Terahertz wave generation from hyper-Raman lines in two-level quantum systems driven by two-color lasers

0.23 MB·English

by  Wei Zhang

Terahertz wave generation from hyper-Raman lines in two-level quantum systems
driven by two-color lasers
∗
Wei Zhang , Shi-Fang Guo, Su-Qing Duan, and Xian-Geng Zhao
Institute of Applied Physics and Computational Mathematics, P. O. Box 8009(28), Beijing 100088, China
(Dated: January 24, 2012)
Basedonspatial-temporalsymmetrybreakingmechanism,weproposeanovelschemeforterahertz
(THz)wavegenerationfromhyper-Ramanlinesassociatedwiththe0thharmonic(aparticulareven
harmonic)inatwo-levelquantumsystemdrivenbytwo-colorlaserfields. Withthehelpofanalysisof
quasi-energy,thefrequencyofTHzwavecanbetunedbychangingthefieldamplitudeofthedriving
laser. By optimizing the parameters of the laser fields, we are able to obtain arbitrary frequency
2
radiation in the THz regime with appreciable strength (as strong as the typical harmonics). Our
1
proposalcanberealizedinexperimentinviewoftherecentexperimentalprogressofeven-harmonics
0
2 generation by two-color laser fields.

n PACSnumbers: 42.65.Ky,42.50.Hz,78.20.Bh
a
J

3 IntroductionTerahertz(THz)radiation,electromag- of the incident laser)19. Here is one observationthat the
2 neticradiationwithtypicalfrequencyfrom0.1THzto10 shiftofthehyper-Ramanlinetotheassociatedharmonic

THz-lying inthe spectrum gapbetweenthe infraredand could be small and could be tuned by the external field,
]
s microwaves, has varieties of applications in information even though the fundamental frequency is large. Then
c and communication technology, biology and medical sci- one may use the 0th hyper-Raman line (that associated
i
t ences,homelandsecurity,andglobalenvironmentalmon- with the 0th harmonic) to generate the low-frequency
p
itoring etc.1. The generation of THz wave is a challenge (THz) wave. There have been some studies on the har-
o
problem due to the lack of appropriate materials with monics and hyper-Raman lines. In many cases, there
.
s small bandgaps in the usual optical approach. Great ef- are only odd harmonics due to the particular symmetry
c
i fort has been made to the design of THz sources2–15. andtheassociatedhyper-Ramanlineisofhighfrequency
s
THz wave can be obtained by both electronic and opti- and/or weak intensity. Therefore one has to solve these
y
h cal methods. The uni-travelling-carrierphotodiode6 and two problems to generate THz wave by using the hyper-
p the quantum cascade laser9–13 are two examples. The Raman lines in the nonlinear optical processes.
[ electronic approach is limited in the low frequency end Inthis article,weadoptaneffective opticalmethodto

of the THz regime, and the optical approach usually fo- obtain THz wave in typical two-level quantum systems,
1
v cuses onthe cases ofsmallenergy gap. Technologicalin- which solves the two problems mentioned above. Based
9 novation in photonics and nanotechnology has provided on the generalized spatial-temporal symmetry principle
2 us more new ways for THz radiation generation. Re- we developed recently20, we propose a novel mechanism
7 cently an interesting approach based on semiconductor to obtain the 0th hyper-Raman lines in THz regime by
4 nanostructure driven by acoustic wave was suggested16, introducing additional laser with frequency 2kω (k =
. 0
1 and electrically pumped photonic-crystal THz laser was 1,2,3,...). Using analysis of quasienergy and optimiza-
0 developed17. THz wave generation by up conversion tion of laser fields, we are able to obtain considerably
2 method throughhigh-order harmonics generation(HHG) intense THz radiation with desired frequency. With the
1 in semiconductor quantum dot(QD) was also suggested help of our optimization method, it is quite likely that
:
v in our previous work18. our proposal can be realized in experiment considering
i the recent experimental progress of even-harmonics gen-
X One challenge of THz wave generation in optical ap-
eration by two-color laser.
r proach in usual system is to generate low frequency ra-
a diation from a quantum system with a large energy gap. Theoretical formulism Our two-level quantum sys-
temdrivenbylaserfields[seetheschematicdiagramFig.
