# A telecom band single-photon source using a grafted carbon nanotube coupled
to a fiber Fabry-Perot cavity in the Purcell regime.

Antoine Borel  Théo Habrant-Claude  Federico Rapisarda  Laboratoire de
Physique de l’ENS, Université PSL, CNRS, Sorbonne Université, Université Paris
Cité, Paris, France  Jakob Reichel  Laboratoire Kastler-Brossel, ENS, Sorbonne
Université, Paris, France  Steeve Doorn  Center for Integrated
Nanotechnologies, Materials Physics and Applications Division, Los Alamos
National Laboratory, Los Alamos, NM, USA  Christophe Voisin  Yannick
Chassagneux<EMAIL_ADDRESS>Laboratoire de Physique de l’ENS,
Université PSL, CNRS, Sorbonne Université, Université Paris Cité, Paris,
France

###### Abstract

We report on the coupling of a reconfigurable high Q fiber micro-cavity to an
organic color center grafted to a carbon nanotube for telecom wavelength
emission of single photons in the Purcell regime. Using three complementary
approaches we assess various figures of merit of this tunable single photon
source and of the cavity quantum electrodynamical effects : the brightening of
the emitter is obtained by comparison of the count rates of the very same
emitter in free-space and cavity coupled regimes. We demonstrate a fiber
coupled single-photon output rate up to 20 MHz at 1275 nm. Using time-resolved
and saturation measurements, we determine independently the radiative quantum
yield and the Purcell factor of the system with values up to 30 for the
smallest mode volumes. Finally, we take advantage of the tuning capability of
the cavity to measure the spectral profile of the brightness of the source
which gives access to the vacuum Rabi splitting $g$ with values up to
$25\;\mu$eV.

###### keywords:

Single photon source, fibered fabry perot cavity, Purcell effect, telecom
wavelengths, carbon nanotube

##  1 Introduction

A range of quantum technologies, including quantum telecommunications or all
optical quantum computing units rely on the availability of high quality
single-photon sources which make up the ultimate flying quantum-bits. Among
condensed-matter nano-emitters, self-assembled III-V quantum dots 1, 2, 3,
point defects in host matrices 4, 5, 6 or 2D materials are promising
candidates 7, 8. However, few of these systems emit photons in the telecom
bands (around 1.3 $\mu$m and 1.55 $\mu$m) where the powerful architecture of
silicon based photonics and the fiber networks infrastructures could be
exploited 9, 10, 11. Single-wall carbon nanotubes (SWCNTs) are among the most
advanced systems in this respect 12, with the recent discovery of
$\text{sp}^{3}$ diazonium grafting of organic color centers that unlocked the
realization of room temperature, highly antibunched, telecom band emission of
single-photons 13, 14. Nevertheless, the only way to harness the full power of
such emitters is to efficiently couple them to photonic structures capable of
controlling their emission properties through cavity quantum electrodynamics
(cQED) effects and to funnel efficiently the photons into the working channel.
Here, we demonstrate the efficient coupling in the Purcell regime
($\text{F}_{\text{P}}\sim$ 30) of individual grafted carbon nanotubes to an
open fiber Fabry-Perot cavity. We reach a single mode fiber-coupled stream of
anti-bunched photons of $2\text{\times}{10}^{7}\text{\,}\mathrm{s}^{-1}$ at a
wavelength of $1.3\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. Exploiting
the unique versatility of our experimental setup, we were able to compare the
luminescence properties of the same individual emitter in free-space (FS) and
upon its coupling with a spectrally and spatially matched cavity. We could
also vary continuously the key parameters of the cavity (such as the resonant
wavelength or the mode volume) to carry an in-depth investigation of cQED
effects. Using three independent approaches, we quantitatively assess the
efficiency of the emitter/cavity coupling. We first compared the brightness of
the same nano-emitter before and after the reversible coupling to the cavity,
showing not only spectral filtering of the emission but also a strong
brightening (x20). Furthermore, using time-resolved measurements we could
directly observe the Purcell acceleration of the radiative recombination by
comparing the lifetime of the same nano-emitter with or without the cavity and
by tuning the mode volume of the latter. Finally, we were able to investigate
an original feature of these mixed 0D/1D nano-emitters, by exploiting the
strong acoustic phonon sidebands 15, 16 to tune the working wavelength of the
single-photon source 17. By analysing the variation of the single photon
source efficiency as a function of detuning, we propose an third independent
estimate of the strength of the cavity/emitter coupling. Finally, we discuss
the strength and limitations of each of these approaches to qualify the
relevance of each figure of merit with respect to applications.

##  2 Samples and setup

The sample consists in CoMoCAT (7,5) SWCNTs wrapped in a polymer (PFO-BPy) and
dispersed in a toluene solution containing polystyrene (5 % mass) before being
spin-coated on a high reflectivity Bragg mirror, resulting in isolated SWCNTs
in a 160 nm thick layer of polystyrene (PS). Carbon nanotubes were then
functionnalized with 3,5 dichlorobenze diazonium salt by the dip doping method
13. The 3,5 dichlorobenzene ion is covalently attached to the SWCNT surface by
hybridization of $\text{sp}^{2}$ C-C bound into $\text{sp}^{3}$. Such
hybridization introduces a very localized 100-200 meV deep potential well for
the center of mass of excitons, resulting in efficient trapping of the photo-
generated excitons in the nanotube. This trapping leads to a red-shifted
photo-luminescence and to a robust photon antibunching up to room temperature
18, 13.

The experimental setup consists of a modified confocal microscope designed to
observe the very same nano-emitter either in free space or coupled to a cavity
19. The cavity is depicted in Fig. 1 (b). It consists of a flat Bragg mirror
and a concave mirror engineered by CO2 laser ablation at the apex of an
optical fiber. The typical curvature radius is 10 $\mu$m while the cladding is
pencil shaped by multiple laser impacts in order to access the shortest
possible cavity length. The microscope objective is an aspheric lens at the
centre of which a 300 µm diameter hole is drilled to pass the optical fiber.
The flat mirror and the fiber apex were coated by similar dielectric stacks to
reach a theoretical reflection coefficient $R\simeq 0.9995$ for the flat
mirror and $R\simeq 0.9997$ for the fibered mirror in the wavelength range
1250 - 1500 nm.

