# Channeling of fluorescence photons from quantum dots into guided modes of an
optical nanofiber tip

Resmi M, Elaganuru Bashaiah, and Ramachandrarao Yalla  School of Physics,
University of Hyderabad, Hyderabad, Telangana, India-500046<EMAIL_ADDRESS>
###### Abstract

We demonstrate the channeling of fluorescence photons from quantum dots (QDs)
into guided modes of an optical nanofiber tip (ONFT). We deposit QDs on the
ONFT using micro/nano fluidic technology. We measure the photon-counting and
emission spectrum of fluorescence photons that are channeled into guided modes
of the ONFT. The measured emission spectrum confirms the deposition of QDs on
the ONFT. We perform numerical simulations to determine channeling efficiency
($\eta$) for the ONFT and a single dipole source (SDS) system. For the
radially oriented SDS at the center of the facet of the ONFT, we found the
maximum $\eta$-value of 44% at the fiber size parameter of 7.16, corresponding
to the ONFT radius of 0.71 $\mu m$ for the emission wavelength at 620 $nm$.
Additionally, we investigate the SDS position dependence in transverse
directions on the facet of the ONFT in view of keeping experimental
ambiguities. The present fiber inline platform may open new avenues in quantum
technologies.

  * January 2024

Keywords: Optical nanofiber tip, Single photons, Quantum dots.

##  1 Introduction

The ultimate control of light-matter interaction at the single photon level
and the implementation of such a system are challenging tasks in contemporary
quantum physics, with potential applications in the field of quantum
technologies[1, 2]. Single photons are ideal information carriers for quantum
information science including quantum networks. Moreover, with the present
technology, the information can be readily encoded into the photon degrees of
freedom for exchange between distant quantum processing nodes, leading to
quantum entanglement, another key ingredient of quantum technology. Many
protocols have been proposed and implemented to achieve efficient manipulation
of single photons using various techniques. Examples would include high
finesse optical cavities in free-space[3], photonic crystal cavities[4],
micropillar cavities[5], diamond nanobeams[6], and silicon nitride alligator
waveguides[7].

However, the crucial challenge lies in efficiently directing single photons
emitted by a single photon source into single-mode fibers (SMF). To address
this challenge, recently optical nanofibers (ONF) [8] and optical nanofiber
tips (ONFT) [9, 10] are proven to be promising candidates. ONFs are tapered
optical fibers with a sub-wavelength diameter at the waist region. ONFTs are
tapered optical fibers with a sub-wavelength diameter at one edge. A
substantial difference in effective refractive index between the core (silica)
and clad (air) of the ONF/ONFT leads to properties such as strong optical
confinement, large evanescent field, strong field enhancement, etc. [11, 12].
Due to the large evanescent field, ONFs are indispensable components in the
fields of quantum optics, quantum information science [13], and sensing [14,
15]. Furthermore, the intensity distribution is modified due to the silica
core and vacuum clad [16]. Channeling of the dipole emission into a step-index
fiber with respect to different orientations and positions has been modeled by
Matthew R. Henderson et al. [17]. They demonstrated that the significant
refractive index contrast between the core and the clad of the SMF is highly
effective for channeling single photons.

Various groups worldwide have made notable progress in enhancing channeling
efficiency ($\eta$). Efficiently channeling single photons emitted by a single
photon source becomes achievable when the source is positioned in the vicinity
of the ONF, as supported by theoretical studies [18]. The decay rate into each
mode of a multi-mode waveguide when a single photon emitter is placed close to
its surface and changes in the $\eta$-value with respect to the guide’s radius
and dipole orientations have been studied [19]. Solid state single photon
emitters [20] such as quantum dots (QD), nitrogen-vacancy (NV) centers, and
silicon-vacancy (SiV) centers are being used for experiments. In some
experiments ions, dye molecules, and defect centers in diamond crystals are
used as single photon emitters [21, 22, 23, 24, 25, 26]. Also, single photons
are generated in a controlled way from a strongly coupled atom-cavity system
[27] and a single trapped two-level atom [28].

A cesium atom in the vicinity of the ONF can achieve a remarkable $\eta$-value
of up to 28% [29, 30]. A 22% of channeling was demonstrated by placing QDs on
the surface of the ONF [8]. Also 7.4% of total photons emitted from a single
QD coupled to the ONF has been demonstrated [31]. Single molecules have been
efficiently coupled to the ONF with the maximum $\eta$-value upto 30% [32].
Laser cooled atoms interfaced with ONFs have also been demonstrated [33].
Using a flat silica ONFT, it has been numerically demonstrated that up to 38%
of photons emitted from a single quantum emitter can be directly channeled
into an SMF [9]. Though one-sided channeling, the ONFT offers higher
$\eta$-value compared to two-sided channeling in ONFs.

