Theory of Rabi interaction between an infrared-active phonon and cavity-resonant modes

0.13 MB·English

by  Hadley M. Lawler

Theory ofRabiinteraction between aninfrared-active phonon and cavity-resonant
modes
7
Hadley M. Lawler
0 ∗
Chemistry Division, Naval Research Laboratory, Washington, DC 20375, USA
0
2
Jason N. Byrd
n
Department of Physics, Universityof Washington, Seattle, Washington 98195, USA
a
J
5 Gavin K. Brennen
InstituteforQuantum OpticsandQuantumInformationoftheAustrianAcademyofSciences,

] 6020, Innsbruck, Austria
i
c
s
- Wepresent the theory of interaction between apolar vibration inasemiconductor and an electromagnetic
l
r modeofasurroundingcavity.Tuningthecavityfrequencynearthetransverseopticalphononfrequencycouples
mt thephonon-inducedpolarizationfieldwithinthedielectricandtheelectromagneticfieldinthecavity.Depending
onengineeringparameters,wepredictthatthiscavityquantumelectrodynamicinteractionmayreachthestrong
t. coupling regime. The resonances of the system are in the terahertz spectral region, and while spectroscopic
a measurementsareapossiblerouteforthedetectionofsuchasystem,weemphasizethepossibilityofmeasuring
m
theRabioscillationdirectlyinthetimedomainwithfemtosecondopticalpulses.
-
d
PACSnumbers:71.36.+c,42.50.Pq
n
o
c
I. INTRODUCTION stratebehaviorwhichisinterestingfromthestandpoint
[
ofnarrow-bandterahertztechnologyandquantumop-

2 tics, andwhichpresentsfavorableengineeringparam-
Weisbuch et al. initiated a new field of study with
v
eters. Forinstance,aswillbediscussed,acavity-GaAs
7 the demonstration of Rabi splitting between excitons
apparatus should exhibit a Rabi splitting of approxi-
8 and electromagnetic modes in nanoengineered semi-
6 conductor heterostructures1, and similar phenomena mately 30 K. This value roughly corresponds to the
2 havenowbeendemonstratedinquantum-dotsystems2. bulk longitudinalphononfrequencyenhancementand
1 the Kramers-Kronig-accompanied transverse phonon
Fundamental and technologically promising develop-
6 oscillator strength. This robust Rabi coupling may
mentsareongoing,includingprospectsforapplication
0 beaccessiblewiththerelativelyliberalrequirementof
tolight-emittingdiodes,probesof thequantumstatis-
/
at tical limit3, and investigation of bosonic many-body several-micron-scaleengineering.
m effects4. The Rabi interactionwe are postulatingis formally
- Coincidentally, time-resolved studies of fast inter- and physically similar to time-domain beating ob-
d actions in semiconductors have emerged with fem- served in two other coherent phonon systems. At
n tosecond laser technology5,6,7. With new technol- high pump fluences, a plasmon-mediated beating has
o ogy allowing time-domain, optical observations of been demonstrated13, and more recently the phonon
c
: fundamental excitations in crystals, coherent optical has been Rabi-coupled to superlattice-tunedexcitonic
v phononoscillationshavebeenreadoutinseveralsys- states14. Theintrinsiclifetimeofthephononmodeand
Xi tems,suchasGaAs8 andbismuth9. Experimentshave ourestimateforthelifetimeofthecavitymodeareeach
reported intriguing light-lattice interactions, such as greater than the fast and fluence-dependentdephasing
r
a phonon squeezing10, pulse-controlled spatiotemporal time of the plasmon, and than the excitonic transition
latticecoherence11,andexoticphonondynamics12. lifetime.Weshalldiscussourviewthatthephononand
cavity-mode lifetimes will limit the coherent oscilla-
Weareproposingameasurementwhichmayreveal
tionwepropose. Forthephotonmediatedcaseweare
a Rabi splitting between a coherentopticalphononin
addressing, this suggests a more pronounced demon-
a polar crystal and a far-infrared cavity mode. We
strationofstrong-couplingbehavior,andagreaterdu-
mean”coherent”phonontobetakeninthecontempo-
rationoftheoscillation.
