Laser-driven parametric instability and generation of entangled photon-plasmon states in graphene and topological insulators

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by  Mikhail Tokman

Laser-driven parametric instability and generation of entangled photon-plasmon states
in graphene and topological insulators
Mikhail Tokman,1 Yongrui Wang,2 Ivan Oladyshkin,1 A. Ryan Kutayiah,2 and Alexey Belyanin2
1Institute of Applied Physics, Russian Academy of Sciences
2Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843 USA
(Dated: April 11, 2016)
We show that a strong infrared laser beam obliquely incident on graphene can experience a
parametric instability with respect to decay into lower-frequency (idler) photons and THz surface
plasmons. The instability is due to a strong in-plane second-order nonlinear response of graphene
which originates from its spatial dispersion. The parametric decay leads to efficient generation of
THzplasmonsandgivesrisetoquantumentanglementofidlerphotonsandsurfaceplasmonstates.
A similar process can be supported by surface states of topological insulators such as Bi Se .
2 3
6
1
0 I. INTRODUCTION to spatial dispersion in a hot plasma. Further enhance-
2 ment of χ(2) occurs at resonance between the pump fre-
quencyandtwicetheFermienergy: ω =2(cid:15) /(cid:126). Finally,
r Nonlinear parametric decay of a pump laser photon p F
p theefficiencyofparametricdown-conversionisenhanced
intotwolower-frequencyphotons(usuallycalled”signal”
A when one of the generated fields is not a photon but a
and ”idler”) in a nonlinear crystal possessing a second-
surface plasmon mode supported by a massless 2D elec-
8 order nonlinearity is the most popular method of gener-
tron layer. A non-zero value of the nonlocal in-plane
atingentangledphotonstates[1]. Athigherpumpinten-
] sities the parametric process can experience gain which χ(2) and plasmon enhancement of the nonlinear signal
h were pointed out before for second-harmonic generation
leads to the instability and exponential amplification of
p
[5,6](whichonlyincludedintrabandtransitionsinafree-
- coupled signal and idler fields. Stimulated parametric
t decay enables optical parametric amplifiers and oscilla- carriermodel)andfordifference-frequencygeneration[7].
n
Here we develop the first theory of the parametric decay
a tors as popular tunable sources of long-wavelength ra-
u diation from near- to far-infrared [21]. They typically ingraphene, whichincludesfullyquantumdescriptionof
q employ bulk transparent crystals under phase-matching the nonlinear response and quantization of all fields.
[
conditions for frequencies and wave vectors of the fields

3 participating in a three-wave mixing interaction: detector
v  (THz) p~ ~i
6 ω =ω −ω ; k =k −k , (1) ~ ~
2 s p i s p i ki s
9
where the subscripts s, p, and i represent signal, pump 
7 i
andidler,respectively. Inviewoftheserequirements,the
0
1. veryideaofparametricamplificationsupportedbyjusta p kp q (1/cm)
monolayerofmaterialseemsunrealistic. Surprisingly,we
0  q graphene
findthatstimulatedparametricdecayoflaserphotonsis  s
6
feasible in 2D systems of massless Dirac electrons.
1
: Any surface has anisotropy between in-plane and out-  substrate
v
of-planeexcitations,andgrapheneisnoexception. How-
i
X ever,thesecond-ordersusceptibilityχ(2) associatedwith
ijk
r this surface anisotropy is very small in graphene [3] and FIG.1. Schematicoftheparametricdecayofthepumppho-
a
we don’t even consider it below. A much stronger non- tonintoanidlerphotonandasurfaceplasmon,whichsatisfies
linear response is expected when all fields and electron conservationofenergyandin-planecomponentofmomentum.
excitations lie in-plane. This is obvious already in the Inset shows matching of the signal frequency tothe real part
of surface plasmon frequency obtained by solving Eq. (2).
