Table Of ContentCarrier-wave Rabi flopping signatures in high-order harmonic generation for alkali
atoms
M. F. Ciappina1,∗ J. A. P´erez-Hern´andez2, A. S. Landsman3,
T. Zimmermann4, M. Lewenstein5,6, L. Roso2, and F. Krausz1,7
1Max-Planck Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany
2Centro de L´aseres Pulsados (CLPU), Parque Cient´ıfico, E-37008 Villamayor, Salamanca, Spain
3Max Planck Institute for the Physics of Complex Systems Nothnitzer Straße 38, D-01187 Dresden, Germany
4Physics Department, ETH Zurich, CH-8093 Zurich, Switzerland
5 5ICFO-Institut de Ci`ences Fot`oniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
1 6ICREA-Instituci´o Catalana de Recerca i Estudis Avanc¸ats,
0 Lluis Companys 23, 08010 Barcelona, Spain and
2 7Department fu¨r Physik, Ludwig-Maximilians-Universit¨at Mu¨nchen,
n Am Coulombwall 1, D-85748 Garching, Germany
a (Dated: January 19, 2015)
J
We present the first theoretical investigation of carrier-wave Rabi flopping in real atoms by em-
5
ploying numerical simulations of high-order harmonic generation (HHG) in alkali species. Given
1
theshort HHGcutoff,related tothelow saturation intensity,weconcentrateon thefeatures of the
third harmonic of sodium (Na) and potassium (K) atoms. For pulse areas of 2π and Na atoms, a
]
h characteristicuniquepeakappears,which,afteranalyzingthegroundstatepopulation,wecorrelate
p with the conventional Rabi flopping. On the other hand, for larger pulse areas, carrier-wave Rabi
- floppingoccurs, andisassociated withamorecomplex structureinthethirdharmonic. Thesenew
m
characteristics observed in K atoms indicate the breakdown of the area theorem, as was already
o demonstrated undersimilar circumstances in narrow band gap semiconductors.
t
a
.
s It is well known that semiconductors, when modeled periodisequaltothedrivenlightperiod. Evenwhenthe
c
as a two-level system, develop a periodic oscillation of envelope area for this case is Θ=4π, it is clear that the
i
s the population inversion when interacting with constant Bloch vector does not return to the south pole, as may
y
light, a phenomenon predicted by I. I. Rabi in the 30s, be expected. To the contrary, a more chaotic behavior
h
p called Rabi flopping [1]. Rabi flopping has also been ob- is observed in the motion of the Bloch vector, resulting
[ servedwhen using ultrafast opticalpulses e.g. [2, 3]. For in a more complex shape in the spectrum of the optical
these pulses, peculiar behavior emerges when the driven polarization. Furthermore, the well-knownarea theorem
1
v light intensity is so high that the period of one Rabi os- of the nonlinear optics fails when this parameter regime
1 cillation is comparable with that of one cycle of light. is reached. Note that multipeak splitting of the reso-
2 In this case, the area theorem has been shown to break nance fluorescence spectrum by short pulses in the stan-
0
down [3], and a new phenomena, known as carrier-wave dard Rabi flopping regime was predicted in Refs. [5–8],
4
Rabi flopping (CWRF), emerges. These features can be although this effect is due to a complex temporal inter-
0
. schematicallyobservedin the so-calledBlochsphere (for ference effect, rather than CWRF.
1
the definition and more details see e.g. [4]), presented in
0
5 Fig. 1. (a) w (b) w
1 In particular, Fig. 1(a) depicts the conventional Rabi
v: flopping on a Bloch sphere. For this case the Rabi pe-
i riod is much larger than the driven light period and the
X
Bloch vector, formed by the real (u) and imaginary (v) v v
r
parts of the optical polarization and the population in-
a
u u
version(w)ofatwo-levelsystem,spiralsupstartingfrom
1 1
the south pole (corresponding to all the electrons in the
ground state), reaches the north pole and returns to its w t w t
initialpositionforthecaseofsquared-shapedpulseswith
anenvelope areaofΘ=2π. Here opticaloscillationsare -1 -1
mappedtoanorbitoftheBlochvectorparalleltotheuv
FIG. 1. Sketch of the Bloch Sphere showing the different
orequatorialplane. Additionally,oscillationsofthepop-
regimes. (a)schematicshowingthetravel oftheBlochvector
ulationinversionaregivenbythemotionintheuwplane. for conventional Rabi flopping for a pulse with an envelope
The corresponding spectrum of the optical polarization pulseareΘ=2π. (b)sameforcarrier-waveRabifloppingfor
would exhibit then two peaks centered around the two- a pulse with an envelope pulse area of Θ =4π. The bottom
level transition frequency. On the other hand, Fig. 1(b) panels show the evolution of the population inversion w (see
thetext for more details).