rarysense,referringtoamodeexhibitingtime-domain
phaseandamplitudeinformation. IfsuchaRabisplit- With the above in mind, the qualitative features of
ting can be probed through a time-domain quantum our model system take shape. The model possesses
beat, the phenomenon may be interpreted as a co- an in-planetranslationalinvariance,so thatthepump-
herent oscillation of energy between a polar mechan- driven optical phonon should maintain phase over in-
ical mode in the crystal and a vacuum cavity reso- planelengthsatleastcomparabletothedielectricthick-
nance. As such, the system we suggest may demon- ness. Theeffectivelyone-dimensionalsystemremain-
2
ingconsistsofanodelesselectromagneticcavitymode resonanceenergyincreaseslinearlywiththeinversedi-
superposed upon a uniform polarization, which fol- mension. ThusFig. 2bearsresemblancetothepolari-
lows a step function over the dielectric length. The tondispersionofbulkGaAs.
polarization is modeled as the combined effect of N Accordingly, Savona et al., in a study of excitonic
chargedharmonicoscillators,oneforeachGaAsionic cavitypolaritons,generalizedHopfield’sbulkpolariton
pair. Each oscillator has a natural frequency and a theorytoallowforavacuumcavity,andfortheexten-
mass correspondingto the GaAs reducedmass, and a sionoftheelectromagneticfieldbeyondtheboundaries
charge corresponding to the Born-effective charge of ofaconfinedcondensed-mattersystem23.Inthiswork,
bulkGaAs. wedeveloptheanalogoustheoryforthephononiccav-
Phononicoscillatorstrengthshavebeenphenomeno- itypolariton.
logically understood for many years15,16, and it is no
surprise that a coherently excited phonon may radi-
ate. Terahertz radiation from pump-excited coherent II. DERIVATIONOFINTERACTION
phonons has indeed been observed17. The radiation HAMILTONIAN
spectra include an emission peak at the longitudinal-
optical phonon frequency and a broadband response TheformalismtofollowwillemulateSavona’slight-
fromthetransientdynamicsofphotocarriers18. Weare matterinteractiontheoryforexcitonpolaritons,which
suggesting the possibility that a coherent phonon, in motivatedit. Oursystemiscomprisedofarectangular,
proximity to a cavity of well-defined resonance, can semiconductingcrystal, of thickness dd, inside a cav-
undergoa hybridizationof its quantum state with the ityboundbyfar-infraredreflectivesurfaces. Thehalf-
electromagneticresonance,asisthecaseinatomicand waveresonancelength,includingthedielectricandthe
excitonicsystems. Intheweak-couplinglimit,sponta- cavity,isdc. Inrealizingsuchasystem,aconvenience
neousphotonemission and absorptionare influenced, ispresentedbythehighreflectivityofsimplemetalsin
and in the strong-couplinglimit emission and absorp- thefarinfrared. ThispropertymakestheuseofBragg
tion can occur in coherent succession. It may then reflectors and other advanced micro-fabrication tech-
be possible to control and enhance the amplitude and niques, which are required at optical frequencies, un-
phasecoherenceoftheexcitations. Infact,ithasnow necessary in the far-infrared. Thesemiconductorpro-
beenexperimentallyconfirmedthatengineeredgeome- vidinglatticemotionmustlackaninversionsymmetry,
tries can enhance the terahertz radiation of coherent so that it possesses an optically active phonon. The
phonons by two orders of magnitude19, and that the simplest, and best studied such material is GaAs, and
polaritonicspectrum can be tuned by varyingelectro- sowewilladoptitinthetheorytofollow.Thegeome-
magneticboundaryconditions20. tryandrelevantlengthsareillustratedinFig.1.
The far-infrared frequenciesare much smaller than
Ifthephenomenawedescribecanbeobserved,and
the fundamental electronic gap of GaAs, and so, ne-
optimally controlled, they may provide a promising
glectinganyphotocarriers,ourHamiltonianneedonly
specimenfortheinvestigationofmacroscopicquantum
considerthebarepieceH whichincludesthephonon
phenomena.Theymayalsoproverelevanttoimportant 0
field in the lattice and the electro-dynamic field, and
new terahertz technologies. For instance, the system
theirinteraction,i.e.