Figure 1: (a) Overview of the experimental setup producing a fibered stream of
$2.1\text{\times}{10}^{7}\text{\,}$ single-photons per second at a wavelength
of 1.3 $\mu$m. (b) Zoom on the fibered micro-cavity with excitation and
collection schemes.

The fiber-lens assembly system is mounted on an inertial nano-positionner for
uniaxial displacement up to 2 mm on the z axis (Fig. 1 (a,b)). This
displacement is used to open and close the cavity at will (and hence to switch
from the free-space to the cavity configurations), to adjust the mode volume
(by changing the cavity length by $\lambda/2$ steps) and to finely tune the
cavity resonance to match each individual nano-emitter. In addition, a 1 µm
course piezoelectric actuator is used for fast modulation of the cavity
length. The Bragg flat mirror is clamped on two 2 mm range inertial nano-
positionners for displacement in the (x,y) flat in order to adjust the spatial
coupling of the nano-emitter to the cavity mode.

Excitation of the nano-emitter was set to 1064 nm with either a
Titanium:Sapphire cw laser (Spectra Physics 3900s) or with the seed of a
Supercontinuum pulsed laser (NKT Photonics Fianium 4W, $\tau_{p}=8$ ps,
$f_{rep}=$38.26 MHz). The photoluminescence (PL) is collected through the
drilled aspheric lens in FS configuration, while in cavity configuration it is
collected either through the flat mirror or at the output port of the
manufactured fiber. The PL is analyzed with a dispersive spectrometer and an
InGaAs photodetector array for spectral measurements or with broadband fibered
superconducting single photon detectors (Scontel SSPD, 32 ps jitter) for time-
resolved measurements.

The typical low-temperature PL spectrum of an individual grafted nanotube is
displayed in Fig. 2 (a). In addition to the main line (hereafter called zero-
phonon line (ZPL)), the PL spectra consistently show well developed sidebands
which result from the efficient coupling of the localized exciton to the 1D
acoustic phonon bath of the CNT 15, 16. The Debye-Waller factor (ratio of the
ZPL to the integrated intensity) is typically 0.6-0.8. The intensities of the
sidebands are directly related to the Bose-Einstein occupation probabilities
of the phonon modes. Thus, at cryogenic temperature, the blue sideband (phonon
absorption) is reduced with respect to the red one (phonon emission). In
addition, the lines are broadened by pure dephasing and spectral diffusion.
The latter is minimized by the use of a polystyrene matrix and by using a
minimal excess energy in the photo-excitation process 20.

Using a pair of fast SSPD and a correlation board, we performed a Hanbury-
Brown and Twiss experiment to measure the intensity correlations of the photon
stream, as shown in Fig. 2 (d). The data show a clear anti-bunching behavior
with $g^{(2)}(0)\leq 0.2$ and a bunching behavior in the 10 ns range (better
seen in the cw measurement shown in Fig. 2 (e). This bunching possibly
reflects the fast time-scale intermittency of the emission related to the
existence of a dark triplet state, or a fast spectral diffusion exceeding the
spectral width of the cavity.

Figure 2: (a) Free space emission spectrum. (b) 2D plot of the emission
spectrum recorded as a function of the cavity length. Color scale indicates
the intensity. (c) Spectrum obtained at the spectral matching of the cavity
mode with the emitter ZPL (Note that the cavity spectrum is broadened due to
vibrations). Second order correlations in cavity (Fabry-Pérot mode p=6) under
pulsed excitation, $g^{2}(0)=0.18$. (d), and under cw excitation,
$g^{2}(0)=0.4$ (down to 0.36 once corrected by the response function), (e). A.
clear bunching is observed under cw excitation and is attributed to
intermittency in the emission with typical time scale of 10 ns.

##  3 Brightening of the emitter through cQED effects

In a high quality factor micro-cavity the spectral density of electromagnetic
modes can be deeply modified, yielding an increase of the radiative decay rate
(the so-called Purcell effect) when a nano-emitter is placed in the field
anti-node 21 of a resonant mode. In a simple picture, the coupling to the
cavity opens a new decay channel for the emitter. At resonance, the total
decay rate of the emitter reads 22
$\gamma_{\text{cav}}=\gamma_{\text{fs}}+DWF_{\text{P}}\gamma^{\text{R}}$,
where $\gamma_{\text{fs}}=\gamma^{\text{NR}}+\gamma^{\text{R}}$ is the free-
space decay rate with $\gamma^{\text{NR}}$ and $\gamma^{\text{R}}$ the non-
radiative and radiative decay rates respectively. $F_{\text{P}}$ is the
Purcell factor which is defined as the ratio of the radiative decay rate
towards the resonant cavity mode to the FS radiative decay in the ZPL. The
ratio of the emission in the ZPL to the entire spectrum is given by the Debye-
Waller factor $DW$. The Purcell factor reads 22 :

$F_{\text{P}}=\frac{3}{4\pi^{2}}\frac{(\lambda/n)^{3}}{V_{\text{eff}}}Q_{\text{eff}}\
,$ (1)

with $V_{\text{eff}}=\frac{\int|nE|^{2}d^{3}r}{|nE|^{2}_{\text{max}}}$ the
effective mode volume and
$Q_{\text{eff}}=(Q_{\text{cav}}^{-1}+Q_{\text{em}}^{-1})^{-1}$ the effective
quality factor.

