Matter Coupling to Strong Electromagnetic Fields in Two-Level Quantum Systems with Broken Inversion Symmetry

0.16 MB·English

by  O. V. Kibis

Matter Coupling to Strong Electromagnetic Fields in Two-Level Quantum Systems
with Broken Inversion Symmetry
O.V. Kibis1,∗ G.Ya. Slepyan2, S.A. Maksimenko2, and A. Hoffmann3
1Department of Applied and Theoretical Physics, Novosibirsk State Technical University,
Karl Marx Avenue 20, 630092 Novosibirsk, Russia
2Institute for Nuclear Problems, Belarus State University, Bobruyskaya St. 11, 220050 Minsk, Belarus
3 Institut fu¨r Festk¨orperphysik, Technische Universit¨at Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany
We demonstrate theoretically the parametric oscillator behavior of a two-level quantum system
with broken inversion symmetry exposed to a strong electromagnetic field. A multitude of reso-
nance frequencies and additional harmonics in the scattered light spectrum as well as altered Rabi
frequency are predicted to be inherent to such systems. In particular, dipole radiation at the Rabi
9
frequencyappearstobepossible. SincetheRabifrequencyiscontrolledbythestrengthofcoupling
0
electromagneticfield,theeffectcanserveforthefrequency-tunedparametricamplificationandgen-
0
eration of electromagnetic waves. Manifestation of the effect is discussed for III-nitride quantum
2
dots with strong build-in electric field breaking the inversion symmetry. Terahertz emission from
n arrays of such quantum dots is shown to beexperimentally observable.
a
J
PACSnumbers: 42.50.Hz,47.20.Ky,78.67.Hc,85.60.Jb

2
1
The resonant interaction of quantum systems with and |bi as excited and ground states, respectively. Let
]
h strong electromagnetic field [1, 2] is permanently in the the system interacts with a classical linearly polarized
p focus of interest, both due to high methodological value monochromatic electromagnetic field. The system is de-
- of arising problems and their direct relation to current scribed by the Hamiltonian Hˆ = Hˆ +Hˆ . The free-
t 0 int
n applied projects,such as new generationof high-efficient particle Hamiltonian, written in the basis of these two
a lasers [3], laser cooling of atoms [2], development of ba- states, is Hˆ = ~ω σˆ /2, where σˆ is the Pauli matrix
u 0 0 z z
sis for quantum information processing [4], etc. One of and ω is the resonant frequency of the two-level sys-
q 0
[ bright manifestations of the strong field-matter coupling tem. The interaction Hamiltonian is expressed in terms
is the Rabi effect [2]: oscillations of the level population of the amplitude E and frequency ω of the driving field
2 in a quantum system exposed to a monochromatic elec- byHˆ =−Edˆcos(ωt),wheredˆ istheelectricdipolemo-
v int
0 tromagnetic wave. The simplest physical model leading ment operator and dij =hi|dˆ|ji are its matrix elements.
9 to harmonic Rabi oscillations is a two-level symmetrical Thecriticalassumption,whichdistinguishesthe systems
0 quantum system placed in a given classical single-mode being considered from standard ones, is the violation of
4
electromagnetic field [2]. Incorporation into this sim- the inversionsymmetry. Since in that case the states |ai
.
9 plest model additional physical factors results in many and|bidonotpossessa certainspatialparity,the diago-
0 nontrivialeffects. For example, accountingfor the quan- nalmatrixelementsofthedipolemomentoperatorprove
8 tum nature of light leads to the concept of radiation- to be nonequivalent, d 6= d , dictating thus physical
0 aa bb
: dressedatoms[1]andthe‘collapse-revivals’phenomenon effects described hereafter.
