Cavity-funneled generation of indistinguishable single photons from strongly dissipative quantum emitters

4 MB·English

by  Thomas Grange

Cavity-funneled generation of indistinguishable single photons from strongly
dissipative quantum emitters
Thomas Grange,1,2 Gaston Hornecker,1,2,3 David Hunger,4 Jean-Philippe
Poizat,1,2 Jean-Michel G´erard,1,3 Pascale Senellart,5 and Alexia Auff`eves1,2
1Universit´e Grenoble-Alpes, “Nanophysics and Semiconductors” joint team, 38000 Grenoble, France
2CNRS, Institut N´eel, “Nanophysics and Semiconductors” joint team, 38000 Grenoble, France
3CEA, INAC-SP2M, “Nanophysics and Semiconductors” joint team, 38000 Grenoble, France
4Fakult¨at fu¨r Physik, Ludwig-Maximilians-Universita¨t, Schellingstr. 4, 80799 Mu¨nchen, Germany
5CNRS, Laboratoire de Photonique et de Nanostructures, 91460 Marcoussis, France
(Dated: February 9, 2015)
5
1
We investigate theoretically the generation of indistinguishable single photons from a strongly
0
dissipative quantum system placed inside an optical cavity. The degree of indistinguishability of
2
photonsemittedbythecavityiscalculatedasafunctionoftheemitter-cavitycouplingstrengthand

b thecavitylinewidth. Foraquantumemittersubjecttostrongpuredephasing,ourcalculationsreveal
e that an unconventional regime of high indistinguishability can be reached for moderate emitter-
F cavity coupling strengths and high quality factor cavities. In this regime, the broad spectrum of
the dissipative quantum system is funneled into the narrow lineshape of the cavity. The associated
6
efficiency is found to greatly surpass spectral filtering effects. Our findings open the path towards

] on-chip scalable indistinguishable-photon emitting devices operating at room temperature.
l
l
a
h Indistinguishable single photons are the building typicallyseveralordersofmagnitudelargerthanthepop-
-
blocks of various optically-based quantum information ulationdecayrate(typicallyrangingfrom3to6ordersof
s
e applications such as linear optical quantum computing magnitude) [12, 14, 21, 22, 24, 26–29, 31]. Hence the in-
m
[1, 2], boson sampling [3–7], quantum teleportation [8] trinsic indistinguishability given by Eq. 1 is almost zero.
. or quantum networks [9]. Indistinguishable photons are A possible way to enhance the indistinguishability is to
t
a usually generated either using parametric down conver- spectrally filter the emitted photons a posteriori. How-
m
sion [10], or alternatively directly from a single two-level ever, this linear-filter strategy leads to a very low effi-
- quantum emitter such as atoms, color centers, quantum ciency. Engineeringofbothefficiencyandindistinguisha-
d
dotsororganicmolecules[11–20]. Parametricdowncon- bility are possible by placing the dissipative QE in an
n
o version is presently the most mature technology avail- opticalcavity[15,27,32–44]. Ausualstrategyisthento
c able, but the usual low efficiency of the nonlinear pro- use the Purcell effect to enhance the spontaneous emis-
[
cesses is a severe limitation to the scalability of such sion, as in Eq. 1 an increase in γ results in an increase

2 sources. On the other hand, sources based on single of I. However, reaching Purcell factors larger than γ∗/γ
v solid-state quantum systems have been greatly devel- for room-temperature solid-state systems appears to be
1 opped in the last decade, as they hold the promise to well beyond the present experimental state of the art.
3
9 combine indistinguishable, on-demand, energy-efficient, In this letter we propose a realistic and robust way to
0 electrically drivable and scalable characteristics. How- generate highly indistinguishable photons from strongly
0 ever, except at cryogenics temperature, solid-state sys- dissipative QE (i.e. for γ∗ (cid:29) γ). The idea is to exploit
1. tems emitting single-photons are subject to strong pure a cavity-quantum-electrodynamics (cavity-QED) regime
0 dephasing processes [21–29], making them at first view oflowcavitylinewidthandmoderatecavity-emittercou-
5 inappropriateforquantumapplicationsrequiringphoton pling,inwhichthebroadspectrumofthedissipativeQE
1 indistinguishability. is funneled into the narrow emission line of the cavity.
:
v Atwo-levelquantumemitter(QE)coupledonlytovac- In this regime, high indistinguishability is predicted to-
i uum fluctuations should emit perfectly indistinguishable gether with efficiencies orders of magnitude higher than
X
photons. However, as soon as pure dephasing of the QE spectral filtering. Insights into the full quantum calcu-
r
a occurs, the degree of indistinguishability of the emitted lation are gained by semiclassical derivations of indistin-
photons is reduced to [30] guishability in limiting cases of dissipative cavity QED.