Cavities and hybrid systems are introduced on the ONF and the ONFT for
enhanced $\eta$-value. It has been numerically demonstrated that a maximum
$\eta$-value of $\sim$75% when an NV center is placed on a hybrid system of
the ONF and a diamond nanowire [34]. Single nanodiamonds containing NV centers
deposited on the ONF enhances the coupling directly to an SMF [4]. The
coupling of SiV centers in nanodiamonds with the ONF has been demonstrated
[35]. It has been quantitatively demonstrated that a charged exciton for a
single QD deposited on the ONF and cooled down to cryogenic temperatures acts
as an excellent quantum emitter [36, 37]. Emission properties of single QD
deposited on the ONF is enhanced in the presence of a single gold nanorod,
leading to efficient coupling of single photons into guided modes of the ONF
[38]. QDs deposited near gold nanorods on the surface of the ONF creates an
enhanced single photon source [39]. Regarding ONFTs, the enhancement in the
$\eta$-value in the presence of a gold nanoparticle near the ONFT has been
predicted using numerical simulations [40]. The maximum evanescent coupling of
30% has been coupled to a tapered fiber tip from another excitation taper
[41]. Improved antibunching using single photon sources based on statistical
ensembles at the tip of the ONFT is predicted numerically and theoretically
[42]. Excitation of surface plasmons on gold-coated ONFTs and coupling them
into guided mode of optical fibers have been shown experimentally [43].

Although channeling of single photons into guided modes of the ONF has been
widely studied, no experimental study in this regard utilizing the ONFT has
been reported. Also the fluorescence photon counts and emission spectrum
measurement of the single photon emitter on the ONFT using free space and
guided mode excitatioins have not been reported yet. The systematic
$\eta$-value dependence study is not extended to multi-mode regime. Also, the
systematic study to demonstrate the variation in the $\eta$-value according to
the single quantum emitter position on the facet of the ONFT has not been
reported yet.

Regarding the photon emitter, we use QDs in our experiments. QDs are
semiconductor particles of a few nanometers in size [44, 24]. Upon absorbing
light, electrons transition from the valence band to the conduction band,
generates an electron-hole pair known as an exciton. Upon recombination of the
electron and hole, energy is emitted in the form of a photon. Since the QD
particle size falls under the Bohr radius, the energy required to create an
exciton increases, leading to the effect quantum confinement. QDs display
distinctive optical characteristics due to alterations in their band gap
energy induced by quantum confinement effects [45]. It has been experimentally
shown that CdSe/ZnS QDs exhibits a perfect antibunching at room temperature
under continuous or pulsed excitation [46].

In this paper, we demonstrate the channeling of fluorescence photons from QDs
into guided modes of the ONFT. We deposit QDs on the ONFT using micro/nano
fluidic technology. We measure the photon-counting and emission spectrum of
fluorescence photons channeled into guided modes of the ONFT using a single
photon counting module (SPCM) and a spectrometer (SP), respectively. The
emission spectrum peak at $\sim$614 $nm$, with the full width at half maximum
(FWHM) $\sim$22 $nm$, confirms the deposition of QDs on the ONFT. The ONFT is
fabricated using a chemical etching technique employing hydrofluoric (HF)
acid. We determine the $\eta$-value for the ONFT and a single dipole source
(SDS) system by performing numerical simulations. We found a maximum
$\eta$-value of 44%, at the fiber size parameter ($k_{0}a$) of 7.16,
corresponding to the ONFT radius of 0.71 $\mu m$. The SDS orientation and
wavelength are set at radial and 620 $nm$, respectively. The SDS is placed at
the center of the facet of the ONFT. Additionally, we found that sweeping the
SDS position in transverse directions on the facet of the ONFT does not
significantly affect the $\eta$-value.

##  2 Simulation procedure

Figure 1: (a) and (b) show conceptual schematics for the SDS on the surface
and inside the ONF, respectively. (c) and (d) show conceptual schematics for
the SDS inside and on the facet of the ONFT, respectively. ONF, ONFT, SDS, M,
and a denote optical nanofiber, optical nanofiber tip, single dipole source,
monitor, and radius of the ONF/ONFT, respectively.

The conceptual schematics are shown in Figs. 1. Numerical simulations are
performed to determine the $\eta$-value for the ONF/ONFT and the SDS system
using a Finite Difference Time Domain (FDTD) (FDTD package, Ansys) method [47,
48]. We investigate various possibilities of positioning the SDS on the
ONF/ONFT and the $\eta$-value variation for each case. The system is enclosed
in a three-dimensional simulation area of $6$$\times$6$\times$$25$ $\mu
m^{3}$. The refractive index of a cylinder is chosen to be that of silica. The
refractive index of the surrounding medium is set to be 1. For simulations
with the ONF, the length of the silica cylinder is set at 30 $\mu m$ along the
z-axis so that it crosses the perfectly matched layer (PML) boundaries and is
considered infinitely long. In the case of the ONFT, the length is set at 25
$\mu m$ such that one end of the silica cylinder does not cross the PML
boundary and is treated as the tip. The SDS is placed 10 $nm$ away from the
surface of the ONF/ONFT to make it distinguishable for the simulation mesh and
for considering the size of the QD. Note that the QD size is around 6-7 $nm$.
The mesh type is set to be auto non-uniform so that the near-field SDS pattern
is taken into account and boundary conditions are satisfied. The SDS
wavelength is set at 620 $nm$, corresponding to the wavelength of QDs used in
the present experiment. The orientation of the SDS is set along
radial/azimuthal/axial direction of the ONF/ONFT. We place a 2D power monitor
(M) 15 $\mu m$ away from the SDS to determine transmission ($T$) and Purcell
factor ($PF$). Then, we determine $\eta$=$\frac{T}{PF}$ [49, 29, 8]. The
simulations are performed by varying the ONF/ONFT radius (a) from 0.062 $\mu
m$ to 1.24 $\mu m$, which corresponds to the $k_{0}a$-value from 0.62 to 12.56
($a/\lambda$ varies from $0.1$ to $2$). For reference, we perform simulations
for the following four cases