v inthepopulationdynamicsofasystemexposedtocoher- We shall seek the solution of the Schr¨odinger equa-
i
X ent light. Time-domain modulation of the field-matter tion with the Hamiltonian described above in the form
r couplingconstant[5],local-fieldeffectsinnanostructures of|ψi=Ca(t)|ai+Cb(t)|bi. Substituting this expression
a [6,7]andphonon-induceddephasing[8]providenewpos- intotheSchr¨odingerequation,wearriveattheequations
sibilities for the control of the Rabi oscillations. Many
interesting effects manifest themselves in more complex i~C˙ = ~ω0 −Ed cos(ωt) C −Ed cos(ωt)C , (1)
a aa a ab b
systems,suchastwocoupledRabioscillators[9],Rabios- h 2 i
cillators based on superconducting electrical circuits [4], i~C˙ = −~ω0 −Ed cos(ωt) C −Ed cos(ωt)C .(2)
b bb b ba a
and systems where Rabi oscillations are strongly influ- h 2 i
encedby intrabandmotionof quasi-particles[10]. In the
In order to solve these equations with respect to C
given Letter we investigate the role in the Rabi effect of a
and C , we first rewrite them for modified amplitudes
a new physical factor — violation of the inversion sym- b
c = C exp[±iω t/2−iφ (ω,t)], where φ (ω,t) =
metry. Inherent in many quantum systems, the factor is a,b a,b 0 a,b j
Ed sin(ωt)/~ω, and signs ± correspond to indexes a
ignored by the conventional physical model [2] of Rabi jj
and b, respectively. The system of equations (1)–(2) is
oscillations. Meanwhile,ourtheoreticalanalysispredicts
then reduced to the form as follows
pronouncedmanifestation of the violation in a set of ob-
servable effects in different physical systems. ~c˙ =iE∗ d cos(ωt)c eiω0t, (3)
a eff ab b
Let us consider a two-level quantum system with |ai ~c˙ =iE d cos(ωt)c e−iω0t, (4)
b eff ba a
2
where the effective electric field strength is it is suffice to consider the process in the framework of
classical electrodynamics. It allows analyzing electronic
∞
E (t)=Ee−iκsin(ωt) =E J (κ)einωt, (5) subsystem, in respect to irradiation of electromagnetic
eff n
waves, as a classical dipole with the oscillating dipole
n=X−∞ moment d(t) = hψ|dˆ|ψi. For definiteness, we identify
J (κ) is the Bessel function of the first kind, and initial time as that correspondingto system being in the
n
κ = E(d − d )/~ω is the symmetry violation pa- excited state, i.e., C (0) = 1 and C (0) = 0. Then,
bb aa a b
rameter. Let us stress that Eqs. (3)–(4) are analogous substituting wave function |ψi with coefficients (6)–(7)
to standard equations of two-level system [2] with the into expression for the dipole moment and omitting the
only difference that the driving field amplitude E is re- time-independent terms, we arrive at the expression
placed by the effective amplitude (5). That the effec-
ttievme atmopbleitcuodmeeEaeffp(at)raims teitmriec-doespcielnladteonrtwciatuhsefsultlhseestyos-f d(t)=(daa−dbb)4ΩΩ2R2eiΩt−dabΩ2ΩR
∞
itnotrrsi.nsIinc pparortpiceurtliaers,,aunmuuslutaitlufdoerosftarnesdoanradntRafrbeiquoesnciclileas- × Jm−n(κ)einωt(cid:20)∆Ω + 12(cid:18)1− ∆Ω(cid:19)e−iΩt
n=X−∞
ω = ω /n, n = 1,2,3,... appears in the system. Fur-
0
1 ∆
therweshallassumethatthedrivingfieldfrequencyω is − 1+ eiΩt +c.c. (9)
2(cid:18) Ω(cid:19) (cid:21)
in the vicinity of the frequency ω /m (m-th resonance)
0
and the conditions |Ed mJ (κ)/κ~(ω −mω)|&1 and
ab m 0 for the dipole moment in the vicinity of m-th resonance.
|Ed nJ (κ)/κ~(ω − nω)| ≪ 1 (n 6= m) are fulfilled,
ab n 0 As follows from Eq. (9), the radiation spectrum consists
allowingusto neglectinteractionofthe two-levelsystem
of a singlet at the frequency Ω, and an infinite sequence
withallharmonics(5)otherthanharmonicsm±1. Then
oftripletswiththefrequenciesnω, nω±Ω(seeFig.1). It
Eqs. (3)–(4) resume the form analogous to equations for
should be emphasized that the frequency multiplication
symmetric two-levelsystems and can easily be solved by
is the general property of systems with broken inversion
invokingthe rotating-waveapproximation[2]. Assuming
symmetryandcanbeusedforthehighharmonicgenera-
thenondiagonaldipolematrixelementstobereal-valued,
tion[12]. Amplitudesofharmonicsofthetripletsrapidly
we obtain
Ωt i∆ Ωt
C (t)= C (0) cos − sin
a a
(cid:26) (cid:20) (cid:16) 2 (cid:17) Ω (cid:16) 2 (cid:17)(cid:21)
Ω Ωt
+i RC (0)sin e−imωt/2eiφa(ω,t), (6)
b
Ω (cid:16) 2 (cid:17)(cid:27)
Ωt i∆ Ωt
C (t)= C (0) cos + sin
b b
(cid:26) (cid:20) (cid:16) 2 (cid:17) Ω (cid:16) 2 (cid:17)(cid:21)
Ω Ωt
+i RC (0)sin eimωt/2eiφb(ω,t), (7)
a
Ω (cid:16) 2 (cid:17)(cid:27)
where the parameters ∆ = ω −mω, Ω = Ω2 +∆2, FIG. 1: Schematics of the light scattering in a two-level sys-
0 R tem with brokeninversion symmetryin thevicinityof oneof
and the Rabi frequency p
thepossible resonances ω=ω0/m.