presentsresultsforamuchshorterpulse,wherethe Rabi
2
Experimentsonnarrowbandgapsemiconductorshave sideringthelow-orderharmoniccutoffdevelopedinalkali
shownaclearsignatureofCWRF,whichmanifesteditself atoms, closely related to their low saturation intensity).
as a split in the third harmonic of the emitted light into To create the conditions for CWRF, we used an atomic
theforwarddirection[9]. Recently,Rabifloppingandthe systeminwhichtheperiodofaRabioscillation[13](cor-
consequentcoherentpulsereshapinghasbeenexperimen- respondingto thetransitionbetweenthe groundandthe
tallyobservedinaquantumcascadelaser[10],suggesting first excited states [14, 15]) is similar to one period of
anewpromisingapproachtoshortpulsegeneration. One the laser light. For the usual (Ti:Sa) laser sources such
of the advantages of atoms (relative to semiconductors) system was provided by K atoms, with a transition en-
is the possibility to employ longer laser pulses and, as a ergy between the ground and the first excited state of
consequence,to exploreabroaderrangeoflaserparame- 1.61eV(hence closeto the lasersourcephotonenergyof
ters, as well as provide analternative to carrierenvelope 1.55 eV).
phase (CEP) characterization. In addition, it has been Alkali atoms, due to their atomic structure (gas no-
shownthatinsemiconductorstheCoulombinteractionof ble structure plus only one external electron), are well-
carriersin the bands gives rise to an enhancementof the suited to be described by the single active electron ap-
external laser field and consequently the envelope pulse proximation (SAE). We therefore focus on such atoms
area by as much as a factor of two, considerably compli- to avoid the possible role of electron-electron correla-
cating the observedinterpretationofCWRF phenomena tions, which have been found to have an important,
[2, 9, 11]. yet still poorly understood role in HHG spectra [16].
When the atom was simplistically modeled as a two- Based on SAE approximation, we use the atomic poten-
level system, conventional Rabi flopping behavior and tial reported in [17] to describe K and Na atoms. Using
CWRF features were observed (see e.g. [4, 12] and ref- a Hartree-Fock-based method we set the two parame-
erences therein). However, it is well known that the ters Vc and Ve in the following generic potential form,
two-level approximation breaks down when strong elec- VK,Na(r) = (r+Vrc0)2 − (r+Ver0), where Vc accounts for the
tricfieldsareapplied,inparticularintheCWRFregime. effectoftheatomiccore(nucleusplusallcompleteshells),
AnimportantquestionemergesastowhatextendCWRF V represents the external potential and r = Vc. Using
e 0 Ve
could potentially be observed in real atoms. In this Let- this method we find the ground, 3s, and the first ex-
terwedemonstrateforthe firsttime howthe CRWFsig- cited state, 3p, of K, as well as the 4s and 4p for Na,
natures show up in the high-order harmonic generation with precisionof∆E ≈±0.0084eV(for details see Table
(HHG) spectra of real atoms. In particular, using a ro- I).Inaddition, we alsocompute numericallythe element
busttheoreticalapproachthataccuratelymodelsboththe transition dipole dns→np = hψns|z|ψnpi for both atoms
groundandexcitedstatesofKatomscombinedwithreal- (n=3 for Na and n=4 for K) and compare them with
isticlaserparameters,weobserveclearlydistinctfeatures the experimental values reported in Ref. [18] (see Table
in the third harmonic and correlate it with the behavior I).Ascanbededuced,ourtheoreticaldipolevaluesshow
of the ground state population. The latter approach is excellentagreementwithexperimentalmeasurements,in-