γ T As depicted in Fig.1, we consider a two-level QE sys-
I = = 2 , (1) tem {|ψ (cid:105),|ψ (cid:105)} coupled to a cavity mode whose Fock
γ+γ∗ 2T g e
1
states are denoted {|0(cid:105),|1(cid:105),...}. All the dissipative terms
where γ = 1/T is the population decay rate, γ∗/2 = are assumed to be described within the Markov approx-
1
1/T∗ the pure dephasing rate, and 1/T =2/T +1/T ∗ imation [2, 46]. The relevant parameters are: the QE
2 2 1 2
the total dephasing rate. For solid-state QE emitting decayrateγ (whichmayincluderadiativeaswellasnon-
photonsatroomtemperaturesuchascolorcenters,quan- radiative components), the cavity decay rate κ, the pure
tum dots or organic molecules, pure dephasing rates are dephasing rate γ∗ ; g is the dipolar coupling between
2
emitters verifying γ∗ (cid:29) γ. Without any cavity, the de-
gree of indistinguishability would then amount to 10−4
accordingtoEq.1. InFig.2(a),I isplottedasafunction
∗ 𝛾𝛾 of the coupling g and the cavity linewidth κ while γ and
𝛾𝛾 γ∗ arefixed. Tworegionsofhighindistinguishabilityare
found, which are discussed below. The one in the upper-
𝜅𝜅
right corner corresponds to very large couplings g and
𝑔𝑔
broad cavities such as g > γ∗ and κ > γ∗, which is ex-
FIG. 1. Schematic of the system under study: a dissipative tremely challenging to reach experimentally for strongly
two-level emitter coupled to an optical-cavity mode. dissipative emitters (i.e. for large values of γ∗). The
other region of high indistinguishability, in the lower-left
corner, appears for good cavities for κ<γ together with
the QE and the cavity mode (see Fig.1). The emitter-
moderate or small coupling values g. As these values are
cavitydetuningissettozero(i.e. perfectresonance). For
within experimental reach, this unconventional regime is
simplicity, we assume an instantaneous excitation of the
highly attractive for the generation of indistinguishable
QE,sothatonlyonequantumofexcitationcanbetrans-
photons.
ferred to the cavity. Within the rotating-wave approxi-
To get a physical insight into the calculated indistin-
mation, it is therefore sufficient to investigate the dissi-
guishability, we divide the (κ,g) space into three regimes
pativequantumdynamicsinthetwo-dimensionalHilbert
labelled from “1” to “3” in Fig. 2 and study the cor-
spaceformedby{|ψ ,0(cid:105),|ψ ,1(cid:105)}. Thedegreeofindistin-
e g responding limiting cases below. First, we can distin-
guishability of photons can be defined by the probability
guish between the QE-cavity coherent regime occuring
of two-photon interference in a Hong-Ou-Mandel exper- for 2g > κ + γ + γ∗ and the incoherent regime 2g <
iment [47]. For a single-photon emitter, this indistin- κ+γ+γ∗. Inthecoherentregime,labelled“1”inFig.2,
guishability figure of merit reads [30, 48]:
thedynamicsconsists ofdampedRabioscillationswhich
(cid:82)∞dt(cid:82)∞dτ|(cid:104)aˆ†(t+τ)aˆ(t)(cid:105)|2 evolves into an incoherent population of the two polari-
I = 0 0 (2) ton modes (i.e. mixed QE-cavity state). In the limit
(cid:82)∞dt(cid:82)∞dτ(cid:104)aˆ†(t)aˆ(t)(cid:105)(cid:104)aˆ†(t+τ)aˆ(t+τ)(cid:105) where 2g (cid:29) κ+γ +γ∗, the indistinguishability degree
0 0
reads [49]:
where a(†) are the ladder operators of the EM mode in
(γ+κ)(γ+κ+γ∗/2)
which the photons are emitted. This equation imposes
I = . (6)
the necessary condition for perfectly indistinguishable cc (γ+κ+γ∗)2
photons that time correlations of the EM field decay the Asthisexpressionisindependentofg,increasingg alone
same way as the intensity, i.e. that photons are Fourier- does not allow to reach arbitrarily large indistinguisha-
transform limited. The calculation of the above quanti- bility, as seen on Fig. 2. On the contrary, nearly-perfect
ties can be separated into two steps (see the Supplemen- indistinguishability occurs in the coherent regime only if
talMaterial[49]). First,wecalculatetheevolutionofthe 2g ≥κ(cid:29)γ∗.
density matrix ρˆ(t) by solving the Lindblad equation. In the incoherent limit (2g (cid:28) κ + γ + γ∗), the dy-
namics of the system can be described in terms of inco-
ρˆ(t)=e−iLˆt|ψe,0(cid:105)(cid:104)ψe,0| (3) herent population transfer with an effective transfer rate
between the QE and the cavity given by [2, 46]:
where Lˆ is the total Liouvillian of the system [49]. Sec-
ondly, we calculate the retarded Green’s function, which 4g2
R= . (7)
reads in the {|ψ ,0(cid:105),|ψ ,1(cid:105)} basis: κ+γ+γ∗
e g
Within this incoherent regime, we can further define a
GˆR(ω)=(cid:18)ω+iγ/2+iγ∗/2 g (cid:19)−1. (4) bad-cavity regime for κ>γ+γ∗ (labelled “2” in Fig. 2)
g ω−δ+iκ/2 and a good-cavity regime for κ < γ +γ∗ (labelled “3”
in Fig. 2) [2, 50]. In the bad-cavity limit κ (cid:29) γ +γ∗,
The two-time correlator of the cavity field can be ex-
the cavity can be adiabatically eliminated, and its sole
pressedasaproductoftheretardedpropagatorGˆR(τ)=
−i(cid:82) dωe−iωtGˆR(ω) and the density matrix [49]: effect is to add a new channel of irreversible radiative
emission at a rate R. Reabsorption by the QE is then
(cid:104)a†(t+τ)a(t)(cid:105) =(cid:104)ψ ,1|GˆR(τ)ρˆ(t)|ψ ,1(cid:105). (5) negligible. Thedynamicsofthecoupledsystemcanthen
g g be described by the one of an effective QE with a decay
rate γ+R. Applying Eq. 1 to this effective QE leads to
In Fig. 2, we report calculations for strongly dissipa-
tive emitters verifying γ∗ =104γ. This is a typical value an indistinguishability of
for a solid-state QE at room temperature, and the re- γ+R
I = . (8)
sultsarequalitativelysimilarforanystronglydissipative bc γ+R+γ∗
3
0
1 0 6 Within this regime of incoherently-coupled bad cavity,
g(cid:1) = the usual strategy to increase indistinguishability is ba-
1 0 5 1 0.3 0.7 0.9 sically to maximize R. This can be done by increasing g
0.6 0.8 and/orminimizingκ. However,fromEq.S38,near-unity
1 0 4 2 g = g + g*+ k 0.4 0.5 indistinguishability requires R (cid:29) γ∗ and consequently
0.2
1 0 3 2 2g (cid:29) γ∗.It is found that the coupling strength g has to
exceed γ∗ by nearly one order of magnitude in order to
g / 1 0 2 0.1 0.1 R = (cid:1) 01.9 reach an indistinguishability value of I = 0.9. Reaching
g1 0 1 0.7 0.3 3 00..78 suchcouplingistechnologicallyextremelychallengingfor
0.9 0.5 0.2 0.6 solid-state emitters under ambient temperature.