Usually, in the emission spectrum of a quantum system,
1(a)] is described by the Hamiltonian
there are harmonics, as well as the associated hyper-
Ramanlines. Hyper-Ramanlinesarecausedbythetran-
sitions between the dressed bound states. Nth hyper- 2
Ramanlines aredefinedasthoseassociatedwiththe nth H =XEi|iihi|+G(t)(|1ih2|+|2ih1|), (1)
harmonics(they appearwiththe correspondingharmon- i
icssimultaneouslyduetothesamesymmetryreason,and
locate near nω , where ω is the fundamental frequency where G(t) = F(t)e·µ12, is the Rabi frequency caused
0 0 by periodic laser field E = F(t)e, F(t) = F(t+T), e a
unit vector. µ = h1|er|2i is the dipole between state
12
|1i and state |2i. The energy spacing between the two
∗Authortowhomanycorrespondenceshouldbeaddressed,Email: states[withenergiesEi(i=1,2)]issetas△E =E1−E2.
zhang [email protected] The dynamics of our system is described by the equa-
2
tion of motion of the density matrix21 (a) -6 (d)
E
1
∂ρ i n(cid:90),n(cid:90)(cid:114)(cid:39)
=− [H,ρ]−Γ·ρ, (2) (cid:19) (cid:19)
-11
∂t ¯h
(cid:39)
where H is the Hamiltonian, the last term describes E
2
possible dissipative effects (such as spontaneous phonon -16
emission) and we set ¯h = 1 in the following. We nu- -6 (b) -60 1 2 3 (e) 4
merically calculate the photon emission spectra of the a.u.)
two-level quantum systems by solving the density ma- S)(-12 -11
trix in Eq.(2) through Runge-Kutta method with the g(10
o
time step of 0.002/ω , total steps of 1200000 and the L
0
electron initially setting at the lower level. The average -18 -16
0 1 2 3 4 5 6 7 8 0 1 2 3 4
dipolecanbecalculatedasD(t)= µ ρ (t). Weuse -6 (c) -6 (f)
Fourier transformation to obtain thPeijemiisjsiiojn spectrum u.)
a.
¯hSω(ν0)(ω=0|tRhedtderxivpin(−gifiνetl)dDfr(et)q|u2.enIcny)thaestchaelcuunlaittioonfewneersgeyt. g(S)(10-12 -11
o
We choose ω = 100THz as an example, yet in general L
0
onecanalsouseotherfrequency(muchlargerthanTHz) -18 -16
0 1 2 3 4 5 6 7 8 0 1 2 3 4
laser as will be discussed later. The two-level quantum (cid:81)(cid:18)(cid:90) (cid:81)(cid:18)(cid:90)
(cid:19) (cid:19)
systems canbe realizedin many systems,suchas atoms,
molecules, and semiconductor quantum dots. Here we
FIG. 1: (a) The schematic diagram of our two-level sys-
use the typicalparametersofaquantumdot: the energy
tem driven by an incident laser. The emission spectrum
spacing∆E =10¯hω0,thedipolemomentµ12 =0.5e·nm, contains the higher-order harmonics and the accompanied
phonon emission coefficient Γij =3.4×10−3ω0. hyper-Raman lines; (b)-(d) Emission spectra in the presence
We explore the emission spectra using above numer- of various driving fields F(t). (b) F(t) = F1cos(ω0t); (c)
ical approach and the theory on the selection rule in F(t) = F2cos(2ω0t); (d) F(t) = F1cos(ω0t)+F2cos(2ω0t);
high-orderharmonicgenerationbasedonthe generalized (e)F(t)=F1cos(ω0t)+F2cos(3ω0t);(f)F(t)=F1cos(ω0t)+
spatial-temporalsymmetry. Asdescribedinourprevious F2cos(4ω0t). F1 =4.4×109V/m,F2 =2.2×109V/m.