wedescribemayallowmoredetailedstudythephonon
signal limiting electro-optic sampling, and is related
H=H +H. (1)
0 I
to the control of terahertz polaritonic waveforms and
phonon-polaritonphotonics21,22. Theamplitudedecay Generally,thelatticetermisasetofharmonicoscilla-
mayalso help to illuminatethe natureof the coherent tors,eachwithawavevectorwithintheBrillouinzone,
state, and probe interactions among lattice, electronic andtherearethreebranchindexvaluesperatominthe
andelectromagneticdegreesoffreedom. primitive unit cell. Similarly, the photon field can be
We will refer to the coupled phonon-vacuumreso- expandedintocomponentsofgivenwavevectorandpo-
nance excitation as a phononic cavity polariton. Tra- larization,suchthatthereflectiveboundaryconditionis
ditionally, the polariton was considered as a bulk maintained.Inthecrudestapproximation,welimitour
excitation9, but there is a conceptualresemblance be- consideration of the electro-dynamic degrees of free-
tweenthecanonicalpolaritondispersionandacavity- domtothelowest-energymodesatisfyingthereflective
resonant system. If one imagines a level (avoided) boundarycondition,thehalf-waveresonance.Thegen-
crossingdiagramrepresentingthevariationoftheme- eral multimode case is treated in the Appendix. Fur-
chanical and electromagnetic energies with inverse thermore, we limit the lattice-light interaction to lat-
cavity dimension, the resultis notunlike the standard ticemodesofvanishingcrystalmomentum,relativeto
polaritondispersion,withthecorrespondenceofwave Brillouin zone dimensions. This is justified from en-
vector to inverse cavity dimension. In this picture, ergyandmomentumconservationintheemissionand
themechanicalresonanceisconstantatthetransverse- absorptionof photonsby lattice vibrations24. Finally,
optical phonon frequency, while the electromagnetic boththephononandphotonpolarizationsaretakento
3
d w . The x-projection of the normalized, local, har-
d TO
monic oscillator displacement field can be expressed
throughthephononoperatorsabove:
1 h¯
! x(r)= (a+a†), (5)
P xˆ √Ns2µw TO
where there are N atomic pairs within the dielectric.
zˆ The commutator in the interaction Hamiltonian, then,
is:
yˆ
dc 1[x(r),Hp]= i h¯w TO(a a†). (6)
ih¯ −s 2Nµ −
Now, the normalized vector potential is approxi-
FIG. 1: Schematic of a dielectric crystal with polarization mated as a planar, standinghalfwave, polarizedalong
ccup r/r2ewntT~PO=anPdyˆdidns<idedca≪cavlixt,yl.yTwhheelreenglxth,ysacraeletshesattriasnfysvdecrs≈e xzˆ,. aTnhdeisphaapsperogxriamdiaetnetdibsyaliotsngletnhgethi-natverefraacgeednovramluael,,
dimensionsofthedielectric.
q=p /d . Ifthedielectricandthecavityshareacross-
c
sectionalareaS,then,thexˆ-polarizedcomponentofthe
be in the xˆ-direction, which is within the plane of the vectorpotentialis:
interfaceandreflectivesurfaces.
Withtheseapproximationsinmind,thephotonand A(r )= 2p h¯c 1/2cos[qr ](b+b†). (7)
phononcontributionscanbewrittenexplicitlyinterms z Sd q√e z
(cid:18) c (cid:19)
ofthefieldcreationandannihilationoperators,
ThefirsttermintheinteractionHamiltonianmaybe
p h¯c re-writtenas:
H =h¯w (a†a+1/2)+ (b†b+1/2)+H. (2)
TO √e dc I Z(cid:229) A(r)1[x(r),H ] = iZ(cid:229) 2p h¯c 1/2 h¯w TO
Thefirsttermontherighthandsideisthelatticeterm, −c r ih¯ p c r Sp √e 2Nµ
andindicatesthetransverseopticalphononfrequency. cos[qr(cid:16)z](a−(cid:17)a†)(qb+b†).
(8)
Thesecondterm isthephotonterm, andindicatesthe
speed of light and the effective dielectric constant e , The sum overatomic pairs, each occupyinga volume
W , may be approximated with an integral over the z
whichcharacterizesthecompositesystemofcavityand
dielectric.Areasonableapproximationtoe ,whichwill direction:
beassumedhere,isavolumeaverageofthebulksemi- (cid:229) S dd/2
cos[qr ]= dr cos[qr ]. (9)
conductor dielectric constant and the permittivity of z W Z z z
freespace;i.e. r −dd/2
ThefinalformofthefirsttermintheinteractionHamil-
e =1+(e 0 1)dd/dc. (3) tonianfollows,
−
Amoreaccuratevaluemaybeobtainedthroughmeet- Z(cid:229) A(r)1[x(r),H ] = ihw p w TO dc
ing Maxwell’s boundary conditions at the interfaces −c r ih¯ p p 3/2 c√e √dd
andreflectivesurfaces,butsuchacorrectiononlyshifts sin[p2qdddc](a−a†)(b+b†)
thecavityresonance,andwearetuningnearthetrans- (10)
verseopticalphononfrequency,w TO,inanyevent. In Where NW =Sdd, and the conventional definition of
thesecondtermontheright,thecreationanddestruc- the nuclear plasma frequency is substituted, w p =
tionoperatorscorrespondtophotonswithpolarization, Z(4p /µW )1/216. The second term in the interaction
again,alongthexˆ-axis. Hamiltonianis:
The derivation of our interaction Hamiltonian (cid:229) A2(r) = (cid:229) Z2 2p h¯c cos2[qr ](b+b†)2
closelyresemblesSavonaetal’sexcitonictheory. The r r 2µc2Sqdc√e z
general form of the interaction, with electromagnetic = w 2p h¯ d +dcsin p dd (b+b†)2.