1 0 0 0.8 0.4 00..45 On the other hand, the incoherent good-cavity regime
0.6 0.3 (labelled “3” in Fig. 2) occurs for κ < γ +γ∗. In this
1 0 -1 g * 000..12 g*/g rsgec*ga/gilme ec,omthpearcaabvlietytocanorstloonregetrhethpahnottohnesQwEithdinepahatsiimnge
1 0 -2 k = time. The cavity itself then acts as an effective emitter
( a ) 1 0 -2 1 0 -1 1 0 0 1 0 1 k 1 0 2g 1 0 3 1 0 4 1 0 5 1 0 6 incoherently pumped by the QE [49], so that the cavity
/
field correlations read
1 0 6
(cid:104)a†(t+τ)a(t)(cid:105) =ρ (t)e−Γcτ/2, (9)
cc
1 0 5 1
where ρ (t) is the population of the cavity mode and Γ
cc c
1 0 4 2 g = g + g*+ k isthelinewidthofthecavity-likeeigenstate. FromEq.4,
0 .9 9 9 2 one can derive that Γ =κ+R, which is the sum of the
1 0 3 0 .1 0 0 c
3 0 .9 9 0 cavity decay rate κ into EM modes plus the incoherent
1 0 2 R = (cid:1) 0 .3 0 0 0 .7 0 0 0 .9 0 0 0 .8 0 0 0 .6 0 0 reabsorption rate R between the cavity and the QE. By
gg / 1 0 1 0 .5 0 0 0 .4 0 0 0 .2 0 0 01..90000 stholevirnegsutlhtiengpocpauvliatytiodnynraamteicesquρat(iot)nsinanEdqsb.y9palungdgEinqg.i2n,
0 .0 1 0 0 0.800 cc
0.700 one finds an indistinguishability of [49]:
1 0 0 1 .0 0 E -0 3 0.600
0.500
0.400 γ+ κR
11 00 --21 kg = kg = * 0000....01230000000 Igc = γ+κκ++R2R (10)
( b ) 1 0 -2 1 0 -1 1 0 0 1 0 1 k 1 0 2g 1 0 3 1 0 4 1 0 5 1 0 6 Consequently, large indistinguishability o√ccurs in this
/ regime for κ < γ and R < γ (i.e. g < γγ∗/2) , in
1 0 6 agreement with the full calculation shown on Fig. 2(a).
This can be understood by noting that two ingredients
1.0
1 0 5 are involved in the degradation of indistinguishability in
1
10 thisgood-cavityregimewherethecavityactsastheeffec-
1 0 4 2 g = g + g*+ k
tiveemitter. Thefirstpointisthattheinitialincoherent
2
1 0 3 feeding of the cavity occurs on a time scale 1/γ, produc-
1.0E + 03 ing a time uncertainty in the population of the cavity.
1 0 2 R = (cid:1) 3 Hence κ has to be kept small compared to γ in order to
g / 100 1.0E +04 prevent such time-jittering effect, in analogy to the inco-
g1 0 1
1.0E +03 herent pumping of QE via high energy states [48]. The
10
1 0 0 1.0 100 second point is that, after the initial filling of the cav-
ity, incoherent exchange processes between the QE and
1 0 -1 10 the cavity can still occur. However, back and forth inco-
1 0 -2 kg = kg = * 1.0 herent hopping between the cavity and the QE leads to
the dephasing of the photons emitted by the cavity. To
( c ) 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6
k g
/ prevent such detrimental hopping, R<γ is required.
We now discuss the efficiency of the photon emission
FIG. 2. (a) Indistinguishability figure of merit (I), (b) ef- from the cavity mode, i.e. the probability to have emis-
ficiency β and (c) funneling ratio F, plotted as a function sion by the cavity mode per initial excitation of the QE.
