Temperature tunability of quantum emitter - cavity coupling in a photonic wire microcavity with shielded sidewall loss

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by  M. Bernard

Temperature tunability of quantum emitter - cavity coupling in a photonic wire
microcavity with shielded sidewall loss
M. Bernard1,2 and M. Ghulinyan1,∗
1Centre for Materials and Microsystems, Fondazione Bruno Kessler, I-38123 Povo, Italy
2Deptartment of Physics, University of Trento, I-38123 Povo, Italy
Recent technological advancements have allowed thefrequentlyutilizedones. Diskandsphereresonatorsoffer
to implement in solid-state cavity-based devices phe- Q’s in excess of millions but their large mode volumes make
nomena of quantum nature such as vacuum Rabi themimpracticalforcavityQEDstudies. PhCnanocavities,
splitting1,2, controllable single photon emission3,4 which offer mode volumes as small as 0.04 µm3 along with
and quantum entanglement5,6. For a sufficiently reasonably high active Q’s of 20,000, require careful geome-
6strong coupling between a quantum emitter and a try engineering and precise technological control27.
1
cavity, large quality factors (Q) along with small Multilayered FP microcavities, on one side, demand rela-
0
modal volume (V ) are essential2,7,8. Here we show tively simple and controllable technology, and, on the other
2 eff
that by applying a 5nm Al coating to the sidewalls of side,areeasilydescribedfromtheoreticalpointofview28. In
n
a submicrometer-sized Fabry-Pérot microcavity, the thesedevicestinymodalvolumesmayalsobeachievedwhen
a
Jcavity Q can be temperature-tuned from few hun- lateral dimensions squeeze down to several wavelengths.
dreds at room temperatures to 2×105 below 30 K. Here, however, sidewall losses due to scattering and poor
0
2This is achieved by, first, a complete shielding of the modeconfinementspoilthecavityQseverely1,29. Therefore,
sidewall loss with ideally reflecting lateral metallic a strategy to maintain high Q’s in small-sized FP microcav-

]mirrors and, secondly, a dramatic decrease of the ities is challenging and could be beneficial to cavity QED
s
ccavity’s axial loss for small-sized devices due to the experiments.
tilargely off-axis wavevector within the multilayered Intheabsenceofmaterialabsorption, aFPmicrocavityis
p
structure. Our findings offer a novel temperature- characterized by two main channels of passive loss (Fig. 1a):
o
tunableplatformtostudyquantumelectrodynamical (i) the loss, γ , due to the finite reflectivity of dielectric
. DBR
sphenomena of emitter-cavity coupling. We demon- mirrors and (ii) the lateral loss, γ , which accounts for both
c L
istratethataRabisplittingof2g=24GHz(0.142nm) the electromagnetic (EM) wave scattering on the imperfect
s
can be readily achieved at 40 K in a 0.8µm-sized de- boundaries and radiation from dielectric cavity towards the
y
hvice, which has an Veff ≈ 0.0845 µm3, comparable to environment. Whilethemodevolumedropswithdecreasing
pbest 2D photonic crystal (PhC) nanocavities2. the device size, L, the lateral loss grows exponentially. This
[ leadstoasituationwheretheratioQ/(cid:112)V ,andhencethe
The physics of strong coupling between a quantum emit- eff
1 couplingrategpassesamaximumatanoptimaldevicesize1.
terandacavityconstitutesthebackboneofcavityQuantum
v An efficient isolation of the EM mode from the environ-
Electrodynamics(QED)9. Thepioneeringstudiesofcoupling
1 ment would be ideal to prevent the mode Q from degrada-
between electronic (quantum emitter) and photonic (cav-
5
tion. This, for example, can be achieved by applying highly
4ity) states10–13 have triggered the expansion of cavity QED
reflectingmirrorstothesidewallsoftheFPcavity(Fig.1b).