vectorpotentialApolarizedalongxˆ,is: 8p c√e d p dc
(cid:16) (cid:17)(cid:16) h i(cid:17) (11)
HI= −cZ(cid:229) A(r)i1h¯[x(r),Hp]+2Zµc22(cid:229) A2(r). (4) terWmiitnhththeeHseamexilptolinciiatne,xtphreewsshioonlesHfoarmtihlteoniinatnerfaocrtitohne
r r
systemis,
Thissumistakenoverthelatticesitepositions,where
each Ga and As atomic pair are interpreted as har- H = h¯w TO(a†a+1/2)+√h¯ecdp c(b†b+1/2)
monically oscillating dipoles, characterized by Born- +ih¯w (b+b†)(a a†) (12)
AP
effective charge, Z, reduced mass, µ, and frequency +h¯w (b+b†)2. −
AA
4
Theinteractioncoefficientsaretakenfromabove, First,werequirethepolaritontobesomesuperposition
of phonon and photon fields; and second, we require
w = w p w TO dc sin[p dd], thatthecommutatorofthepolaritonoperatorwiththe
AP p 3/2 c√e √dd 2dc (13) Hamiltoniantakethestandardharmonicform.
w AA = w8p2p qc√1e dd+dpcsin[pddcd] . Theformerissatisfiedbywritingthepolaritonfield
(cid:16) (cid:17)(cid:16) (cid:17) operatorasa linearcombinationofcreationandanni-
Physically, this interaction corresponds to a uniform hilationoperatorsofthephononandphotonfields,
polarizationcurrentcouplingtothepositiveandnega-
tivewavevectorsofthelowestfrequencystandingwave B=Wa+Xb+Ya†+Zb†. (14)
thatmatchestheboundaryconditions.
Eachofthecoefficients,W,X,Y,andZ,implicitlyde-
pends on the cavity dimension, d , through the cou-
c
III. NEARRESONANTBEHAVIOR plings,h¯w andh¯w . Thesecondrequirementisex-
AP AA
pressedas,
Simplistic physical considerations lead to the con-
clusion that any eigenstate of our system requires a [B,H]=EB, (15)
mixture of lattice and electromagnetic character. A
phonon will radiate, and conversely a photon will be where on the right-handside the energyeigenvalueis
absorbed by the crystal, so that neither one of these indicated.
states alone is stationary. To find an eigenstate of the Makingthesubstitutions,D=h¯w ,andC=h¯w ,
AA AP
Hamiltonian, which we refer to as the cavity polari- the above two equations may be combined in matrix
ton, we define its operator subject to two conditions. form:
√h¯epdcc +2D−E −iC −2D −iC W
iC h¯w E iC 0 X
 2D TOiC− h¯p c− 2D E iC  Y =0
 −iC −0 −√e dc −iC − −h¯w−TO−E  Z 
Theeigenvaluesareobtainedbysolvingforthecondi- GaAs are w =36 microns, w =51 microns, and
TO p
tionthattheabove4 4matrixhavezerodeterminant. e =10.89. The crystalwidth to cavity width ratio is
0
×
The exact expressions for positive branch of energies takentobefixedatd =d /50.
d c
thatexhibitananticrossingis:
E = h¯ (w 2+4w w +w 2 (w 4+8w w 3 IV. DEPHASINGANDPROSPECTSFOR
± +h16w 2AAw 2−AA2w 2TOwT2O−±8w AAw 2TOwAA MEASURINGRABIOSCILLATION
1/2
+16w 2APw TOw +w 4TO)1/2)/2 Whenacavity-modeisweaklycoupledtoanexcita-
i (16) tionofnon-zerooscillatorstrength,spontaneousemis-
wherew =cp /√e d . Theseenergies,asafunctionof sion and absorption rates can be influenced by tun-
c
d , areplottedinFig. 2. Whenthecavityistunedfar ing the cavity through the resonance. When the cou-
c
fromw ,thetwobrancheshavenegligiblecurvature, pling is increased, and the inverse energy difference
TO
withtheslopeofthephoton-likebranchproportionalto of the avoided crossing surpasses a characteristic dis-
c, and the slope of phonon-likebranchapproximately sipation rate, coherent interaction may be observed.
