Near-field intensity correlations in parametric photo-luminescence from a planar microcavity

0.54 MB·English

by  Davide Sarchi

Near-field intensity correlations in parametric photoluminescence from a planar
microcavity
Davide Sarchi∗ and Iacopo Carusotto
CNR-INFM BEC Center and Dipartimento di Fisica,
Universita` di Trento, via Sommarive 14, I-38050 Povo, Trento, Italy
(Dated: January 26, 2010)
We study the spatio-temporal pattern of the near-field intensity correlations generated by para-
metricscatteringprocesses inaplanaroptical cavity. A generalized Bogolubov-deGennesmodelis
usedtocomputethesecondorderfieldcorrelationfunction. Analyticapproximationsaredeveloped
0
tounderstandthenumericalresultsinthedifferentregimes. Thecorrelation patternisfoundtobe
1
robust against a realistic disorder for state-of-the-art semiconductor systems.
0
2
PACSnumbers: 71.36.+c,42.50.Ar,42.65.Yj,71.35.Lk

n
a
J I. INTRODUCTION complex geometries that in many aspects mimic the be-

6 havior of quantum fields in curved space-times21.
2
Quantum correlations in many-body and optical sys- Our approach is based on a generalized, non-

] tems areplayinga crucialrolein a varietyof fields, from equilibrium Bogolubov-de Gennes approximation in
r the microscopic study of novel states of matter in quan- which weak fluctuations around the classical coher-
e
h tum fluids1,2, to the control and suppression of noise in ent field are described in terms of a quadratic
t optical systems3–7, to the exploration of analog models Hamiltonian14,22–24. Solving in frequency space the cor-
o
of gravitationaland cosmologicalsystems8–10. respondingquantumLangevinequationsallowstoobtain
.
at In this perspective, non-linear optical systems exhibit predictions for the spatial and temporal dependence of
m a rich variety of phenomena11,12, involving the interplay the in-cavity intensity correlations. These then directly
of parametric scattering with losses, non-linear energy transfertothenear-fieldcorrelationsoftheemittedlight.
-
d shifts,andthepeculiardispersionoflightinconfinedge- The study of the simplest planar geometry is extended
n ometries. As a most significant example of this physics, to the more realistic case of weakly disordered systems:
o anexampleofphasetransitionoftheBose-Einsteinclass inthisregime,theeffectofdisorderisshownnottoqual-
c
[ has recently started being investigated in the optical itatively modify the peculiar correlation pattern.
parametric oscillation in planar geometries13,14. In Section II, we review the theoretical formalism
2
Theoretical work has suggested that further insight in based on the Bogolubov approximation and we discuss
v
this physics can be obtained from the spatial and tem- how correlations transfer from the in-cavity field to the
3
0 poral pattern of correlations of the emitted light15,16: as emitted radiation. In Section III.A we present the nu-
2 thecriticalpointforparametricoscillationisapproached, merical results for a one-dimensional, spatially uniform
4 the correlation length and time of the parametric emis- case. Approximate analytical calculations are presented
0. sion show a divergence that is closely related to the one in Section III.B and used to physically interpret the nu-
1 of thermodynamical phase transitions. merical results. Numerical results for a two-dimensional
9 To this purpose, semiconductor microcavities in the disordered system are discussed in Section IV. Conclu-
0 strong coupling regime appear as most favorable sys- sions are drawn in Section V.
:
v tems13,14,17 as they are intrinsically grown with a pla-
i nar geometry, and nonlinear interactions between the
X
dressed photons – the so-called polaritons – are remark-
r
a ably strong. First experiments demonstrating quantum
II. FORMALISM
correlations have recently appeared18–20.
In the present work, we investigate the physics of the
intensity correlations that are generated by parametric In this section we briefly review the formalism which
scattering processes. Depending on the pump frequency will be used to calculate the intensity correlations of the
and intensity, different regimes can be identified. In ad- emitted light. As we are restricting to the case of small
dition to the usual short-distance correlations that are fluctuations around a strong coherent field, a Bogolubov
generallypresentinany interactingsystem, spontaneous approachisadaptedtodescribethedynamicsofquantum
parametric emission processes are responsible for addi- fluctuations. In the simplest case of a spatially homoge-
tional long-distance ones, whose correlation length di- neous geometry the Bogolubov equations can be worked
verges as the optical parametric oscillation threshold is outtoanalyticalexpressionsforthephysicalobservables.
approached. Inadditiontotheirintrinsicinterest,anex- Inthe generalcase,numericalresultscanbeobtainedby
perimental study of intensity correlationin this simplest invertingtheBogolubovmatrixforthespecificgeometry
system will open the way to the investigation of more under investigation.
2
A. The system Hamiltonian fields are split into a strong coherent –classical– compo-
nent and weak quantum fluctuations
Wedescribetheexcitonandthephotonquantumfields
Ψˆx(r) and Ψˆc(r) as scalar Bose fields. The Hamiltonian Ψˆ (r,t)=e−iωpt Φ (r)+δψˆ (r,t) . (8)
x(c) x(c) x(c)
is the sum of terms describing the free propagation of
excitons and photons Hˆ , their dipole coupling Hˆ , the h i
0 R
two-body exciton-exciton interaction Hˆ , saturation of ThisdecompositionistheninsertedintotheHamiltonian
X
the exciton-photon coupling Hˆ and the external pump- (1) and all terms of third and higher order in the fluctu-
s
ing Hˆ 14,25: ations are neglected. This leads to a quadratic Hamilto-
p
nian for the fluctuation field that can be attacked with
Hˆ =Hˆ +Hˆ +Hˆ +Hˆ +Hˆ . (1) available theoretical tools.