Grafted carbon nanotubes are quantum emitters that suffer from important non
radiative decay channels. Even if room temperature quantum yields up to $\sim
20$% have been reported 23, the low-temperature behavior is not documented. We
show below that it remains very weak at the single emitter level so that non-
radiative decay amounts for more than $90\%$ of the total decay rate of our
emitters in free space. Thus, the Purcell effect, which modifies the radiative
rate only, should only weakly modify the total decay rate of the emitter on
the one hand but on the other hand it should significantly increase its
brightness. The ratio of the decay rates and emission flux in FS vs cavity
read :

$\displaystyle\frac{\phi_{\text{cav}}}{\phi_{\text{fs}}}$
$\displaystyle=\frac{DWF_{\text{P}}}{1+\eta_{QY}DWF_{\text{P}}},$ (linear
regime) (2) $\displaystyle\frac{\phi_{\text{cav}}}{\phi_{\text{fs}}}$
$\displaystyle=DWF_{\text{P}},\quad$ (saturation regime) (3)
$\displaystyle\frac{\gamma_{\text{cav}}}{\gamma_{\text{fs}}}$
$\displaystyle=1+\eta_{\text{QY}}DWF_{\text{P}},$ $\displaystyle\text{(linear
regime)},$ (4)

with $\eta_{\text{QY}}=\gamma^{\text{R}}/\gamma_{\text{fs}}$. Note that
$\phi_{fs}$ is the total photon flux, including the phonon sidebands.

The emitter-cavity spectral matching is optimized using a longitudinal sweep
of the cavity length (Fig. 2 (b)). The resonance appears as a bright spot at
the crossing of the cavity dispersion and the emitter emission energy. Setting
the cavity length at resonance, we record the emission spectrum shown in Fig.
2 (c). Note that the excitation power may be slightly modified in the cavity
due to interferences arising from the finite reflectivity of the mirrors at
1064 nm, which prevents direct comparisons of the count rates in the linear
regime. However, measurements at saturation are immune to this effect.

We first performed saturation measurements on the same nano-emitter in free
space and in cavity at the lowest reachable mode volume (Fabry-Pérot mode
$p=6$, $p$ being the longitudinal order, i.e. the cavity length in units of
$\lambda/2$ which can be extracted from the free spectral range), under pulsed
or cw excitation at an excitation wavelength of 1064 nm. Fig. 3 shows the raw
count rates obtained after integration over the whole spectrum as a function
of the excitation power in FS and in cavity (flat mirror port) configurations.
The saturation plateau is higher in cavity than in free space, which is a
direct signature of the brightness enhancement.

After careful calibration of all optical parts of the setup (see SI), we show
that, at resonance, one count on the CCD corresponds to $41$ photons exiting
the cavity from the flat mirror port and simultaneously $44$ photons emitted
in the fiber port. At saturation, the detector count rate ($4.7\times 10^{5}$
counts/s) thus corresponds to a single mode fiber coupled photon flux of
$2.1\times 10^{7}$ photons/s. Compared to the FS configuration, this flux
corresponds to a brightening of the source ”in the first lens” of about 16.

To quantify the genuine emission enhancement (that is, at the level of the
emitter) and hence the Purcell factor, we need to correct the previous count
rates from the extraction efficiencies for both cavity outputs and for the FS
configuration. To this aim, we used numerical simulations (see SI). With very
similar dielectric coatings on the fiber and on the flat mirror, the exit
probabilities are expected to be rather balanced. However, due to the high
quality factor reached in this cavity, the absolute exit probabilities are
extremely sensitive to any small internal losses (scattering in the dielectric
stack due to residual roughness, in the polystyrene layer…), which constitute
pure losses (in contrast to the transmission losses which are ”useful
losses”). In order to assess these pure losses, we compared the simulated and
measured Q factors of the cavity as a function of the cavity length within the
cavity stability range (cavity modes from $p=6$ to $p=9$). In particular, due
to the pencil-shape of the fiber apex, additional spill-out losses are
observed in the simulations that yield an almost constant quality factor for
all the modes $Q_{\text{cav}}\simeq 5.5\pm 0.5\times 10^{4}$ (see SI). In
addition, the numerical simulations showed that the photons exiting through
the optical fiber do not perfectly couple to the fiber core but can be lost in
the cladding due to the important curvature of the mirror. Experimentally, we
measured the cavity quality factor at the emitter wavelength by ring-down
measurements and obtained a constant value $Q_{\text{cav}}\simeq 1\pm
0.1\times 10^{4}$ for all the longitudinal modes from $p=6$ to $p=9$. The
significant mismatch between the simulated and experimental values is assigned
to the internal losses, which amount to 1300 ppm. Then, we deduce effective
exit probabilities of 7.8% through the flat mirror and 6.0% through the fiber
mirror for the cavity mode $p=6$. 111If the origin of these internal losses
could be identified and suppressed, we could expect a coupling as high as 31%
into the single mode fiber and 40% through the flat mirror, assuming that the
losses towards the cladding are unchanged. We deduce the ratio of the
collection efficiencies in FS and in cavity configurations (flat mirror
output) to be $\eta_{\text{fs}}/\eta_{\text{cav}}=4.9\pm 0.2$. We thus obtain
$\phi_{\text{cav}}/\phi_{\text{fs}}=19\pm 2$ at saturation, which yields from
eq. 3 and $DW=0.65\pm 0.05$ a first estimate of the Purcell factor
$F_{\text{P}}=29\pm 4$. Note that the count rate ratio is the same (within the
error bars) in the linear regime, which is consistent with
$\eta_{\text{QY}}DWF_{\text{P}}\ll 1$, as will be confirmed using lifetime
measurements.

Figure 3: Spectrally integrated emission count rates as a function of the
excitation power (log-log scale) for the FS configuration (blue symbols) and
cavity configuration (red symbols) for cw excitation (open symbols) and pulsed
excitation (full symbols). (The excitation power for the FS pulsed line has
been rescaled by a factor 0.59 so that the low linear part of the curve
matches the cw one (excitation misalignment). Inset : same data in linear
scale.

.