zero.Inthislimit,thecavitymodeandlatticevibration Forinstance,aRabioscillationcantakeplacebetween
are decoupled. When the cavity mode is tuned near modes of primarily lattice character, and of primar-
w ,however,thereisasignificantcoupling,andthere ily electromagnetic character. The fundamental dis-
TO
isevidentcurvatureinthetwobranches. tinction between the strong- and weak-coupling con-
The Rabi frequency at maximum coupling can be ditions is governed by the relative magnitudes of the
seen in Fig. 2 as the minimum value of the energy near-resonantoscillatorstrengthoftheexcitation,and
differencebetween the two branches. The figure also themodeandexcitationlifetimes.
indicatesthe cavity detuningrangeoverwhichsignif- Nearcavityresonance,thestrongcouplingcondition
icant coupling may be expected. The parameters for isw > g g /4,whereg ,g arethefull-width
AP c TO c TO
| − |
5
Acknowledgments
We aregratefultoSanjivShresta, whosecomments
andinsightsledtoourconsiderationofthework. H.L.
O
T received support from the National Research Coun-
ω
! cil through a NRC-NRL Resident Research Associ-
/
E ateship and from the Office of Naval Research both
directly and through the Naval Research Laboratory.
JNB thanks the NSF REU program. GKB received
supportfromtheAustrianScienceFoundationandthe
InstituteforQuantumInformation.
cπ/√ǫωTOdc
APPENDIXA
FIG.2:Variationofthepositiveenergyeigenstatesofthepo-
laritonHamiltonianasafunctionoftheinversecavitylength
dc. The dielectric is taken to be a GaAs crystal of width In the following, we present a full multimode cal-
dd =dc/50. Theresonanceoccursneardc p c/√ew TO= culation of the the polariton Hamiltonian and the dis-
∼
16.45µm.
persion relation for a dielectric crystal embeddedin a
cavity. The bareHamiltonianforoursystem is (upto
anoverallconstant):
at half-maximum linewidths for the cavity mode and
phonon25. Ford d thisimplies
wpd ≪dd/cdc> g c g TO /2. (17) H0=~(cid:229)q,s h¯w TOa~†q,s a~q,s +h¯√ce |~q|b~†q,s b~q,s . (A1)
| − |
The bulkGaAps phononlinewidth is well-studied24,
andisinthethousandsofmicrons. Thecavitylifetime Here a~q,s ,a~†q,s are the annihilation and creation op-
is a less fundamental quantity, and depends on engi- erators for the phonon field at the frequency w TO.
neeringofthecavity.State-of-the-artetalontechniques The phonon wave vector~q =(~q ,qz) is taken to be
can increase the cavity lifetime at the expense of the small (so thatwave lengthsare m⊥acroscopic),and the
tuningrange.Withatwo-microntuningrange,acavity phonon polarization vector is eˆs . Similarly, b~q,s ,b~†q,s
lifetimewhich is againin the thousandsofmicronsis aretheannihilationandcreationoperatorsforthepho-
possible with commerciallyavailable technology. We ton field. The dielectric constant is the volume av-
seethanthattheaboveinequalityissatisfiedtoanorder eraged value e =1+(e 1)d /d . In the radiation
0 d c
−
ofmagnitude.Thisstrongcouplingisaconsequenceof gauge, the Hamiltonian describing the interaction be-
thetwo-dimensionalemitterattheinterface. tweenthefieldsis:
Weconcludethenthatstrongcouplingislikely. The
Rabi period is also greater than the temporal resolu- H = 1 d3x~A ~P˙+ Z2 d3x~A ~A
tion of ultrafast techniquesusing femtosecondoptical I −cZcrystal · 2W µc2Zcrystal ·
pulses.Presumingthatthedependenceofthereflectiv- (1) (2)
= H +H .
ityonlatticestrainisdistinguishablefromtheeffectof I I
(A2)
the electromagnetic field, the Rabi oscillation should whereW isthevolumeperlatticecellofatomicpairs,
beobservablebypumpingtheopticalphonon,andper-
µ is the reducedmassof the pair, Z isthe Born effec-
formingtime-domainreflectivitymeasurements.
tive charge, and c is the vacuum speed of light. The
macroscopicpolarizationcurrentis
V. SUMMARYANDCONCLUSIONS
~P = h¯Z2 (cid:229) (a ei(qyy+qzz)
Weexpressthequantuminteractionbetweenanop- q2W µw TOSddqy,qz qy,qz (A3)
tically active phonon and a surrounding cavity’s res- + a†qy,qze−i(qyy+qzz))xˆ.