0 R x s p
The classicalcomponent Φ (r) is obtained from the
x(c)
The free propagationterm has the form Heisenberg equations of motion of the quantum field
Ψˆ (r) by factorizing out the multi-operator averages.
x(c)
Hˆ = d2rΨˆ†(r)[E (~2/2m ) 2+U (r)]Ψˆ (r) This leads to the following pair of generalized Gross-
0 x x− x ∇ x x Pitaevskii equations:
Z
+ d2rΨˆ†(r)[ǫ ( i~)+U (r)]Ψˆ (r) (2)
c c − ∇ c c ~2 2
Z ~ω Φ (r) = ∇ +U (r) i~γ /2+g Φ (r) 2
p x x x x x
whereEx istheexcitonenergy,mx itseffectivemass,and (cid:18)− 2mx − | |
ǫ is the photon dispersion in the planar cavity; U are + 2g Re Φ∗(r)Φ (r) )Φ (r)
c x,c s { x c } x
the externalpotentials acting onrespectivelythe photon + ~Ω +g Φ (r) 2 Φ (r), (9)
R s x c
andtheexciton. Thedipoleexciton-photoncouplinghas | |
the form ~ωpΦc(r) = (cid:0)ǫc( i~)+Uc(r) i(cid:1)~γc/2 Φc(r) (10)
− ∇ −
+ h~Ω +g Φ (r) 2 Φ (r)i+~F eikp·r,
Hˆ =~Ω d2r[Ψˆ†(r)Ψˆ (r)+h.c.] (3) R s | x | x 0
R R x c
Z (cid:0) (cid:1)
Note that the classical field Φ (r) that is obtained
c,x
andisquantifiedbytheRabifrequencyΩ . Theeffective
R from this equation does not include the correction due
two-body exciton-exciton interaction term
to the backaction of fluctuations onto the coherent com-
ponent28. Totakethis effectintoaccount,oneshouldin-
1
Hˆx = 2gx d2rΨˆ†x(r)Ψˆ†x(r)Ψˆx(r)Ψˆx(r) (4) clude the contribution hδψˆx†(r)δψˆx(r)i of the fluctuating
Z fields to the density and then iteratively solve (9-10) up
models both Coulomb interactionand the effect of Pauli toconvergence. Asinthepresentpaperweareinterested
exclusion on electrons and holes26,27. inthecorrelationpropertiesofthefieldfluctuations,this
effect can be safely neglected in what follows.
Hˆ =g d2r[Ψˆ†(r)Ψˆ†(r)Ψˆ (r)Ψˆ (r)+h.c.] (5) Withintheinput-outputformalism12,thequantumdy-
s s c x x x
namics of fluctuations can be written in terms of quan-
Z
tum Langevin equations of the form22,23,29
is the term modeling the saturation of the exciton oscil-
lator strength26. The external pump term has the form
i~∂ δΨ(r,t)=Mˆ δΨ(r,t)+~f(r,t), (11)
t
Hˆ =~ d2r[Ψˆ†(r)F (r,t)+h.c.]. (6)
p c p
where
Z
In the following of the paper, we assume that the sys-
temisdrivenbyacontinuous-wavemonochromaticpump δΨ=(δψx,δψx†,δψc,δψc†)T (12)
with a plane wave spatial profile
is the four-component quantum fluctuation field. The
F(r,t)=e−iωpteikp·rF0. (7) matrixMˆ isobtainedbylinearizingtheHeisenbergequa-
tionofmotionfor the quantumfield aroundthe classical
Excitons and photons decay in time with a rate γ and
x component. In our case, it has the form:
γ , respectively.
c
Tˆ i~γx Σxx Ω˜ Σxc
B. Bogolubov-de Gennes formalism Mˆ = −x(−ΩΣ˜x12x)2∗ −TˆxΣ−1x2ci~γ2x Tˆ−(ΣRx1i2c~)γ∗c −Ω0˜12R  ,
To make the problem analytically tractable, we per-  (ΣRxc)∗ Ω˜12 c−0 2 Tˆ i~γc 
 − 12 − R − c− 2 
form the Bogolubov approximation24: the two quantum  (13)
3
with and making use of (21), simple expressions for the two-
operator averagesare obtained
~2 2
Tˆ (r) = ∇ +U (r) ~ω +2g Φ (r) 2
x x p x x
− 2m − | | 4
x
+ 4gsRe{Φ∗x(r)Φc(r)}, (14) gx(r,r′,ω)= dshs,l|Mˆω−1|r′,1i∗hs,l|Mˆω−1|r,1iΓl,
Tˆ(r) = ǫ ( i~)+U (r), (15) Xl=1Z
c c c (27)
− ∇
Σxx(r) = g Φ2(r)+2g Φ (r)Φ (r), (16)
12 x x s x c
4
Ω˜R(r) = ~ΩR+2gs Φx(r) 2, (17) g (r,r′,ω)= ds s,l Mˆ−1 r′,3 ∗ s,l Mˆ−1 r,3 Γ ,
Σx12c(r) = gsΦ2x(r). | | (18) c Xl=1Z h | ω | i h | ω | i l
(28)
The quantum Langevin force
4
f =(fx,fx†,fc,fc†)T (19) mx(r,r′,ω)= dshs,l|Mˆω−1|r′,2i∗hs,l|Mˆω−1|r,1iΓl,
describes the zero-point fluctuations in the input field. Xl=1Z
(29)
As the field dynamics takes place in a small frequency
window around ωp and the input field is assumed to be 4
in the vacuum state, the spectrum ofquantum Langevin m (r,r′,ω)= ds s,l Mˆ−1 r′,4 ∗ s,l Mˆ−1 r,3 Γ .
c h | ω | i h | ω | i l
forcecanbeapproximatedbyawhitenoiseinbothspace l=1Z
X
and time, (30)
where Γ=2π~2(0,γ ,0,γ )T and the matrix
f (r,t)f†(r′,t′) =γ δ δ(r r′)δ(t t′). (20) x c
h ξ χ i χ ξ,χ − −
Mˆ =Mˆ ~ω1 (31)
Thisapproximationisaccurateinthepresentcasewhere ω −
no other radiationis incident onto the cavityin addition is evaluated in the basis (r,j) . Here, r spans the real
tothedrivingatω andthelight-mattercoupling~Ω is | i | i
p R space and the label j = 1,...,4 refers to the block form
small as compared to the natural oscillation frequencies
of the Bogolubov matrix (13).