classical free-carrier limit because of an extreme band
nonparabolicity [4]. However, graphene is a centrosym-
metric system for low-energy in-plane excitations, which The schematic of the nonlinear process is shown in
shouldprohibitanysecond-orderresponse. Nevertheless, Fig. 1. An obliquely incident pump photon decays into
a non-zero χ(2) appears beyond the electric dipole ap- an idler photon and a ”signal” plasmon of a much lower
proximation when one includes the dependence of χ(2) frequency ω = ω −ω (cid:28) ω but a comparable wave
s p i p,i
on the in-plane photon wave vectors, i.e. the spatial dis- vector q ∼ q . The second of phase matching condi-
s p
persion. In this case the isotropy of graphene is effec- tions in Eqs. (1) is replaced by its in-plane projection
tively broken by the wave vector direction. The spatial q = q −q . In addition, the signal frequency should
s p i
dispersion effects turn out to be quite large because of match the real part of surface plasmon dispersion ω(q)
a large magnitude of the electron velocity v , similarly shown in the inset to Fig. 1: ω =ω(q ). Note that both
F s s
2
positiveandnegativeprojectionsoftheidlerwavevector we choose the normalization constant E (in quasi-
s0
q arepossible,wherethepositivedirectionisassumedto electrostatic approximation) as:
i
the right. In particular, negative values of q give access
i
to larger plasmon wave vectors qs = qp−qi = |qp|+|qi| |Es0|2 =(cid:126)(Re(∂χs/∂ω))−1. (6)
and frequencies.
The effect of dissipation of a plasmon field (within
ω (cid:29) γ ) and its nonlinear interaction with other
s s
fields can be taken into account within the Heisenberg-
II. QUANTIZED SURFACE PLASMON FIELD
IN GRAPHENE Langevin approach [9]. For quasi-monochromatic wave
fields,itisconvenienttoconsiderawavepacketofsurface
plasmon modes with frequencies and wave vectors con-
Consider a geometry of Fig. 1, i.e. a 2D layer of mass-
centrated in a narrow spectral range ∆ω (cid:28)ω , ∆q (cid:28)q
less Dirac electrons in z = 0 plane between two me- s s
dia with dielectric constants (cid:15) and (cid:15) . The plasmon near a central component ∝ eiqsr(cid:107)−iωst [10–13]. Within
1 2
thisapproachweintroducetheannihilationandcreation
frequency ω and in-plane wave vector q are related
s s operators aˆ (r ,t) and aˆ†(r ,t) that are slowly varying
through the dispersion relation for a TM-polarized sur- s (cid:107) s (cid:107)
in time and space relative to ω and q . Their com-
face mode [7]: s s
mutator is equal to the number of quantized modes per
4πχ + (cid:15)1 + (cid:15)2 =0, (2) unit area Lx ×Ly = 1 within the spectral interval ∆ω:
s p p ∆ω
1 2 [aˆ ,aˆ†] = , where v is the group velocity of a
s s 2πL v s
(cid:113) y s
where p1,2 = qs2−(cid:15)1,2ωc2s2. At the THz frequencies surface plasmon which determines its spectral density of
smaller than twice the Fermi energy 2(cid:15) /(cid:126) one needs states and Ly is the aperture size of the beam.
F
Equationsforaslowlyvaryingfieldamplitudeofasur-
only to take into account the intraband contributions to
face plasmon wave packet can be obtained in the same
the linear 2D susceptibility χ (ω ,q ), which in the limit
s s s
wayasforthepropagatingopticalfields;seee.g. [11–13]:
of strong degeneracy is given by [7]
∂aˆ ∂aˆ i
χs(ωs,qs)= 2πe(cid:126)22EωF (ω(vs+qi)γ2s) ×[1− ∂ts +vs ∂xs +γsaˆs = (cid:126)Pˆs(2)Es∗0+Fˆs, (7)
s F s
ωs+ωsiγ++iγvsFqs(cid:114)1+ ωs+2iγvFs−qsvFqs(cid:21) (3) Lwahnegreevγisn=no(cid:126)i−se1,(Iamn[dχsP]ˆ)s(|E2)s20=|,Fxˆs0(Pˆt)s(2i)setiqhsex−opiωesrta+toHro.cf.thies
the second-order nonlinear component of the operator of
whereEF istheFermienergyandγs isthedecayrateof the polarization.