onantelectromagneticmode. Whenthetwoaretuned,
wedemonstrateaRabisplitting,whichisevidentinthe The vector potential must satisfy appropriate bound-
calculatedenergylevelsforvaryingcavitydimension. ary conditions along the cavity axis zˆ. We
Furthermore, we review various experimental studies assume perfectly reflecting mirrors at the cavity
andanalyzethestrong-couplingcriteriatosupportour boundaries meaning the momentum components of
view that a time-domain Rabi oscillation may be ob- the field along zˆ satisfy q = np /d for n Z.
z c
∈
servedinthesystemwedescribe. The suitably normalized vector potential is then
6
Hereafterwesuppressthepolarizationindexs asitis
~A = 2p h¯c (cid:229) (cid:229) sin(2np z/dc) understoodthatonlyxˆpolarizedfieldscontributetothe
+ q√ceoSsd((c2qnx+,q1y)np∈z/Zd(cid:16)c)(|q~⊥|2+(2np /dc)2)1/4 tianitnertahcetifounl.leUxspirnegsstihoenrfeolrattihoeni~nP˙te=ra−ctih¯io[n~P,teHr0m],s:we ob-
(q~ 2+((2n+1)p /dc)2)1/4
× (cid:229) |s⊥(|b~q,s ei~q⊥·~x+b~†q,s e−(cid:17)i~q⊥·~x)eˆs .
(A4)
HI(1) = ih¯w pq4cS2wdTcOdd√e Zcrystald3xqy,(cid:229)q′y,q′zn(cid:229)∈Z(cid:16)(q2yc+os((((22nn++11))pp/zd/cd)c2))1/4(bqy,(2nd+c1)p aq′y,q′zei(qy+q′y)yeiq′zz
+− b(qq2yy+s,i(n(2(2n2d+ncnp1p)/pzd/acd)†qc2′y)),1q/′z4e(ib(qqyy−,2qdn′ycp)yaeq−′y,iqq′z′zezi+(qyb+†qqy′y,)(y2ned+icq1′z)pza−q′yb,qq′zye,2−dncpi(aqy†q−′y,qq′y′z)eyie(qiqy−′zzq−′y)ybe†q−y,iq(2′znzd+c+1)pba†q†qy,′y,2dqnc′zpea−qi′y(,qqy′z+e−q′yi)(yqey−−qiq′y′z)zy)eiq′zz
− b†qy,2dncp a†q′y,q′ze−i(qy+q′y)xe−iq′zz)
= q(cid:229)y,q′zn(cid:229)∈Z(cid:16)−C(oqy,2dncp ),q′z(bqy,2dncp(cid:17)a−qy,q′z+bqy,2dncp a†qy,q′z+b†qy,2dncp aqy,q′z+b†qy,2dncp a†−qy,q′z)
H(2) =+ iCh(¯eqw y2p,(2nd+c1)p ),q′z(db3qxy,(cid:229)(2nd+c1)p(cid:229) a−qy,q′z−bqcyo,(s2(nd+(c21)mp a+†qy,1q)′zp+z/bd†qyc,)(2cnod+cs1()p(a2qny+,q′z1−)pbz†q/y,d(2cn)d+c1)p a†−qy,q′z)(cid:17)
I 4c√e dcSZcrystal qy,q′ym,n∈Z(cid:16)(q2y+((2m+1)p /dc)2)1/4(q2y+((2n+1)p /dc)2)1/4
× (bqy,(2md+c1)p eiqyy+b†qy,(2md+c1)p e−iqyy)(bqy,(2nd+c1)p eiq′yy+b†qy,(2nd+c1)p e−iq′yy)+(q2y+(2sminp(2/mdcp)z2/)d1/c4)(sqin2y(+2n(2pnzp/d/cd)c)2)1/4
× (bqy,2dmcp eiqyy+b†qy,2dmcp e−iqyy)(bqy,2dncp eiq′yy+b†qy,2dncp e−iq′yy)
= (cid:229)qy m(cid:229),n∈Z(cid:16)Do(qy,2dmcp ),2dncp (bqy,2dmcp b−qy,2dncp +bqy,2dmcp b†qy,2dncp +b†qy,2dmcp bqy,2dncp +b†qy,2dmcp b†−qy,2dncp )
+ De(qy,(2md+c1)p ),(2nd+c1)p (bqy,(2md+c1)p b−qy,(2nd+c1)p +bqy,(2md+c1)p b†qy,(2nd+c1)p +b†qy,(2md+c1)p bqy,(2nd+c1)p +b†qy,(2md+c1)p b†−(Aqy5,()2nd+c1)p )(cid:17).