5from fundamental research9 towards its solid-state imple-
0mentationinlaboratorydevices1–6,14,15toquantumphotonic For this, we studied numerically a FP multilayered micro-
. cavity, which has a finite lateral dimension L and is in-
1circuits16,17. The rate g at which the cavity and the emitter x
finite in the third dimension (see Supplementary Methods
0exchange photons is crucial to determine if the system is in
6 section for details). In our approach, 5 nm thick Al coat-
weak or strong coupling regime. In the first case, following
1
Fermi’sgoldenrule,thefree-spacespontaneousemissionrate
:
voftheemittercanbealteredwhenitradiateswithinanelec-
Xitromagneticcavity11,18. Strongcouplingmanifestswhenthe
emitter and the cavity start to exchange photons coherently
r
athroughaprocessknownasvacuumRabioscillations8. This
happens when g2 ≥ (γ −γ )2/16, where γ are the cavity
c e c,e
and emitter linewidths, respectively13. The strong coupling
regime has gained a particular interest since it constitutes
the basis of a rich spectrum of cavity QED phenomena8.
The strong coupling requires several conditions to be sat-
isfied. First, along with a narrow linewidth, γ , the emitter
e
has to possess large oscillator strength f, in order to pro-
√
videnecessarycouplingrates(g ∼ f). Secondly,thecavity FIG.1. Schematic representation of the multilayered FP-
needstohavelargemodalQ(∼γ−1)andsmallvolumeV , type photonic wire microcavity. (a)Thesystemlossisdom-
c eff inatedbythedielectricmirrorlossesforlargelateralsizeL and
since g scales as Ve−f1f/2. Among integrated solid-state cavity bythelaterallossγL whenLx/λ(cid:46)1. (b)AsidewallAl-covxerage
structures Fabry-Pérot-type (FP) micropillars3,19, PhC-slab acts as a temperature-tunable mirror, which hinders the lateral
nanocavities2,20,21,microdisks22–24andmicrospheres25,26are loss at low T’s and boost the cavity Q by a factor of 1,000.
2
FIG. 2. Size and temperature-dependent characteristics of FP microcavities with metallic sidewall mirrors. (a) FEM-
calculated power reflection, transmission and absorption of 5 nm thick Al plate. (b) The spectral shift of the cavity resonance as a
functionofsizeisalmostinsensitivetotemperature-dependentreflectionfrommetallicmirrors. (c)The2DmapofcalculatedQ-factor
(log-scale) as a function of L and T shows significant variations for small lateral size. (d) The extracted from (c) Q(L )-trends for
x x
a series of selected temperatures and the corresponding analytical curves, calculated according to the model Eq.(1). (e) The power
reflectancemapofaninfinite1DFPmicrocavityunderE -polarizedplane-waveexcitationatobliqueangleshasbeencalculatedusing
z
transfer-matrixformalism. ThescattereddataareFEMcalculationresultsfortheFPcavitywithfinitelateral-sizeL ,whichisrelated
x
toΘi throughλ0/2Lx =tan(arcsin(sinΘi/n)). (f)Thegradualincreaseofadielectricmirror’sEz-wavereflectanceattheFPresonance
wavelength λ(L ) explains the cavity linewidth narrowing at larger Θ (smaller L ).
x i x
ings are applied to the sidewalls of the cavity. The lateral Quitesurprisingly,weobservethatinthepresenceofside-
loss is thus substituted by the loss, γ , of the metallic mir- wall mirrors the system temperature has an unprecedented
M
ror. The reflectivity of Al mirrors is temperature-tunable effectonthecavityQ.Inageneralpicture,foragivenlateral
owing to a significant increase of the electrical conductivity, size,theQgrowsmonotonicallyasthetemperaturedecreases
σ(T),byfiveordersofmagnitudewhenthematerialiscooled (Fig.2c). Thisimprovement,however,iscriticallydependent
down from room temperatures to few kelvins (see Supple- onthecavitysize(seealsoSupplementaryFig.S2). Inpartic-
mentary Fig. S1). In particular, we find from finite-element ular,whileforrelativelylargecavities(L /λ>3)thechange
x
method (FEM) calculations that the power reflection of a in Q amounts to a factor of two, we find an improvement of
near-infrared wave (λ=1.55 µm) from a 5nm-thick Al sheet up to three orders of magnitude when the lateral size be-
is R=94% at T=300 K, while the remaining power is com- comes smaller than the wavelength (L /λ<1). In fact, the
x
pletely absorbed within the metal (Fig. 2a). The reflectance Q-factorofthecavitywithL =0.8µm(Fig.2d)boostsfrom
x
at lower temperatures grows rapidly following to a good ap- 200to200,000whenthestructureiscooleddownfrom300K
(cid:16) (cid:17)1/2 to5K.WenotethattheQ(L )dependenceatroomtemper-
proximation the trend R ≈ 1− 2(cid:15)0ω , where ω is the x
πσ(T) atureisbothquantitativelyandqualitativelyverysimilarfor
EM wave frequency and (cid:15) is the vacuum permittivity30,31.
