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Adiabatic stabilization of excitons in intense terahertz laser

Adiabatic stabilization of excitons in intense terahertz laser
Adiabatic stabilization of excitons in intense terahertz laser

a r X i v :c o n d -m a t /0109038v 1 4 S e p 2001Adiabatic stabilization of excitons in intense terahertz laser

Ren-Bao Liu and Bang-Fen Zhu

Center for Advanced Study,Tsinghua University,Beijing 100084,People’s Republic of China

High-precise calculation of near-infrared absorption and transient spectra in bulk semiconductors

irradiated by an intense terahertz (THz)laser shows that,the ionization rate of the ground-state

exciton,probed through the dynamic Fano resonance,may decrease with the increase of the THz

laser intensity.This counterintuitive e?ect indicates the excitons are stabilized against the ?eld-

ionization.The much lower Rydberg energy and “atomic unit”of laser intensity for excitons,and the

possibility of creating excitons from the “vacuum state”allow observing the excitonic stabilization

in experiments,in contrast to the case of atomic stabilization.

PACS numbers:https://www.doczj.com/doc/897845185.html,,78.20.Jq,32.80.Fb,42.50.Md In accordance to Einstein’s theory of the photoelectric e?ect,an electron can be stripped from its atom by light beams with su?ciently high frequency.As the intensity of a light beam is increased,the stripping probability of the electron increases owing to the increasing number of photon impacts.Surprisingly,since the late 1980’s,theoretical investigations and numerical simulations [1–6]have predicted that when the electric ?eld,associated with a laser of su?ciently high intensity and frequency,approaches or exceeds the electrostatic ?eld between the electron and the ion in an atom,the wave function of the irradiated atom may be distorted adiabatically into a distribution with two well separated peaks.The peak spacing increases with increasing the ?eld intensity,thus the atomic electron spends more time far away from the nucleus,and the ionization rate slows down dramatically till almost totally suppressed.In other words,the adia-batic stabilization of the atom occurs.The observation of this atomic stabilization e?ect is,however,extremely di?cult.Firstly,for the atomic ground state,the high-frequency and high-intensity con-dition requires an extreme ultraviolet laser with intensity

larger than an atomic unit (~3.51×1016W/cm 2),which

seems not available in the near future.Secondly,and

more vitally,su?ciently slow turn-on is demanded for

the laser pulse to adiabatically drive the atomic ground

state into a stable dressed state;during the rise time of

the super-intense laser pulse,however,substantial ion-

ization has already been inevitable [7],which prevents

the e?ect from observation.Some researchers even ar-

gued that it was impossible to observe such an adiabatic

stabilization of atoms [8,9].In fact,except for a few dis-

putable indications for the stabilization of high Rydberg

states [10],so far there is no unambiguous experimental

evidence for the atomic stabilization.

It is compelling to settle the debate whether this e?ect

exists or not.Apart from the pure curiosity in basic re-

search,the atomic stabilization is also of importance in

potential application,such as the high harmonic genera-tion [11]and quantum computation [12],as it can quench the undesired ionization and dephasing caused by the in-tense laser that is used to drive electrons or to manipulate a qubit in an atom.In this Letter,we report on the adiabatic stabiliza-tion of hydrogen-atom-like excitons in semiconductors.The excitonic stabilization (XS),in contrast to its atomic counterpart,is de?nitely an observable e?ect,owing to the basic characteristic of excitons that they are created from “vacuum”of carriers by the interband optical exci-tation.In our study,the dressed exciton states are pre-pared by weak near-infrared (NIR)laser pulses in semi-conductors in the presence of a quasi-continuous tera-hertz (THz)?eld,as in most recent experiments on the dynamical Franz-Keldysh e?ect [13–15].The ionization rate or lifetime of the dressed excitons can be directly ex-tracted by monitoring either the linewidth of absorption peaks or the decay time of transient signals.Thus the “turn-on”problem is safely circumvented as long as the pulsed interband excitation and the following dephasing process of excitons are well covered by the ?attop region of the THz laser pulse.Moreover,the present laboratory gear is already ready for observing the XS e?ect.Due to the small e?ec-tive mass μand large dielectric constant ε,an exciton has much smaller binding energy E B than the hydrogen atom does,and the “excitonic unit”for laser intensity (∝μ4/ε5.5)in a typical semiconductor like GaAs is about 10orders of magnitude smaller than the atomic unit.So,the “high-intensity and high-frequency”requirement for the XS can readily be ful?lled by the free-electron lasers operating with MW/cm 2power and THz frequency [16].Our investigation is based on the high-precise calcula-tion of interband optical spectra,which includes properly the Coulomb interaction,the nonperturbative ac-?eld,and the contribution of continuum states.To avoid un-necessary complexity,the excitonic state is treated within the simple e?ective-mass approximation.Then the mo-tion equation for the relative motion of an electron-hole