surface plasmons. The Langevin noise source ensures a correct expres-
The standard quantization procedure in the limit of sionforthecommutatoroftheplasmonfieldinthepres-
ωs (cid:29)γ leads to ence of its interaction with a dissipative reservoir. It is
convenient to define the properties of the noise source in
(cid:88)
Eˆ = Es(z)aˆseiqsr(cid:107)−iωst+H.c. (4) termsofitsspectralcomponentsFˆs =(cid:82) Fˆsωe−iωtdω and
qs Fˆ† = Fˆ . Assuming a dissipative reservoir in ther-
sω s;−ω
mal equilibrium and adjusting for the 2D geometry, we
wherer =(x,y)andaˆ ,aˆ† areannihilationandcreation
(cid:107) s s can write [9]
operators of surface plasmon modes. The z-distribution
of the field Es(z) is [7]: (cid:68)Fˆ† (x(cid:48))Fˆ (x)(cid:69)= γsNT(ωs)δ(ω−ω(cid:48))δ(x−x(cid:48)); (8)
(cid:18) (cid:19) ω(cid:48) ω πL
iq y
Es(z)= x0±z0p1,s2 Es0e∓p1,2z, (5) (cid:68)Fˆ (x)Fˆ† (x(cid:48))(cid:69)= γs(NT (ωs)+1)δ(ω−ω(cid:48))δ(x−x(cid:48)),
ω ω(cid:48) πL
y
wheretheupperandlowersignscorrespondtoz >0,z <
0,respectively. Similarlytothecaseofpropagatingfields wherethe(cid:104)...(cid:105)meansaveragingoverbothaninitialquan-
[8, 9], the energy of the plasmon field inside a volume V tum state in the Heisenberg picture and the statistics of
can be written as the dissipative reservoir, NT(ωs) = (cid:0)e(cid:126)ωs/(kBT)−1(cid:1)−1.
Hˆ = 1 (cid:88)(aˆ†aˆ +aˆ aˆ†)(cid:90) (cid:18)E∗∂(ω(cid:15)˜)E +B B∗(cid:19)d3r, Ionf Ethqes.ab(7s)e,n(c8e)oinftthheenloimnliitneγarx/pvola→riza∞tiocno,rtrhesepsoonludtsiotno
8π s s s s s ∂ω s s s s s
s V thermal equilibrium:
where (cid:15)˜is the dielectric permittivity tensor and the pro- (cid:10)aˆ†aˆ (cid:11)→(cid:10)aˆ†aˆ (cid:11) = NT(ωs)∆ω. (9)
jection of the quantization volume onto the 2D layer s s s s T 2πL v
y s
is equal to 1. After substituting Eqs. (2) and (5), the
last equation is reduced to a standard Hamiltonian for Eq. (9) corresponds to a general property of thermal
the plasmon field, Hˆ = (cid:80) (cid:126)ω (q )(cid:0)aˆ†aˆ +1/2(cid:1), if emission: its power received by a matched antenna
s qs s s s s
3
≈ L v (cid:126)ω (cid:10)aˆ†aˆ (cid:11) does not depend on the size and difference frequency generation of surface plasmons in
y s s s s T
shape of the aperture [15]. graphene by a bichromatic quantum field. For classical
fields this process has been predicted in [7] and observed
in [16]. The operator Gˆ describes the creation of plas-
mons by a parametric decay of the pump photons.
III. PARAMETRIC INSTABILITY IN Theoperator-valuedEq.(11)hasastationarysolution
GRAPHENE given by
(cid:34) (cid:35)
Consider parametric second-order interaction of Gˆ−γ
aˆ = exp ( s)x (12)
surface plasmons with an s-polarized bichromatic s v
s
pump+idler field incident from the z >0 half-space:
 (cid:34) (cid:35)−1 
Eˆ = (cid:88) Eˆj; Eˆj =y0Ej0cˆje−ikjz+iqjx−iωjt+H.c., ×aˆs(0)+(cid:90) xexp (Gˆ−v γs)x(cid:48) (Jˆ+Fˆs)dvx(cid:48),
j=p,i 0 s s
wherethenormalizationfields|Ej0|2 =2π(cid:126)ωj/n21 arede- Here we will only deal with a coherent classical pump
fined for a unit quantization volume, cˆj are Heisenberg field at frequency ωp. The field at the idler frequency
operators of slowly varying amplitudes corresponding to ω is present only as a quantum and/or thermal noise.