work20, for a quantum system described by Hamiltonian
(1), if there exists one symmetric operation Q, which is
the time shift θ : t → t+T/2 (T = 2π/ω ) combined
with another operation Ω in spatial/spectr0um domain, +Xe−i[(ε2−ε1)+(n−m)ω0]ta2a∗1hφm1 |Pˆ|φn2i
i.e. Q = Ω·θ , such that the initial condition and the m,n
Hamiltonian (up to a sign) are invariant, and the dipole +Xe−i[(ε1−ε2)+(n−m)ω0]ta1a∗2hφm2 |Pˆ|φn1i. (3)
operator Pˆ has a definite parity, then the emission spec- m,n
trum contains no odd/even component if operator Pˆ is
even/odd. For our two-level quantum system driven by Ifthesystempossesasymmetrygeneratedbytheoper-
a monochromatic laser, i.e. E(t)=F1cos(ω0t), there ex- atorQ=Ω·θ andtheFloquetstate|φα(t)ihasadefinite
ists one symmetric operation Q1, which is the time shift parity under Q, i.e., Q|φα(t)i = ±|φα(t)i, then we have
θ combined with spatial operation Ω1: c1 → −c1 (here Ω|φmαi = ±(−1)m|φmαi. Then we have hφmα|Pˆ|φnβi = 0
andinthefollowingcj,j =1,2,referstotheannihilation (α,β =1,2)forn−meven/oddnumberandstatesα,β of
operator for state |ji), and the Hamiltonian is invariant same/differentparity. Quiteoftentheharmonicsandthe
and the dipole operator Pˆ is odd. Then odd harmonics associatedhyper-Ramanlinesappearsimultaneouslydue
alonearegeneratedbecauseofthespatial-temporalsym- to the same symmetry properties of the Floquet states.
metry. Other types of spatial-temporal symmetry may Inthecasewithamonochromaticlaser,wehaveoddhar-
lead to many interesting emission patterns20. monics and the associated hyper-Raman lines. While in
Here we give some analysis on the hyper-Raman lines other cases with symmetry broken, more harmonics and
and their relation to the harmonic components. In our the associated hyper-Raman lines are generated.
systems driven by a periodic field, the quasienergy state Emission patterns Let’s first look at the emission
has the form |ψα(t)i = e−iεαt|φα(t)i, where the Floquet spectrum of a system driven by a monochromatic inci-
state |φα(t)i = |φα(t+T)i can be written in the form dent laser with frequency ω0. As seen in Fig.1(b), there
|φα(t)i=Pme−imω0t|φmαi. Weconsiderastate|ψ(t)i= areoddharmonicsaswellastheassociatedhyper-Raman
a1|ψ1i+a2|ψ2i = a1e−iε1t|φ1i+a2e−iε2t|φ2i. Then we lines due to the spatial-temporalsymmetry propertiesof
have the Floquet states as analyzed above. It is natural that,
for the incident laser with frequency 2ω , the emission
hψ|Pˆ|ψi= 0
spectrum contains components with frequencies of 2ω ,
0
Xe−i(n−m)ω0t[|a1|2hφm1 |Pˆ|φn1i+|a2|2hφm2 |Pˆ|φn2i] 6ω0 and 10ω0...(odd orders of incident frequency 2ω0)
as shown in Fig.1(c). One notices that there is no 0th
m,n
3
hyper-Ramanline in this case. If we use two-colorlasers
with frequencies ω and 2ω , interesting phenomena ap- 0.10 (a) 30 (b)
0 0
pear. As seen from Fig.1(d), even harmonics and the
ainsgsotchiaetesdechoynpderl-aRsearmfianeldli.nesEasrpeecgieanlleyraateldowbyfirnetqruoednuccy- S(a.u.)0.05 15 peak 1 peak 2
radiation, 0th hyper-Raman radiation is generated. We
wouldliketopointoutthattheemissionspectrumforthe
case with driving field F(t)=F cos(ω t)+F cos(2ω t)
1 0 2 0
0.00 0
is not the supposition of those driven by F1cos(ω0t) and 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0

F cos(2ω t). Forexample,theharmonic4ω inFig. 1(d)
2 0 0
neither appears in Fig.1(b) nor in Fig.1(c). It is also not
FIG. 2: (a) The intensity of hyper-Raman line associated
the frequency summation or difference of the harmonics
with 1th harmonic for F(t)=F1cos(ω0t), F1 is related to ν,
of systems driven by monochromatic incident laser with the frequency of the hyper-Raman line; (b) The intensities
frequencyω0 and2ω0. Infact,theappearanceofalleven of hyper-Raman lines associated with 0th (peak 1) and 1th
components (and the associated hyper-Raman lines) in (peak 2) harmonics for F(t) = F1cos(ω0t) + F2cos(2ω0t),
Fig. 1(d) is the consequence of symmetry breaking as F1 =4.4×109V/m,F2 is related to ν.
discussed above.