  * •

Case (i): The SDS is placed on the surface of the ONF

  * •

Case (ii): The SDS is placed inside the ONF

  * •

Case (iii): The SDS is placed inside the ONFT

  * •

Case (iv): The SDS is placed on the facet of the ONFT

Schematics for the four cases are shown in Figs. 1 (a)-(d). Although
channeling occurs to both sides of the ONF, we show only one sided channeling
for reference. In contrast, the channeling of photons occurs only to one side
of the ONFT. To check the validity of simulations, the SDS is placed on the
surface of the ONF, and the position of the SDS is varied along the radial
direction such that the SDS moves away from the surface of the ONF to $1$ $\mu
m$ away. Considering experimental ambiguity in placing QDs at the center of
the facet of the ONFT, we perform simulations to determine the variation in
the $\eta$-value when the SDS position is swept along the axial ($z$-axis) and
transverse ($x$ and $y$-axes) directions on the facet of the ONFT.

##  3 Experimental procedure

###  3.1 Fabrication of the ONFT

The ONFT is fabricated using a two-step chemical etching technique employing
HF acid (Avra, ASH2565) [50, 51, 52]. We use 30% and 24% HF acid for the first
and second steps, respectively. An etching time of 60-100 minutes for the
first and 30-90 minutes for the second steps are followed. We use a silica SMF
(1550B-HP, Coherent) of length 50 $cm$ for fabricating the ONFT. The etching
time is controlled to produce the desired ONFT diameter. The fabricated ONFT
is characterized optically and morphologically using laser and field emission
scanning electron microscopy (FESEM) (Sigma 360 VP, Zeiss), respectively.
Also, a quality check of the ONFT can be asserted by optical characterization.
If serious scattering is observed, we don’t use the sample for further
experiments. Details of the fabrication procedure and characterization are
reported elsewhere. The fabricated ONFT is affixed to a metal holder (MH)
using a UV-curable glue for further experiments.

###  3.2 Deposition of QDs on the ONFT

Figure 2: (a) A conceptual diagram for the deposition of QDs on the ONFT. LD,
M, VND, FC, SMF, MH, PS, ONFT, and QD denote laser diode, mirror, variable
neutral density filter, fiber coupler, single mode fiber, metal holder,
precision stage, optical nanofiber tip, and quantum dot, respectively. (b) The
experimental picture of the scattering observed in the QD solution during
deposition.

As QDs are proven to be an excellent single photon emitter [53, 54], we use
CdSe/ZnS core-shell type colloidal QDs (900219, Sigma Aldrich) for our
experiments. The original QD solution has a concentration of 5 $mg/ml$ in
toluene. The original QD solution is diluted to 5%. 5 $\mu l$ of the original
QD solution is added to 95 $\mu l$ of toluene to produce 100 $\mu l$ of the
diluted QD solution. QDs are deposited on the ONFT using micro/nano-fluidic
technology. The experimental schematic of the deposition procedure is shown in
Fig. 2 (a). The fabricated ONFT is connected to a laser diode (LD) (DL-G-5,
Holmarc) at a wavelength of 532 $nm$ through two mirrors (M1 and M2), a
variable neutral density filter (HO-VND-N50, Holmarc), and a fiber coupler
(FC) (PAF2-7B, Thorlabs). For precise movement, the MH is mounted on a high-
precision stage (PS) (TS 65, Holmarc). The PS is adjusted such that the ONFT
is slowly dipped in the diluted QD solution. When the ONFT reaches the QD
solution, one can readily observe a scattering, which serves the confirmation
of the deposition, as shown in Fig. 2 (b). Note that the scattering comes from
light transmitted by the ONFT and gets enhanced when the ONFT is dipped in the
QD solution. Then, the ONFT is withdrawn from the QD solution for the
characterization.