i
afinitespectralwidth∆ω [11–14]. Thenonlinear2Dpo- In this case, and for (cid:126)ω (cid:28) k T, the term Jˆ can be
s B
larization at frequencies ω generated in the graphene
p,i,s neglected as compared to the Langevin noise term, and
plane z =0 is given by the operator Gˆ can be replaced by a c-number:
PPˆˆis((22)) ==xy00χχ(y(xisyy,,2xy2))EEˆis†∗0Eˆaˆp†s+EˆpHe−.ciq.,sx+iωst+H.c., G≈|Γ|2nc1χx(sy,y2)Tχi(yciy,o2xs)∗θ1(cid:10)icˆ†pcˆp(cid:11). (13)
Pˆ(2) =y χ(p,2)E aˆ Eˆeiqsx−iωst+H.c., (10)
p 0 yyx s0 s i Taking the thermal noise as a boundary condition and
taking into account Eqs. (8) and (12) one can get
where Eˆ are the ∝exp(−iωt) parts of the electric field
p,i
ogpraeprahteonrespaltanteh.eTphuemyparaenrdelaidtleedrtforetqhueefineclidesopωepr,iationrsthine aˆ†saˆs =exp(cid:20)2Re[Gv]−γsx(cid:21)(aˆ†saˆs)T (14)
theupperandlowerhalf-spacesthroughstandardbound- s
(cid:20) (cid:18) (cid:20) (cid:21)(cid:19)(cid:21)
γ Re[G]−γ
ary conditions. The second-order susceptibilities at cor- × 1+ s 1−exp −2 sx ,
responding frequencies are χ(s,2) = χ(2) (ω = ω −ω ), Re[G]−γs vs
xyy xyy s p i
χ(i,2) =χ(2) (ω =ω −ω ), χ(p,2) =χ(2) (ω =ω +ω ).
yyx yyx i p s yyx yyx p i s where the operator (aˆ†aˆ ) corresponds to the thermal
Index α in χ(2) (ω =ω(cid:48)∓ω(cid:48)(cid:48)) corresponds to the polar- fieldandhasanaveragesvsalTuegivenbyEq.(9). Notethat
αβγ
ization of the field at the mixing frequency ω, and the there is a 1/v dependence in the gain factor in Eq. (14)
s
index β corresponds to the polarization of the field at a which describes the enhancement in the gain for slowly
larger of the two frequencies ω(cid:48),ω(cid:48)(cid:48). moving plasmons as compared to photons.
Using the nonlinear polarizations and boundary con- From Eq. (14) one can obtain an important result,
ditions for the fields, Eq. (7) becomes namely the criterion for parametric instability:
∂∂aˆts +vs∂∂aˆxs +(γs−Gˆ)·aˆs =Jˆ+Fˆs, (11) Re(χ(xsy,y2)χ(yiy,2x)∗)>0, (15)
(cid:104) (cid:105)
where Re[G]≈|Γ|2Re χx(sy,y2)χ(yiy,2x)∗ Ip n21 >γ , (16)
√ c2(cid:126)ω T cosθ s
2π ω ω p i 1i
Jˆ=Γχ(s,2)cˆ†cˆ , Γ=i i pT T E∗ ,
xyy i p n2 i p s0 where I is the incident pump intensity.
1 p
Gˆ =|Γ|2n1 (cid:32)χ(xsy,y2)χ(yiy,2x)∗cˆ†pcˆp − χ(xsy,y2)χ(ypy,x2)cˆ†icˆi(cid:33). neTedo tcoalcsuulbastteittuhteemthaegnciotmudpeonoefntthseopfatrhame esterciocngda-ionrdweer
c T cosθ T cosθ
i 1i p 1p susceptibility tensor. Their derivation is straightforward
but cumbersome, so we keep it in the Supplemental Ma-
Here T = 2n cosθ /(n cosθ + n cosθ ) are
p,i 1 1p,i 1 1p,i 2 2p,i terial;seeEq.(S10). Theirsalientfeatureisthepresence
Fresnel transmission coefficients for s-polarized pump
of resonances when one of the three frequencies involved
and idler fields with incidence angles θ1p,i and refraction in three-wave mixing is close to 2(cid:15) = 2(cid:126)v k . This
F F F
anglesθ . Eq.(11)wasderivedneglectingthetermsof
2p,i is a weaker resonance than the one that would exist in
the order α|χ(2)|2 and |χ(2)|3 where α=e2/(cid:126)c. coupled quantum wells [17] where χ(2) would scale as a
The terms Jˆ and Gˆ in Eq. (11) include all possi- product of two Lorentzians. Still, it enhances the value
ble three-wave mixing processes. The term Jˆ describes of χ(2) by a factor of ω/γ. A similar resonance exists
4
in the third-order nonlinear response of graphene [18]. field θ . Negative angles correspond to negative projec-
1i
Closetoresonanceonehastoincludetheimaginarypart tions of q . For the plot we took n = 1 and n = 2,
i 1 2
of the frequency which describes the decay rate of the γ = 1012 s−1, the pump beam at a 10-µm wavelength
p,i
optical or plasmon polarization. We will take the same and incidence angle of π/4. The pumping intensity was
values for the imaginary part γ for both pump and idler assumedtobeI =1GW/cm2,whichcorrespondstoin-
p
frequencies. Furthermoreweassumeω (cid:29)ω (cid:29)γ and tensities in the experiment [16]. The gain is only weakly
p,i s s
consider strongly degenerate graphene with a frequency dependent on the idler emission angle except for a nar-
of the pump field close to 2v k . In addition to res- row range around θ =π/4 where q (cid:39)q and therefore
F F 1i p i
onant enhancement of the nonlinearity, this eliminates q ,ω → 0. In this range the gain becomes negative;
s s
interband absorption losses for the plasmons and re- however, the approximation ω (cid:29)γ becomes invalid, so
s s
duces electron scattering. Under these conditions, when this case requires a separate investigation.