closely related to the description of semiconductors and dicating that the potential givenaboveaccuratelyrepre-
atoms modeled as two-level systems. In order to further sents K and Na atoms. For finding optimal laser pa-
supportourconclusionsofCWRF-likebehaviorinK,we rameters to experimentally observe CWRF in atoms, an
also compute HHG of Na atoms, in which the transi- accurate dipole moment is essential since Rabi flopping
tion energy between the ground and first excited state frequency is linearly proportional to the dipole moment
is not resonant with the driven light, and, as a conse- as well as the electric field strength [10, 13]. In addition,
quence, a conventional HHG spectra, i.e. single peaks HHG spectra, which is the focus of our investigation,
at odd harmonics of the driven frequency, is observed. is believed to be particularly sensitive to the details of
Our predictions can be tested experimentally using cur- electrondynamicsinsideatomsandmolecules,makingit
rently available ultrashort laser pulses of a Ti:Sa laser crucial to not only accurately describe the ground state
with wavelengthscentered in the range of 750−800 nm. (as many atomic potentials in the literature do), but, in
We start by describing our theoretical approach, thecaseofresonanttransitionsduetoRabiflopping,also
putting special emphasis on the choice of the atomic po- the excited state [16, 19].
tentials,suchthatourresultsforbothgroundandexcited To compute HHG spectra, we numerically solve the
statesareinexcellentagreementwithexperimentalmea- three dimensional Time Dependent Schr¨odinger Equa-
surements (see Table I). To find CWRF signatures, we tion(3D-TDSE)inthelengthgaugeandusingtheatomic
focus on the HHG spectra. Since the HHG spectra is potential,V (r),givenaboveforKandNaatoms,re-
K,Na
proportional to the electron dipole moment, we can es- spectively. Theharmonicyieldfromasingleatomisthen
tablish a one to one correspondence between the media, proportionaltotheFouriertransformofthedipoleaccel-
modeledasacollectionofoscillators,andsingleatomsil- eration of its active electron and can then be obtained
luminatedbyastronglaserfield(anymacroscopiceffect, from the electronic wave function after time propaga-
such as phase matching, could be safely neglected, con- tion. Our code is based on an expansion of spherical
3
(a) (d)
u.) -6 u.) -6
arb. -7 -10 × 2 arb. -7
eld ( -8 eld ( -8
onic yi -9 -11 onic yi-1-90
arm-10 2.5 3 3.5 arm-11
of h-11 of h-12
og -12 K og -13 Na
l l
-13 -14
(b) (e)
arb. u.) -6 -9 × 2 arb. u.) --76
d ( -7 d ( -8
el el
onic yi -8 -10 onic yi-1-90
arm -9 2.5 3 3.5 arm-11
h h
og of -10 K og of --1132 Na
l l
-11 -14
(c) (f)
arb. u.) -6 -9 × 2 arb. u.) --76
d ( -7 d ( -8
el el
onic yi -8 -10 onic yi-1-90
arm -9 2.5 3 3.5 arm-11
h h
og of -10 K og of --1132 Na
l l
-11 -14
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
harmonic order harmonic order
FIG. 2. 3D-TDSE harmonic spectra in K for the corresponding laser intensities I = 3.158×1011 W/cm2 (panel a), I =
5.6144×1011 W/cm2 (panel b) and I =1.108×1012 W/cm2 (panel c). Panels (d), (e) and (f) represent the HHG in Na for
thesamelaser parameters. Theinsetsof panels(a), (b)and (c)show azoom of thethird harmonicω/ω0 =3(seethetextfor
details).
Sodium Experimental (NIST) Numerical field peak amplitude (E0 = pI/I0 with I0 = 3.5×1016
(present) W/cm2), ω = 0.0596 a.u. (λ = 765.1 nm), N the total
0
3s 5.139 eV 5.135 eV number of cycles in the pulse and φ the CEP. Further-
3p 3.036 eV 3.038 eV more, T defines the laser period T = 2π/ω0 ≈ 2.5 fs.