###  3.3 Characterization of QDs deposited ONFT

We characterize QDs deposited on the ONFT in two excitation schemes. One is a
free space excitation and the other is a guided mode excitation. The
experimental sketch for free space excitation is shown in Fig. 3 (a). The
characterization is performed using the SP (STS-VIS, Ocean Optics) and the
SPCM (SPCM-AQRH-14-FC, Excelitas). The laser light is directed to the FC as
explained in section 3.2 and then to a collimator using an SMF. The laser
light is focused on the ONFT using a microscope objective lens (OL) (Plan C N,
40X, Olympus) for exciting QDs. The PS is used to align the ONFT to the focal
point of the OL. The end of the ONFT is connected to a fiber port (FP) and
then passes through a long pass filter (FEL 600, Thorlabs) of cut-on
wavelength 600 $nm$, and to another FP. A multi-mode fiber (MMF) is used to
connect the FP to the SPCM for more signal collection. The fluorescence
photons channeled into guided modes of the ONFT are measured using the SPCM
connected to a photon counter (C8855-01, Hamamatsu). The emission spectrum of
fluorescence photons is measured using the SP. In free space excitation, a
laser power of 0.02 $\mu W$ at the SMF in Fig. 3 (a) is focused on the ONFT
using the OL.

The experimental sketch for guided mode excitation is shown in Fig. 3 (b). The
laser light is guided through the ONFT using a 50:50 splice fiber (SF)
(TW670R5A2, Thorlabs). QDs are excited through guided modes of the ONFT. The
emitted fluorescence is detected through the same SF. Note that the guided
mode excitation is robust against alignment compared to free space excitation.
The LD is connected to one end of the SF same as for the free space
excitation, mentioned as ‘A’ in Fig. 3 (a). The other end of the SF is
connected to the SPCM/SP same as for the free space excitation, mentioned as
‘B’ in Fig. 3 (a). In guided mode excitation, a laser power of 0.008 $\mu W$
at the SMF in Fig. 3 (b) is used. Here, the laser light is focused on QDs
through the ONFT of diameter 312 $nm$.

Figure 3: (a) and (b) are conceptual diagrams of free space and guided mode
excitations to characterize QDs deposited on the ONFT, respectively. LD, M,
VND, FC, CL, OL, QD, ONFT, MH, PS, FP, F, SMF, MMF, SP, and SPCM denote laser
diode, mirror, variable neutral density filter, fiber coupler, collimator,
objective lens, quantum dot, optical nanofiber tip, metal holder, precision
stage, fiber port, long pass filter, single-mode fiber, multi-mode fiber,
spectrometer, and single photon counting module, respectively.

##  4 Simulation results

Figure 4: Dependence of channeling efficiency ($\eta$) as a function of fiber
size parameter ($k_{0}a$). (a) and (b) are for single dipole sources (SDS) on
the surface and inside the optical nanofiber, respectively. (c) and (d) are
for the SDS inside and on the facet of the optical nanofiber tip,
respectively. Black circles, blue squares, and red triangles denote radial,
azimuthal, and axial orientations of the SDS, respectively.

The simulation predicted results for cases (i)-(iv) are summarised in Figs. 4
(a)-(d). The horizontal axes denote $k_{0}a$-values and the vertical axes
denote $\eta$-values, respectively. For the case (i), the $k_{0}a$-value
varies from 0.62 to 4.65. For case (ii), case (iii), and case (iv), the
$k_{0}a$\- value varies from 0.62 to 12.56. Black circles, blue squares, and
red triangles denote radial, azimuthal, and axial orientations of the SDS,
respectively. One can readily observe that maximum $\eta$-value occurs for the
radially orientated SDS compared to azimuthal and axial orientations for all
cases. The radial and azimuthal orientations are same for Figs. 4 (b)-(d).
Note that $\eta$-values are for one side of the ONF. For the case (i) and
(ii), the peak $\eta$-value of 15% and 27% occurred for the radial orientation
at the $k_{0}a$-value of 1.44 and 1.76, which corresponds to the ONF radius of
0.14 $\mu m$ and 0.17 $\mu m$, respectively. For case (ii), one can observe
that the maximum $\eta$-value is almost twice that of case (i).

For the case (iii), the peak $\eta$-value of 30% occurred at the
$k_{0}a$-value of 1.88, corresponding to the ONFT radius of 0.18 $\mu m$. It
is obvious from the results that when the ONFT is used instead of the ONF,
there is an increase in the $\eta$-value. For the case (iv), one can readily
see four peaks. We denote the first, second, third, and fourth peaks as P1,
P2, P3, and P4, respectively. The maximum $\eta$-value corresponds to P3 in
Fig. 4 (d). For P3, the maximum $\eta$-value of 44.5% is obtained at the
$k_{0}a$-value of 7.16, which corresponds to the ONFT radius of 0.71 $\mu m$.
The P1, P2, and P4 $\eta$-values of 37%, 43.5%, and 43% occurred at
$k_{0}a$-values of 2.01, 4.39, and 10.05, which corresponds to the ONFT radius
of 0.2 $\mu m$, 0.43 $\mu m$, and 1 $\mu m$, respectively.

Next, we investigate the SDS position dependence on the $\eta$-value from the
surface along the radial direction of the ONF. The corresponding simulation
predicted result is shown in Fig. 5 (a). The $k_{0}a$-value of the ONF is set
at 1.44. The horizontal axis denotes distance along the radial direction
($d_{r}$) and the vertical axis denotes $\eta$-value. Black circles, blue
squares, and red triangles represent radial, azimuthal, and axial orientations
of the SDS, respectively. Data points are fitted with a single exponential
decay function, denoted by red solid line. As the SDS moves away from the ONF,
the $\eta$-value decreases exponentially.