|ω −2v k |<γ, we obtain In Fig. 3 we show the pumping intensity I needed to
p F F p
reach the parametric instability threshold, Re[G] = γ ,
s
χ(s,2) =χ(i,2)∗ ≈ 3e3vF2 qp . (17) as a function of plasmon decay rate γs, for the same nu-
xyy yyx 16π(cid:126)2ω ω2γ merical parameters as in Fig. 2 and for the idler emitted
i s
atθ =20degrees. Inthiscasethephasematchingcon-
1i
dition is satisfied when the plasmon frequency ω /2π is
s
equal to 1 THz (see Fig. 2).
3 12 Themagnitudeofthegaincanbefurtherincreasedby
non-Bernal stacking of multiple graphene layers, which
)
1 ) will reduce the threshold intensity.
− 1
1s2 8 −s Low-energysurfacestatesofa3Dtopologicalinsulator
10 12 Bi2Se3 are massless 2D Dirac fermions described by the
1 0
( 1 effective Hamiltonian H = vF((cid:126)σ × p(cid:126))z [19], where vF
(
n is two times smaller than in graphene. The states have
gai1 4 ωs different chirality as compared to those in graphene but
thesamematrixelementsoftheinteractionHamiltonian
andthesamestructureoftheopticalresponse. Proceed-
0 0
ing in the same way, one can show that the parametric
−50 0 50 100
θ (degree) gain for a thin Bi2Se3 film (i.e. two surfaces) will have a
1i magnitude lower by a factor of ∼ 26 due to a two times
lower v and two times lower degeneracy.
FIG. 2. The gain Re[G] (solid blue line) and the plasmon F
frequencycorrespondingtophasematchingconditions(green
dashed line) as a function of the angle θ between the direc-
1i
tion of the idler wave vector in medium 1 and the normal.
IV. GENERATED IDLER FIELD FLUX
4 The outgoing flux of idler photons generated as a re-
10
sultofparametricdecayofthepumpcarriesinformation
on the intensity of generated surface plasmon field and
2) the surface nonlinearity. To calculate the average flux of
m 3
10 the idler photons on the detector we calculate first the
c
/
W operatoroftheidlerfieldgeneratedbythenonlinearcur-
M rent. UsingEqs.(10)andstandardboundaryconditions,
(102 we arrive at
h
p,t
I n Γχ(i,2)
δEˆ =y E cˆ(2)eikiz+iqix−iωit+H.c., cˆ(2) = 1 yyx aˆ†cˆ .
101 i 0 i0 i i ccosθ1i s p
10 11 12
10 10 10
γs(s−1) Oernaeteadlsnounmebeedrsotfopclaalscmuloanteqtuhaenatave(cid:10)raˆa†gaˆe v(cid:11)al,uewhoficthheisgceonl--
s s D
lectedbythedetectorfromlengthL . UsingEq.(14)we
x
FIG. 3. The pumping intensity I needed to reach the para-
p obtain
metric instability threshold, Re[G] = γ in Eq. (16), as a
s
function of the plasmon decay rate γs. (cid:10)aˆ†aˆ (cid:11) = 1 (cid:90) x(cid:10)aˆ†aˆ (cid:11)dx≈(cid:10)aˆ†aˆ (cid:11) eΞ−1; (18)
s s D L s s s s T Ξ
Figure2showsthegain(left-handsideofEq.(16))and x 0
Re[G]−γ
the plasmon frequency corresponding to phase matching Ξ=2 sL .
conditionsasafunctionoftheemissionangleoftheidler vs x
5
The resulting average flux of the idler photons on the parametric decay of the pump, and demonstrated their
detector of transverse area A is given by entanglement.