0 FPdeviceswithorwithoutmetalcoverage. Theexplanation
The calculated R is larger than 99% already at 55 K and
is that in one case the Q is limited by the EM wave absorp-
rapidlyreachesunity(R=99.998%)atliquidheliumtemper-
tioninthemetalwhileinthesecondcaseitisspoiledbythe
atures (T=5 K).
lateral loss from the cavity.
The presence of metallic mirrors has imperceptible effect The remarkable result of linewidth narrowing in small-
on the fundamental mode wavelength, λ, of the FP cav- sized cavities at lower temperatures necessitates an in-depth
ity. Our calculations show that, as expected29, the resonant understandingoftheunderlayingphysics. Asitappearsfrom
(cid:113)
wavelength scales as λ(Lx) = λ0/ 1+(λ0/2nLx)2, where Fcriegass.in2gc,tdr,enbdeltoowanTi=n3cr0eKasitnhgeoQne(Lgxa)incinugrvaesfaicntvoerrotft1h0e00daet-
λ = 2nd is the center wavelength of the laterally infinite
0
the smallest size of 0.8µm. This implies that at low temper-
planar cavity with mode’s effective refractive index n and
atures cavity losses decrease drastically – a fact, which can
thickness d. Here, for simplicity, materials dielectric func-
not be explained by only close-to-unity reflectance of side-
tionshavebeenconsideredtemperature-independent. Figure
wall mirrors. To understand fully the complex mechanism
2bplotsaseriesofλ(L )-curvesforlateralsizesintherange
x
underlaying this phenomenon we refer to the definition of
from 5.2 µm to 0.8 µm, calculated at various temperatures.
themodequalityfactorintheformQ=λ(L )/(γ +γ )
It shows that the different curves are essentially coinciding, x DBR M
and analyze separately the two loss contributions.
which means that, for a given size L , the cavity resonance
x
wavelength is rather robust against temperature variation. We find that the dielectric mirror’s loss γ decreases
DBR
3
FIG. 3. QD-microcavity strong coupling and Rabi splitting. The QD spontaneous emission map as a function of temperature
and spectral detuning for photonic wire sizes (a) L =0.8µm, (b) 2.0 µm and (c) 5.2 µm. At a fixed oscillator strength the coupling
x
coefficient g scales with V−1/2 and is 71 pm, 43 pm and 26 pm for different L ’s, respectively. Panels (d)-(c) show the SE spectra at
eff x
selectedtemperatures,while(g)-(i)comparethecavitylineshapeatextremetemperaturesofT=5Kand300Ktothatoftheemitter.
as the lateral size L shrinks. The explanation of this un- fitting constants. Analytical curves are shown in Fig. 2d as
x
expected result is based on the fact that a normal-incident solid lines. The various trends have been excellently repro-
plane E -polarized wave experiences a tilted angle propaga- duced by plugging in Eq.(1) only the corresponding value of
z
tion within the multilayered dielectric structure. This angle the Al mirror reflectance at a given T (seeFig. 2a).
is defined as Θ = arctan(k /k ), where k = π/L and
c x y x x Further calculations show that, since the cavity mode vol-
k = 2πn/λ are the size-quantized and axial wavevectors, (cid:112)
y 0 umedropslinearlywithL ,theratioQ/ V growsmono-
x eff
respectively. According to the definition, Θ increases con-
c tonically as the cavity gets smaller. As noted before, this
tinuouslyasthelateralwavevectork growsatsmallerL . In
x x propertymakesthemetal-coatedFPcavityanexcellentplat-
terms of wave transmission, this is equivalent to the behav-
form for cavity QED experiments with an added value of
ior of an infinite (L −→∞) 1D FP microcavity under an
x,y temperaturetunability. Figure3a-ccomparesthecalculated
E -wave excitation at an external incidence angle Θ , which
z i spontaneous emission (SE) maps of three cavities with lat-
is related to Θ according to Snell’s law sin(Θ )=nsin(Θ ).
