Table Of ContentAtomically thin semiconductors as nonlinear mirrors
Sina Zeytinoglu, Charlaine Roth, Sebastian Huber, and Atac I˙mamo˘glu
Institute for Quantum Electronics, ETH Zu¨rich, CH-8093 Zurich, Switzerland.
(Dated: January 31, 2017)
We show that a transition metal dichalcogenide monolayer with a radiatively broadened exci-
ton resonance would exhibit perfect extinction of a transmitted field. This result holds for s- or
p-polarized weak resonant light fields at any incidence angle, due to the conservation of in-plane
momentum of excitons and photons in a flat defect-free two dimensional crystal. In contrast to
extinction experiments with single quantum emitters, exciton-exciton interactions lead to an en-
hancementofreflectionwithincreasingpowerforincidentfieldsthatarebluedetunedwithrespect
7
to the exciton resonance. We show that the interactions limit the maximum reflection that can be
1
achieved by depleting the incoming coherent state into an outgoing two-mode squeezed state.
0
2
n Monolayersoftransitionmetaldichalcogenides(TMD)
a
such as MoSe or WSe constitute a new class of two
J 2 2
dimensional (2D) direct band-gap semiconductors [1–4].
8
Lowest energy elementary optical excitations in TMDs
2
intheabsenceoffreeelectronsorholesareexcitonswith
] an ultra-large binding energy of ∼ 0.5 eV [5]. Remark-
l
l ably, recent experiments have demonstrated predomi-
a
h nantly spontaneous-emission limited exciton transition
- linewidths in clean MoSe flakes embedded in hexagonal
2
s
boron nitride (hBN) layers [6]. Since radiative broad-
e
m ening dominates over disorder induced inhomogeneous
broadening, TMD monolayers can be considered as ideal
.
t
a two-dimensional (2D) optical materials.
m InthisLetterweshowthataTMDmonolayeractsasa
- perfect atomically-thin mirror for radiation that is reso-
d
nantwithradiativelybroadenedexcitonicresonances. In
n
the limit of weak resonant incident laser field, destruc-
o
c tive interference between the transmitted field and the
[ field generated by the TMD excitons leads to perfect ex-
tinction. In-plane momentum conservation ensures that
1
v the transmitted field vanishes for any incidence angle as FIG. 1: (a) The schematic of the experimental setup. A col-
limated coherent laser field is incident on a transition metal
8 long as the generated 2D excitons have a perfect overlap
dichalcogenide (TMD) monolayer. In addition to coherent
2 with the incident field polarization: this is the case for
2 transmitted and reflected fields, a two-mode squeezed-state
an incident s-polarized field. Remarkably, p-polarized
8 (TMSS)isgeneratedduetoexciton-excitoninteractions. Un-
0 fields also yield perfect extinction since the generated like the coherent fields, the TMSS is emitted into a large
. longitudinally-polarized exciton couples exclusively to p- solid-angle. Electromagneticfieldcanbecharacterizedascon-
1
0 polarized outgoing radiation. On the other hand, any sisting of right (left) propagating input rin (lin) and output
7 superposition of s- and p-polarized fields will have finite rout (lout) modes. (b) The reflection from the 2D TMD layer
for weak near resonant drive. Destructive interference be-
1 transmission: this is a consequence of finite energy split-
tween the directly transmitted field and the field generated
: tingofthetransverseandlongitudinalexcitonresonances
v by the TMD excitons lead to a suppression of transmission
i induced by the electron-hole exchange interaction [7]. and an enhancement of reflection. For perfect coupling effi-
X
Theexciton-excitoninteractionsensurethattherewill ciency η=1 between the input light and the excitons on the
r be a non-zero transmitted field as the intensity of the TMD layer, the system constitutes a perfect mirror. Imper-
a
fectcouplingefficiencyduetothenon-radiativelifetimeofthe
fieldisincreased: thisistheanalogofsaturationinduced
excitons reduces the reflection maxima to η2.