Ω =2Ed mJ (κ)/κ~ (8)
R ab m
decreasewith increasing n or m, while the singlet Ω am-
are written in the vicinity of m-th resonance. Since we plitude depends on neither n nor m. Radiation of the
consider the strong coupling regime, the driving field E dipole (9) in the vicinity of each resonance is character-
is assumed to be sufficiently strong, Ω τ ≫ 1, to ne- ized by its own Rabi frequency (8) decreasing with the
R
glect impact of the linewidth ~/τ. Note that the Rabi resonancenumber m increase. In systems with inversion
frequency in systems with broken inversion symmetry symmetry diagonal elements of the dipole moment are
(8) shows nonmonotonic dependence on the driving field identical. As a result, higher resonances (m>1), higher
strength and even can turn zero at those values of E triplets (n > 1), and the singlet vanish. In that case,
which correspondto roots of the Bessel function J (κ). coefficients(6)–(7)andexpressionforthedipolemoment
m
It is wellknownthat insymmetricaltwo-levelsystems (9) coincide with the solution of the problem presented
the Rabi effect manifests itself in the power spectrum in Ref. [2].
of the scattered light by peaks centered at the incident Further we confine consideration to systems with a
light frequency ω and at the displaced frequencies ω±Ω weak violationof the inversionsymmetry, when the con-
(Mollow triplet, [11]). In order to reveal peculiarities of dition|κ|≪1holdstrue. Insuchsystemstheasymmetry
the scattered light induced by the symmetry violation, effect is expected to be most pronounced in the vicinity
3
ofthefirstresonancem=1(ω ≈ω ). Ifthenwerestrict of different types of QDs, nitride-based confined struc-
0
ourselvestothe mostinterestingcaseofresonantdriving tures seem to be most promising for the effect observa-
field, when ∆ = 0 and Ω = Ω , we reduce (9) to the tion (for parameters of III-nitrides AlN, GaN, and InN
R
expression seeRef. [13]). Indeed, GaNandsimilarIII-groupnitride
alloys have a hexagonal (wurtzite) structure. As a con-
d −d
d(t)= aa bb cos(Ω t)−d sin(ω t)sin(Ω t), (10) sequence of giant piezoelectric effect inherent in hexag-
R ab 0 R
2 onal crystals, QDs based on structures AlN/GaN and
InN/GaN have a strong built-in strain-induced electric
where Rabi frequency (8) takes its conventional form
Ω = Ed /~. In typical quantum systems the Rabi field with strength E of several MV/cm [14, 15]. Due to
R ab
the strong electric field, conduction-band electrons and
frequency is much less then the driving field frequency
valence-band holes in the QDs get spatially separated
ω = ω . Therefore, in addition to the high-frequency
0
[16, 17], and the effective dipole moment of a III-nitride
harmonics ω ±Ω (second term in the right-hand part
0 R QDisgivenbysimplerelation|d −d |∼el,wherelis
ofEq. (10)),violationoftheinversionsymmetryleadsto aa bb
theQDheight. Then,fortypicall ofseveralnanometers,
theirradiationoflow-frequencyelectromagneticwavesat
the effective dipole moment of nitride QDs is estimated
Rabi frequency (first term in the right-hand part of Eq.
as |d −d |∼ 10Debye, what is tens thousands times
(10)) with the time-averagedradiation intensity aa bb
aslargeastheeffectivedipolemomentofhydrogenatom
|d −d |2 in the same electric field.
IR = aa12c3bb Ω4R. (11) Since the Rabi frequency depends on the driving field
strengthE, the brokeninversionsymmetry-inducedlow-
Thequantity|daa−dbb|isfurtherreferredtoaseffective frequency singlet ΩR in the dipole emission spectrum
dipole moment. Note thatthe frequencyofthe scattered can be used for the tunable generation of electromag-
radiation ΩR depends only on the driving field E and netic waves at Rabi frequency with intensity (11). This
does not depend on both the frequency of the scattering problem is especially challenging for frequency ranges
system ω and the frequency of incident light ω. where traditional methods either fail or inefficient, such
0
The simplest quantum system devoid of the inversion as terahertz domain. Since this domain lies between ra-
center is a Rydberg hydrogen atom imposed to a homo- dio and optical frequency ranges,neither optical nor mi-
geneous static electric field E. Assuming the atom to be crowavetechniques aredirectly applicable for generating
in ground state, we find that the Rabi frequency is ex- THz waves. Therefore, a search for effective THz ra-
pressed in terms of the Bohr radius a = ~2/m e2 by diation sources is one of most excited fields of modern
B e
Ω = 4(2/3)5ea E/~. Correspondingly, for the elec- applied physics [18, 19, 20]. One of the latest trends to
R B
tric field E much less than the intra-atomic electric field fill the THz gapis using nanostructures as THz emitters
e/a2,theintensityoftheradiationattheRabifrequency and detectors. The quantum cascade THz transitions in
B
is determined by Eq. (11) with the effective dipole mo- QD systems [21, 22] and different electron mechanisms
ment given by |d −d | = (1/8)(4/3)11a3E. This ex- ofTHzemissionfromcarbonnanotubes[23,24,25]have
aa bb B
pressionisapplicablenotonlytohydrogenatombutalso been proposed and are being actively studied. Thus, the
canbe usedfor estimating parametersofthe emissionat proposedmechanismofTHzemissionfitswellthecurrent
the Rabi frequency in arbitrary quantum systems with tendencies in nanophotonics.
