) . u 0.22
n c
mi 0.2 Emax=0.02
(cid:12)(cid:12)(cid:12) E ato 0.18 Emax=0.03
( Emax=0.04
2 0.16
a) )|
0] 1 (t 0.14
≡ 10
L p(u) |Z 0.12
E 0.95 10
TDSEp[10 0.9 ppp(1((1a0pd0))) 0.1 -4 -2 t 0(fs) 2 4
/ 10
E]L 0.85 pT10DSE FIG. 4. The squared modulus of the dipole transition ma-
[0 trix element between dressed states ϕ(p)(t) and ϕ(p)(t) ,
p1 0.8 (cid:12)(cid:12) 0 E (cid:12)(cid:12) 1 E
as defined by Eq. (17). Emax is the(cid:12)peak value of(cid:12)the elec-
-3 -2 -1 0 1 2 3 4 tric field of the laser pulse. The modulus of the laser field is
τ (fs) schematically shown with black dots.
probe
1.2
0] 1 b) tions. The overall decrease of p with τ in Fig. 2b,
≡ 01 probe
L p(u) as well as the increase of p10 in Fig. 3b are clearly due
E 0.8 10
E[ p(1a0) to the depletion of the first excited state by the laser
DS 0.6 p(d) field. The periodic modulations present in all the fig-
Tp10 p(1p0) ures deserve more consideration. In principle, they may
/ 10
]L 0.4 pTDSE be a consequence of two effects: the Stark shift, which
E 10
[ detunes the transition frequency away from the central
10 0.2
p frequency of the XUV pulse, and the modulation of the
0 dressed transition matrix element (17). In the present
-3 -2 -1 0 1 2 3 4
τ (fs) example, we find that the dominant contribution is due
probe
to the effect of the laser field on the dressed matrix ele-
FIG. 3. The effect of the laser field on the transition proba- ments. To see this, we plot |Z (t)|2, evaluated with the
10
bility from state |0i to state |1i. For details, see the caption aid of ϕ(p)(t) , for different intensities of the laser field:
of Fig. 2 and thetext. n
Fig. 4.(cid:12)For thEe central half-cycles of the laser pulse, the
(cid:12)
In Fig. 3, we present similar calculations for a case modula(cid:12)tion depths of |Z10(t)|2 are very close to those of
transition probabilities in Figs. 2 and 3. A strong field
more relevant to possible experiments—in this case, the
tendstoreduce|Z (t)|2 becauseitdisplacestheelectron
XUVpulseexcitesourmodelatomfromitsgroundstate. 10
wavefunction in the first excited state, while the ground
The figure depicts the probability to find the atom in
state in the present example has a very small polariz-
state |1i after the interaction with the XUV and laser
ability. As a consequence, the overlap between the two
pulses. In contrast to Fig. 2, the unperturbed states
dressedstates decreases,which results in a smaller value
ϕ(u)(t) fail to describe the modulation of the transi-
1 of the dipole transition matrix element between them.
(cid:12)(cid:12)tion proEbability by the laser field. Both ϕ(1a)(t) and The relatively fast oscillations of |Z10(t)|2 at the tail
(cid:12) of the laser pulse are mainly due to the multiphoton ex-
ϕ1(p)(t) yieldtransitionprobabilitiesthata(cid:12)(cid:12)greewiEththe citationofstate |3ifromstate |1i. The energydifference
(cid:12)
(cid:12)TDSE cEalculations. The correspondingdiscrepanciesare betweenthesetwostatesisǫ(0)−ǫ(0) =−0.113+0.408≈
(cid:12) 3 1
(cid:12)comparableforthemoreintenselaserpulse(Fig.3b),but 0.29 atomic units; since the transitions from both these
using ϕ(p)(t) clearly gives more accurate results if the states to |0i are dipole-allowed, this leads to quantum
1
beats with a period of 2π/0.29 = 21 at. u. = 0.5 fs.
deplet(cid:12)ion of sEtate |1i is weak (Fig. 3a). Thus, in most
(cid:12) These quantum beats do not, however, appear in the
cases,(cid:12)the projected dynamical states ϕ(p)(t) , defined
n probabilities of the XUV-induced transitions presented
byEq.(24),bestsatisfytherequiremen(cid:12)tsthatEweplaced in Figs. 2 and 3 because the XUV pulse, which is reso-
(cid:12)
on dressed bound states. (cid:12) nant with the transition between states |0i and |1i, does
Having identified the most promising definition of not haveenough bandwidth to drive transitions between
dressed bound states, we can interpret the effect of the states |0i and |3i. This fact is successfully accounted for
laser pulse on the probabilities of XUV-induced transi- by our analytical theory.
7
CONCLUSIONS AND OUTLOOK freeboundstates. Finally,fromthe projecteddynamical
states ϕ(p) welearnthat the field-free continuouspartof
the spectrum influences the dynamics,but does not play
We have theoretically investigated how an ionizing
a crucial role for the computation of transition matrix
few-cycle near-infrared laser pulse affects single-photon
elementsbetweenboundstates. Thesefourexamplesare
bound-bound transitions driven by an attosecond pulse
clearlynottheonlypossiblechoices. Forexample,rather
of extreme ultraviolet radiation. Our approach is based
thanrestrictingtoafew field-freeboundstates,onemay
ontheassumptionthatevenwhenthelaserfieldisstrong
use adiabaticordynamicstates confinedto abox,which
enoughtosignificantlyionizeaquantumsystem,onecan
would admit virtual continuum states without the need
describeboundelectronswithsomedressed bound states,
for a full solution of the TDSE, as in the projected dy-
whichweintroduce inanabstractwaybya setofsimple
namical states.
requirements listed at the beginning of section . With
We have thus shown that also in situations where sig-
such dressed states, we have obtained relatively simple
nificant and strongly time-dependent ionization occurs,
analytical results for the perturbative interaction with a
dressed bound states allow for a simple analysis of ex-
probe pulse, such as Eq. (16). For a particular model
perimentally observable dynamics and carry our picture
problem, we have systematically comparedfour different
of state-depletion and perturbative transitions between
kinds of dressed bound states and found that the most
the dressed states into extreme time scales and intensity
accurateresultsaregenerallyobtainedwiththeprojected
regimes. This finding is conceptually important. Prag-
dynamical states (24). These states are easily evaluated
matic considerationshow to further improve the dressed
from numerical simulations of the interaction with the
states and possibly find a compromise between accuracy
laser pulse, and they make no assumptions about the
and computational simplicity are left to future work.
adiabaticity of the response to the laser pulse.
As the mainobservableinournumericalexamples,we
havechosentheprobabilitytofindasingle-electronatom
ACKNOWLEDGMENTS
in a certain state after the interaction with light pulses.
This was done to test our analyticaltheory in a possibly
simpleandconvincingway. Experimentally,itiseasierto Theauthorsacknowledgeilluminatingdiscussionswith
measure the transient absorption of an attosecond pulse J. Gagnon, E. Goulielmakis, and F. Krausz. This work
in the presence of an ionizing laser field [10]. It should wassupportedbytheDFGClusterofExcellence: Munich
be straightforward to employ dressed bound states to Centre for Advanced Photonics.
theoretically investigate attosecond transient absorption
measurements. In this paper, we have only made a first
step in this direction by writing an explicit expression
for the dipole response associated with an XUV-driven
∗
[email protected]
laser-dressedbound-bound transition: Eq. (19).
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