According to our theory, the symmetry of the quan-
tum system generated by Q is broken by introducing
1
0.5
the second laser with frequency 2kω (k =1,2,3...), and
0
it is notbrokenby introducing the secondlaser with fre-
0.4
quency (2k−1)ω (k = 1,2,3...). These predictions are
0
verified by our numerical results shown in Fig.1(e) (the 0.3
second laser of frequency 3ω ) and Fig.1(f) (the second
0
0.2
laser of frequency 4ω ). It is clear that the 0th hyper-
0
Raman line does not appear in the cases with symmetry
0.1
(seeFigs.1(b)(c)(e)). Interestingly,thesecondlaserfield
0.02
with frequency 4ω leads to the appearance of 2nth har- 0.0
monics(evenforn0=1)andtheassociatedhyper-Raman 0 2 F1 ( 1409 V/m) 6 8
line(inparticularthe0thhyper-Ramanline)aspredicted
FIG. 3: The quasienergy difference (solid line) between two
by our theory, since the second laser with frequency 4ω
0 quasi-eigenstates and the frequency (dots) of the 0th hyper-
breaks the spatial-temporal symmetry generated by Q .
1 Raman line versus the magnitude of external field F1. F2 =
Using this mechanism, we can explain the experimental 2.2×109V/m.
results of observing even harmonics in helium or plasma
plumes(containingnanoparticles,carbonnanotubes,etc)
drivenbyatwo-colorlaser,22 wherethesecond-harmonic
driving term, despite being very small, breaks the sym-
wave Now we discuss the optimization scheme of THz
metry and thus allows additional strong even harmonic
wave generation with driving field F(t) = F cos(ω t)+
1 0
components. Ourtheoryalsonaturallyexplainsthe gen-
F cos(2ω t) in our proposal. From equation (3), we see
2 0
eration of even harmonics and associated hyper-Raman
that the difference of the quasienergy is related to the
lines by breaking the spatial symmetry23.
shift of the hyper-Raman line form the corresponding
As seen from equation (3), the frequency of hyper- harmonics,orthefrequencyofthe0thhyper-Ramanline.
Raman line is determined by the quasienergy, which is This relation is verified by our numerical calculation as
tunable. One may ask whether one can use the hyper- shown in Fig. 3. Moreover from Fig. 3, one sees that
Raman line associated with 1th harmonic to generate the 0th hyper-Ramanline with very smallfrequency can
the low frequency/THz wave? Actually, as seen from be obtained in some parameter regimes, which can be
Fig. 2(a) the intensity of the hyper-Raman line asso- used for THz wave generation. By applying this rela-
ciated with the 1th harmonic decreases dramatically as tion ∆ = |ǫ − ε | = ¯hν (ν the frequency of the 0th
1 2
the frequency decreasing. To make this pointclearer,we hyper-Raman line), any designated frequency THz wave
compare the low frequency components associated the canbe obtainedbytuning the magnitudesofthe twoex-
1th harmonic and 0th harmonic for the case with two- ternal fields F and F . For example, THz radiation of
1 2
color laser shown in Fig. 2(b). One sees that in the ev- 2THz (0.02ω ) can be obtained by tuning the external
0
ery low frequency regime, the intensity ofthe 0th hyper- field as shown in Fig. 3 [intersection points of the solid
Raman line (associated with the 0th harmonic) is much straight-line (ν = 0.02ω ) and the line for quasienergy].