broken inversion symmetry. For that aim, the Bohr ra- Parameters of the THz emission from III-nitride QDs
dius a should be replaced by the characteristic linear can be evaluated in the following way. Using the esti-
B
extension of the system. Therefore, the radiation inten- mate d ∼ 10Debye for the QD interband dipole mo-
cv
sity(11)riseswiththeincreaseofthesystemsize. Inthat ment [13], we obtain that the Rabi frequency turns out
connection, confined semiconductor nanostructures with to be lying in the THz range at relatively weak strength
discrete energy spectrum and linear extension multiply of the driving field, E ∼ 105V/cm. Then, substitut-
exceeding the atomic size, quantum dots (QDs) [3], can ing the previously obtained value of the effective dipole
serve as prospective systems for the effect observation. moment |d −d | ∼ 10Debye into Eq. (11), for the
aa bb
In quantum dots, driving electromagnetic field trans- typical QD area ∼ 10−12 cm2 we arrive at the radia-
fers electrons from valence band into conduction band; tive intensity per single QD ∼10−11 W/cm2 in the THz
what is why in QDs ground state |bi corresponds to range. This estimate multiples as N2 for array of N
the absence of free carriers while first excited state |ai identical QDs with the lateral extension less than the
is the state with electron in the conduction band and Rabi wavelength λ = 2πc/Ω . In such an array, all
R R
hole in the valence band. In that case, ~ω is approx- QDs emit waves in phase. The THz-range wavelength
0
imately equal to the semiconductor bandgap, and the λ ∼ 10−2 cm restricts the array lateral size. Taking
R
Rabifrequencyisdeterminedbythestandardexpression into account that the typical density of nitride QDs is
Ω = Ed /~ with the dipole matrix element d cor- ∼1011cm−2,wecanestimatethenumberofQDsinsuch
R cv cv
responding to interband transitions. Among a variety anarraybyN ∼107−108. Therefore,the THz emission
4
power from the submillimeter-sized QD array may ap- effect of the carrier-wave Rabi flopping. Such an effect
proach the micro-Watt level, which is characteristic for is observed in semiconductors when the Rabi frequency
theresonant-tunnelingdiodesbasedoncarbonnanotubes becomes comparable to the band gap frequency [30].
[26]. Certainly, state-of-the-art THz quantum cascade
lasers demonstrate essentially larger output power [27]. Toconclude,letusstressthatviolationoftheinversion
However, array of QDs with broken inversion symmetry symmetry is common property of quantum oscillators of
providessmoothfrequencytuningbythevariationofthe different physical origination and, consequently, the ef-
driving field intensity. The reducing impact of inhomo- fect predicted is expected to manifest itself in spectral
geneous broadening is beyond of the estimate and will characteristics of different optical processes, e.g. reso-
be consideredelsewhere. As to the homogeneousmecha- nant fluorescence of molecules.
nismsofthelinebroadening,likephononscattering,they
The research was partially supported by the INTAS
are accounted for phenomenologically in the linewidth
~/τ and do not impose any specific restrictions on the project 05-1000008-7801, IB BMBF (Germany) project
BLR08/001,theEUFP7TerACaNprojectFP7-230778,
observabilityoftheeffect,butjustinfluencethecriterion
RFBR (Russia) project 08-02-90004, ‘Development of
of the strong light-matter coupling regime. Thus, the
Scientific Potential of Russian Higher Education’ Pro-
upper estimate presented of the radiation output allows
gram(project2.1.2/2115),andBRFFR(Belarus)project
proposingthe brokeninversionsymmetry-inducedmech-
F08R-009. The work of O.V.K. and S.A.M. was par-
anism of the radiationfor the development of novel-type
tially carried out during the stay at the Institut fu¨r
THz emitters based on QDs avoid inversion symmetry.
Festko¨rperphysik, TU Berlin, and supported by DAAD
Obviously, parametric amplification of THz radiation in
and DFG, respectively.
QD arrays is also possible and can be applied for THz
detecting.
Once more type of asymmetrical two-level quantum
systemswherethe discussedeffectcanbe observedis su-
perconducting quantum qubits [4, 28]. Away from the ∗
Electronic address: [email protected]
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