Figure 5: The dependence of channeling efficiency ($\eta$) as a function of
the single dipole source (SDS) position: (a) is for the radial direction of
the optical nanofiber. Black circles, blue squares, and red triangles
represent radial, azimuthal, and axial orientations of the SDS. (b), (c), and
(d) are for $z$, $x$, and $y$-directions on the facet of the optical nanofiber
tip, respectively. Black squares, stars, and triangles represent $k_{0}a$\-
values of 2.01, 4.39, and 7.16, respectively. (b), (c), and (d) correspond to
the radial orientation of the SDS. Insets show the position of the SDS.

Additionally, we sweep the SDS position along $z$, $x$, and $y$ directions
from the center of the facet of the ONFT. The simulation predicted results are
summarised in Figs. 5 (b)-(d). The horizontal axes denote distance along the
$z$-direction ($d_{z}$)/$x$-direction ($d_{x}$)/$y$-direction ($d_{y}$) and
the vertical axes denote $\eta$-values. Black squares, black stars, and black
triangles represent $k_{0}a$\- values 2.01 (P1), 4.39 (P2), and 7.16 (P3),
respectively. All the results correspond to the radial orientation. The
simulation predicted results for sweeping the SDS position along the
$z$-direction of the ONFT are shown in Fig. 5 (b). Data points are fitted with
a single exponential decay function, denoted by the red solid line. As the SDS
goes away from the ONFT, the $\eta$-value decreases. The simulation result
corresponding to sweeping the SDS position along the $x$-axis on the facet of
the ONFT is shown in Fig. 5 (c). One can readily observe that the $\eta$-value
does not vary significantly for the $k_{0}a$-value of 7.16. The
maximum/minimum $\eta$-value is 44.5%/21%. From the center of the facet to 0.5
$\mu m$ away along the $x$-direction, the $\eta$-value of 40% is being
maintained. The $\eta$-value variation for $k_{0}a$-values 2.01 and 4.39 is
significant. For the $k_{0}a$-values of 2.01 and 4.39, the maximum/minimum
$\eta$-values are 37%/13% and 43.5%/17%, respectively. The simulation result
corresponding to sweeping the SDS position along the $y$-axis on the facet of
the ONFT is shown in Fig. 5 (d). One can readily observe that the $\eta$-value
does not vary significantly for the $k_{0}a$-value of 7.16. The
maximum/minimum $\eta$-value is 44.5%/26%.

##  5 Experimental results

Figure 6: A typical field emission scanning electron microscopy image of an
optical nanofiber tip diameter of $\sim$312 $nm$.

A typical FESEM image of a fabricated ONFT is shown in Fig. 6. The diameter of
the ONFT is found to be $\sim$0.31 $\mu m$. The measured optical transmission
is $\sim$50%. We performed the photon counting experiment and spectrum
measurement for 15 samples. Out of these, only typical spectrum and photon
counts are shown here. A typical observed photon counting as a function of
time using free space excitation is shown in Fig. 7 (a). The horizontal axis
denotes time in seconds, and the vertical axis denotes counts (c/s). Black,
red, and green traces denote the dark count rate of the SPCM, background, and
fluorescence photon counts from QDs into the ONFT, respectively. The
corresponding histogram for the three is shown in Fig. 7 (b). An average of 70
($\pm$25) c/s and 350 ($\pm$60) c/s are observed as the dark count rate of the
SPCM and background, respectively. An average of 1312 ($\pm$120) c/s is
observed as the fluorescence photon counts when QDs on the ONFT are focused.
The background corrected fluorescence spectrum observed using free space
excitation is shown in Fig. 7 (c). It is fitted with a Lorentzian peak
function. It exhibits a peak wavelength at 614 $nm$ with FWHM 22 ($\pm$1)
$nm$.

Figure 7: (a) and (d) are measured photon counts as a function of time using
free space excitation and guided mode excitation, respectively. (b) and (e)
are histograms corresponding to (a) and (d), respectively. (c) and (f) are
background-corrected fluorescence spectra measured using free space excitation
and guided mode excitation, respectively. Black, red, and green traces/ bars
denote the dark count rate of the single photon counting module, background,
and fluorescence photon counts from quantum dots into guided modes of the
optical nanofiber tip, respectively.

A typical observed photon counting as a function of time using guided mode
excitation is shown in Fig. 7 (d). The horizontal axis denotes time in seconds
and the vertical axis denotes counts (c/s). Black, red, and green traces
denote the dark count rate of the SPCM, background, and fluorescence photon
counts from QDs into the ONFT, respectively. The corresponding histogram for
the three is shown in Fig. 7 (e). An average of 70 ($\pm$25) c/s, 498
($\pm$75) c/s, and 1171 ($\pm$110) are observed as the dark count rate of the
SPCM, background, and fluorescence photon counts from QDs on the ONFT,
respectively. The background-corrected fluorescence spectrum observed using
guided mode excitation is shown in Fig. 7 (f). It is fitted with a Lorentzian
peak function. It exhibits a peak wavelength at 617 $nm$ with FWHM 22 ($\pm$1)
$nm$.