of the cavity linewidth κ/γ and the emitter-cavity coupling
The efficiency of photon emission in the cavity mode is
strengthg/γforafixedratioγ∗/γ =104. Thefunnelingratio
given by
is defined as F = βIγ∗/γ. Solid lines delimit three different
regimes discussed in the text: coherent coupling (“1”), inco- (cid:90) ∞
herent coupling and bad cavity regime (“2”), and incoherent β =κ (cid:104)a†(t)a(t)(cid:105). (11)
coupling and good cavity regime (“3”). 0
4

In Fig. 2(b) the efficiency β is plotted as a function of
) 1 .0
the cavity linewidth κ and the emitter-cavity coupling b (
y 3 0 0
strengthg. Near-unitary(i.e. on-demand)efficienciesare nc 0 .8 I
ie
orebgtiaminee,dwienfitnhdeupper-rightcorner. Intheweak-coupling d effic 0 .6 2 5 0 (F)
Efficiencies largerβth=anκR0.+5 γκa(Rrκe+tyRp)ically obtained(1fo2r) uishability (I) an0 00.0..141 (cid:1) F 112 050 000 Funneling ratio
g
R > γ and κ > R. This is compatible with high indis- tin
tinguishability in the region of high g and high κ values dis1 E -3 5 0
In
(i.e. right-upper corner in Fig. 2), but not in the good 1 E -4
0
cavityregime(i.e. region“3”inFig.2). Nevertheless, as 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6
discussed in the following, the product of efficiency and C a v ity lin e w id th k /g
indistinguishabilityβI inthegoodcavityregimecanstill
be way above the one obtained by any linear spectral fil- FIG.3. IndistinguishabilityfigureofmeritI (fullcalculation
in black solid line), efficiency β (blue solid line) and cavity
tering technique. Let us consider a linear spectral filter,
funneling ratio F (red solid line) for a fixed emitter-cavity
with a narrow spectral range ∆ν , through which the
f coupling of g = 10γ. The full calculation for indistinguisha-
spectrumofthebroadQEissent. Weassume∆ν (cid:28)γ∗.
f bility is compared with analytic expressions in two different
The output efficiency is bounded by βf ≤ ∆νf/γ∗. Due limitingcases: incoherently-coupledgoodcavityregime(dot-
totheFourier-transformcondition,thecorrespondingin- ted line, Eq. S47) and coherently-coupled bad cavity regime
distinguishability is bounded by I ≤γ/∆ν . Hence the (dashed line, Eq. S38). The analytical expression for β (dot-
f f
efficiency-indistinguishability product for spectral filter- ted line, Eq. 12) overlaps perfectly with the full calculation.
ing cannot exceed β I ≤ γ/γ∗. In order to compare
f f
β×I in the present cavity-QED scheme with the upper
limit for spectral filtering, we define a cavity-funneling
factor µeV [51]. Assuming (cid:126)γ = 60 µeV and (cid:126)γ∗ = 7 meV for
γ∗ an InAs/GaAs QD at 300K [21, 52], we predict I=0.72,
F = βI, (13)
γ β=0.088 and F=7.3. Secondly, we consider a single sili-
con vacancy (SiV) center in a nano-diamond coupled to
such that F values larger than unity necessarly indicate
a fiber cavity. For SiV at 300K, we take γ = 2π× 160
a spectral cavity-funneling effect. F indicates the min- MHz and γ∗ = 2π× 550 GHz [53]. Coupling SiV with a
imum enhancement ratio of β ×I with respect to any
fiber cavity with g=2π×1.0 GHz and κ=2π×30 MHz is
spectral-filteringeffect. Inpracticethisenhancementwill
within experimental reach [39], for which our calculation
be larger since light emitted from a cavity can be very
predicts I=0.81, β=0.035 and F=99. These predicted
efficiently collected [15], in contrast to free-space spon-
degrees of indistinguishability at room temperature are
taneous emission. In Fig. 2(c), the funneling F is plot-
comparable with state of the art values obtained from
ted in the same parameter range (κ,g) as previous plots.
low temperature single-photon sources under incoherent
Only the values satisfying the cavity-funneling condition
pumping [15], with efficiencies far beyond any spectral
of F > 1 are shown. It appears clearly that almost-
filtering technique.
perfect indistinguishability in the good cavity regime is
compatible with cavity funneling. In Fig. 3, I, β and F In summary, for strongly dissipative emitters we pre-
are plotted as a function of κ for a fixed value of g. The dict an unconventional regime of high indistinguishabil-
fullcalculationisfoundtobeingoodagreementwiththe ity in which the broad spectrum of the quantum emitter
above formulae for the incoherent regime. It illustrates is funneled into a narrow cavity resonance. For typi-
the necessary trade-off between indistinguishability and cal room-temperature quantum emitters, the associated
efficiency in the good-cavity regime, where a clear maxi- efficiency can surpass any spectral filtering schemes by
mumofthefunnelingfactoroccurs. Thelargecalculated orders of magnitude. This strategy opens the road to-
values for F are signatures of a very efficient redirection wards the generation of indistinguishable single photons
of the QE spectrum into the unperturbed cavity spec- from solid-state quantum emitters under ambient tem-
trum of linewidth κ. perature.
Finally, we propose two experimental realizations of
this unconventional regime at room temperature. We This work has been funded by the European Union’s
firstconsiderasingleself-assembledquantumdotcoupled Seventh Framework Programme (FP7) under Grant
to a photonic crystal cavity. State of the art photonic AgreementNo. 618078(WASPS).JasonSmithandJohn
crystal cavities can provide (cid:126)g =120 µeV and (cid:126)κ=20 Rarity are acknowledged for fruitfuil discussions.
5
[25] T. Grange, Phys. Rev. B 80, 245310 (2009).
[26] S. Bounouar, M. Elouneg-Jamroz, M. d. Hertog,
C. Morchutt, E. Bellet-Amalric, R. Andr´e, C. Bougerol,
[1] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, Y. Genuist, J.-P. Poizat, S. Tatarenko, et al., Nano Lett.
46 (2001). 12, 2977 (2012).
