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Supplemental material: Tapered fiber coupling of single photons emitted by
a deterministically positioned single nitrogen vacancy center
Lars Liebermeister,1, a) Fabian Petersen,1 Asmus v. Münchow,1 Daniel Burchardt,1 Juliane Hermelbracht,1
Toshiyuki Tashima,1 Andreas W. Schell,2 Oliver Benson,2 Thomas Meinhardt,3, 4 Anke Krueger,3, 4 Ariane
Stiebeiner,5 Arno Rauschenbeutel,5 Harald Weinfurter,1, 6 and Markus Weber1, 6, b)
1)
Fakultät für Physik, Ludwig-Maximilians-Universität München, 80799 München Germany
Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin Germany
3)
Institut für Organische Chemie, Universität Würzburg, 97074 Würzburg Germany
4)
Wilhelm Conrad Roentgen Research Center for Complex Materials Systems, Universität Würzburg,
97074 Wuerzburg Germany
5)
Atominstitut, Technische Universität Wien, 1020 Wien Austria
6)
Max-Planck-Institut für Quantenoptik, 85748 Garching Germany
2)
(Dated: 25 November 2013)
For an NV-center hosted in a diamond nano-crystal, in
principle, the orientation of the two orthogonal dipoles
p1 and p2 can be determined via polarization dependent
excitation measurements and polarization analysis of the
emitted fluorescence light1 . In our experiments the linear polarization angle α of the excitation laser (λ=532
nm) was varied and the broadband NV-fluorescence was
detected polarization insensitive. We therefore could determine the orientation of the NV symmetry axis, given
by the Euler angles φ and θ (see Fig.1, b), but not the
orientation of these dipoles (given by the Euler angle γ)
in the plane perpendicular to the NV symmetry axis1 .
However, from knowledge about the orientation of the
NV symmetry axis with respect to the nano-fiber and
with the help of numerical FDTD simulations we can give
a lower and upper bound for the predicted and measured
nano-fiber coupling efficiency β.
As a first step we measured the countrate of confocally
detected photons (scattered by the single NV-center on
the nano-fiber) as a function of the excitation polarization angle α (see Fig.1, a) and fitted the measured data
with
P (α)
,
I(α) = k
P (α) + Psat
for a fixed saturation power Psat = 1.17mW. This model
takes into account saturation of the two coupled dipole
transitions at room temperature2,3 and for the polarization dependence
P (α) = P0 [sin2 (α − φ) + cos2 (α − φ) cos2 θ]
of exciting the two dipoles p1 and p2 in the weak excitation regime1 . In the model, θ is the angle between the
NV symmetry axis and the optical axis of the confocal
a) email:
b) email:
lars.liebermeister@physik.uni-muenchen.de
markusweber@lmu.de
z( )
b)
a)
NV
nanofiber
y (∥)
x (⟂)
FIG. 1. (a) Measured confocal countrate of an NV-center on
a nano-fiber as function of the excitation polarization angle α
and corresponding fit (excitation power 0.5 mW). (b) Euler
angles (θ, φ, γ) used to describe the spatial orientation of the
two orthogonal dipoles p1 and p2 of the NV-center on the
surface of an optical nano-fiber. The projections of pi onto
the eigenaxes of the nano-fiber (⊥, k, ρ) allows to determine
the nano-fiber coupling efficiency.
microscope (z-axis) whereas φ is the Euler angle corresponding to a rotation around the z-axis. The mentioned
fit yields the desired Euler angles
φ = (36.3 ± 3.0)◦
θ = (70.7 ± 0.3)◦ ,
which determine the orientation of the NV symmetry
axis.
Now we can derive a lower and upper bound of the
expected nano-fiber coupling efficiency. A single, arbitrarly oriented dipole pi (φ, θ, γ) of the NV-center can be
decomposed with the help of an Euler transformation4
into the components p⊥ , pk and pρ in the nano-fiber
eigenbasis {⊥, k, ρ}. As we know the coupling efficiency
of a dipole (positioned 10 nm above the surface of the
nano-fiber) oriented along the respective eigenaxis from
FDTD simulations (0.275, 0.156 and 0.349) we can calculate the nano-fiber coupling efficiencies β1 and β2 of the
two dipoles p1 and p2 :
q
βi = (0.275pi,⊥ )2 + (0.156pi,k)2 + (0.349pi,ρ )2
For an NV-center the nano-fiber coupling efficiency β is
2
then given by
β=
β1 + β2
.
2
scope objective is then given by
q
ηicoll = (0.0515pi,⊥)2 + (0.0705pi,k)2 + (0.00429pi,ρ)2 ,
and for an NV-center
η coll =
By variation of γ, which takes into account the unknown
orientation of the two dipoles in the plane perpendicular
to NV symmetry axis, and for fixed angles φ = 36.3◦ and
θ = 70.7◦ we derive an upper and lower bound for the
expected coupling efficiency
η1coll + η2coll
.
2
Again by variation of the angle γ (for fixed angles φ =
36.3◦ and θ = 70.7◦ ) we can derive an upper and lower
bound for η coll , yielding
0.040 ± 0.002 ≤ η coll ≤ 0.044 ± 0.002
(28.78 ± 0.03)% ≤ β ≤ (29.22 ± 0.03)%
Besides the determination of the expected nano-fiber
coupling efficiency the performed FDTD simulations enable also to estimate the decay rate Γf ree into free-space
modes, an important part of the determination of the
measured fiber coupling efficiency. On the basis of FDTD
simulations we also determined the fraction of light scattered by a dipole (aligned parallel to one of the eigenaxes
(⊥, k, ρ) of the nano-fiber) into the effective numerical
aperture (NA∗ = 0.32 ± 0.01) of the microscope objective (0.0515, 0.0705, 0.00429). For an arbitrarily oriented
dipole pi the fraction of photons collected by the micro-
With these values and by taking into account all losses
of our confocal microscope the emission rate of the NVcenter into free-space modes is (1.7 ± 0.1 ≤ Γf ree ≤
1.8 ± 0.1) Mcts/s. Combined with the total radiative
decay rate into the fiber mode Γnf = (194 ± 2)kcts/s the
measured nano-fiber coupling efficiency is bounded by
(9.5 ± 0.6)% ≤ βmeas ≤ (10.4 ± 0.7)%
1 D.
Zheng, PhD-thesis, Ecole Normal Superieure de Cachan (2010).
P. Mayer Alegre, C. Santori, G. Medeiros-Ribeiro, and R. G.
Beausoleil, Phys. Rev. B 76, 165205 (2007).
3 K.-M. C. Fu, C. Santori, P. E. Barclay, L. J. Rogers, N. B. Manson,
R. G. Beausoleil, Phys. Rev. Lett. 103, 256404 (2009).
4 Y. Y. Hui, Y.-R. Chang, N. Mohan, T.-S. Lim, Y.-Y. Chen, and
H.-C. Chang, J. Phys. Chem. A 115, 1878 (2011).
2 T.