reductioninextinctionobservedinresonantlydrivensin-
gle quantum emitters. Unlike the latter however, strong
extinctionofthemeantransmittedfieldispossibleinthe
TMD case by tuning the incident field to blue (red) side tion of the mean excitonic field due to interactions. The
oftheresonanceforrepulsive(attractive)exciton-exciton transmitted field in this regime is a broadband squeezed
interactions. Wefindthatforlaserdetuningsandintensi- coherent state.
ties where the exciton system approaches bistability, the Figure 1 (a) depicts the experimental system we ana-
extinction of the mean field is only limited by the deple- lyze. We assume near-resonant monochromatic light in-
2
cidentonamonolayerTMD,whoseexcitonicexcitations t ↔ t > t. The input-output relation allows us to
0 f
have the ladder operators x . We model the electromag- expresstheoutputfieldsintermsoftheinputfieldsonce
k
netic environment that the excitonic field couples to in we solve the equation of motion of x in terms of the
k
terms of right and left propagating field modes whose input noise operators [8]
ladder operators are denoted as r and l , respectively. We first consider a non-interacting exciton system
k k
Weassumeadefect-freeflatTMDmonolayerandalarge where H (k) = ω (k)x†x , with ω (k) = ω +
TMD exc k k exc exc
excitation spot such that in-plane momentum k is con- k2/(2m )−ω ,whereω ,istheexcitonenergyinthe
exc p exc
served, and the total Hamiltonian can be written as lab frame. We then obtain the following solution to Eq.
(4)
(cid:88) (cid:88)
H = H(k)= [HTMD(k)+Hbath(k)+Hint(k)], x (ω)=−i(cid:0)e−iθ√γ(cid:1)G (ω,k)(cid:2)rin(ω)+lin(ω)(cid:3), (7)
k k 0 k k
(1)
where we introduced the free propagator
G (ω,k) ≡ 1 . Using Eq. (6), the
where 0 γ−i[(ω−ωexc(k)]
solution for the right outgoing field can be written as
(cid:90) ∞ (cid:104) (cid:105)
Hbath(k)= dωω rk†(ω)rk(ω)+lk(ω)lk(ω) , (2) rkout(ω)=[1−γG0(ω,k)]rkin(ω)−γG0(ω,k)lkin(ω). (8)
−∞
(cid:90) ∞ The quantity γG (ω,k) can also be understood as the
H (k)= dωκeiθ/2x†[r (ω)+l (ω)+h.c.], (3) 0
int k k k reflection coefficient of the TMD system. Remarkably,
−∞
at resonance [ω =ω (k)], the propagator is purely real
exc
and HTMD is to be specified. To simplify the expres- and equal to the density of states G0(ωk,k) = γ−1.
sions, we have set (cid:126) = 1 and expressed frequencies in Hence,theexpectationvalueofrout(ω)overarightmov-
k
a frame rotating with the incident coherent field fre- ing coherent field is zero and the reflection is perfect, cf.
quency ωp. Parameters κ and θ/2 are the strength and Fig. 1 (b). For each k (cid:28) (ω+ωp)/c, Hint(k) describes
the phase of the coupling, respectively. The bath opera- a single harmonic oscillator mode coupled to a one-
tors are properly normalized using the appropriate den- dimensional radiation field reservoir with ρ (k,ω) (cid:39)
(cid:112) env
sity of states [e.g., rk(ω) = ρenv(k,ω)rk(kz) ]. The 1/c. Thetransmisison/reflectionproblemisthereforefor-
density of states of the environment depends on the in- mally equivalent to the corresponding problem for a sin-
plane momentum and frequency in the rotating frame of gle quantum emitter coupled to an optical or microwave
ωp as ρenv(k,ω) = 1c√(ω+ωω+p)ω2p−(ck)2. Sim(cid:112)ilarly the cou- wfiealvdegwuhiedree[a9n],hianrmthoenicliimtyitcaonf abewneeagkleicntceidd.ent coherent
plingconstantκ(ω)alsoincludesafactor ρ (k,ω). In
env Theeffectofcouplingtoadditionaldecaychannelscan
the following, we only consider in-plane wave-vectors for
be taken into account by γ →γ¯. In this case, we obtain
whichtheelectromagneticbathcanbetreatedasMarko-
at resonance
vian.