[2] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. [27] R. Albrecht, A. Bommer, C. Deutsch, J. Reichel, and
Dowling,andG.Milburn,Rev.Mod.Phys.79,135(2007). C. Becher, Phys. Rev. Lett. 110, 243602 (2013).
[3] J. B. Spring, B. J. Metcalf, P. C. Humphreys, [28] R. Albrecht, A. Bommer, C. Pauly, F. Mu¨cklich, A. W.
W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, Schell, P. Engel, T. Schro¨der, O. Benson, J. Reichel, and
N.Thomas-Peter,N.K.Langford,D.Kundys,etal.,Sci- C. Becher, Appl. Phys. Lett. 105, 073113 (2014).
ence 339, 798 (2013). [29] L. J. Rogers, K. D. Jahnke, T. Teraji, L. Marseglia,
[4] M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, C.Mu¨ller,B.Naydenov,H.Schauffert,C.Kranz,J.Isoya,
S.Aaronson,T.C.Ralph,andA.G.White,Science339, L. P. McGuinness, et al., Nat. Commun. 5 (2014).
794 (2013). [30] J. Bylander, I. Robert-Philip, and I. Abram, Eur. Phys.
[5] A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. J. D 22, 295 (2003).
Galva˜o,N.Spagnolo,C.Vitelli,E.Maiorino,P.Mataloni, [31] B. Lounis and W. Moerner, Nature 407, 491 (2000).
and F. Sciarrino, Nat. Photonics 7, 545 (2013). [32] J.-M. G´erard, B. Sermage, B. Gayral, B. Legrand,
[6] M.Tillmann,B.Daki´c,R.Heilmann,S.Nolte,A.Szameit, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81,
and P. Walther, Nat. Photonics 7, 540 (2013). 1110 (1998).
[7] N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, [33] L. C. Andreani, G. Panzarini, and J.-M. G´erard, Phys.
A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ram- Rev. B 60, 13276 (1999).
poni, P. Mataloni, et al., Nat. Photonics 8, 615 (2014). [34] E.Moreau,I.Robert,J.-M.G´erard,I.Abram,L.Manin,
[8] D. Fattal, E. Diamanti, K. Inoue, and Y. Yamamoto, and V. Thierry-Mieg, Appl. Phys. Lett. 79, 2865 (2001).
Phys. Rev. Lett. 92, 037904 (2004). [35] J.Reithmaier,G.Sek,A.Lo¨ffler,C.Hofmann,S.Kuhn,
[9] H. Kimble, Nature 453, 1023 (2008). S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke,
[10] M. Eisaman, J. Fan, A. Migdall, and S. Polyakov, Rev. and A. Forchel, Nature 432, 197 (2004).
Sci. Instrum. 82, 071101 (2011). [36] S. Varoutsis, S. Laurent, P. Kramper, A. Lemaˆıtre,
[11] C. Santori, D. Fattal, J. Vuˇckovi´c, G. S. Solomon, and I. Sagnes, I. Robert-Philip, and I. Abram, Phys. Rev. B
Y. Yamamoto, Nature 419, 594 (2002). 72, 041303 (2005).
[12] A. Kiraz, M. Ehrl, T. Hellerer, O¨. Mu¨stecaplıog˘lu, [37] K. Hennessy, A. Badolato, M. Winger, D. Gerace,
C. Bra¨uchle, and A. Zumbusch, Phys. Rev. Lett. 94, M.Atatu¨re,S.Gulde,S.Fa¨lt,E.L.Hu,andA.Imamog˘lu,
223602 (2005). Nature 445, 896 (2007).
[13] R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. [38] Z.Di,H.V.Jones,P.R.Dolan,S.M.Fairclough,M.B.
Ritchie, and A. J. Shields, Nat. Photonics 4, 632 (2010). Wincott, J. Fill, G. M. Hughes, and J. M. Smith, New J.
[14] R. Lettow, Y. Rezus, A. Renn, G. Zumofen, E. Ikonen, Phys. 14, 103048 (2012).
S. Go¨tzinger, and V. Sandoghdar, Phys. Rev. Lett. 104, [39] H. Kaupp, C. Deutsch, H.-C. Chang, J. Reichel, T. W.
123605 (2010). Ha¨nsch,andD.Hunger,Phys.Rev.A88,053812(2013).
[15] O.Gazzano,S.M.deVasconcellos,C.Arnold,A.Nowak, [40] J. Riedrich-Moller, C. Arend, C. Pauly, F. Mucklich,
E.Galopin,I.Sagnes,L.Lanco,A.Lemaˆıtre,andP.Senel- M. Fischer, S. Gsell, M. Schreck, and C. Becher, Nano
lart, Nat. Commun. 4, 1425 (2013). Lett. 14, 5281 (2014).
[16] Y.-M. He, Y. He, Y.-J. Wei, D. Wu, M. Atatu¨re, [41] P. Pathak and S. Hughes, Phys. Rev. B 82, 045308
C. Schneider, S. Ho¨fling, M. Kamp, C.-Y. Lu, and J.-W. (2010).
Pan, Nat. Nanotechnol. 8, 213 (2013). [42] T.Close,E.M.Gauger,andB.W.Lovett,NewJ.Phys.
[17] M. Mu¨ller, S. Bounouar, K. D. Jo¨ns, M. Gla¨ssl, and 14, 113004 (2012).
P. Michler, Nat. Photonics 8, 224 (2014). [43] P.Kaer,P.Lodahl,A.-P.Jauho,andJ.Mork,Phys.Rev.
[18] L. Monniello, A. Reigue, R. Hostein, A. Lemaitre, B 87, 081308 (2013).