##  6 Discussions

As seen in Fig. 4 (a), the result indicates that the maximum $\eta$-value is
15%. The predicted results are in good agreement with previously reported
results [8, 29]. When the SDS is placed inside the ONF, there is a significant
increase in the $\eta$-value to 27% as shown in Fig.4 (b). It is consistent
with previously reported results [55]. This clearly implies that when the SDS
is positioned inside the ONF instead of on the surface, the $\eta$-value is
almost doubled. Although the $\eta$-value is higher than that of the case (i),
it is less than 30% when the ONF is used. The $\eta$-value crossed 30% for the
case (iii) as shown in Fig. 4 (c). This simulation result implies that the
$\eta$-value for the ONFT is higher than the ONF. The present $\eta$-value of
44% is achieved when the SDS is positioned on the center of the facet of the
ONFT as shown in Fig. 4 (d). This signifies that, in this configuration, a
substantial number of photons emitted from the SDS effectively channels into
guided modes of the ONFT. Due to the interference between the radiation from
the SDS and that from the mirror image of the SDS on the facet of the ONFT,
the $\eta$-value is enhanced compared to the cases where the SDS is on the
surface or inside the ONF/ONFT [9].

As seen in Fig. 4 (d), the single mode condition for the wavelength 620 $nm$
is satisfied when $k_{0}a$$<$ $2.28$. The first peak $\eta$-value ($k_{0}a$=
$2.01$) corresponds to the single-mode regime, and the successive peaks
correspond to the multi-mode regime. For the first peak, the contribution is
only from the fundamental mode ($HE_{11}$) and for the other three peaks
contribution is from the fundamental mode and higher order modes, leading to
an increase in the $\eta$-value. The maximum $\eta$-value of 44.5% occurred at
the $k_{0}a$-value of 7.16, which corresponds to the ONFT radius of 0.71 $\mu
m$. This result indicates that the SMF is not necessarily tapered to sub-
wavelength scales to realize the maximum $\eta$-value. This implies that
simulated results can be feasible for experiments. The present guided mode
excitation experiments are performed in the single-mode regime since the ONFT
diameter of 0.31 $\mu m$ corresponds to the $k_{0}a$-value of 1.57.
Experiments in the multi-mode regime can be done as it is easy to fabricate
thicker ONFTs.

As seen in Fig. 5 (a), as the SDS moves away from the ONF, the $\eta$-value
decreases exponentially as expected. As the SDS moves away from the ONF/ONFT,
evanescent field gets decayed and hence $\eta$-value follows exponential
decay. The predicted results are in good agreement with previously reported
results [29, 16]. As seen in Fig. 5 (b), if another single photon source is
used instead of QDs, its size variation can be accounted. One can observe that
the $\eta$-value pattern for the P1 in Figs. 5 (c) and (d) are almost the
same. This is due to the operational condition in the single-mode regime. In
contrast, for the other two peaks, P2 and P3, which are in the multimode
regime, there is an oscillatory behavior. This may be due to the asymmetry in
the multimode field distribution along x and y-directions [19]. Note that, in
both cases, the SDS is oriented along the radial direction. As seen in Fig. 5
(c), from the center of the facet to 0.5 $\mu m$ away along $x$-direction, the
$\eta$-value of 40% is being maintained for the $k_{0}a$-value of 7.16. This
implies that the SDS is not necessarily to be placed at the center of the
facet of the ONFT. From the experimental point of view, it gives the freedom
to deposit QDs anywhere along the $x$-direction on the facet. But the SDS
position along the $y$-direction has to be determined carefully as seen in
Fig. 5 (d).

Though irregularities exist on the ONFT shown in Fig. 6, its size is much
smaller than the wavelength detected, 620 $nm$. Therefore it does not
seriously affect the light confinement. As seen in Fig. 7 (a), the measured
dark count rate of the SPCM is 70 ($\pm$25) c/s, which is consistent with the
value quoted by the company. As seen as red traces, 350 ($\pm$60) c/s is
observed as the background photon counts, which includes the dark counts of
the SPCM. As seen as green traces, an average of 1312 ($\pm$120) c/s is
observed as the fluorescence photon counts. One can understand that only a
total of $962$ ($\pm$134) c/s is the fluorescence photon counts from QDs that
are channeled into guided modes of the ONFT. In a similar fashion, as seen in
Fig. 7 (d), an average of 673 ($\pm$133) c/s is observed as the fluorescence
photon counts. Note that the ONFT used for free space and guided mode
excitations are different. Also considering all losses through the guided mode
excitation, the observed counts is much less than the actual fluorescence
photons counts. The guided mode excitation undergoes high coupling loss
comapared to the free space excitation. Note that keeping the ONFT aligned
with the focal point of the OL throughout the experiment is challenging.
Hence, guided mode excitation is preferred over free space excitation to avoid
mechanical instability in the system. As the simulation predicted, a
significant amount of photons are channeled into the ONFT. The peak wavelength
in the emission spectrum of fluorescence photons from QDs channeled into
guided modes of the ONFT confirms the deposition of QDs on the ONFT. The peak
matches with the company-quoted emission wavelength 620 ($\pm$10) $nm$. Hence,
we demonstrated the channeling of fluorescence photons from QDs into guided
modes of the ONFT.