(cid:20) (cid:21)
The Heisenberg-Langevin equation of motion for x γ γ
k rout(ω )≈ 1− rin(ω )− lin(ω )
evolving under the influence of such a bath is given by k k γ¯ k k γ¯ k k
≡(1−η)rin(ω )−ηlin(ω ), (9)
i k k k k
x˙ (t)= [H ,x (t)]
k (cid:126) TMD k
where η ≡ γ/γ¯ is the coupling efficiency of the TMD
(cid:26) (cid:27)
− ie−iθ√γ(cid:2)rin(t)+lin(t)(cid:3)+ 2γx (t) , (4) excitonstotheelectromagneticmodes,withtheidentical
k k 2 k in-planemomentum,k. AsshowninFig. 1(b),thepeak
reflectivity at resonance is given by η2.
where γ ≡ 2πκ2 is the radiative decay rate of TMD ex- When the input coherent field is incident on the TMD
citons into the right (left) moving modes in the Markov layeratanangleθ (cid:54)=0thepolarizationoftheincoming
inc
approximation. In Eq. (4), the right (left) moving input light becomes important. In particular, the polarization
noise operator, rin (lin) is defined as componentperpendiculartotheTMDlayerdoesnotcou-
k k
ple to x (t). As a result, the coupling constant for the
(cid:90) ∞ k
rin(t)= e−iω(t−t0)r (ω)dω, (5) p polarized light is reduced with respect to the s polar-
k k
−∞ ized light by a factor cos2(θinc). The role of polarization
can be described, by using an additional polarization la-
witht <t. Theinput-outputrelationinfrequencyspace
0 bel (cid:15)=s,p describing polarization of the incoming light.
is [8]
Then the input output relation is written as
√
rkout(ω)=rkin(ω)−ieiθ γxk(ω), (6) rkou,(cid:15)t(ω)≡[1−γ(cid:15)G0(ω,k)]rkin,(cid:15)(ω), (10)
where ω is the frequency label of the bath modes, and where again we assumed that the left moving input
r (ω) is defined the same way as in Eq. (5) but with modes are in the vacuum state. Moreover, electron-hole
out
3
exchange interactions in a TMD monolayer ensure that
theexcitonicexcitationsthatcoupletosandppolarized
lightarespectrallydistinctandleadtoparabolic(linear)
dispersion for s (p) polarized excitons [7]. Even though
both s and p polarized incident light would exhibit per-
fect extinction on their respective resonances, the reflec-
tivity is below unity for any ω when the incoming light
p
is in a superposition of the s and p polarizations.
Theeffectsofexciton-excitoninteractionscanbetaken
into account by
(cid:34) (cid:35)
(cid:88) g
Hint = ω (k)x†x + x†x†x x
TMD exc k k 2 0 0 0 0
k
(cid:48)
g (cid:88)
+ (x†x† x x +x†x†x x +4x†x x†x ),
2 k −k 0 0 0 0 k −k 0 0 k k
k
(11)
where we kept the highest order terms that would dom-
inate the dynamics in the limit of a large coherent am-
plitude in the driven exciton mode k = 0. We assumed
repulsive contact interactions g > 0 and excluded the
k =0 mode in the sum (cid:80)(cid:48). We first obtain the density FIG. 2: (a) The reduction of reflection due to depletion of
k the k = 0 coherent state for three different values of g while
of particles at k = 0 when the system is driven by a co- keeping gβ2 = 3.5 constant. All parameters are plotted in
herent planewavethat is normally incident onthe TMD the units of γ. In the limit that g → 0 (orange line), the
√
plane [(cid:104)rin (ω = 0)(cid:105) = β γ]: solving the Heisenberg- depletion does not modify the reflection maximum, since in
k=0
Langevin equation and substituting in the mean field at this limit the excitonic density at resonance is infinite. For
g =1.75 (g =0.35) [green (blue) line], the depletion reduces
k =0, we obtain
the reflection maximum down to ≈ 0.9 (≈ 0.97). The green
(cid:104)x (cid:105)=−i(cid:0)e−iθγβ(cid:1)G¯(0,0) (12) shaded area indicates the region where the mean field treat-