D
(cid:68) (cid:69) n2|Γ|2|χ(i,2)|2I L ∆ω (cid:18)eΞ−1 (cid:19)
Πˆ(2) = 1 yyx p x N (ω )+1 ,
i 2πc2v (cid:126)ω cosθ Ξ T s
s p 1i
VI. SUPPLEMENTAL MATERIAL
cA
where Πˆ(2) = Dcˆ†cˆ.
i n i i A. Second-order nonlinear susceptibility in
1
Close to the parametric instability threshold, when graphene
eΞ−1
∼1, the idler photon flux is
Ξ In this Supplemental Material we provide the general
expressions for the components of the second-order sus-
(cid:68) (cid:69) ∆ωγ L
Πˆ(2) ∼ s xN (ω ), (19) ceptibility tensor that are relevant for the parametric
i 2πv T s
s three-wave mixing in graphene.
The Hamiltonian of graphene near the Dirac point K
i.e. itisoftheorderofthethermalfluxatamuchlower
is
surface plasmon frequency collected from the length
equal to the plasmon decay length, Lx ∼vs/γs. H =v σ·pˆ=v (cid:18) 0 pˆx−ipˆy(cid:19), (21)
F F pˆ +ipˆ 0
x y
where pˆis the momentum operator relative to K and σ
V. PLASMON-PHOTON ENTANGLEMENT is a 2D vector of Pauli matrices. The eigenenergies are
(cid:15) (k)=±(cid:126)v k, and eigenstates are
± F
The total idler field propagating away from the (cid:18) (cid:19)
1 s
graphene layer to the detector consists of the reflected (cid:104)r |s,k(cid:105)= √ exp(ik·r ) , (22)
noise field and the generated parametric field calculated (cid:107) 2A (cid:107) eiφ(k)
above:
where s = 1 for conduction band, s = −1 for valence
band, A is the area of graphene, and φ(k) is the angle of
n Γχ(i,2)
cˆ ≈R cˆ + 1 yyx aˆ†cˆ , (20) the wave vector k with the x-axis.
r i i ccosθ s p
1i The interaction Hamiltonian between graphene and
the optical field which has an in-plane component of the
n cosθ −n cosθ
where R = 1 1i 2 2i is the Fresnel reflec- electric field can be written as
i n cosθ +n cosθ
1 1i 2 2i e e
tion coefficient for the S-polarized field. Here we ne- Hˆop =v σ·A= vˆ·A, (23)
glected thermal noise at high frequencies, N (ω ), and int Fc c
T p,i
absorption in monolayer graphene ∼ πα. Calculating
wherevˆ=v σisthevelocityoperator,andAisthevec-
F
quantum-mechanicalaveragesofthequantitiesquadratic
tor potential of the optical field, which is related to the
with respect to the reflected field, it is easy to see that
electric field by E = (−1/c)∂A/∂t. Using this Hamil-
Eq. (20) corresponds to an entangled plasmon-photon
tonian, the evolution equation for the density matrix is
state. In particular, for a classical pump field and an
given by
initial state in the form Ψ = |0 (cid:105)|0 (cid:105), one can show
0 i s
[13] that parametric decay leads to the state of the kind ∂ e
i(cid:126) ρ =((cid:15) −(cid:15) )ρ + (vˆ·A) (ρ −ρ )
Ψ=α|0i(cid:105)|0s(cid:105)+β|1i(cid:105)|1s(cid:105). ∂t mn m n mn c mn nn mm
Ingeneral,thecalculationsofquantum-mechanicalav- e (cid:88)
+ ((vˆ·A) ρ −ρ (v·A) ), (24)
erages of any physical quantities are much easier to per- c ml ln ml ln
formintheHeisenbergpictureusingEq.(20)forHeisen- l(cid:54)=m,n
berg operators, without converting to the Schr¨odinger
wherebothlinearandnonlineareffectsareincluded. We
picture. In particular, it is obvious from Eq. (20) that
calculate the field-induced current in second order with
for a given spectrum of the pump field any physical ob-
respectto the opticalfield, as a quantum-mechanical av-
servable for a surface plasmon field can be related to a erageofthecurrentoperatorjˆ=−ev σwiththedensity
corresponding observable for the idler field at frequency F
matrix.