##  4 Lifetime measurements

Comparison of the lifetimes in FS and in cavity configurations brings a direct
insight into the Purcell acceleration of the radiative decay. Time-resolved
measurements were performed using superconducting photodetectors (SSPD)
coupled to a PicoHarp 400 aquisition card. Spectral selection was ensured by
dichroic bandpass filters. The PL transient in FS shows a biexponential
behavior with a short time $\tau_{1}=23\pm 5$ ps and a long time
$\tau_{2}=256\pm 4$ ps (Fig. 4 (a)). At the smallest reachable cavity mode
($p=6$), a biexponential fitting of the transient yields
$\tau_{2}^{\text{cav}}=216\pm 15$ ps and $\tau_{1}^{\text{cav}}=28\pm 2$ ps.
The relative weight of the long component ($A_{2}\tau_{2}/(\sum
A_{i}\tau_{i})$) being larger than 80%, we define the lifetime ratio as
$\tau_{2}^{\text{fs}}/\tau_{2}^{\text{cav}}=1.19\pm 0.09$. We repeated such
lifetime measurements for different mode volumes. The data is shown in Fig. 4
(c-d) as a function of $\lambda^{3}/V_{\text{eff}}$. For each mode this mode
volume was computed by finite element simulation (see SI). The emitter
lifetime decreases as the inverse of the cavity mode volume in agreement with
theory (eq. 1 and 4), as shown in Fig. 4 (b).

Using the count rates and recombination rates ratios
$\phi_{\text{cav}}/\phi_{\text{fs}}$ and
$\gamma_{\text{cav}}/\gamma_{\text{fs}}$ we are able to extract both the
Purcell factor $F_{\text{P}}=29\pm 4$ (in line with the estimate based on
brightening of the previous section) and the emitter quantum yield
$\eta_{\text{QY}}=1\pm 0.4\ \%$. This value is confirmed by a direct
experimental estimate. In fact, in pulsed FS saturation measurements, the
saturation intensity reads
$I_{\text{sat}}=\eta_{\text{coll}}\eta_{\text{QY}}f_{rep}$, where
$\eta_{\text{coll}}$ is the collection efficiency of the optical setup in free
space, $f_{\text{rep}}$ is the pulse laser repetition rate. Hence, we deduce
$\eta_{\text{QY,sat}}=0.7\pm 0.2\ \%$. This radiative quantum yield is
relatively low compared with value reported at room temperature 23, but
similar to the values reported at low temperature for intrinsic (non
functionalized) nanotubes. This effect, which has to be confirmed on a larger
data set, could be related to a dark state trapping the population at
cryogenic temperatures.

Figure 4: (a) Free space and cavity (p=6, $\lambda^{3}/V_{\text{eff}}=0.4$)
lifetime comparison and their respective biexponential fits. The IRF is
represented in grey line. (b) Experimental cavity decay rate acceleration
factor (blue open circle) as a function of $\lambda^{3}/V_{\text{eff}}$ and
the fit according to equations 1 and 4. The error bars are calculated from
repeated lifetime measurements. (c) Lifetimes as a function of the inverse of
the cavity mode volume. (d) Relative contribution of the long time component
extracted from the biexponential fits over the signal integral
$A_{2}\tau_{2}/(A_{1}\tau_{1}+A_{2}\tau_{2})$.

##  5 Phonons assisted CQED effects

Interestingly, the FS spectrum $S_{\text{NT}}(\omega)$ of the emitter deviates
significantly from a Lorentzian with side-bands that result from acoustic
phonon assisted emission processes 16. When the cavity is detuned from the
ZPL, the output signal remains sizeable over a significant spectral range,
which results from the coupling of the emitter to the cavity through phonon
assisted processes. A factor of merit of the tuning capability of our system
is the spectral profile of the brightness $\beta(\omega_{\text{cav}})$, which
describes the emission probability in the cavity mode for each single
excitation of the emitter as a function of the cavity resonance frequency
$\omega_{\text{cav}}$.

The expression of $\beta(\omega_{\text{cav}})$ has been derived in 24 as a
function of the spectral and dynamical properties of the emitter in free space
and reads :

$\beta(\omega_{\text{cav}})=\frac{g^{2}\tilde{S}^{\text{emi}}(\omega_{\text{cav}})/\gamma}{1+g^{2}\tilde{S}^{\text{emi}}(\omega_{\text{cav}})/\gamma+g^{2}\tilde{S}^{\text{abs}}(\omega_{\text{cav}})/\kappa},$
(5)

where
$\tilde{S}^{emi}(\omega_{\text{cav}})=(S_{\text{NT}}*\mathcal{L})(\omega_{\text{cav}})$
is the FS emission spectrum convoluted to a Lorentzian profile
$\mathcal{L}(\omega)=[(\pi\kappa/2)(1+(2\omega/\kappa)^{2})]^{-1}$. It
represents the overlap of the cavity mode with the FS emission spectrum (cf
SI). The relationship with the usual Purcell picture is that
$\frac{\gamma_{\text{cav}}}{\gamma_{\text{fs}}}=1+g^{2}\tilde{S}_{\text{max}}/\gamma_{\text{fs}}$,
which connects $\tilde{S}_{\text{max}}$ to the Debye-Waller factor $DW$ factor
through $\tilde{S}_{\text{max}}=4DW/(\gamma^{*}+\kappa)$.

Note that in our system $\kappa\simeq 30\gamma$ so that the re-absorption
contribution $g^{2}\tilde{S}^{\text{abs}}/\kappa$ can be neglected. The
dynamical parameters $\gamma$ and $\kappa$ are deduced from time-resolved
measurements. Hence, we can in principle deduce the value of the Rabi coupling
$g$ from measurements of the brightness spectral profile.