0 mentisexpectedtobenotreliableforthischoiceofg. Here,
where G¯(ω,k) = 1 with ω¯ = ω (k)+(2− adepletionofthecoherentexcitondensityexceeding10%is
γ−i[ω+ω¯(k)] k exc usedasabenchmark. Theblackarrowsindicatethehystere-
δ )g|ψ |2, and |ψ |2 ≡(cid:104)x†x (cid:105) is the number of excitons sis curve. The faint blue curve is the same as Fig. 1(b) for
k,0 0 0 0 0
at mode k = ω = 0. The excitonic density obeys the η = 1. The anti-squeezing (b) and squeezing (c) spectra at
∆=gβ2 =3.5, in units of γ. For this detuning and coherent
following self-consistency equation
exciton density, the fluctuations on top of the coherent field
γ2 haveaBogoliubov-likespectrumofaninteractingbosonicsys-
|ψ0|2 =γ2|G¯0|2|β|2 = γ2+ω¯2|β|2, (13) tem. Thebandwidthofsqueezingisdeterminedbythedecay
0 rate of the excitons. While there is no upper bound for anti-
squeezing in the g → 0 limit, the largest squeezing that can
which is third order in |ψ |2 and results in bistable solu-
√ 0 beachievedislimitedto≈3dBsincetheexcitonicsystemis
tions for (ω−ω0)> 3γ [10] . coupled to two separate baths.
Inthemean-fieldlimitwherethesecondlineofEq.(11)
is neglected, the condition for perfect extinction is
ω¯ = 0. As expected, perfect extinction entails that
0
the exciton flux through the TMD layer equals the pho-
both g and the detuning is taken to zero. Figure 2 (a)
ton influx. That is,
shows the mean-field reflection spectrum (yellow) as ω
p
ω −ω (k =0) istunedacrosstheresonanceforgβ2 =3.5: theexcitonic
γ|ψ˜ |2 =γ|β|2 =γ p exc , (14)
0 g system is bistable for this choice of parameters and the
unityreflectioncanbeobtainedasω istunedacrossthe
p
where |ψ˜0|2 is the density of excitons at perfect extinc- bare excitonic resonance towards higher frequencies.
tion. We emphasize that in this mean-field limit, perfect
extinction is possible for any positive (negative) value To take into account the effect of quantum fluctua-
of g in the TMD monolayer as long as the pump is blue tions on top of the coherent excitonic field ψ , we solve
0
(red)detunedbyanamountdeterminedbyEq.(14)from theequationsofmotionforthesefluctuationsuptolinear
the bare excitonic resonance. Naturally, the expression order. In our analysis, we assume a single parabolic dis-
inEq. (14)interpolatestothenon-interactingcasewhen persion (s-polarized) excitonic dispersion for simplicity.
4
The solution for the k (cid:54)= 0 excitonic operators are Evaluating the expectation values, and defining the re-
flection coefficient r self-consistently gives us the set of
x (ω)=−i(cid:0)e−iθ√γ(cid:1)G(ω,k) equations sufficient to solve for r and γ¯
k
×(cid:8)nin(ω)+U[G¯(−ω,−k)]∗(nin )†(−ω)(cid:9), (15)
k −k |r|2−Re(r)+D =0 (20)
γ
where nin(ω) ≡ rin(ω) + lin(ω), U ≡ −igψ2, and the r− =0 (21)
k k k 0 γ¯−i[ω (0)+g|r|2β2]
dressed propagator is exc
where
[G¯−1(−ω,−k)]∗
G(ω,k)≡ G¯−1(ω,k)[G¯−1(−ω,−k)]∗−|U|2. 2D ≡ 1 (cid:90)(cid:90) ∞ d2kdω|vk(ω)|2 (22)
2π3 γβ2
−∞
Eq. (15) allows for a clear interpretation when con-
is the total quantum depletion into the right and left
trasted with the solution of the non-interacting problem
propagating output modes. The parameter γ¯ = γ¯ > γ
in Eq. (7). The first difference between the two is the D
extra noise term U[G¯(−ω,−k)]∗(nin )†, which describes is the renormalized radiative decay rate of the coherent
−k densityofexcitonsatk =0,andensuresthattheoptical
the particle number violating coupling between the exci-
theorem in Eq. (20) is satisfied.
tonsandthefluctuationsoftheelectromagneticvacuum.