A. Martinez, R. Grousson, and V. Voliotis, Phys. Rev. [44] P. Kaer and J. Mørk, Phys. Rev. B 90, 035312 (2014).
B 90, 041303 (2014). [2] A.Auffeves,J.-M.G´erard,andJ.-P.Poizat,Phys.Rev.A
[19] A. Sipahigil, K. D. Jahnke, L. J. Rogers, T. Teraji, 79, 053838 (2009).
J.Isoya,A.S.Zibrov,F.Jelezko,andM.D.Lukin,Phys. [46] A. Auff`eves, D. Gerace, J.-M. G´erard, M. F. Santos,
Rev. Lett. 113, 113602 (2014). L. Andreani, and J.-P. Poizat, Phys. Rev. B 81, 245419
[20] Y.-J. Wei, Y.-M. He, M.-C. Chen, Y.-N. Hu, Y. He, (2010).
D.Wu,M.Kamp,S.Hoefling,C.-Y.Lu,J.-W.Pan,etal., [47] C. Hong, Z. Ou, and L. Mandel, Phys. Rev. Lett. 59,
Nano Lett. (2014). 2044 (1987).
[21] P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. [48] A. Kiraz, M. Atatu¨re, and A. Imamog˘lu, Phys. Rev. A
Sellin, D. Ouyang, and D. Bimberg, Phys. Rev. Lett. 87, 69, 032305 (2004).
157401 (2001). [49] See supplemental materials.
[22] M. Bayer and A. Forchel, Phys. Rev. B 65, 041308 [50] A. Auff`eves, B. Besga, J.-M. G´erard, and J.-P. Poizat,
(2002). Phys. Rev. A 77, 063833 (2008).
[23] A.Berthelot,I.Favero,G.Cassabois,C.Voisin,C.Dela- [51] Y. Arakawa, S. Iwamoto, M. Nomura, A. Tandaechanu-
lande, P. Roussignol, R. Ferreira, and J.-M. G´erard, Nat. rat,andY.Ota,IEEEJ.Sel.Top.Quant.18,1818(2012).
Phys. 2, 759 (2006). [52] R. Heitz, I. Mukhametzhanov, A. Madhukar, A. Hoff-
[24] S. Kako, C. Santori, K. Hoshino, S. Go¨tzinger, Y. Ya- mann,andD.Bimberg,J.Electron.Mater.28,520(1999).
mamoto, and Y. Arakawa, Nat. Mater. 5, 887 (2006). [53] E. Neu, D. Steinmetz, J. Riedrich-Mo¨ller, S. Gsell,
6
M. Fischer, M. Schreck, and C. Becher, New J. Phys. 13,
025012 (2011).
1
Supplemental Materials
HAMILTONIAN AND MASTER EQUATION CALCULATION OF TWO-TIME CORRELATORS
TheHamiltonianofatwo-levelquantumemitter(QE) In order to conveniently express the two-time correla-
interacting with a quantized mode of an optical cavity tors, we make use of the non-equilibrium Green’s func-
can be written as: tionformalism[S3]. Herethelesser,greaterandretarded
Green’s function are respectively defined by:
Hˆ =(cid:126)ω cˆ†cˆ+(cid:126)ω cˆ†cˆ+(cid:126)g(cˆ+cˆ†)(aˆ+aˆ†), (S1)
QE cav
where cˆ† and cˆ are fermionic creation and annihilation (cid:18)(cid:104)cˆ†(t+τ)cˆ(t)(cid:105) (cid:104)cˆ†(t+τ)aˆ(t)(cid:105)(cid:19)
operators for the QE while aˆ† and aˆ are bosonic cre- Gˆ<(t+τ,t)= , (S10)
(cid:104)aˆ†(t+τ)cˆ(t)(cid:105) (cid:104)aˆ†(t+τ)aˆ(t)(cid:105)
ation and annihilation operators for the cavity. ω is
QE
the QE frequency, ω is the cavity frequency, and g is
cav
the QE-cavity coupling strength. We consider up to one (cid:18)(cid:104)cˆ(t)cˆ+(t+τ)(cid:105) (cid:104)aˆ(t)cˆ+(t+τ)(cid:105)(cid:19)
excitationinthesystemsothatwithintherotating-wave Gˆ>(t+τ,t)= (cid:104)cˆ(t)aˆ+(t+τ)(cid:105) (cid:104)aˆ(t)aˆ+(t+τ)(cid:105) ,
approximation the dynamics involved only the states
(S11)
{|g,0(cid:105),|e,0(cid:105),|g,1(cid:105)}. As no coherent coupling occurs be-
tween |g,0(cid:105) and the other states, it is sufficient to study
the dynamics within the basis formed by {|e,0(cid:105),|g,1(cid:105)}, GˆR(τ)=Θ(τ)(cid:2)G>(t+τ,t)+G<(t+τ,t)(cid:3), (S12)
in which the Hamiltonian reads
where τ is a time difference. Note that for simplicity
(cid:18) (cid:19) we have dropped in the above definitions the factor i
0 g
Hˆ = , (S2) involved in standard definitions [S3] . The retarded and
g δ
lesser self-energies describing the dissipative terms are
where δ = ω −ω is the detuning between the QE expressed as:
cav Qe
and the cavity. The density matrix can be written as
ρ(t)=(cid:18)ρee(t)=(cid:104)cˆ†(t)cˆ(t)(cid:105) ρec(t)=(cid:104)cˆ†(t)aˆ(t)(cid:105)(cid:19) (S3) ΣˆR(t+τ,t)=δ(τ)(cid:18)(γ+γ∗)/2 0 (cid:19) (S13)