To finely measure $\beta(\omega_{\text{cav}})$ despite mechanical vibrations
and spectral diffusion, we chose to modulate the cavity detuning (at a
frequency of 65 Hz) over a large spectral range in order to obtain directly
the envelope of the output spectrum $\text{E}_{\text{mod}}(\omega)$ as shown
in Fig. 5 (a) and (b). The coupling to the cavity yields a drastic enhancement
of the relative weight of the phonon sidebands compared to the ZPL, enlarging
considerably the working spectral range of the single-photon source compared
to its free space bandwidth. This envelope profile is closely related to the
brightness spectral profile and in the limit of an incoherent pumping of the
cavity by the nanotube, the relation reads (cf SI and 25):
${\text{E}_{\text{mod}}(\omega)=c(\beta*\mathcal{L})(\omega)\simeq c\
a\stackrel{{\scriptstyle\approx}}{{S}}_{\text{NT}}/(1+a\stackrel{{\scriptstyle\approx}}{{S}}_{\text{NT}})}$,
where $c$ is a constant, $a=g^{2}/\gamma$ and the double tilde notation refers
to a double convolution with $\mathcal{L}$. Here, the cavity spectrum is five
times narrower than that of the emitter, hence the convolution with
$\mathcal{L}$ has a small effect but cannot be completely neglected.

Figure 5: (a) Principle of the measurement of the brightness spectral profile:
the envelope of the output spectrum is measured as a function of the cavity
energy $\omega_{\text{cav}}$, either point by point for each cavity detuning
(black plain lines) or dynamically by modulating the cavity frequency
resonance $\omega_{cav}$. The latter method is used in the following. The
normalized envelope $\text{E}_{\text{mod}}$ is depicted in red; it is closely
related to $\beta$. (b) Experimental normalized envelopes
$\text{E}_{\text{mod}}$ for several longitudinal cavity modes and compared
with the free space spectrum $S_{\text{NT}}$ (black). (c) Spectral profiles
obtained after numerical inversion of $\text{E}_{\text{mod}}$ to retrieve
$\stackrel{{\scriptstyle\approx}}{{S}}_{\text{NT}}$ with $g^{2}/\gamma$ as
free parameter (cf SI). (d) Values of $g^{2}$ (resp. $g$, right nonlinear
scale) deduced from the previous analysis as a function of
$\lambda^{3}/V_{eff}$ together with a linear fit (grey)

To rule out any spurious broadening effect, we took advantage of the fact that
the expression of $\text{E}_{\text{mod}}$ can easily be inverted, so that the
free-space emission spectrum can be recovered from the envelope modulation
profile:
${\stackrel{{\scriptstyle\approx}}{{S}}_{\text{NT}}\simeq\text{E}_{\text{mod}}/(a\
c-a\ \text{E}_{\text{mod}})}$. Hence, we can perform this simple algebraic
transformation for each longitudinal mode with $g$ as the only free parameter.
The outcome of this transformation is displayed in Fig. 5 (c) where the
transformed envelope modulation profile obtained for each longitudinal mode
nicely falls back on the free-space spectrum convoluted by the cavity profile
$\stackrel{{\scriptstyle\approx}}{{S}}_{\text{NT}}$, proving the cQED origin
of the apparent spectral broadening. The inset of Fig. 5 (c) shows the values
of $g^{2}$ as a function of the inverse of the mode volume for each
longitudinal mode, together with a linear fit. The values obtained for the
smallest mode volumes reach up to $25\;\mu$eV. This linear dependence is in
agreement with theory and simply reflects the increasing fluctuations of the
vacuum electric field when reducing the mode volume and hence the increasing
dipolar coupling. Compared to previous reports 17, we obtain a significantly
larger Rabi splitting (up to a factor of 4), which mostly reflects that the
ratio $\lambda^{3}/V$ mechanically increases when shifting the working
wavelength to the telecom bands while keeping a similar geometrical mode
volume.

In principle, the three cQED effects investigated in this work (brightening
effect, lifetime reduction and spectral enlargement of the brightness pofile)
reflect similarly the enhancement of light-matter coupling encoded in $g$.
However, deriving a quantitative relationship between these quantities is not
straightforward due to the interplay between spectral diffusion and pure
dephasing in the linewidth of the emitter 17. For instance, when neglecting
spectral diffusion with respect to pure dephasing in a first approach, the $g$
values deduced from the lifetime measurements
($2g=\sqrt{\gamma^{*}\Delta\gamma/DW}$) are consistently smaller than that
deduced from the spectral method (by a factor of $\sim 3$), showing that fast
spectral diffusion remains a significant contribution to the linewidth even
for the thinnest ZPL profiles ($\sim 200\mu$eV). This is consistent with the
strong bunching effect observed in the intensity correlation measurements
(Fig. 2 (e)) on a time-scale of a few nanoseconds 20. Hence, we emphasize that
the three approaches presented here are complementary and yield accurate
estimate of their respective cQED figures of merit ($\Phi/\Phi_{0}$,
$F_{\text{P}}$ or $g$) which are connected to each other in a non trivial way
in complex condensed-matter emitters combining non-radiative recombination,
pure dephasing and spectral diffusion.

##  6 Conclusion

In conclusion, we have coupled a single organic colored center grafted on a
carbon nanotube to a tunable fiber microcavity, achieving a bright, telecom
wavelength tunable single-photon source, with a fibered output count rate up
to 20 MHz. Using the unique versatility of our setup we could explore
independently different figures of merit of cQED effects. We demonstrated a
brightening effect of about 20 in the first lens, even at low pumping rate,
which reflects the strongly beneficial Purcell effect for dim emitters (low
quantum yield). Time-resolved measurements have brought a direct insight into
the acceleration of the lifetime of the emitter, yielding independent
estimates of the Purcell factor ($F_{\text{P}}\simeq 30$ ) and the intrinsic
quantum yield of the emitter at low temperature which remains surprisingly low
for a colored center. Finally, using the original open geometry of the setup,
we could tune the working wavelength over a broad spectral range and observe
cQED effects on the phonon side-bands of the emitter. We observed a strong
broadening and warping of the brightness spectral profile, which is a unique
signature of cavity enhanced phonon assisted emission mechanisms. This
approach yields an independent estimate of the Rabi splitting of the system
(up to $25\mu$eV). This strong enhancement of light-matter interaction paves
the way to reaching the strong coupling regime with a single emitter, a
requirement for many advanced quantum technology applications.