c i c eral sizes L =0.8µm, 2.0 µm and 5.2 µm, respectively (see
x
Figure2eshowsthereflectionspectrumofaninfiniteFPmi-
details in Supplementary Section 3). The spectra are calcu-
crocavity under Ez-polarized plane wave excitation mapped latedfixingtheemitterlinewidth14 toγ =30pm(∼20µeV)
forarangeofincidenceanglesfromnormalto60◦. Thecav- e
and considering that the emitter and the cavity are always
ity resonance is at λ = 1.55 µm at normal incidence and
0 resonant, λ = λ(L ). The emitter-cavity coupling is thus
e x
drifts gradually towards shorter wavelengths as Θ grows. (cid:112)
i defined by the Q(T)/ V ratio for the cavity of given
eff
Forcomparison, FEMresultsforλ(L )(Fig.2b)areplotted
x size L . These results demonstrate that a whole range of
x
in the same graph as scatter data. In this case Θ and L
i x quantum couplings – from weak to strong – can be observed
are related according to the definition. The good agreement
within a unique device by appropriately choosing the work-
between these results is in strong support to our explana-
ing temperature. For example, a strong Rabi splitting of
tion. Finally, in Fig. 2e we show how the reflectance of the
2g = 142 pm can be readily achieved below 40 K in the
laterallyinfinitedielectricmirrorattheresonantwavelength
smallest devices, which provide with mode volumes as small
λ(Θi) grows progressively with the angle (decreasing size). as 0.0845 µm3.
In fact, Lorentzian fits to the resonant lines show that a
Figures 3d-f compare the characteristic features of the SE
higherreflectionfrommultilayeredmirrorsleadstoanexpo-
spectra at 300 K, 40 K and 5 K. In particular, for L =
nential narrowing of linewidths, i.e. growing Q-factors (see x
0.8µmsize(Fig.3d),theSEspectrumisessentiallyidentical
Supplementary Fig. S3).
tothebareemitter’slineat300K.Thisisbecausethecavity
By considering that the metallic mirror loss γ =
M
−λ2ln(R(T))/2πLxiscavitysize28andtemperaturedependent, with Q∼200 has minute effect on photon confinement and
g (cid:28) γ /4 leads to weak coupling (Fig. 3g). Contrary to
we model the total Q as c
this, at 5 K the cavity linewidth is much narrower than γ .
e
2πn (cid:20) λ2lnR(T) (cid:21)−1 Here, g > γ /4 and, thus, the strong coupling manifests as
Q= =2πnk (γ −Ae−Lx/B)k2− k2 , e
kγ s y 2πL x twodistinctlypeakedSEspectrumwithaRabisplittingthat
tot x
(1) is tenfold larger than the cavity resonance width γ . Similar
c
where the two loss contributions are weighted for the EM comparisonsaremadealsoforL =2µmand5.2µmcavities
x
field intensity in axial (y) and lateral (x) directions. Here, inFigs.3e,handFigs.3f,i,respectively. Inparticular,wesee
γ is the saturation loss at very large L , while A and B are thatforthelargestcavitytheemitter-cavitysystemremains
s x
4
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Supplementary Information
Methods
The dielectric structure
The multilayered structure is constituted by a stack of
two quarter-wavelength dielectric materials A and B, with
refractive indices n = 1.456 and n = 3.632 and thick-
A B
nesses d = λ /(4n ) and d = λ /(4n ), respectively.
A 0 A B 0 B
The laterally infinite planar cavity has a center wavelength
λ = 1550 nm. The FP microcavity is formed by sand-
0
wiching a half-wavelength layer AA between two 5.5-period
distributed Bragg reflectors BABABABAB (DBR).