One can quantitatively measure the $\eta$-value by observing the photon
counting rate for guided modes and radiation modes simulatenously. The
$\eta$-value of 22% is already experimentally demonstrated using the ONF and a
single QD system [8]. Therefore, it should be possible to quantitatively
measure the $\eta$-value with the ONFT. As described in 3.2, we dip the ONFT
in the QD solution, leading to ambiguity in the position of deposition of QDs.
We do further experiments to investigate the exact location of QDs deposited.
The ongoing experiments to realize the exact location of QDs deposited on the
ONFT and the number of QDs deposited will be discussed elsewhere. The
deposition of a single QD on the surface of the ONF has been realized and
reproduced theoretical predictions [8]. Therefore, we believe that a single QD
on the ONFT is also possible to realize and reproduce the present simulation
results. The ongoing experiments regarding this will be reported elsewhere.
Although we use QDs in our experiments, the study can be extended for atoms
and defects in nanodiamonds.

##  7 Summary

In summary, we demonstrated the channeling of fluorescence photons from
quantum dots (QDs) into guided modes of an optical nanofiber tip (ONFT). We
deposited QDs on the ONFT using micro/nano fluidic technology. We measured the
photon-counting and emission spectrum of fluorescence photons channeled into
guided modes of the ONFT using two excitation schemes- free space and guided
mode. The measured emission spectrum confirms the deposition of QDs on the
ONFT. We determined channeling efficiency ($\eta$) for the ONFT and a single
dipole source (SDS) system by performing numerical simulations. The maximum
$\eta$-value of 44% occurred at the fiber size parameter of 7.16,
corresponding to the ONFT radius of 0.71 $\mu m$ for the emission wavelength
at 620 $nm$. Additionally, we investigated the SDS position dependence in
transverse directions on the facet of the ONFT in view of keeping experimental
ambiguities. The automatic fiber coupling scheme is a unique aspect of the
present scheme. This cavity-free and operating at room temperature method is a
unique development to realize a quantum network node. The present work will
have significant applications in modern physics, chemistry, biology and
industry. The quantum functionalized optical fibers can be readily used to
create the quantum internet. The present fiber inline platform may open new
avenues in quantum technologies.

## Acknowledgments

RM acknowledges the University Grants Commission (UGC) for the financial
support (Ref. No.:1412/CSIR-UGC NET June 2019). RRY acknowledges financial
support from the Science and Engineering Research Board (SERB) for the Core
Research Grant (CRG) (File No. CRG/2021/009185), Institute of Eminence (IoE)
grant at the University of Hyderabad, Ministry of Education (MoE) (File No.
RC2-21-019), and the Scheme for Transformational and Advanced Research in
Sciences (STARS) grant from Ministry of Human Resource Development (MHRD)
(File No. STARS/APR2019/PS/271/FS).

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Epoch-wise double descent is the phenomenon where generalisation performance improves beyond the point of overfitting, resulting in a generalisation curve exhibiting two descents under the course of learning. Understanding the mechanisms driving this behaviour is crucial not only for understanding the generalisation behaviour of machine learning models in general, but also for employing conventional selection methods, such as the use of early stopping to mitigate overfitting. While we ultimately want to draw conclusions of more complex models, such as deep neural networks, a majority of theoretical conclusions regarding the underlying cause of epoch-wise double descent are based on simple models, such as standard linear regression. To start bridging this gap, we study epoch-wise double descent in two-layer linear neural networks. First, we derive a gradient flow for the linear two-layer model, that bridges the learning dynamics of the standard linear regression model, and the linear two-layer diagonal network with quadratic weights. Second, we identify additional factors of epoch-wise double descent emerging with the extra model layer, by deriving necessary conditions for the generalisation error to follow a double descent pattern. While epoch-wise double descent in linear regression has been attributed to differences in input variance, in the two-layer model, also the singular values of the input-output covariance matrix play an important role. This opens up for further questions regarding unidentified factors of epoch-wise double descent for truly deep models.

§ INTRODUCTION

The bias-variance trade-off implies a U-shaped generalisation error, as a function of model complexity. It suggests that generalisation will improve with an increasing model complexity only to a certain point, after which the model will show signs of overfitting, and from which it never recovers. The double descent phenomenon contradicts this idea, showcasing how generalisation performance can improve even beyond the point where the model perfectly fits the training data [4, 18], leading to a generalisation curve following a so called double descent pattern.