##  7 Supplementary informations

###  7.1 Cavity characteristics

We measured the geometry of the concave micro-mirror with an AFM. We simulated
the electric field of the cavity in cylindrical coordinates. In figure 6 (a),
the dielectric interfaces are represented in black solid lines. The Bragg
mirrors consist in alternating high and low refractive index layers resulting
in a stop band in the range 1.25 µm to 1.55 µm. The top layer of the fiber is
a high refractive index layer made of $TiO_{2}$, and the top layer of the
planar mirror is the of low refractive index $SiO_{2}$. The polystyrene layer
on top of the planar mirror is also considered.

Figure 6: Finite element numerical simulation of the electric field, $|E|$ in
the cavity, computed at a wavelength of 1.275 $\mu$m (a). The blue, red and
magenta lines represent the surfaces through which the losses are computed and
reported in (b) as a function of the longitudinal mode order ”p”.

The electric field magnitude reaches a maximum of 58 in arbitrary units. In
order to visualize the losses channels of the system, we use a saturated $|E|$
colour scale (max at $10^{-2}$ arbitrary units). The losses are computed
through the flux across the surfaces shown in blue, red and magenta in figure
6.(a). The losses were computed for several Fabry-Pérot longitudinal modes
ranging from 6 to 11 at a wavelength of 1.275 µm, as shown in figure 6 (b).

We observe that spill-out losses are no more negligible as soon as $p>6$,
which may be due to a tip effect of our pencil shape geometry. They even
overcome the other losses by an order of magnitude when $p>9$ leading to a
restricted cavity stability length of approximately 6 µm, significantly lower
than the theoretical value given by the radius of curvature ($\mathcal{R}=10$
µm) of the concave mirror. In addition we observe losses occurring through the
cladding of the fiber, which are comparable in magnitude to the collection
channels $S_{subs}$ and $S_{fiber}$, meaning a substantial share of photons
are lost through the cladding. Using these simulations, we computed the
effective mode volume and the quality factor. These computed values $Q_{cav}$
are reported in Table 1 and compared to the experimental ones obtained by
ring-down measurements.

The difference between the computed and the measured quality factors is
attributed to residual scattering/absorption losses arising from the PS layer
(eg : due to the surface roughness, impurities…). We assume these losses are
independent of the mode order. Finally, taking into account all the losses
channels we computed the exit probabilities for the photons emitted by the
nanotube either through the fiber or through the back mirror as a function of
the cavity mode order as shown in Table 1.

p |  $V_{eff}$ ($\lambda^{3}$) | $Q_{cav,th}(\times 10^{4})$ | $Q_{cav,exp}(\times 10^{4})$ | $P_{cav\rightarrow subs,exp}(\%)$ | $P_{cav\rightarrow fiber,exp}(\%)$
---|---|---|---|---|---
6 | 2.49 | 5.69 | 1.12 | 7.85 | 6.03
7 | 2.86 | 4.92 | 1.05 | 5.32 | 4.40
8 | 3.53 | 6.02 | 0.99 | 4.89 | 3.74
9 | 4.23 | 5.45 | 1.04 | 4.64 | 3.14
Table 1: Mode volume,$V_{eff}$, computed cavity quality factor $Q_{cav,th}$
and measured quality factor $Q_{cav,exp}$ as a function of the longitudinal
mode order $p$. Exit probabilities for the photons through the substrate
($P_{cav\rightarrow subs,exp}$) , or through the fiber $P_{cav\rightarrow
fiber,exp}$. The total differs from 100% due to losses.

###  7.2 Assessment of optical losses in the optical paths

The optical transmission coefficients of the optical parts or
collection/detection efficiencies of the instruments were either measured
directly using a 1.31 µm laser diode for the lenses and mirrors, or estimated
numerically for the collection of the custom objective in free-space
configuration and for the output probabilities of the cavity (see Table 1).
They are reported in Table 2. The extraction efficiencies take in account the
numerical aperture of the objectives and the radiation diagram of the emitter
both in cavity and free space.

In cavity, the photon beam exiting through the planar mirror is collimated by
a 0.16 numerical aperture aspheric lens with a 5 mm diameter. In addition, a
plano-convex lens is installed onto the back of the 6 mm thick planar mirror
to improve the collection by the aspheric lens. However, these optical parts
being inside the cryostat, their position cannot be finely adjusted and the
effective collection efficiency has to be calibrated empirically.

| Free-space | Cavity : planar mirror port (p=6) | Cavity : fiber port (p=6)
---|---|---|---
Extraction in the 1st lens | 19% | 5.6% | 6.0%
Beamsplitter | 61.2 % | 98 % | –
Optical parts in the cryostat | – | 34 % | –
Other parts along the path | 58% | 74% | 19%
Spectrometer + CCD | 9.8% | 9.8% | –
SSPD | – | – | 78%
Table 2: Extraction yield ”in the first lens” (or in the collecting fiber)
estimated from numerical simulations. Transmission coefficient of optical
parts along the collection path for free-space and cavity configurations
deduced from measurements at $\lambda=1310$ nm. Efficiency specifications for
the spectrometer + CCD detector and for the SSPD detector.

To this aim, we recorded simultaneously the flux of the emitter at 1.275 µm in
the cavity mode (p=6) with the spectrometer for the planar mirror output and
with the SSPD for the fiber output with an cw excitation power of 150 µW. The
count rates are shown in Figure 7. Using Table 2, we can experimentally
evaluate the ratio of photons exiting the cavity through the fiber over the
one exiting the cavity through the planar mirror. The experimental ratio is
2.3, which means, using the values of Table 2 that the collection efficiency
of the optical parts in the cryostat is about 0.33. Finally, using this
calibration we estimate the collection efficiency ratio for the free-space vs
cavity mode through the planar mirror : $\eta_{\text{fs}}/\eta_{\text{cav-
plan}}=4.9$.

| Free-space | Cavity (p=6) | Cavity (p=6) |
---|---|---|---|---
|  | planar mirror port | fiber port |
Extraction in the 1st lens | 19 % | 5.6 % | 6.0 % | simulations
Transmission along the path |  |  |  |
and detector efficiency | 3.5 % | 2.4 % | 15 % | experimental
Overall efficiency | 0.66 % | 0.135 % | 0.90 % |
Table 3: Summary of useful extraction and overall collection efficiencies in
all the configurations of the experimental setup.