ω . For example, if the pump field spectrum is much
i We will seek the response at the sum frequency ω +
narrower than the spectrum of the plasmon fluctuations 1
ω to the bichromatic optical field with in-plane electric
then the spectrum of surface plasmons is related to the 2
fields at frequencies ω directed along unit vectors η
spectrum of idler photons. 1,2 1,2
:
In conclusion, we showed the feasibility of stimulated
parametric decay of photons of a strong laser pump 1 1
obliquely incident on graphene. We calculated the flux A= 2η1A(ω1)ei(q1·r(cid:107)−ω1t)+ 2η2A(ω2)ei(q2·r(cid:107)−ω2t)+c.c.
of surface plasmons and idler photons generated by (25)
6
The result will be applicable to the difference frequency in powers of q ,q . The integral over the term of zeroth-
1 2
process by choosing either positive or negative frequen- order in q vanishes, as expected from symmetry. We
cies, with the corresponding change in q for a given ω. will keep the terms linear in q. Also we have to evalu-
Thesecond-orderdensitymatrixelementsatthesumfre- ate separately the intraband contribution s = s = s
l m n
quency ω +ω are evaluated to be andalltypesofmixedinterband-intrabandcontributions:
1 2
s = s = −s , s = s = −s , and s = s = −s .
m n l m l n n l m
ρ(2)(ω +ω )= 1(cid:16)e(cid:17)2 A(ω1)A(ω2) Here we give only the component of the second-order
mn 1 2 2 c (cid:126)(ω +ω )−((cid:15) −(cid:15) ) nonlinear conductivity tensor which gives the main con-
1 2 m n
× (cid:88) (cid:0)(vˆ·η1)eiq1·r(cid:107)(cid:1)ml(cid:0)(vˆ·η2)eiq2·r(cid:107)(cid:1)ln tribution to the signal:
l(cid:54)=m,n
(cid:20) (cid:21) e3v2 1
× (cid:126)ω(2ρ−nn(−(cid:15)lρ−ll(cid:15))n) − (cid:126)ω(1ρl−l−((cid:15)ρmm−m)(cid:15)l) σx(2y)y(ω1+ω2;ω1,ω2)=−s((cid:15)F)2π(cid:126)F2ω12ω22(ω1+ω2)
1
+{1↔2}. (26) ×
(ω2−4v2k2)(ω2−4v2k2)((ω +ω )2−4v2k2)
1 F F 2 F F 1 2 F F
Thematrixelementsenteringtheaboveexpressionare ×(cid:2)4(v k )2ω ω (ω +ω )2(q ω2+q ω2)
F F 1 2 1 2 1 2 2 1
given by
+4(v k )4(q ω4−(6q +4q )ω ω3
F F 1 2 1 2 1 2
(cid:0)(vˆ·η)eiq·r(cid:107)(cid:1)mn = 12vF (cid:2)(ηx−iηy)smeiφn −8(q1+q2)ω12ω22−(4q1+6q2)ω13ω2+q2ω14)
+(η +iη )s e−iφm(cid:3)δ . (27) − 16(vFkF)6(q1ω2(ω2−2ω1)+q2ω1(ω1−2ω2))(cid:3).
x y n km,kn+q (30)
The average of the corresponding Fourier harmonic of
the induced current with the density matrix can be cal- Here s((cid:15) ) = ±1 depending on whether the Fermi level
F
culated as is in the conduction or valence band. The result for the
(cid:88)(cid:16) (cid:17) difference frequency can be obtained from Eq. (30) by
J(2)(ω1+ω2)=−e vˆe−i(q1+q2)·r(cid:107) nmρ(m2n)(ω1+ω2). flipping the sign of ω2 and q2.