ρ (t)=(cid:104)aˆ†(t)cˆ(t)(cid:105) ρ (t)=(cid:104)aˆ†(t)aˆ(t)(cid:105) 0 κ/2
ce cc
Its evolution is described by the following Linbladt
master equation [S1, S2]: (cid:18)γ∗G< (t+τ,t) 0(cid:19)
Σˆ<(t+τ,t)=δ(τ) e,e (S14)
∂ρˆ 0 0
=L[ρ] (S4)
∂t
For such time-independent Hamiltonian the retarded
Green’s function depends only of one variable. It can be
i
Lˆ[ρ]= [ρˆ,Hˆ]+Lˆ +Lˆ +Lˆ (S5) expressed in angular frequencies as:
(cid:126) QE cav deph
(cid:90)
wherethedissipativetermsdescribingrespectivelythe GˆR(ω)=−i dωeiωtGˆR(τ). (S15)
QE decay, the cavity decay and pure dephasing reads in
the {|e,0(cid:105),|g,1(cid:105)} basis
The Dyson’s equation for the retarded Green’s function
reads:
(cid:18) (cid:19)
ρ ρ /2
Lˆ [ρˆ]=−γ ee ec (S6)
QE ρ /2 0
ce
GˆR(ω)=(ω−H −ΣR(ω))−1 (S16)
(cid:18) (cid:19)
0 ρ /2
Lˆ [ρˆ]=−κ ec (S7)
cav ρ /2 ρ
ce cc
(cid:18)ω+iγ/2+iγ∗/2 g (cid:19)−1
GˆR(ω)= (S17)
(cid:18) 0 ρ /2(cid:19) g ω−δ+iκ/2
Lˆ [ρˆ]=−γ∗ ec (S8)
deph ρ /2 0
ce From Kadanoff-Baym equations, which describe the
An initial excitation of the QE is assumed, i.e. ρˆ(0)= equationsofmotionoftheGreen’sfunctions[S3],onecan
|e,0(cid:105)(cid:104)e,0|. The evolution of the density matrix is then easily derive the following relation in the case of Marko-
computed using vian self-energies for τ >0:
ρˆ(t)=eLt|e,0(cid:105)(cid:104)e,0| (S9) Gˆ<(t+τ,t)=GˆR(τ)Gˆ<(t,t)=GˆR(τ)ρˆ(t). (S18)
2
From this relation we can extract the cavity correla- derivethedegreeofphotonindistinguishabilityineachof
tions these cases. We assume a perfect resonance (i. e. δ =0)
between the QE and the cavity.
(cid:104)aˆ†(t+τ)aˆ(t)(cid:105)=(cid:104)g,1|GˆR(τ)ρˆ(t)|g,1(cid:105)
(S19)
=GR(τ)ρ (t)+GR(τ)ρ (t)
cc cc ce ec
Hence the calculation of the two-time correlators is
splitted into the calculation of two one-time operators,
namely the density matrix ρˆ(t) on one hand and the re-
tardedGreen’sfunctionGˆR(τ)ontheotherhand. GˆR(τ)
Coherent coupling regime
can be computed by Fourier transforming Eq. S17, or by
solving its equation of motion which reads
Coherent-couplingregimebetweentheQEandthecav-
∂ (cid:104) (cid:105)
i GˆR(τ)=iδ(τ)ˆI+ Hˆ −iΣˆR(0) GˆR(τ). (S20) ityoccursif2g >κ+γ+γ∗. Inthelimit2g (cid:29)κ+γ+γ∗,
∂τ
wederivebelowananalyticalexpressionfortheindistin-
guishability. In this limit, the coherent part of the dy-
INDISTINGUISHABILITY IN LIMITING CASES namics (i.e. Rabi oscillations) is much faster than the
OF CAVITY-QED incoherent part (i.e. population and phase decay). An
approximate solution of the dynamics is obtained by de-
In the following discuss three limiting cases of dissipa- coupling these two timescales. The density matrix then
tive cavity quantum electrodynamics (cavity-QED) and reads:
ρ(t)(cid:39)e−(γ+κ)t/2(cid:34)e−γ∗t/2(cid:32)cos2(gt) −sin2(i2gt)(cid:33)+(1−e−γ∗t/2)(cid:18)21 0(cid:19)(cid:35), (S21)
sin(2gt) sin2(gt) 0 1
2i 2
where the first term describes the initial damped Rabi oscillations, while the second term accounts for the incoherent
part of the dynamics after dephasing. The retarded Green’s function reads
(cid:18) (cid:19)
GR(τ)(cid:39)e−(γ+κ+γ∗)τ/4× cos(gτ) −isin(gτ) . (S22)
−isin(gτ) cos(gτ)
Byaveragingoverthefastcoherentdynamics,thecavitypopulationandthecavitycorrelationsaregivenrespectively
by
1
<ρ (t)>= e−(γ+κ)t/2, (S23)
cc 2
1(cid:104) (cid:105)e−(γ+κ+γ∗)τ/2
<|(cid:104)aˆ†(t+τ)aˆ(t)(cid:105)|2 >= e−(γ+κ+γ∗)t+e−(γ+κ)t , (S24)
4 2
where the <> indicates an average over the fast rotating terms. It gives a photon indistinguishability of
(cid:82)∞dt(cid:82)∞dτ <|(cid:104)aˆ†(t+τ)aˆ(t)(cid:105)|2 > (γ+κ)(γ+κ+γ∗/2)
I = 0 0 = . (S25)