E and ǫ 30,31.
x c
D. Intensity correlations
C. Evaluation of the correlation functions
The present paper is focussed on the spatio-temporal
In the presentcaseof a monochromaticpump, tempo-
correlations of the intensity fluctuations of the in-cavity
ralhomogeneityguaranteesthatthecorrelationfunctions
photon field. As usual in generic planar microcavities,
onlydependonthetimedifferencet t′anddifferentfre-
the in-cavity intensity correlations directly reflect into
−
quency components are decoupled:
the near-field intensity correlation of the emitted light
~ωδΨ(r,ω)=MˆδΨ(r,ω)+~f(r,ω). (21) outsidethe cavity. Anexplicitcalculationofthe relation
between the in-cavity and the emitted fields in the spe-
The two-operator averages of the fluctuation fields can cificcaseofsemiconductorDBRmicrocavitiesisgivenin
be written as the Appendix.
g˜(1) (r,t;r′,t′)= dωe−iω(t−t′)g˜(1) (r,r′,ω), (22)
x(c) x(c)
Z
The four operator correlation function describing the
and
correlation of the intensity fluctuations of the in-cavity
m˜(1) (r,t;r′,t′)= dωe−iω(t−t′)m˜(1) (r,r′,ω). (23) photon field is defined as usual as:
x(c) x(c)
Z G(2)(r,t;r′,t′)= Ψˆ†(r,t)Ψˆ†(r′,t′)Ψˆ (r′,t′)Ψˆ (r,t) .
where we have defined h c c c c (i32)
At the level of the Bogolubov approximation,three- and
g˜x(1()c)(r,r′,ω)= dω′hδψx†(c)(r,ω)δψx(c)(r′,ω′)i, (24) four-operatoraveragesof the fluctuation field can be ne-
Z glected and the correlation function (32) can be written
in terms of two-operatoraverages as
m˜(1) (r,r′,ω)= dω′ δψ (r,ω)δψ (r′,ω′) . (25)
x(c) h x(c) x(c) i G(2)(r,r′,t) = Φ (r)2 Φ (r′)2+ Φ (r′)2 ψˆ†(r)ψˆ (r)
Z | c | | c | | c | h c c i
By introducing the frequency-domain correlation func- + |Φc(r)|2hψˆc†(r′)ψˆc(r′)i
tions + 2Re Φ (r′)∗Φ (r) ψˆ†(r,t)ψˆ (r′,0)
{ c c h c c i}
hfξ(r,ω)fχ†(r′,ω′)if =2πγχδξ,χδ(r−r′)δ(ω−ω′) (26) + 2Re{Φc(r′)∗Φc(r)∗hψˆc(r,t)ψˆc(r′,0)i}(3.3)
4
A B C D
4
V) 2 D
e 0.6
m
E} (k−02 eV)
Re{ −4 (a) (c) (e) (g) 2 (m 0.4
0 |x
Φ
V) |x
e g
m 0.2
} (k−0.1 C
E
{ B
m (b) (d) (f) (h) A (i)
I−0.2 0
−5 0 5 0 2 4 0 2 4 −5 0 5 0 20 40 60
k (µm−1) k (µm−1) k (µm−1) k (µm−1) |F |2 (ps−2 µm−2)
0
FIG.1: Real(a-g)andimaginaryparts(b-h)oftheBogolubovdispersionasafunctionofthewavevectork,forthefourregimes
A-Dmarkedinpanel(i). Theexciton interaction energyisg|Φx|2 =10−4 meV(a,b),g|Φx|2 =0.04 meV(c,d),g|Φx|2 =0.08
meV (e, f) and g|Φx|2 =0.61 meV (g, h). (i) Bistable loop in thein-cavity exciton field intensity as a function of theincident
pumpintensity. The red dashedline representstheunstablesolutions. The pointslabeled with theletters A-Dmark thefour
regimes considered in panels (a-h).
A particularly useful quantity is the reduced correlation l 1 µm. This corresponds to effective 1D nonlin-
1D
function: ear c∼oupling constants g = 1.5 10−3meV µm and
x
g =0.5 10−3 meV µm. Such geo×metries are presently
G¯(2)(r,t;r′,t′)= G(2)(r,t;r′,t′) 1, (34) usnder ac×tive experimental investigation32. Extension to
nc(r)nc(r′) − the 2D case does not qualitatively affect the physics and
will be discussed in Sec. IV in connection to disorder
scaled by the intensity of light
issues.
Weconsideraconfigurationwherethepumpwavevec-
n (r)= Φ (r)2+ δψˆ†(r)δψˆ (r) , (35)
c | c | h c c i torkp =2.2µm−1 iscloseto the magicwavevector,and
thepumpenergyis~ω ǫ (0)= 8.7meV.Thisconfig-
which can be written in the following simple form: p− c −
uration allows to study the most significant regimes by
simply varying the pump intensity. In Fig. 1 (i), we plot
G¯(2)(r,r′,t) 2Re Φ (r′)∗Φ (r) ψˆ†(r,t)ψˆ (r′,0)
≃ c c h c c i the corresponding exciton field intensity as a function of
n the pump intensity F 2: the hysteresis cycle typical of
+ Φc(r′)∗Φc(r)∗hψˆc(r,t)ψˆc(r′,0)i × bistable systems is a|pp0a|rent25,33,34.