mn
Afterconvertingthenonlinearconductivitytothenon-
(28)
linear susceptibility according to
Next, we transform from summation to integration
over k-states, introduce the corresponding occupation iσ(2)(ω +ω ;ω ,ω )
numbers f(s,k) of the momentum states in each band, χ(2)(ω +ω ;ω ,ω )= ijk 1 2 1 2 ,
apply the momentum conservation in a three-wave mix- ijk 1 2 1 2 ω1+ω2
ingprocess,andtakeintoaccountspinandvalleydegen-
eracy. The result is one can verify that in the absence of dissipation all com-
ponents of the nonlinear susceptibility tensor that we
J(2)(ω +ω )=− e3vF2 A(ω )A(ω ) calculated satisfy permutation relations originated from
1 2 16π2c2(cid:126)2 1 2 symmetry properties; see e.g. Ch. 2.9 in [20]:
(cid:88) (cid:90) 1
d2k
(ω +ω )−v (s |k+q |−s |k−q |)
sm,sn,sl 1 2 F m 1 n 2 χ(ij2k)(ω3 =ω1+ω2)=χ(j2ik)(−ω1 =−ω3+ω2)
(cid:20)
f(s ,|k−q |)−f(s ,|k|)
× n 2 l =χ(2)(−ω =−ω +ω ), (31)
ω −v (s |k|−s |k−q |) kji 2 3 1
2 F l n 2
(cid:21)
f(s ,|k|)−f(s ,|k+q |)
− l m 1 where in-plane wave vectors have to be permuted to-
ω −v (s |k+q |−s |k|)
1 F m 1 l gether with frequencies.
(cid:104) (cid:105)
× (η −iη )s eiφ(k)+(η +iη )s e−iφ(k+q1) Thesecond-orderresponsegoestozerowhentheFermi
1x 1y m 1x 1y l
energy(cid:15) goestozero,andismaximizedwhenoneofthe
(cid:104) (cid:105) F
× (η2x−iη2y)sleiφ(k−q2)+(η2x+iη2y)sne−iφ(k) three frequencies involved in three-wave mixing is close
(cid:104) (cid:105) to 2(cid:15)F/(cid:126) = 2vFkF. Close to resonance with 2(cid:15)F/(cid:126) one
× (xˆ+iyˆ)sme−iφ(k−q2)+(xˆ−iyˆ)sneiφ(k+q1) hastoincludetheimaginarypartofthefrequencywhich
comes from the omitted relaxation term −γρ in the
+{1↔2}. (29) mn
density-matrix equations. This amounts to substituting
This equation can be integrated numerically for any ω1 → ω1 +iγ1, ω2 → ω2 +iγ2, ω1 +ω2 → ω1 +ω2 +
given geometry of incident fields and electron distribu- iγ3. Note that if we flip the sign of ω2 the sign of +iγ2
tion. WeconsiderthelimitoftheFermidistributionwith remains the same. If dissipation is included, one cannot
a strongdegeneracy, direct allin-plane photonwave vec- use permutation relations Eq. (31) and has to evaluate
tors along x-axis, and expand the integrand in Eq. (29) each component of χ(2) independently.
ijk
7
B. A coupled oscillators model for the parametric the plasmon and idler fields:
gain
∂E
s +γ E =iζ E E∗, (32)
∂t s s s p i
∂E∗
i +γ E∗ =−iζ∗E∗E , (33)
∂t i i i p s
where
The instability condition Eq. (16) can be easily inter-
preted and understood within the classical model of two 1 (cid:20) (cid:18)∂χ (cid:19)(cid:21)−1
ζ = χ(s,2) Re s ,
parametrically coupled oscillators. Consider a classical s 2 xyy ∂ω
pump beam of amplitude E and ω incident on a non- π
p p ζ = ω χ(i,2)∗,
linear 2D layer in vacuum. The pump field decays into i lcosθ i yyx
i
a surface plasmon field within a unit area A = 1 and
s
an idler photon field at frequency ω within a volume of γ =c/l istheeffectivedecayrateoftheidlerfieldinthe
i i
a cylinder of length l oriented at an angle θ with re- 0D model. Equations (32) and (33) have an exponen-
i
spect to the normal to area A . In this mean-field zero- tially growing solution for both parametrically coupled
s
dimensional(0D)modelonecanderivethefollowingcou- waves [21] if Re(ζ ζ∗)|E |2 > γ γ , which coincides with
s i p s i
pled differential equations for the complex amplitudes of Eq. (16), if we use Eq. (6) and assume n =n =1.
2 1
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