We notice, that any different choice for material refrac-
tive indices does not alter the findings and conclusions of
this paper. In fact, all results can be reproduced for any
other contrast (n −n ) of refractive indices by appropri-
B A
ately choosing the number of DBR periods.
Numerical calculations of finite size FP microcavities
Finite-elements method – The resonant frequency and the
cavity Q were calculated resolving the wave equation using
the COMSOLTM commercial software’s Eigenfrequency Do-
mainSolvermodule. ThetemperaturedependenceofAlmir-
rors conductivity σ(T) (Fig. S1) has been considered in the
wave equation
(cid:18) (cid:19)
σ(T)
∇×(∇×E)µ−1−k2 (cid:15) − =0, (S1)
r 0 r ω(cid:15)
0
where (cid:15) and µ are the material relative permittivity and
r r
permeability, respectively.
FIG. S2. Calculated spectral lineshapes of cavity resonances at
varioustemperaturesforthedevicelateralsizeof(a)L =5.2µm,
x
(b) 2 µm and (c) 1 µm. Solid lines are constructed Lorentzians
using Eigenfrequency domain results (center frequency and Q),
while scattered data come from Frequency analysis where the
plane wave transmission is calculated at free-running frequency
values. Insets show the distribution of the electric field E (nor-
z
mal to the screen plane) of the cavity mode.
thepowertransmissionoftheFPmicrocavityiscalculatedby
running the normal incident plane wave frequency ω around
theexpectedresonance. Anexcellentagreementbetweenthe
Eigenfrequency and Frequency Domain calculations is evi-
FIG. S1. The conductivity vs temperature of an Al thin film
(from Ref.32). dent. The electric field distributions (mode shape and spa-
tial extension) are shown as insets in corresponding graphs
In Fig. S2a-c the fundamental modes spectral lineshape of Fig. S2.
at selected temperatures is calculated for three different
cavity sizes (L = 5.2 µm, 2 µm and 1 µm). The solid
x
lines are based on constructed Lorentzian lineshapes follow- Linewidth narrowing in planar infinite FP cavity at
ing | Γ/2 |2, where the eignefrequency ω and the loss oblique incidence
ω−ω0−Γ/2 0
Γ=ω /Q were obtained from Eigenfrequency analysis.
0
Thescattereddatapoints(◦)inthesamegraphsarethere- Transfer matrix calculations – The planar microcavities
sultsfromFrequencyDomaincalculations. Inthisapproach, transmissionspectrawerecalculatedforthesamelayerstack
6
was calculated in a broad spectral range and, secondly, a
Lorentzian fit was applied to the resonant transmission line.
Finally, the extracted Q’s were plotted against the incident
angle and fitted using an exponential function (Fig. S3).
Spontaneous Emission spectrum calculations
We have calculated the Spontaneous Emission (SE) spec-
tra using Eqs. 9,10 from Ref.13:
SE(ω,T)∼(cid:12)(cid:12)(cid:12)Ω+−ω0+ιγ2c − Ω−−ω0+ιγ2c(cid:12)(cid:12)(cid:12)2, (S2)
(cid:12) ω−Ω ω−Ω (cid:12)
+ −
(cid:115)
FIG. S3. The transfer-matrix calculated Q of the infinite FP Ω (T)=ω ± g2−(cid:18)γc−γem(cid:19)2−ιγc+γem (S3)
microcavityresonance((cid:3))againsttheincidenceangleΘi andthe ± 0 4 4
exponential fit curve (solid line).
where ω is the center frequency of the cavity resonance,
0
γ and γ are the cavity and emitter linewidths, respec-
c em
tively, g is the coupling rate. In strong coupling conditions,
BABABABAB−AA−BABABABAB usingahomemade
when g2 ≥ (γ −γ )2/16 and Ω describe the doublets of
numeric code. c e ±
LorentziansseparatedbytheRabifrequency2g andindivid-
In order to extract the cavity Q at oblique angles of ual effective linewidths of (γc+γem)/4.
incident plane TE-polarized wave, the following procedure TheSEspectraarestronglydependentonthesystemtem-
has been adopted: first, the power transmission spectrum perature via ω (T) and γ (T).
0 c