0
largerthan that of the 1th hyper-Ramanline. Therefore Fig. 4(a) shows the values for F and F under which
1 2
oneshouldusethe0thhyper-Ramanlinetogeneratethe the low frequency is ν =0.02ω . There are disconnected
0
THz radiation effectively. parameterregimesofdrivingfieldsforgenerationofTHz
Tuning of the frequency and intensity of THz radiation. Furthermore, the intensity of the THz wave
4
In our method, there is another tunable parameter, the
-6
8 (a) -3 (b) phase difference between two incident lasers. The main
9 10 V/m)6 (6.9,6.2) S)(a.u.) -1-029.00 0.05 0.10 phUysnicliakleppircetuviroeursesmtuadiniesstohne snaomn-elifnoerardidffreirveinntgptwhaos-leesv.el
(F 24 Log(-109 sgyesnteermatsi2o5n,19t,h26e–2h9,ypoeur-rRascmhaenmelinoefsgaennderalotiwngfrleoqwuenfrcey-
2
quency(THz)waveisbasedonthesymmetryprinciplein
0 -15 typical two-level quantum systems of large energy spac-
2 F41 ( 109 V/m6) 8 0 1 2 3 4 ing (in the order of eV). It can be easily obtained from
naturalsystems(suchasatoms,molecules,andsemicon-
FIG. 4: (a) The parameters of F1 and F2 for the 0th hyper- ductors)andartificialstructures(suchasquantumdots).
Ramanlinewithfrequency2THz=0.02ω0. Thetriangleindi- The large energy spacing makes it robust against ther-
cates theoptimal parameterfor themaximalintensity of0th
malfluctuationandalsoinsensitiveto the initialstate20.
hyper-Ramanline. (b)Thecorrespondingemission spectrum
Our mechanism based on the symmetry principle is dif-
for the case with optimized external field.
ferent from those (based on four-wave mixing8 or tran-
sit current30) used before. Our non-perturbative anal-
ysis and calculation show that one only needs to tune
can be optimized. One can find the proper parameters the intensity of the two-color laser for typical two-level
forthemaximalintensitywithdesiredfrequency[thetri- quantum systems, which is very convenient in experi-
angle in Fig. 4(a)]. The corresponding emission spec- ments. We may be able to obtaintunable THz radiation
trum is shown in Fig.4(b). It is seen that the intensity due to the availability of stable, tunable driving sources
of THz radiation has been increased 200 times after op- abovetheTHzregime1,22. Moreover,peoplehavealready
timization and we are able to obtain appreciably intense observed even harmonics in experiments with two-color
THz wave (as strong as typical harmonics). Here we es- laser22,24. Therefore it is quite likely that the 0th hyper-
timate the emission power based on a system of arrays Raman line (in THz regime) associated with 0th har-
of semiconductor quantum quantum dots. The emission monic canbe obtainedin experimentif ouroptimization
power ofeachdot is P ∼|µ |2ν4/3c3 ∼10−22W. There method is used.
12
are about N ∼ 108 dots in the regime of size of wave-
Summary A novel mechanism based on spatial-
length which emit wave in phase. Therefore, the total
temporal symmetry breaking is proposed to obtain THz
emission power from a sample of submillimeter size is
wave from 0th hyper-Raman line in two-color pumped
around N2P ∼µW.23
two-level quantum system. Quasienergy is calculated to
In our approach, the typical driving field intensity
determine the parameters of the incident laser to obtain
is in the order of 1012 ∼ 1013W/cm2, which is lower
radiation of arbitrary frequency from 0.1THz to 10THz.
than the typical driving field intensity (in the order of
Upon optimization of the driving fields, we are able to
1014W/cm2 ∼ 1015W/cm2) used in other methods for
obtainTHz wavewithdesiredfrequencyandappreciable
THz generation and/or HHG by two-color laser8,22,24.
intensity (as large as that of the typical harmonics).
One should also use a long driving laser pulse [much
longer than its period(2π/ω )] as that used in Ref. [22]. Acknowledgments This work was partially sup-
0
The emitted hyper-Raman line frequency can not only ported by the National Science Foundation of China un-
be tuned by the driving field intensity as seen in Fig. der Grants No.10874020, 11174042 and by the National
3, but also by the frequency of the incident laser. If Basic Research Program of China (973 Program) un-
we use the incident laser with frequency in the regime der Grants No. 2011CB922204,CAEP under Grant No.
ω ∼ 50THz−500THz, we may obtain the 0th hyper- 2011B0102024, and the Project-sponsored by SRF for
0
Raman line with frequency 1 ω ∼ 1THz − 10THz. ROCS, SEM.
50 0
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