The quantum depletion is significant in the vicinity
Microscopically, this is allowed due to the interactions
of the excitonic bistability threshold [10], where the real
mediatedbytheexcitoniccoherentstate. Secondly,both
part of the pole of the dressed propagator [G(k,ω)] ap-
noise terms are now propagated by the dressed propa-
proaches zero, causing v (0) defined in Eq. (18) to di-
gator G(ω,k). We note that the imaginary part of the k
verge. However, in the limit where g is taken to zero
poles of G(ω,k) gives the excitation spectrum for the
while keeping gβ2 constant, the region where the deple-
dissipative excitonic condensate, while the real part of
tion is significant shrinks down to a point at the bista-
the poles gives the dissipation rate of the corresponding
bility threshold. This effect can also be understood as
modes [see Fig. 2 (b-c)]. We note that the functions
G(k,ω) and G(k,ω)U[G¯(−ω,−k)]∗ can be thought of as a consequence of the γβ2 term in the denominator of
Eq. (22). The reduction of reflectivity for two differ-
the normal and anomalous propagators for the weakly
ent values of g with appropriately scaled pump power is
interacting excitonic condensate coupled to a Markovian
shown in Fig. 2 (a). In the figure, we also indicate the
bath.
region of the pump frequency ω where the depletion of
Using the input-output relation, we find the solution p
the excitonic condensate is a significant fraction of the
for the right moving output operator
excitonic density (green shaded region); in this param-
rout(ω)=u (ω)rin(ω)+v (ω)(rin )†(−ω) eter range, the mean field treatment is expected to be
k k k k −k unreliable.
+u¯ (ω)lin(ω)+v¯ (ω)(lin )†(−ω), (16)
k k k −k Eq. (16)alsodescribesthesqueezingoftherightmov-
ingoutputnoise. Thecorrelationsbetweenthe±kmodes
where
manifest in the fact that the fluctuations in the output
field are superpositions of input noise with ±k. Using
u (ω)=1−u¯ (ω)=1−γG(ω ) (17)
k k k the expressions above, we can calculate the variance in
vk(ω)=v¯k(ω)=γG(ωk)U[G¯(−ω,−k)]∗. (18) the quadrature fluctuations
1
The form of the solution in Eq. (16) ensures that (∆X2)(k,ω)≡ (cid:104)||(rout)†(−ω)±rout(ω)||2(cid:105),
(cid:104)(rout)†(ω)rout(ω)(cid:105) (cid:54)= 0 even for k (cid:54)= 0. Physically, this ± 2 −k k
k k
means that the input coherent population at k = 0 is
where ||o||2 ≡ o†o. We plot the squeezing and anti-
depleted out into output modes with k (cid:54)=0. As a result,
squeezing spectra at perfect extinction in Fig. 2 (b - c),
the perfect reflection condition in Eq. (14) cannot be
whereweconsiderthelimitg →0whilekeepinggβ2 con-
satisfied even at resonance [i.e., ω¯(0)=0]. Thus, the re-
stant. We observe that at resonance (i.e, ω¯(0) = 0), the
duction of reflection can be viewed as a renormalization
dispersion of the fluctuations follow the equilibrium Bo-
of the decay rate γ of the coherent density of excitons
goliubov dispersion. Because the excitons are coupled
with k = 0.
to both right and left propagating baths, the squeez-
Theconservationofinputandoutputphotonfluxesin
ing properties of the output field is similar to those of a
the absence of absorption yields
non-degenerate optical parametric oscillator (OPO) im-
(cid:90)(cid:90) plemented by a two sided cavity [11, 12].