[Φ (r)2 Φ (r′)2]−1. o (36) Inthe following,wewillconsiderfour differentdensity
c c
× | | | |
regimes, marked on the bistability loop by the points
Inthe nextsection,wewillstudy the spatialandtempo- A-D. Two of them (A, B) correspond to low-density
ral features of this quantity in the most relevant cases. conditions far from any instability, the point C is very
close to the OPO threshold, and point D is in the stable
region above the bistability and parametric oscillation
III. 1D UNIFORM SYSTEM region.
In the present uniform system, the elementary excita-
A. Numerical results tions on top of the steady-state of the pumped system
can be classified in terms of their wavevector k. Exam-
In this section we investigate the case of a spa- ples of their Bogolubov dispersion25 are shown in Fig. 1
tiallyuniformsystemunderaplanewavemonochromatic (a-h) for the four different regimes marked in Fig. 1 (i)
pump. Spatial homogeneity guarantees that the correla- by the points A-D. In the figure, we restrict the field of
tion functions only depend on the relative spatial coor- view to the lower-polaritonregionthatis involvedin the
dinate r =x x′. physics under investigation here.
−
We use typical parameters for a GaAs microcavity For increasing intensities, we observe: i) the low-
with N = 10 quantum wells. In particular, we choose densityparametricregimeA(panelsa,b),wheretheBo-
~Ω =10meV,~γ =~γ =~γ =0.1meV,andwetake golubovmodesreducetothesingle-particledispersion;ii)
R c x
zero exciton-photon detuning at k = 0, E = ǫ (k = 0). the regime B corresponding to moderate densities (pan-
x c
For the sake of simplicity, in the present section we fo- els c, d), where the imaginary parts are modified and a
cus our attention on a quasi-one-dimensional geometry small regionof flattened dispersionappears at the cross-
where polaritons are transversally confined to a length ing of the Bogolubov modes; iii) the regime C close to
5
OPO threshold (panels e, f), where the imaginary part x 10−3
of one mode tends to zero and a flat region in the Bo- 20 (a) 2
golubov dispersion is apparent; iv) the non-parametric
configuration D, for intensities larger than the bistabil- 10 1
itythreshold,wherethenormalandtheghostdispersions
arewellseparatedandtheeigenmodesofthesystemtend s)
p 0 0
againtothesingle-particledispersion,yetblue-shiftedby t (
interactions (panels g, h).
For each of this A-D regimes, we have computed the −10 −1
spatio-temporalpatternofintensitycorrelationswiththe
model described in the previous section. The results are −20 −2
−3
presented in the next subsections. An analytical inter- x 10
pretation will be given later on in Sec. III.B. 5 (b)
0)
=
2)(r,t 0
1. Low-density parametric luminescence (G
−5
−3
x 10
In Fig. 2 (a), we show the spatio-temporal pattern
of the intensity correlation in the low-density regime A ps) 2 (c)
corresponding to Fig. 1 (a, b): this is characterized by = 5
a system of parallel fringes and a butterfly-shaped enve- r, t 0
lope. Thefringeamplitudeisalmostvanishinginsidethe 2)(
cone delimited by the group velocities of the signal and (G−2
the idler modes (represented by the thin solid lines in −30 −20 −10 0 10 20 30
r (µm)
the figure) and largeston the edges of the cone. Further
away in space and time, it decays back to zero as a con-
sequence of the finite polariton lifetime. At zero delay FIG. 2: (a) Pattern of the reduced intensity correlations
t = 0, the central bunching peak around r = 0 shows a G¯(2)(r,t) as a function of the spatial distance r and of the
weak narrow dip [Fig.2(b)]. At larger delays [Fig.2(c)], temporaldelayt,forthelow-densityregimeA,corresponding
toFig. 1(a,b). Thetwothinlines,r=vstandr=vit,high-
thecorrelationsignalisweakerinbetweenthedot-dashed g g
light the group velocities of the signal and idler modes. (b)
lines indicating the edges of the cone.
Cut of the figure (a) along the zero delay t=0 line. (c) Cut
along the t =5 ps line. In panels (b) and (c), the analytical
prediction Eq. (45) is shown as a dashed line. In panel (c),
2. Moderate-density parametric luminescence the two vertical dash-dotted lines delimit the region where
theanalytical result (45) predicts vanishingcorrelations.
In Fig. 3 (a), we show the spatio-temporal pattern
in the moderate density regime B corresponding to the
−3
energy dispersion shown in Fig. 1 (c, d). As compared x 10
20 4
to the low-density case, the qualitative shape of the cor-
relation pattern is qualitatively modified: the system of
parallel fringes extends in a significant way into the in- 10 2
terior of the butterfly shape and the exponential decay
in the external region takes place at a slower rate. This s)
latter effect is a direct consequence of the increased life- p 0 0
time of the Bogolubov modes in the parametric region t (
(see Fig. 1 (d)).
−10 −2
3. OPO critical region −20 −4
−30 −20 −10 0 10 20 30
r (µm)
In Fig. 4 (a), we show the spatio-temporalpattern for
the regime C very close to the OPO threshold. In this
FIG. 3: (a) Pattern of the intensity correlations G¯(2)(r,t),
case, the system of parallel fringes extends to the whole
for the moderate density regime B, still well below the OPO
(r,t) space and correlations are non-vanishing even at
threshold and corresponding to Fig. 1 (c, d). The two thin
very long time and space separations. This is a conse- lines, r = vst, and r = vit, highlight the group velocities of
g g
quence ofthe divergingcorrelationlengthoffluctuations thesignal and idler modes.
in the critical region15. The zero-delay t = 0 cut of the
6
0.02 x 10−4
(a) 20 4
20 (a)
0.01
10 2
ps) 0 0 s)
t ( t (p 0 0
−0.01 −10 −2
−20
−0.02 −20 −4
0.05 x 10−3
(2)G (r,t=0) 0 (b) (2)G (r,t=0)−022 (b)
−4
−0.05 x 10−4
0.05 s)
ps) (c) 5 p 4 (c)
2) (r,t=10 0 (2)G (r,t=−−0450 −25 0 25 50
(G−0.05 r (µm)
−50 −25 0 25 50
r (µm)
FIG. 5: (a) Pattern of the intensity correlations G¯(2)(r,t)
for the high-density regime D, corresponding to Fig. 1 (g,
FIG. 4: (a) Pattern of the intensity correlations G¯(2)(r,t)
h). In this regime, the large blue-shift of the polariton
for the density regime C just below the OPO threshold and
modespreventstheparametricoscillation. Thetwothinlines
correspondingtoFig. 1(e,f). (b)Cutofthefigure(a)along r = ±vmaxt mark the maximal group velocities. (b) Cut of
the zero delay t = 0 line. (c) Cut along the t = 10 ps line. g
the previous figures along the zero delay t = 0 line. (c) Cut
In panels (b) and (c), the dashed line is the result of the
along the t = 5 ps line. In panels (b) and (c), the analyti-
analytical prediction Eqs. (47) and (48).