Transition dipole 2.49 au 2.40 au In the simulations presented here we consider the case
N = 20, corresponding to an intensity envelope of full
Potassium Experimental (NIST) Numerical
(present) width half maximum (FWHM) of 0.36NT, (7.2 optical
cycles≈18fs FWHM), andφ=0(see belowfordetails).
4s 4.340 eV 4.347 eV
4p 2.730 eV 2.725 eV As discussed in the introduction, we set input laser
Transition dipole 2.92 au 2.79 au frequencythe samevalue (inatomic units) asthe energy
corresponding to the transition 4s → 4p of K in order
to observe a CWRF-like behavior. In order to compare
TABLE I. Experimental and theoretical values of the energy
gap between the ground and first excited states, joint with with a conventional situation (meaning the usual condi-
thetransition dipole both for Naand K. tions for HHG), we use the same input laser parameters
with Na atoms, for which the transitionenergy from the
ground to the first excited state, 3s → 3p, corresponds
harmonics, Ym, and takes advantage of the cylindrical to 2.10 eV, and is therefore non-resonant with the laser
l
symmetry of the problem (hence only the m = 0 terms frequency.
need to be considered). The time propagation is based In Fig. 2, we show the harmonic spectra computed
on a Crank-Nicolsonmethod implemented on a splitting fromthe3D-TDSEforbothK(Figs.2(a),2(b)and2(c))
of the time-evolution operator that preserves the norm and Na (Figs. 2(d), 2(e) and 2(f)) atoms. We have cho-
of the electronic wave function. The coupling between senthe laserparametersto coverthree differentregimes,
the atom and the laser pulse in the length gauge, lin- namely,forpanels2(a)and2(d)the envelopepulse area,
early polarized along the z axis, is written as V (z,t) = Θ , is close to 2π. The envelope pulse area is de-
l K,Na
E(t)z, where E(t) is the laser electric field defined by fined as Θ ≈d E ∆t, where d is the dipole
K,Na K,Na 0 K,Na
E(t)=E0sin2(cid:0)ω2N0t(cid:1)sin(ω0t+φ). E0 is the laser electric transition matrix for K or Na (see Table I) and ∆t the
4
(a) 3 3 (d)
Ground state population0000....12468 --01221 Electric eld (10 a.u.)(cid:222)-3 --21012 00001....2468 Ground state population
K -3 -3 Na
0 0
(b) 4 4 (e)
on 1 E 1 on
Ground state populati0000....2468 -022 lectric eld (10 a.u.)(cid:222)-3 -202 0000....2468 Ground state populati
K Na
-4 -4
0 0
(c) 6 6 (f)
Ground state population0000....12468 --02442 Electric eld (10 a.u.)(cid:222)-3 --42024 00001....2468 Ground state population
K Na
-6 -6
0 0
0 5 10 15 20 0 5 10 15 20
Laser periods Laser periods
FIG. 3. Time evolution of the ground state population (red thick line) along the laser pulse (blue thin line) corresponding to
thecases plotted in Fig. 2.
FWHMpulseduration(∆t=18fsi.e.,∆t≈750au). In envelope area significantly exceeds 2π.
particular, for a laser intensity I =3.158×1011 W/cm2
Togetfurther insightintothe physicalmechanismbe-
(E = 0.003 a.u.), Θ ≈ 2π and Θ ≈ 5.4 (for com-
0 K Na hind the complex structure of the HHG spectra in K
parison see the values used in semiconductor GaAs [9]).
atoms, in Fig. 3 we present the time dynamics of the
Panels 2(c) and 2(f) correspond to values of Θ and
K ground state population for all the cases depicted in
Θ , respectively, close to 4π (11.7 for K and 10.1 for
Na Fig. 2. For a two-level system, used as a prototypical
Na), obtained by keeping constant the pulse duration
model for a semiconductor or a simplistic picture for a
and now using a laser intensity I =1.108×1012 W/cm2
realatom, the electrondynamics due to interactionwith
(E =0.0056 a.u.). Panels 2(b) and 2(e) were chosen to
0 laser light can be represented on a Bloch sphere (for de-
haveanintermediatevalueofintensityI =5.6144×1011
tails see e.g [4, 9]). In this case, the ground state popu-
W/cm2 (E = 0.004 a.u.), corresponding to pulse enve-
0 lations and the regular Rabi oscillations can be depicted
lope areas of 8.4 and 7.2 for K and Na, respectively.
asmovingalongthe surfaceofthe sphere(see Fig.1(a)).