The double descent pattern in generalisation error has been observed not only with increasing model size, resulting in model-wise double descent, but also with the number of training epochs, when training models using iterative learning algorithms such as gradient descent, resulting in so-called epoch-wise double descent, see e.g. [3, 18]. While model-wise double descent has been studied extensively ([4, 1, 2, 9, 14, 8] to name a few), it typically involves understanding the asymptotic behaviour of models, assuming infinite training time. Meanwhile, epoch-wise double descent requires understanding the whole trajectory of learning.

Typically, epoch-wise double descent has been studied either mainly  through empirical observation, e.g. [18], or through the theoretical analysis of simple models, such as linear regression [15, 21, 19] or the random feature model [6, 7]. On the one end, [18] connects epoch-wise double descent, similar to model-wise double descent, to the notion of model capacity, attributing the second descent to the model capacity exceeding the number of training samples. On the other end, [15, 19, 7] connect epoch-wise double descent to differences in speed of learning, describing the generalisation error as consisting of overlapping bias-variance trade-off curves, corresponding to fast- and slow-learned features. With this, the eigenvalues of the input covariance matrix is integral to the double descent phenomenon. Moreover, we do not need overparametrisation for the epoch-wise double descent pattern to emerge, something that is also consistent with earlier observations of the phenomenon, see e.g. [3]. However, as the basis for these analyses are models consisting of just  one layer of learnable parameters, while findings are corroborated empirically for deeper models, this leaves an open-ended question of possibly unidentified factors causing epoch-wise double descent in deeper models.

Although [15] admittedly do also study double descent in a two-layer neural network with ReLU activation, the insights of [15] do not connect double descent in two-layer networks directly to the properties of the data, but rather the Jacobian of the network. Meanwhile, another line of work, studying the gradient dynamics of linear two-layer networks, indicate that the singular values of the input-output covariance matrix of the training data are important factors in determining the speed of learning in these models [20, 10, 17, 22]. This suggests another factor important to the double descent phenomenon in two-layer linear neural networks, which has not been attributed as a factor of double descent in single-layer models. However, so far, and to the best of our knowledge, this has not been studied explicitly.

In this paper, we aim to start bridging the gap between epoch-wise double descent in the single-layer linear regression model and multi-layer neural networks, by studying double descent in the two-layer linear neural network. While this model is indeed still linear, the training dynamics are, in contrast to the single-layer model, non-linear. Moreover, the benefits of studying linear models, is that we can obtain closed-form solutions for the training dynamics, at least in the case where weights are decoupled, such that they evolve independently within a transformed space that is defined by the training data. More importantly, it opens up for the possibility of directly connecting epoch-wise double descent to properties of the training data.

For our purposes, we derive the gradient flow for a two-layer diagonal, or decoupled, neural network, similar to what was studied in [22], but with non-isotropic input, and with individual learning rates for the first and second layer. With our formulation, the standard (single-layer) linear regression model is a special case of the dynamics, and hence we can connect our findings also to this special case. For the two-layer decoupled linear neural network, we

* Derive the time-dependent generalisation error  as a superposition of bias-variance curves corresponding to each weight in the decoupled dynamics, assuming that the true model is linear.

* Characterise the behaviour of each individual bias-variance trade-off curve, and use this to find a necessary condition for epoch-wise double descent.

Through this analysis, we identify additional factors of epoch-wise double descent in the two-layer model, not observed in the one-layer model, including a connection to the singular values of the input-output covariance matrix as well as an additional circumstance under which epoch-wise double descent is possible.

§ THEORY

In what follows, we derive decoupled dynamics for the two-layer linear neural networks, acting as a bridge between the dynamics of the one-layer linear model and the decoupled two-layer model with quadratic weights, studied in e.g. [20, 10]. Then, we use these dynamics to study factors of epoch-wise double descent in two-layer linear networks.

§ DISCUSSION

We investigate epoch-wise double descent in decoupled two-layer linear networks, identifying factors of epoch-wise double descent in gradient-based learning. First, apart from differences in the eigenvalues of the input covariance matrix of the training data, also differences in the singular values of the input-output covariance matrix can cause epoch-wise double descent in the two-layer model. Second, by providing a necessary condition for epoch-wise double descent both for the standard linear regression model and for the linear decoupled two-layer network, we reveal an additional circumstance under which epoch-wise double descent can emerge in the two-layer model, not observed for the one-layer model.

There are several venues for further investigating epoch-wise double descent in (linear) neural networks, with the aim to better understand the causes of epoch-wise double descent in deep models. For one, it remains to be investigated if patterns in the training loss, such as the previously observed incremental learning pattern, see e.g [10, 11, 5], could be indicators of epoch-wise double descent. Moreover, several assumptions was made leading up to the dynamics in <ref> and relaxing these assumptions could render new insights into the epoch-wise double descent phenomenon. We shortly discuss two extensions, namely that of deep(er) linear neural networks, as well as that of studying the dynamics of <ref> without decoupling of the weights.

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Supplementary material for
Towards understanding epoch-wise double descent in two-layer linear neural networks