###  7.3 Single mode photon flux in the single mode fiber

To extract the single photon flux in the single mode fiber, we record at the
same time the count rate of photons with the SSPD and the spectrum of the
photons collected through the planar mirror with the spectrometer and CCD
(integration time 1 s), with 100 µW cw excitation power. The count rates from
each output port are shown in Figure 7. We observe a ratio of $6.7\pm 0.3$
between the flux recorded on the SSPD and the one measured on the spectrometer
(after spectral integration over the linewidth).

The flux measured on the spectrometer is typically $2.7\pm 0.1\times
10^{4}$counts/s, for an excitation power of 100 µW. We deduce from CCD
measurements that the fiber-coupled flux at the maximum excitation power of
4.7 mW is $3.1\pm 0.3\times 10^{6}$ counts/s. Correcting from the optical
efficiencies along the optical path towards the CCD, we deduce a typical flux
of $21\pm 2\times 10^{6}$ photons/s coupled into the single mode fiber.

Figure 7: Cavity coupled photon flux recorded as a function of time either
from the fiber output or from the planar mirror. The ratio of the fibered
output to the planar output is $\simeq 6.7$

###  7.4 Extraction efficiency

Figure 8: (a) Calculated FS emission diagram26 for different position of a
dipole in a 160nm thick polystyrene layer deposited on a Bragg mirror. The
$x=0$ position corresponds to the air-PS interface and $x=0.16\mu m$ to the
PS-Bragg interface. The green pattern indicates the angle collected by the
drilled aspheric lens. (b) Comparison of the FS collection efficiency and the
ratio $\epsilon E^{2}/\text{max}(\epsilon E^{2})$ which corresponds to the
variation of the emitter-cavity coupling $g^{2}$ as a function of the vertical
position of the emitter in the PS layer.

In FS configuration, the emission diagram was computed numerically 26 for
several locations of the emitter in the polystyrene layer and the collection
efficiency was computed using the numerical aperture of the aspheric lens
(N.A.=0.7). For a polystyrene layer thickness of 160 nm, the collection
efficiency varies from 19 % if the emitter is close to the Bragg mirror down
to 2 % if the emitter is close to the polystyrene surface due to interferences
with the PL reflected off the Bragg mirror. This effect yields a selection
bias as we mainly chose the brightest emitters. Fortunately, the interference
pattern is the same in the cavity configuration, so that the brightest FS
emitters are also those that are best spatially coupled to the cavity mode.
Hence, as we selected only the brightest emitters, we assume that the the FS
collection efficiency is 19 %. In any case, any possible bias if the estimate
of this collection efficiency would affect similarly the coupling to the
cavity, resulting in a negligible modification in the estimate of the Purcell
factor.

###  7.5 Coupling to phonons

The coupling to phonons can be taken into account following our previous work
24 and 25. In the polaron framework and in the interaction picture, by using a
second order Born approximation, the master equation reads 24:

$\frac{\partial\rho_{I}}{\partial
t}=-\frac{1}{\hbar^{2}}\int_{0}^{t}du[V_{I}(t),[V_{I}(u),\rho_{I}(t)]],$ (6)

where, $V_{I}(t)=i\hbar g(e^{-i\delta
t}e^{i\Omega(t)}a^{\dagger}\sigma^{-}-h.c.)$. We trace out the phonon bath
degree of freedom, using $Tr_{B}(\rho_{th}e^{\mp i\Omega(t)}e^{\pm
i\Omega(u)})=K(t-u)$, where $K(t)$ is the phonon kernel. By letting $t$ go to
$\infty$ in the upper limit of the integral, and by introducing a dephasing
term, one finally gets:

$\frac{\partial\rho}{\partial
t}=\frac{1}{i\hbar}[H_{eff},\rho]+g^{2}\tilde{S}^{\text{emi}}(\omega_{\text{cav}})L[a^{\dagger}\sigma^{-}]+g^{2}\tilde{S}^{\text{abs}}(\omega_{\text{cav}})L[a\sigma^{+}],$
(7)

where $\tilde{S}^{\text{emi /abs}}$ are defined in 24, with the normalisation
$\int d\omega\ \tilde{S}^{\text{emi /abs}}(\omega)=2\pi$. The Lindblad terms
are given by: $L[C]=C\rho C^{\dagger}-\frac{1}{2}(C^{\dagger}C\rho+\rho
C^{\dagger}C)$. The effective hamiltonian (in the polaron framework) is given
by:

$H_{eff}=\hbar(\omega_{x}+\Delta_{x})\sigma^{+}\sigma^{-}+\hbar(\omega_{c}+\Delta_{c})a^{\dagger}a$
(8)

where $\Delta_{x}$ and $\Delta_{c}$ are effective Lamb shift in the presence
of phonons. The effective Hamiltonian is much simpler than the one in 25 since
in a 1D phonon bath, the effective coupling $g^{\prime}=g\langle B\rangle$
goes to zero.

Within this approximation the hamiltonian evolution of the exciton and the
photon are independent and are coupled only through incoherent population
transfer via the Lindblad terms.