cc 1|(cid:82)∞dt<ρ (t)>|2 (γ+κ+γ∗)2
2 0 cc
Incoherent-coupling regime Adiabatic elimination of coherences
From the master equation we have at resonance (δ =
0):
The incoherent limit occurs for 2g (cid:28) κ+γ +γ∗, for
which it is shown below that the coherences can be adi- ∂ρ
abatically eliminated. ∂tee =−γρee+ig(ρec−ρce) (S26a)
3
∂ρ
cc =−κρ +ig(ρ −ρ ) (S26b) For τ >1/κ, only the former term is not vanishing:
∂t cc ce ec
R
(cid:104)aˆ†(t+τ)aˆ(t)(cid:105)(cid:39) GR(τ)ρ (t). (S35)
κ ee ee
∂ρ γ+γ∗+κ
∂tec =− 2 ρec+ig(ρee−ρcc) (S26c) Thelastexpressionindicatesthatinthisregimethecav-
ity correlations do indeed follow the QE correlations,
If γ+γ∗+κ(cid:29)2g, coherences can be adiabatically elim- with:
inated in Eq. S26c by setting ∂ρ /∂t∼0, leading to:
ec
ρ (τ)(cid:39)e−(γ+R)t (S36)
ee
2ig(ρ (t)−ρ (t))
ρ (t)(cid:39) ee cc . (S27)
ec γ+γ∗+κ GR(τ)(cid:39)e−(γ+γ∗+R)τ/2. (S37)
ee
Wearethenleftwiththefollowingrateequationsforthe
As the integral of the correlations (Eq. 2 of the paper) is
populations
dominated by delay times such as τ >1/κ, one finds for
the indistinguishability
∂ρ
ee =−(γ+R)ρ +Rρ , (S28a)
∂t ee cc γ+R
I = . (S38)
bc γ+R+γ∗
∂ρ
cc =−(κ+R)ρ +Rρ , (S28b)
∂t cc ee
Regime of incoherent coupling and good cavity
wheretheexchangeratebetweentheQEandthecavity
reads [S2] Inthelimitγ+γ∗ (cid:29)κ,weshowbelowthatthecavity
correlations are dominated by the terms diagonal in the
4g2 cavitymode. Inthislimit, theprojectionoftheretarded
R= . (S29)
γ+γ∗+κ Green’s function on the cavity mode reads
By solving the coupled rate equations Eq. S28, we find GR(τ)(cid:39)e−Γcτ/2 (S39)
cc
that the efficiency is given by
where Γ is the linewidth of the cavity-like eigenstate
c
(cid:90) κR which can be calculated from from Eq. S17:
β =κ dtρ (t)= . (S30)
cc κR+γ(κ+R)
4g2
Γ =κ+ (cid:39)κ+R. (S40)
c |γ+γ∗−κ|
Regime of incoherent coupling and bad cavity
In Eq. S20, we can adiabatically eliminate GR for τ >
ce
1/(γ+γ∗), which gives
In the bad cavity limit (i.e. κ (cid:29) γ + γ∗), we can
g
eliminate the cavity population (i.e. ∂ρcc/∂t∼0) in the GR(τ)(cid:39)−i e−Γcτ/2, (S41)
above rate equations, leading to ce γ+γ∗
andthusprovidinganupperlimitforthesecondtermin
R
ρcc(t)= κ+Rρee(t) (S31) Eq. S19:
R
On the other hand, approximation on the retarded GR(τ)ρ (t)≤ e−Γcτ/2ρ (t) (S42)
ce ec γ+γ∗ ee
Green’sfunctioninvolvedinEq.S19canalsobederived.
For τ > 1/κ, one can adiabatically eliminate the off- On the other hand, Eq. S26c provides an upper bound
diagonal term ∂GR/∂τ ∼0 in Eq. S20: for ρ :
ce cc
g R
GR(τ)(cid:39)−2i GR(τ), (S32) ρ ≥ ρ (t), (S43)
ce κ ee cc R+κ ee
so that the second term in Eq. S19 reads for τ >1/κ which combined with Eq. S39 gives
GRce(τ)ρec(t)(cid:39) RκGRee(τ)ρee(t). (S33) GRcc(τ)ρcc ≥ RR+κe−Γcτ/2ρee(t). (S44)
On the other hand, the first term in Eq. S19 is given by Eqs. S42 and S44 show that in this regime GRccρcc domi-
nates over GRρ for τ >1/(γ+γ∗):
ce ec
R
GRcc(τ)ρcc(t)(cid:39) κe−κτ/2ρee(t). (S34) (cid:104)aˆ†(t+τ)aˆ(t)(cid:105) (cid:39)GRcc(τ)ρcc(t). (S45)
4
We can hence express the indistinguishability as if the
cavity acts as an effective emitter:
(cid:82)∞dtρ2 (t)(cid:82)∞dτe−Γcτ [S1] F. Petruccione and H.-P. Breuer, The theory of open
I = 0 cc 0 . (S46) quantum systems (Oxford Univ. Press, 2002).
gc 12|(cid:82)0∞dtρcc(t)|2 [S2] A. Auffeves, J.-M. G´erard, and J.-P. Poizat, Phys. Rev.
A 79, 053838 (2009).
By solving the rate equations for populations (Eq. S28), [S3] H.HaugandA.-P.Jauho,Quantumkineticsintransport
and plugging in ρ (t) in the above expression, we find and optics of semiconductors, vol. 123 (Springer, 2008).
cc
after some algebra
γ+ κR
I = κ+R . (S47)
gc γ+κ+2R