pair is

iˉh?tψ(r,t)=(?H?iγ2)ψ(r,t)+χ(t)e?i?0tδ(r),(1)

?H≡E

g ?

ˉh2

4πεr

+eF z cos(ωt),(2)

where E

g

is the band gap,F is the?eld strength of the THz laser associated with frequencyωand linearly polar-ized in the z direction,χ(t)denotes the NIR pulse exci-

tation centered at?0,andγ2is an interband dephasing-rate due to phonon scattering.The THz laser is assumed

in a continuous wave(cw)form,because its microsec-ond duration[16]is much longer than the picosecond interband excitation and dephasing process.This model system is mathematically equivalent to a hydrogen atom in an intense laser except that the electron-hole pair is generated by the NIR laser.

Without the Coulomb coupling,the eigensolution of the time-dependent Hamiltonian(Eq.(2))is the Volkov state[17],which,as the basis set,reads

|k,m =e imωt?ia F k z cos(ωt)+i U p

+U p+mˉhω,denote the sideband index and the wavevector,respectively.Here U p≡e2F2

μω2is the classical ex-

cursion amplitude of an electron in the ac-electric?eld,

and|?k is the Bloch-state with accelerating quasimomen-

tum k?eF

γ2+(??E g+E B)2

,

indicating the occurrence of the DFR.By?tting the

0ωpeak,the ionization rateΓcan be extracted via

γ≈γ2+Γ/2.As shown in Fig.2,γ(hence the ioniza-

tion rate)increases with the laser intensity until F≈16

kV/cm(correspondingly,I≈1.0MW/cm2[21]and

a F≈1.28a B).Beyond that critical?eld strength,the

ionization rate commences to decrease,and approaches

zero at F≈28kV/cm,clearly demonstrating the XS

e?ect.

Γcan also be obtained analytically in the high-

frequency approximation[23].When the wavefunction

dichotomy is negligible for medium a F(say a F<3a B),

a compact result forΓof the ground state[23]can be derived as

Γ≈ nω>E B32π2a F 2 10J2n(a F k n x)dx, where k2n=nω?E B.When?eld is enhanced so that a F k n approaches or exceeds1,due to rapid oscillation of the Bessel function,the integration over x diminishes. Physically,this originates from the destructive interfer-ence of the out-going waves scattered from di?erent part of the initial state.On the other hand,the stretching of the state along the?eld will reduce the wavefunction peak,contributing additionally to the XS.

Fano interference is one of fundamental channels for irreversible decay[24,25],so the XS can also be directly observed in the time domain by monitoring the transient optical signal after a pulsed excitation.The time-resolved interband polarization is calculated for several THz-?eld strengths and plotted in Fig.3.The duration of the NIR pulseτis then set to be2/ω,and the central frequency is chosen for each?eld strength to be resonant with the brightest transition,namely?0?E g=?4,?2,2,5,and 6meV for F=0,8,16,20,and30kV/cm,respectively. To single out the dephasing induced by the DFR only, the reformulated signal P(t)≡P(t)exp(γ2t)is plotted. At?nite?eld,the optical signal roughly oscillates with the period of T≡2π/ω,because the states in the KH po-tential oscillate periodically in the laboratory frame.The modulation beat superimposed on the fast oscillation re-sults from the quantum interference between the Floquet states with di?erent quasienergy.The dynamic Fano in-terference between the discrete state and the continuum opens a new channel for interband dephasing,leading to the decay of the transient signal.The XS e?ect is ver-i?ed by the fact that the dephasing rate calculated for F>16kV/cm begins to decrease,consistent with the variation trend of the ionization rate as shown in Fig.2. For F=30kV/cm(I≈3.6MW/cm2[21]),the decay of the signal almost disappear.To measure the transient signals,nonlinear optical experiments like four-wave mix-ing are usually adopted,in which many-body correlation is important sometimes[26]and may induce new inter-esting e?ects on XS,which,however,is out of the scope of our present study.