(cid:88)
dωd2k(cid:104)(σout)†(ω)σout(ω)(cid:105)−(cid:104)(σin)†(ω)σin(ω)(cid:105)=0. Our results suggest that a TMD monolayer could con-
σ=r,l stitute an interesting source of anti-squeezed light with
(19) favourable properties. Even though small exciton Bohr
5
radiusimpliesthattheinteractionstrengthissmallcom- B. D. Belle, A. Mishchenko, Y.-J. Kim, R. V. Gor-
pared to that of GaAs excitons, the four-wave-mixing bachev,T.Georgiou,S.V.Morozov,etal.,Science340,
process leading to parametric down-conversion is triply 1311 (2013), ISSN 0036-8075, URL http://science.
sciencemag.org/content/340/6138/1311.
resonantimplyingthattheconversionefficiencycouldbe
[5] A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi,
sizeable. Here, translational invariance can be used to
Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hy-
filter out the pump field. Since the nonlinear medium is
bertsen, and T. F. Heinz, Phys. Rev. Lett. 113,
atomically thin, there are no phase- or mode-matching 076802 (2014), URL http://link.aps.org/doi/10.
conditions. Most importantly, the squeezing bandwidth, 1103/PhysRevLett.113.076802.
determinedbyγ, islargerthan250GHz, whichisorders [6] P. Back, M. Sidler, O. Cotlet, A. Srivastava, N. Take-
of magnitude larger than the bandwidth of the squeezed mura, M. Kroner, and A. Imamoglu, arXiv preprint
arXiv:1701.01964 (2017).
vacuum generated in cavity based optical parametric os-
[7] H. Yu, G.-B. Liu, P. Gong, X. Xu, and W. Yao, Nature
cillators.
Communications 5, 3876 EP (2014), URL http://dx.
The Authors acknowledge many useful discussions
doi.org/10.1038/ncomms4876.
with I. Carusotto, C. Ciuti, M. Wouters and M. Van [8] D.WallsandG.Milburn,QuantumOptics,SpringerLink:
Regemortel on squeezed light generation in driven non- Springer e-Books (Springer Berlin Heidelberg, 2008),
linear exciton and polariton systems. This work is sup- ISBN 9783540285731, URL https://books.google.ch/
ported by an ERC Advanced investigator grant (POLT- books?id=LiWsc3Nlf0kC.
[9] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S.
DES) and the Swiss National Science Foundation.
Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.
Schoelkopf, Nature 431, 162 (2004), URL http://dx.
doi.org/10.1038/nature02851.
[10] C. Ciuti and I. Carusotto, physica status solidi (b) 242,
2224(2005),ISSN1521-3951,URLhttp://dx.doi.org/
[1] B.Radisavljevic,A.Radenovic,J.Brivio,V.Giacometti, 10.1002/pssb.200560961.
and A. Kis, Nat Nano 6, 147 (2011), URL http://dx. [11] C. Gardiner and P. Zoller, Quantum Noise: A
doi.org/10.1038/nnano.2010.279.
Handbook of Markovian and Non-Markovian Quantum
[2] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim,
Stochastic Methods with Applications to Quantum Op-
C.-Y. Chim, G. Galli, and F. Wang, Nano
tics, Springer Series in Synergetics (Springer, 2004),
Letters 10, 1271 (2010), pMID: 20229981,
ISBN 9783540223016, URL https://books.google.co.
http://dx.doi.org/10.1021/nl903868w, URL http:
uk/books?id=a_xsT8oGhdgC.
//dx.doi.org/10.1021/nl903868w.
[12] Reynaud, S. and Giacobino, E., J. Phys. Colloques
[3] B. W. H. Baugher, H. O. H. Churchill, Y. Yang, and
49, C2 (1988), URL http://dx.doi.org/10.1051/
P. Jarillo-Herrero, Nat Nano 9, 262 (2014), URL http:
jphyscol:19882112.
//dx.doi.org/10.1038/nnano.2014.25.
[4] L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil,