Intheintegrationwehaveassumedthatℓ p c/w Note that the interaction is invariant under the parity
y TO
≫
so that the Hamiltonian is approximately translation- transformation~q,~q ~q, ~q. The coupling coeffi-
ally invariant along the yˆ direction. The momentum cientsarerelatedby′ →− − ′
(1)
dependent coefficients in HI corresponding to cou- (cid:229) [Ce,o ]2=h¯w De,o . (A8)
plingwithevenandoddparityphotonicfieldsare (qy,qz),q′z TO (qy,qz),qz
q′z
C(eqy,qz),q′z = (h¯qw zpcqos(cddwcddTqdO′z√/e2()qs2z−inq(′z2d)d(1qq2yz+/q22z))1/4× tIwhneethrmeecobomuvleeknrlttiuhmmeita,cpdopdmro→pporndiaectn,etasinndsfiafntoiisrtfeyand|iqienzlefi|cn=tirtie|cqd′zc|io.euleTpclhtirneingc
q cos(d q /2)sin(d q /2)),
Co = −h¯w ′z w TdOz 1d ′z relations(seee.g.16):
(qy,qz),q′z p cdcdd√e (q2z−q′z2)(q2y+q2z)1/4× Ce,o h¯w w TO ,
(q′zcqos(ddq′z/2)sin(ddqz/2) (qy,qz),q′z → p 16c√e 0√q2y+q2z
−(qzcos(ddqz/2)sin(ddq′z/2)). De,o qh¯w 2p .
(A6) (qy,qz),q′z → 8c√e 0√q2y+q2z
(2)
Similarly,forH themomentumdependentcoupling
I ThetotalHamiltonianforoursystemisthen
coefficientsbetweenevenandoddfieldcomponentsis
D(eq,oy,qz),q′z = 4√h¯siwenc2p(d(cqz(−q2yq+′z)qd2zd)/12/)41(q2y+siqn′z(2()q1z/+4q′z)dd/2) H = +(cid:229)qy(cid:229) hnh(cid:229)¯∈wZTh¯O√ace†qyq,qzqa2yqy+,qz(n+p /HdIc()12)b+†qyH,nIp(2/)d.cbqy,np /dc
× qz−q′z ± qz+q′z qz i
(cid:16) (cid:17)(A7) (A9)
7
Since the Hamiltonian is a quadraticpolynomialin Inthebulklimit,foreachmomentumwavevector~qwe
fieldoperators,H canbediagonalizedbyaTyablikov- obtaintheappropriatedispersionrelation16,
Bogoluibov transformation23,26. The Hamiltonian is
translationallyinvariantalongyˆsodifferentq momen-
y
tumcomponentsdonotmix.Toobtainthesolution,we
introduceanewsetofbosonicoperators
Bqy = ℓ+(cid:229)∈(cid:229)Z[X[W(q(yq,ℓ,)qbq)ya,ℓp /dc++YZ((qqy,,ℓq)b)a†−†qy,−ℓp /d].c] (cid:16)E2−(h¯w TO)2(cid:17)(cid:16)E2−(cid:0)h¯√c|e~q0|(cid:1)2(cid:17)−E2(h¯we 0p)2(=A104.)
q′z y ′z qy,q′z y ′z −qy,−q′z
(A10)
When the cavityis tunednearthe resonancecondi-
The coefficients of the expansion are determined by tionatminimalseparation,d p c/w ,thentheeven
c TO
satisfyingthecommutationrelation parityq = p /d modesofth≈efieldarestronglycou-
z c
±
pled to the polarization current. Restricting attention
[B ,H]=EB . (A11)
qy qy to the q =0 modesof the polarizationcurrentwhich
′z
dominatethecouplinganddefiningthecreationopera-
Theconditionthatasolutionexiststhenreducestothe
torfortheevenparityphotonmode:
followingdispersionrelationwhichisanimplicitfunc-
tionofenergy:
E2 (h¯w )2= (cid:229) (fe (E)+fo(E)), (A12)
− TO 2ℓ+1 2ℓ
ℓ Z
∈
b† +b†
where b† 0,−p /dc 0,p /dc, (A15)
≡ √2
fe(o)(E) = 4h¯√ce √q2y+(ℓp /dc)2 (E2 (h¯w )2)
ℓ DE2e−(o()h¯c)2(q2y++(ℓh¯p w/dc)2)(cid:229)/e [(cid:16)Ce(o)− ]2TO ×
(qy,dpcℓ),dpcℓ TOq′z (qy,dpcℓ),q′z (cid:17)
= e (E2−E(h¯2c()h¯2w(qp2y)+2d(dℓ/p 2/ddcc)2)/e ) 1±sinℓ(pℓdp dd/dd/cdc) . anda†≡a†0,0,theinteractingpieceoftheHamiltonian
(A13) thenreducestoEq.12.