calpredictionEq. (52)(reddashedline) andEq. (53)(green
dash-dottedline)arealsoshown. Inpanel(c),thetwovertical
thin dash-dotted lines delimit the region where both the nu-
correlation pattern is shown in Fig. 4 (b) and is charac- mericalresultandtheanalyticalresult(52)predictvanishing
correlations.
terized by a system of fringes centered at r = 0 with a
central bunching peak. The weak dip that was visible in
the weak intensity regime is no longer present.
B. Analytic model and interpretation
To physically understand the numerical results pre-
sented in the previous section, approximate analytic for-
4. Non-parametric regime mulas for the behavior of G¯(2) can be extracted for the
most significant limiting cases. To simplify the analysis,
For even larger values of the pump intensity beyond we restrict our attention to the lower polariton branch.
the bistability threshold, the system is pushed into an Thisapproximationisaccurateaslongasthe Rabisplit-
higher density D regime, where the blue-shift induced ting is much larger than all other energy scales, e.g.
byinteractionsbringsparametricscatteringprocessesfar ~Ω γ, ~Ω g Φ 2, g Φ Φ , and is safely ful-
R R x x s c x
≫ ≫ | | | |
off-resonance [Fig. 1 (g, h)]. As a consequence, the sys- filledinallthecasesthatweareconsideringhere. Inthis
tem of parallel fringes due to parametric correlations al- regime,thedynamicsofthesystemcanbedescribedbya
most completely disappears, as it can be seen in Fig. 5 singlepolaritonfield. Itsdispersion,linewidthγandnon-
(a). The remaining correlation pattern has an opposite linear interaction coefficient g are immediately obtained
shape,withsubstantialcorrelationslimitedtothe region fromthe lineareigenmodesofthe Gross-Pitaevskiiequa-
inside the cone delimited by the maximal group velocity tions (9) and (10) once interactions are neglected14. At
of the polariton dispersion. At zero delay t = 0 [Fig. 5 the same level of approximation, also the k-dependence
(b)] no spatial fringe is visible and the only feature is an of the photon Hopfield factor C C can be neglected.
k
anti-bunching dip at r =0. At finite time delays [Fig. 5 As aresult,the sameHopfieldcoe≃fficient C 4 appearsin
| |
(c)], some spatial fringes appear, but they do not show boththenumeratorandinthedenominatoroftheinten-
any dominating periodicity. This suggests that a con- sity correlation function (34) of the photon field, which
tinuum of processes at different k’s are simultaneously thenreducestothecorrespondingquantityforthelower-
taking place, each of them characterized by a different polariton field.
phase velocity. For a spatially homogeneous system under a
7
monochromaticplanewavepump,equationscanbewrit- In the parametric configuration (see Fig. 1 (a, b)),
teninthemomentumspaceandthereducedsecond-order the dominating contribution to the integral over k is
correlation function reads given by the wavevectors around the signal k and the
s
idler k = k¯ = 2k k modes, where the normal and
i s p s
−
G¯(2)(r,t)=G(2)(r,t)+G(2)(r,t)= the ghost dispersions intersect and parametric processes
1 2
are resonant. We can then approximate the integrand
2 ddk
= Φ2 Re (2π)d e−i(k−kp)·rhδψk†(t)δψk(0)i + baryourenpdlakci=ngkthe energies by their first-order expansions
| | (cid:20)Z (cid:21) s,i
2 ddk
+ Re Φ∗2 ei(k−kp)·r δψk(t)δψk¯(0) ǫ ǫ +~v(s,i)(k k ), (43)
|Φ|4 (cid:20) Z (2π)d h i(cid:21) k ≃ ks,i g − s,i
(37)
with group velocities
where k¯ =2k k. In the presence of quantum fluctua- 1
tions only14,22p,−the normal correlations are vg(s,i) = ~∂kǫ(k)|ks,i. (44)
eiωt In this way, we obtain
δψk†(t)δψk(0) =2π~2γg2 Φ4 dω , (38)
h i | | ∆k(ω) g
Z G¯(2)(r,t) 8π2 Sgn(t)e−γ|t|/2Ssin(Kr Ωt)
2 ≃ ~∆v − ×
and the anomalous correlations are36 g
[θ(tSR )e−κRs θ( tSR (r,t))eκRi(r,t)], (45)
hδψk(t)δψk¯(0)i=−2π~2γgΦ2 dωe−iωt˜ǫk¯ +∆~kω(ω+)i~2γ . where S×=Sgn(s∆vg) and−∆v−g =vgi(s) vg(i).