When the CWRF regime is reached, clear signatures,
Fromthe HHGspectraofKatoms,weobserveadras-
corresponding to the break-down of the area theorem,
ticchangearoundthethirdharmonic,ω/ω =3,aspulse
0
should occur on the Bloch sphere as well (see Fig. 1(b)).
envelope area increases. Our results are directly analo-
gous to the manifestation of CWRF behavior observed Following an analogy with a two-level system, we can
in semiconductors in [9]. Note the marked contrast to distinctly observeCWRF-like behaviorin Figs.3(b) and
the HHG spectra of Na, shown in panels 2(d), 2(e), and 3(c)andthecorrespondingcounterpartintheHHGspec-
2(f), where a characteristic peak is present in the third tra(Figs.2(b)and2(c)). Onthecontrary,aconventional
harmonic regardless of the envelope pulse area. The on- behaviorinthethirdharmonic(Figs.2(a)and3(a)),can
set of this more complex behavior in K atoms for suffi- be correlated with: (i) ordinary Rabi oscillations for the
ciently large envelope pulse areas is in agreement with case of K (Fig. 2(a)), i.e. the ground state is completely
conclusions in [3], where the onset of CWRF behavior depopulated, even though the laser intensity is low and
and the consequent break-downof the area theorem was this would be analogous to a travel of the Bloch vector
predictedforatwo-levelresonantsystemswhenthepulse fromthesouthtothenorthpole[9];(ii)normalbehaviour
5
of atoms in strong field for all the Na cases (Figs. 2(d)- the CEP changes when the driving laser field is a few-
2(f)), i.e. gradual depopulation of the ground state due cycle pulse. Furthermore the CWRF could be used to
to laser ionization (Figs. 3(d)-3(f)). control the laser-induced ionization, known as a crucial
ingredientforharmonicpropagation,viamanipulationof
the ground state population.
(a) = 0
u.) -6 = !/2 × 2 We acknowledge the financial support of the MICINN
d (a. -7 -10 p01ro,jaecntds(FFIISS22001008--1020873844),TEORQCATAAd,vFanISc2ed008G-r0a6n3t68Q-CU0A2--
el
yi -8 GATUA and OSYRIS, the Alexander von Humboldt
c
oni -9 -11 2.5 3 3.5 Foundation (M.L.), and the DFG Cluster of Excel-
m
ar-10 lence Munich Center for Advanced Photonics. This re-
h
of searchhasbeenpartiallysupportedbyFundaci`oPrivada
g -11 Cellex. J.A.P.-H.andL.Rosoacknowledgesupportfrom
o
l-12 K Laserlab-Europe (Grant No. EU FP7 284464) and the
Spanish Ministerio de Econom´ıa y Competitividad (FU-
-6 (b) = 0 RIAM Project FIS2013-47741-R). We thanks Christian
u.) -7 = !/2 Hackenberger for helping us with the artwork.
a.
d ( -8
el
yi -9
c
oni-10
m
ar -11 ∗ [email protected]
h
of -12 [1] I. I. Rabi, Phys.Rev.49, 324 (1936).
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-14
[3] S. Hughes, Phys.Rev.Lett. 81, 3363 (1998).
0 1 2 3 4 5 6 7
[4] M.Wegener,ExtremeNonlinearOptics(Springer-Verlag,
harmonic order
Berlin, 2005).
[5] K. Rzazewski and M. Florjanczyk, J. Phys. B 17, L509
FIG. 4. HHG for K (panel a) and Na (panel b) for different
(1984).
CEPs. ThelaserparametersarethesameasinFigs.2(b)and
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[14] B.SundaramandP.W.Milonni, Phys.Rev.A41, 6571
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nos mediante un l´aser pulsado intenso (Master Thesis,
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Universidad deSalamanca, 2004).
third harmonic does not present appreciable differences.
[18] D. A. Steck, Los Alamos National Laboratory. Los
However the third harmonic of K is strongly affected,
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