(cid:0) (cid:1)
Current address: Department of Physics, University of 10 G.A.Garret,A.G.Rojo,A.K.Sood,J.F.WhitakerandR.
∗
Washington, Seattle, Washington 98195, USA.; Elec- Merlin, Science275, 1638(1997).
tronicaddress:[email protected] 11 T.Feurer,J.C.Vaughan,K.A.Nelson, Science299, 374
1 Weisbuch,C.,M.Nishioka,A.Ishikawa,andY.Arakawa, (2003).
Phys.Rev.Lett. 69, 3314(1992). 12 O.V. Misochko, M. Hase, K. Ishioka, and M. Kitajima,
2 J.P.Reithmaier,G.Sek,A.Lo¨ffler,C.Hoffman,S.Kuhn, Phys.Rev.Lett.92, 197401(2004).
S.Reitzenstein,L.V.Keldysh,V.D.Kulakovskii,T.L.Rei- 13 A.V.KuznetsovandC.J.Stanton, Phys.Rev.B51, 7555
necke, andA.Forchel, Nature432, 197(2004); E.Pe- (1995).
ter,P.Senellart,D.Martrou,A.Lemaˆıtre,J.Hours,J.M. 14 K.Mizoguchi,O.Kojima,T.Furuichi,M.Nakayama,K.
Ge´rard,andJ.Bloch, Phys.Rev.Lett95, 067401(2004). Akahane,N.Yamamoto,andN.Ohtani, PhysRev.B69,
3 G.Khitrova,H.M.Gibbs,F.Jahnke,M.Kira,andS.W. 233302(2004).
Koch, Rev.ofMod.Phys.71, 1591(1999). 15 M. Born and K. Huang, Dynamical Theory of Crystal
4 D.Snoke, Science298, 1368(2002). Lattices (OxfordUniversityPress,Cambridge,1954).
5 W. Ku¨tt, W. Albrecht, and H. Kurz, IEEE J. Quantum 16 J.J.Hopfield, Phys.Rev.112, 1555(1958); D.L.Mills
Electron.28, 2434(1992). andE.Burstein, Rep.Prog.Phys.37, 817(1974).
6 M.C.Beard,G.M.Turner,andC.A.Schmuttenmaer, J. 17 T. Dekorsy, H. Auer, C. Waschke, H. J. Bakker, H. G.
Phys.Chem.B106, 7146(2002); P.Siegel, IEEETrans. Roskos, H. Kurz, V. Wagner and P. Grosse, Phys. Rev.
MicrowaveTheoryTech.50, 910(2002). Lett.74, 738(1995).
7 E.D. Murray, D.M. Fritz, J.K. Wahlstrand, S. Fahy and 18 Y.C.Shen,P.C.Upadhya,E.H.Linfield,H.E.Beere,and
D.A.Reis, PhysRevB72, 060301(2005). A.G.Davies, Phys.Rev.B69, 235325(2004).
8 G.C. Cho, W. Ku¨tt, H. Kurz, Phys. Rev. Lett. 65, 764 19 K. Mizoguchi, T. Furuichi, O. Kojima, M. Nakayama,
(1990). S. Saito, A. Syouji, and K. Sakai, App. Phys. Lett. 87,
9 M.Hase,M.Kitajima,S.Nakashima,andK.Mizoguchi, 093102(2005).
Phys.Rev.Lett.88, 067401(2002). 20 J.Hebling,G.Almasi,I.Z.KozmaandJ.Kuhl,Opt.Expr.
8
10, (1161)(2002). 24 H.M.LawlerandE.L.Shirley, Phys.Rev.B70, 245209
21 K.C.Huang,P.Bienstman,J.D.Joannopoulos,K.A.Nel- (2004).
son,andS.Fan, Phys.Rev.B68, 075209(2003). 25 L.C.AndreaniandG.Panzarini, Phys.Rev.B60, 13276
22 D.W.Ward,J.D.Beers,T.Feurer,E.R.Statz,N.S.Stoy- (1999).
anov,andK.A.Nelson, Opt.Lett.29, 2671(2004). 26 V.M.Agranovich,SovietPhys.JETP37,307(1960).
23 V. Savona, Z. Hradil, A. Quattropani, and P. Schwendi-
mann, Phys.Rev.B49, 8774(1994).