Z −
(39) From this expression, it is immediate to see that the
H~ωerefowrethheavsheiifntetrdosdiuncgeled-ptahretincoletadtiisopnerǫ˜skio=noǫkfp+o2lagr|iΦto|2n−s, saynsdteamwoafvpeavreacltleolrfrKing=eskhasakfreqdueteenrcmyi~nΩed=byǫksth−e~iωnp-
p s p
k and ω respectively indicate the wave vector and the terference of the signal/idler−and the pump mode. The
p p
frequencyofthepump. Thedenominator∆kisdefinedin analytic form of the zero delay t = 0 fringe pattern
terms of the eigenvalues E(1,2) of the Bogolubov matrix resulting from the combination of sin and θ functions
k
is responsible for the dip at r = 0 that is visible in
ǫ˜k i~γ/2 gΦ2 Fig.2(b). The temporal decay of the correlation occurs
Mk = −gΦ∗2 ǫ˜k¯ i~γ/2 , (40) on the same time scale as the bare polariton decay rate
(cid:18) − − − (cid:19) γ. The spatial decay away from the butterfly edges (we
as have set R (r,t) = r v(s,i)t) occurs on a length scale
s,i g
∆k(ω)=(~ω−Ek(1))(~ω−Ek(1)∗)(~ω−Ek(2))(~ω−Ek(2)∗). κ−F1u=rth|∆ervmg|o/rγe,. the an−alytical formula (45) shows that
(41)
no correlation is present inside the cone marked by the
To better compare to the numerical results, we now re-
thin lines in the Fig. 2 and defined (for t > 0) by the
strict our attention to one-dimensional case and we ex-
condition R < 0 and R > 0 (for t < 0 the signs are
s i
tract analytic formulas for the most significant regimes
exchanged). The physics behind this fact is illustrated
considered in the previous subsection.
by the simple geometric construction shown in Fig. 6:
pairs of entangled signal/idler polaritons are generated
at all times and positions by the parametric conversion
1. Low-density parametric luminescence
ofquantumfluctuations intorealexcitations. Signaland
idler polaritons then propagate with group velocities re-
In the low-density limit Φ 0 the Bogolubov eigen- spectively vs,i and transport the correlation to distant
| |→ g
modes tend to the single-particleenergies with a negligi- pairs of points. It is easy to see that the shaded region
ble blue-shift [Fig. 1 (a, b)]. Comparing the expressions inside the cone can never be reached by such a process.
(38)and(39)forrespectivelythenormalandtheanoma- An analogousreasoning was used in Ref.9 to explain the
louscorrelations,itisimmediatetoseethattheformeris correlation signal observed in numerical calculations of
afactorg Φ2/~γ smallerandthereforenegligibleinthis Hawking radiation from acoustic black holes. As polari-
| |
limit. By performing the frequency integral in (37) with tonsdecayata rateγ,the sameconstructionshowsthat
the method of residuals, the spatial correlation function thecorrelationsignalhastodecayinspacewiththechar-
can be written as acteristic length κ−1 =∆v /γ.
g
The perfect agreement between this analytic approxi-
G¯(2)(r,t) 4πgSgn(t)e−γ|t|/2 mation and the numerical result is highlighted in Fig. 2
≃− × (b, c). The lower polariton parameters g, γ, ǫ are cal-
e−iǫkt/~ k
Re dkei(k−kp)r (42) culated from the linear eigenmodes of the GP equations
× (cid:26)Z ǫk+ǫk¯−2~ωp−i~γ(cid:27) (9) and (10). As expected for a Bogolubov theory24, for
8
previous subsection, we perform the frequency integra-
tion in (39) with the method of residuals and we expand
the function in k around k . Then, by retaining only
s,i
the dominant contribution (which, as expected, diverges
exactlyatthresholdγ =γ )andusingtherelations(46),
0
we finally obtain the expressions
2g E t
G¯(2)(r,t) e−(γ−γ0)|t|/2 cos(Kr 0 )
1 ≃ ~ − ~ ×
cos(kR)e−γ1k2|t|/2
dk , (47)
× γ γ +γ k2
Z − 0 1
and
4πg
G¯(2)(r,t) e−(γ−γ0)|t|/2
2 ≃ −~2γ ×
0
FIG. 6: Geometric construction to determine the pairs of eikR−γ1k2|t|/2
Re dk
ppaorinatmseitnricspluacmei-ntiemsceentcheatreagrime ec.orTrehleatsehdadinedtahreealohwi-gdhelnigshittys × (Z γ−γ0+γ1k2 ×
theregionofspace-timewhichisnotcorrelatedwiththepoint (E¯ +∆V(s)k+i~γ /2)e−iKr+iE0t/~ +
(r=0,t=0). Thepoint(r=0,t=0)isparametricallycor- × s g 0
relatedwiththepointα,asaresultoftheparametricemission + h(E¯ +∆V(i)k+i~γ /2)eiKr−iE0t/~ ,
event occurring at the point marked with a star. The same i g 0
holds for the point β. Solid lines mark the motion of signal io(48)
(blue) and idler (red) particles. No parametric event can in-
stead produce correlations between the point (r = 0, t = 0) with R = r V t, E¯ = ǫ˜ E , K = k k , and
− g s,i ks,i ∓ 0 s − p
and the point δ located inside the shaded region. ∆V(s,i) =V v(s,i).
g g g
−
Thisresultcanbefurthersimplifiedbyintroducingthe
approximations E¯ E¯ 0 and ∆V(s) ∆V(i) 0.
a given blue shift g Φ2 the rescaledcorrelationfunction s ≃ i ≃ g ≃ g ≃
| | In this case, the anomalous correlation disappears
is proportional to the nonlinear coupling constant g.
Before proceeding, it is interesting to note that for a 4πg
pump at normal incidence kp =0, one has ks = ki and G¯(22)(r,t) ≃ ~ e−(γ−γ0)|t|/2cos(Kr−E0t/~)×
−
a vanishing relativefrequency Ω=0. The spatialfringes e−γ1k2|t|/2
are then independent on time. From an experimental dksin(kR) =0, (49)
× γ γ +γ k2
point of view, this configuration appears to be the most Z − 0 1
suitableone toobservethe predictedfeaturesasitis less
as the integrand is odd in k, and the spatial correlations
subject to the finite time resolution of photodetectors.
are dominated by the normal contribution G¯(2) G¯(2).
≃ 1
The explicit expression of G¯(2) is given in (47): the
1
Fourier transform of the product of a Gaussian and a
2. OPO critical region
Lorentzian function corresponds, in real space, to the
convolution of an exponential and a gaussian function.