Fig.4displays snapshots of the probability distribu-tion of the exciton wavepacket resonantly excited by a pulse(?0?E g=8.88meV)withτ=20/ω,corre-sponding to the spectral width0.5meV.When the NIR-excitation pulse is over,the wavepacket performs almost perfect periodic swinging along the z?axis with the turn-ing points located at about±?z a F,except for an overall exponential decay due to the ionization.This periodic behavior is veri?ed by the time-dependence of the opti-cal signal(see the inset).Because of the inter-sideband mixing,the wavepacket shape varies drastically within a period,in contrast to the unchanged shape of the period-ically oscillating wavepacket in the KH potential without inter-sideband coupling[1].All these demonstrate that a quasi-bound Floquet state is formed in the strong THz-?eld,and justify our ascribing the suppression of the Fano broadening and dephasing rate to the adiabatic XS.

In summary,the novel excitonic stabilization e?ect,a counterpart of the atomic stabilization e?ect,has been predicted and explored nonperturbatively in a coupled system of semiconductors and intense THz-?elds by high-precise calculation.

This work was supported by the National Science Foundation of China,and R.B.L was also supported by China Postdoctoral Science Foundation.

[17]D.M.Volkov,Z.Phys.94,250(1935).

[18]H. A.Kramers,Collected Scienti?c Papers(North-

Holland,Amsterdam,1956),p.866;W.C.Henneberger, Phys.Rev.Lett.21,838(1968).

[19]R.B.Liu and B.F.Zhu,J.Phys.:Condens.Matter12,

L741(2000).

[20]S.Glutsch,D.S.Chemla,and F.Bechstedt,Phys.Rev.

B54,11592(1996).

[21]Here we use the intensity inside the material with relative

dielectric constantε/ε0=9.0.The vacuum value is larger by about1.3.

[22]U.Fano,Phys.Rev.124,1866(1961).

[23]M.Pont,Phys.Rev.A44,2141(1991);44,2152(1991).

[24]U.Siegner,M.-A.Mycek,S.Glutsch,and D.S.Chemla,

Phys.Rev.Lett.74,470(1995).

[25]C.P.Holfeld,F.L¨o ser,M.Sudzius,K.Leo,D.M.Whit-

taker,and K.K¨o hler,Phys.Rev.Lett.81,874(1998).

[26]D.S.Chemla and J.Shah,Nature411,549(2001). FIG.1.Linear absorption spectrum for several THz-?eld strengths indicated by F.The upper-right inset shows,as an example,the Fano resonance?tting process for F=24 kV/cm,where the squares are the?tted data.The lower-right inset shows schematically the sideband structure of the Flo-quet-state excitonic spectrum.

FIG.2.Fano broadening constant versus the THz-?eld strength.

FIG.3.Real-time dependence of the interband polariza-tion for various THz-?eld strength indicated by F,in which the curves are o?set for clarity.The intensity pro?le of the NIR pulse(the dotted curve)is also shown for comparison. FIG.4.Probability distribution of the NIR-pulse-excited wavepacket at several time instant within a period T for the THz-?eld of F=24kV/cm.The inset shows the semi-logarithmic plots of the intensity of the NIR pulse(dot-ted line)and transient interband optical signal(|P(t)|2,solid line)as functions of time,and the time snatch(marked by the bar)where the snapshots are made.

?-E g (meV)

-1001020

α(?) (a r b . u n i t s )0

8

16

20

24

30

F (kV/cm)

510

24 kV/cm

F (kV/cm)0510********

γ (m e V )

1.0

1.2

1.4

1.6

1.8

ωt /(2π)0102030

|!

(t

)|

2

(

a

r b .

u

n

i t s ) F (KV/cm)=0830

1620

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