####  7.5.1 Emission spectrum

The emission spectrum of the coupled nanotube-cavity system is given by :

$S^{\text{NT in cav}}=\int_{0}^{\infty}dt\int_{0}^{\infty}d\tau
e^{-i\omega\tau}\langle a^{\dagger}(t+\tau)a(t)\rangle+cc$ (9)

The expectation value of the two-time operator can be obtained using the
quantum regression theorem yielding :

$\langle
a^{\dagger}(t+\tau)a(\tau)\rangle=e^{i(\omega_{c}+\Delta_{c})\tau}e^{-\frac{\kappa+g^{2}\tilde{S}^{\text{emi}}+g^{2}\tilde{S}^{\text{abs}}}{2}\tau}\langle
a^{\dagger}(t)a(t)\rangle$ (10)

By neglecting the Lamb shift $\Delta_{c}$ and the additional cavity broadening
$\kappa+g^{2}\tilde{S}^{\text{emi}}+g^{2}\tilde{S}^{\text{abs}}\approx\kappa$
and under a weak continuous excitation of the emitter, the spectrum reads:

$S^{\text{NT in
cav}}\propto\frac{1}{1+\left(\frac{\omega-\omega_{cav}}{\kappa/2}\right)^{2}}\langle
a^{\dagger}a\rangle_{ss}.$ (11)

The steady state $\langle\rangle_{ss}$ population is obtained by solving (in
the steady state regime , with a weak incoherent pump $p$ ):

$\displaystyle\frac{d\langle\sigma^{+}\sigma^{-}\rangle}{dt}=0$
$\displaystyle=$
$\displaystyle-(\gamma+g^{2}\tilde{S}^{\text{emi}})\langle\sigma^{+}\sigma^{-}\rangle+g^{2}\tilde{S}^{\text{abs}}\langle
a^{\dagger}a\rangle+p,$ (12) $\displaystyle\frac{d\langle
a^{\dagger}a\rangle}{dt}=0$ $\displaystyle=$
$\displaystyle-(\gamma+g^{2}\tilde{S}^{\text{abs}})\langle
a^{\dagger}a\rangle+g^{2}\tilde{S}^{\text{emi}}\langle\sigma^{+}\sigma^{-}\rangle.$
(13)

This leads to

$\langle a^{\dagger}a\rangle_{ss}=\frac{p}{k}\beta(\omega_{\text{cav}}),$ (14)

where 24

$\beta(\omega_{\text{cav}})=\frac{g^{2}\tilde{S}^{\text{emi}}(\omega_{\text{cav}})/\gamma}{1+g^{2}\tilde{S}^{\text{emi}}(\omega_{\text{cav}})/\gamma+g^{2}\tilde{S}^{\text{abs}}(\omega_{\text{cav}})/\kappa}.$
(15)

Finally the emission spectrum is given by :

$S^{\text{NT in
cav}}(\omega,\omega_{\text{cav}})\propto\frac{2}{\pi\kappa}\frac{1}{1+\left(\frac{\omega-\omega_{cav}}{\kappa/2}\right)^{2}}\beta(\omega_{\text{cav}}).$
(16)

The equation can be interpreted in the following way : the Lorentzian factor,
which is typically much narrower than the second factor, corresponds to the
spectral filtering by the cavity. It yields an output spectral profile which
is merely the Lorentzian transmission spectrum of the cavity. The second
factor $\beta(\omega_{\text{cav}})$ corresponds to the overall amplitude of
this Lorentzian signal for a given detuning $\delta$ between the cavity and
the ZPL of the emitter. By integrating the output spectrum over $\omega$
(which corresponds experimentally to measuring the output flux for a given
detuning using a photo-diode), we recover $\beta(\omega_{\text{cav}})$.

Reciprocally, if we modulate the detuning and measure the output spectrum over
an integration time much larger than the modulation period, we obtain :

$E_{\text{mod}}(\omega)=\int d\omega_{\text{cav}}S^{\text{NT in
cav}}(\omega,\omega_{\text{cav}})=(\mathcal{L}*\beta)(\omega),$ (17)

where $\mathcal{L}$ is a normalized Lorentzian profile
$\mathcal{L}(\omega)=\frac{2}{\pi\kappa}\frac{1}{1+\left(\frac{\omega}{\kappa/2}\right)^{2}}$.

For the sake of simplicity, we verify numerically that :

$(\mathcal{L}*\beta)\approx\frac{g^{2}\stackrel{{\scriptstyle\approx}}{{S}}_{\text{emi}}/\gamma}{1+g^{2}\stackrel{{\scriptstyle\approx}}{{S}}_{\text{emi}}/\gamma+g^{2}\stackrel{{\scriptstyle\approx}}{{S}}_{\text{abs}}/\kappa}.$
(18)

where the double tilde notation stands for a double convolution with
$\mathcal{L}$. By using the coupling $g$ deduced from the following paragraph,
we check numerically that the approximation (18) does not change the profile
(the standard deviation of the left hand side minus the right hand side of
(18) normalised to 1, is less than $5\cdot 10^{-5}$), but it boils down to a
small vertical stretching of the profile by a factor $\approx 1.05$.

####  7.5.2 Extracting the coupling $g$ from $\text{E}_{mod}$

Naturally, the experimental modulation profile is known up to a global factor,
hence we can only compare normalized profiles. Following this previous
section, and by neglecting the re-absorption processes in (18), the normalized
modulation profile reads :

$\text{E}_{\text{mod}}^{\text{norm}}(\omega)=\frac{1+a\stackrel{{\scriptstyle\approx}}{{S}}_{max}}{a\stackrel{{\scriptstyle\approx}}{{S}}_{max}}\frac{a\stackrel{{\scriptstyle\approx}}{{S}}(\omega)}{1+a\stackrel{{\scriptstyle\approx}}{{S}}(\omega)},$
(19)

where $\stackrel{{\scriptstyle\approx}}{{S}}$ stands for
$\stackrel{{\scriptstyle\approx}}{{S}}_{emi}$,
$\stackrel{{\scriptstyle\approx}}{{S}}_{max}$ is the maximum value of
$\stackrel{{\scriptstyle\approx}}{{S}}$ and $a=g^{2}/\gamma$ is the parameter
that will be extracted by fitting the normalized experimental profile to (19).

The authors thanks X. He for sample preparation. The authors thank C.
Diederichs, E. Baudin, G. Hetet, Z. Said for fruitful discussion as well as T.
Pöpplau for fiber micro-machining. We also thank A. Leclercq and P. Morfin for
help in designing and fabricating mechanical components.

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