For pump intensities just below the OPO threshold,
At zero delay t =0 [Fig. 4 (b)], the Gaussian reduces
g Φ2 . g Φ 2 = ~γ/2, the Bogolubov spectrum
| | | OPO| to a delta-function in space and the correlation signal
strongly differs from the single-particle one [Fig. 1 (e,
showsanexponentialdecayinspacewithacharacteristic
f)]. In particular, the imaginary part of one of the two
length
eigenvalues tends to zero in the vicinity37 of k = k
s
and k = ki = 2kp − ks. In this case, it is easy to ℓ γ1 . (50)
see that for wavevectors in the vicinity of k one has
s,i ≃ γ γ
Re[E(1) ]= Re[E(2) ] and Re[E(1) ]= Re[E(1) ]. r − 0
ks,i+k ks,i+k ki−k − ks+k As expected on the basis of general arguments on phase
Inthis region,the eigenvaluescanbe approximatedwith
transition,andpreviouslyobservedinMonteCarlosimu-
the expression
lations15,thecharacteristiclengthℓdivergesastheOPO
Ek(1,2)−k−→−k−s→,i±E0+~Vg(k−ks,i)−i2~[γ∓γ0±γ1(k−ks,i)2], tphrreessshioonld(4is5)apfoprrothaechwedeaγk0in→tenγs.itIynccaosne,trtahsetzteorothdeeelaxy-
(46) t = 0 fringe pattern now has a cos form with a simple
where V =∂ E /~=∂ E /~ and ~γ =2g Φ2 . bunching peak at r =0.
~γ, thegequaklityk|khsolding kexakc|ktliy at thres0hold Φ| 2| = At longer times, the spatial width of the Gaussian
Φ 2. Followingtheprocedurealreadyadopted| in| the grows as √t, so that for short to intermediate distances,
OPO
| |
9
the dependence is dominated by the Gaussian factor. The most apparent deviation between the analytical
This effect is clearly visible in Fig. 4 (c). The overall form (53) and the numerical result shown Fig.5(c) con-
exponentialdecayintimeoccursonacharacteristictime sists of a tail in the analytic approximationthat extends
up to large distances. This has a simple interpretation:
1
τ . (51) the quadraticapproximationofthe dispersioneliminates
≃ γ γ0 allboundsinthegroupvelocityandpredictscorrelations
−
at any distances. In contrast, the correct dispersion has
which again diverges as the critical point is approached.
an upper bound vmax to the group velocity, which re-
g
stricts the possible correlations to the r < vmax t re-
| | g | |
gion marked by the thin lines in Fig. 5(a) and (c): this
3. Non-parametric regime
intepretationis confirmedby the muchbetter agreement
of the (52) prediction where the group velocity is cor-
The non-parametric regime shown in Fig. 1 (g,h)
rectly taken into account.
corresponds to the case where the polariton and ghost
branches do not intersect, i.e. ǫ˜k + ǫ˜k¯ 6= 0 for all k.
This regime is generally realized when the frequency of IV. TWO-DIMENSIONAL AND DISORDERED
the pump is very much red-detuned with respect to the SYSTEM
renormalized polariton dispersion and, in our configura-
tion,isfulfilledforpumpintensitiesabovethe bistability
In this final section we apply our model to the more
loop.
generalcaseofatwodimensionalinhomogeneoussystem.
In this regime, the Bogolubov eigenmodes can be ap-
Inparticular,we wishto investigatehow the conclusions
proximated by the single particle dispersion blue-shifted
of the previous sections are affected by the presence of
by the interaction, i.e. E ǫ˜ i~γ/2 and an equation
k ≃ k− exciton and photon disorder.
formally equivalent to (42) still holds. However, since
Realistic system parameters for a GaAs microcavity
the denominator remains finite for all k, no pole can be with N = 2 quantum wells are used, with ~Ω = 3.5
R
identified. However, the region of the polariton disper- meV, ~γ = ~γ = ~γ = 0.2 meV, zero exciton-photon
c x
sionthat minimizes the denominatorgives the dominant
detuning. Forthe nonlinearinteractionconstantwetake
contribution. In the case of Fig. 1 (g), this happens at g = 1.5 10−3 meV µm2 and g = 0.5 10−3 meV
x s
k = kp. It is therefore convenient to expand the energy µm2. We×consider a configuration where ×the pump is
dispersionatsecondorderin k−kp. Performingthis ap- orthogonaltothecavityplane,kp =0andwetake~ωp
proximationin the denominator,weobtainthe following −
ǫ (0) = 3 meV. This orthogonal pump configuration
c
formula for the correlationfunction: −
is the most suitable one in view of experiments, as it is
g Φ2 least affected by the temporal resolution of the photon
G¯(2)(r,t) 4g(π | | 2)e−γ|t|/2 (52) detectors.
≃ ǫ˜ − ×
kp Thecorrelationpatternfordifferentvaluesofthepump
eiǫ˜k+kpt/~ intensity,correspondingtothelow-densitylimit,thecrit-
Re dke−ikr .
× (Z 2ǫ˜kp +2upk2+iSgn(t)~γ) iacraelsOhoPwOnrinegFioing.a7nd(at,hbe,nco)nfo-praaratmweotrdiicmceonnsfiiognuarlatsiyosn-
To further simplify this expression, we can expand also temintheabsenceofdisorder. Allthefeaturesdiscussed
the energy exponents, which leads to inthe previoussectionforthe 1Dcasearestillapparent.
In particular, in the low-density case, the correlations
G¯(2)(r,t) ≃ 4g(πg˜ǫ|Φ|2 −2)e−γ|t|/2Re eiǫ˜kpt/~× (53) dbiesianpgptehaergfororudpisvtaenlocceistisems oaflleprotlahraitto|nrmsainx(tth)|e=sigvngat,l±anvdg
kp n idler modes, and they decayexponentially. On the other
dke−ikR eiupk2t/~ . hand, in the vicinity of the OPO threshold, correlations
× Z 2ǫ˜kp +2upk2+iSgn(t)~γ) extendeverywhere. Inthenon-parametricconfiguration,
correlations are non vanishing only for distances smaller
Here, we have set R = r −vg(p)t, vg(p) = ∂k˜ǫk|kp/~ and than r =vgmaxt, vgmax being the largest group velocity.
u =∂2˜ǫ /2.
p k |kp
Theintegraloverk isoftheFresnelkindanddescribes
theinterferenceproducedatthepoint(R,t)bythediffer- We assume white noise disorder for the exciton field
entk-modeswithagappedandquadraticdispersion. For of amplitude 2 meV and a gaussian-correlated disorder
zerotimedelayt=0,thecorrelationshaveatypicalanti- for the photonfield, withamplitude Uc =0.5 meVand
dis
bunching character: they are everywhere negative and a correlation length of ξ = 7 µm. The results of the
c
arestrongestatr =0. Furtheraway,theymonotonically numerical calculations are summarized in Fig. 7 (d-i):
tend to zero with an exponential law of characteristic the profile of the coherent photon field Φ (r)2 in the
c
length κ−1 = up/ǫ˜kp determined by the gap between three cases is shown in panels (g-i) and t|he cor|respond-
the renormalized polariton and ghost branches38. ingspatio-temporalpatternsofcorrelationsareshownin
p
10
x 10−3 x 10−3 x 10−4 the polariton modes is able to prevent parametric oscil-
20 1 2 1
(a) (b) (c) lation.
s) For each regime, we have identified the key features
p 0 0 0 0
t ( of the correlation pattern. In the uniform case, we have
−20 −1 −2 −1 compared our results to the predictions of approximate
x 10−3 x 10−3 x 10−4 analytic models which provide a physical interpretation
20 1 2 1
(d) (e) (f)
totheobservedpatterns. Anorthogonalpumpgeometry
ps) 0 0 0 0 appears as the optimal choice for the experimental ob-
t ( servation as it reduces the required temporal resolution
−20 −1 −2 −1 of photo-detectors. We have verified that the correla-
2y=0)| units)2(g) (h) (i) tdiiosnorpdaetrtearsnlsonargeansotthqeucaolihtaerteivnetlypomlaordiitfioendsbreymaarineadliesltoic-
Φ| (x,c(arb. 01 calTizheedcinonscplaucseio.ns of the present work confirm the ex-
−20 0 20 −20 0 20 −20 0 20
x (µm) x (µm) x (µm) pectation that intensity correlations can be a very pow-
erful tool to study the dynamics of quantum fields in
FIG. 7: Intensity correlation pattern G¯(2)(r,t) as a function condensed matter systems. A study of more complex
ofthespatialcoordinatexandofthetemporaldelaytalonga geometries is presently under way.
y=0linefora2Dsystem. Panels(a-c)areforauniformcase,
while panels(d-f)are foradisordered system. Thepumpin-
tensity is varied from a low value (a, d), to a value close to VI. ACKNOWLEDGEMENTS
theOPOthreshold(b,e)andtoavaluewellabovethebista-
bility loop (c, f) where parametric processes are forbidden.
Panels (g-i) show the coherent photon field profiles |Φc(r)|2 WearegratefultoCristianoCiutiforcontinuousstim-
ulatingdiscussions. D.S.acknowledgesthefinancialsup-
corresponding tothe disordered case of panels (d-f).
portfromthe Swiss NationalFoundation(SNF) through
the fellowship number PBELP2-125476.
panel (d-f). The realization of the disorder potential is
the same for the three values of the pump intensity.
Even if it is responsible for large density modulations, VII. APPENDIX
the considereddisorderisneverabletodestroythe near-
field correlation pattern. Of course the pattern would In this Appendix, we discuss the connection between
eventuallydisappearifamuchstrongerdisorderwascon- the photon field inside the cavity and the emitted light.
sidered,thatisabletofragmentthecoherentfieldindis- In particular, we show that the calculated in-cavity in-
connected parts. However, this latter situation is quite tensitycorrelationsdirectlytransfertothenear-fieldcor-
unusual for state-of-the-art GaAs microcavities. relations of the emitted light.
From the comparison with the corresponding correla- Weconsideraplanarcavityalongthex yplanewitha
−
tion patterns for a clean system [Fig. 7 (a-c)], we can quantumwellplacedatz =0andmirrorswhoseexternal
still appreciate a slight modification in the pattern. Dis- surface is at z = z . In Fourier space, the external
m
orderisresponsibleforthesemodificationsviathreemain field E (k,ω), re±sulting from the transmission across
out
effects: i) the exciton and photon resonances are broad- the mirror is related to the in-cavity field E (k,ω) at
in
ened; ii)k-conservationisbroken,whichsoftensthe con- the quantum well position via the complex transmission
dition for parametric processes; iii) the group velocity is coefficient τ(k,ω)=T1/2eiφ(k,ω):
no longer a well defined concept, so the contour of the
butterfly shape in Fig. 7 (d) is smearedout with respect d2k dω
to panel (a). Eout(r,z,t)= (2π)2 2πei(k·r−ωt)eikz(k)(z−zm)×
Z Z
τ(k,ω)E (k,ω).
in
×
V. CONCLUSIONS
For simplicity, we have neglected here the k- and ω-
dependence of the transmittivity T as we are resticting
Wehavedevelopedaformalismtocomputethesecond
to frequencies well within the stop band of the mirror
order spatial correlation function of polaritons in a pla-
where T is almost constant (see Fig.8(a)).
nar microcavity. This quantity directly transfer into the
For frequencies close to the k=0 cavity frequency ω
near-field intensity correlations of the emitted light. 0
and at low wave vectors k, the phase of the transmis-
Wehavecomputedthe spatio-temporalpatternofcor-
sion coefficient can be accurately approximated by the
relations in different pumping regimes, ranging from the
expansion
parametric luminescence regime, to the critical region
just below the parametric oscillation threshold, and to c
φ(k,ω)=φ(0,ω )+∆ (ω ω ) ∆ k2, (54)
the strong pumping regime where a large blue-shift of 0 t − 0 − 2ω z
0