# Optical probing of the carriers-mediated coupling of the spin of two Co
atoms in a quantum dot

L. Besombes<EMAIL_ADDRESS>Institut Néel, CNRS, Univ. Grenoble
Alpes and Grenoble INP, 38000 Grenoble, France  J. Kobak  Univ. of Warsaw,
Faculty of Physics, ul. Pasteura 5, 02-093 Warsaw, Poland  W. Pacuski  Univ.
of Warsaw, Faculty of Physics, ul. Pasteura 5, 02-093 Warsaw, Poland

###### Abstract

We report on the optical spectroscopy of the spin of two Co atoms in a quantum
dot and interacting with a single exciton. The spectrum of quantum dots
containing two Co atoms are exchange interaction and by the strain at the
location of the magnetic atoms. A wide range of spectrum can be obtained
depending on the relative coupling of each atom to the confined exciton. We
obtained a comprehensive interpretation of the experimental data with a spin
Hamiltonian model. We show that the two Co atoms spins can be orientated by
the injection of spin polarized carriers at zero magnetic field. This induces
a correlation between the two spins that is observed in the intensity
distribution of the emission spectra. The optical absorption in the phonon
sideband of quantum dots doped with two Co reveals resonant absorptions which
strongly depend on a transverse magnetic field. We show that these
characteristic absorptions result from an interplay between the mixing of Co
spin states induced by the presence of in-plane strain anisotropy at the
magnetic atoms location and the transverse field.

##  I Introduction

Individual localized spins in semiconductors, which can feature both
electrical and optical control, hold promise for solid-state quantum
information storage. Optically active semiconductor quantum dots (QDs)
containing an individual carrier offer the possibility of an efficient spin-
photon interface, an essential element towards the establishment of quantum
information networks [1, 2]. Localized spins of transition metal elements can
also be addressed optically when they are inserted in a QD [3, 4, 5, 6, 7].
This spin-photon interface is established by the exchange interaction of
magnetic atoms with the spin of optically created carriers. A wide range of
magnetic atoms can be inserted into semiconductors, offering optical access to
a wide choice of localized electron spin, nuclear spin and orbital momentum
[6].

Semiconductor heterostructures offer in addition the unique possibility of
controlling the exchange interaction between distant magnetic dopants by
varying the carrier density of the host [26]. In the case of a dot containing
two or a few magnetic atoms, the spins are only coupled via a short range
exchange that could be only relevant for the first or second neighbors. The
injection of a single exciton whose coupling with the magnetic atoms is
dominated by the exchange interaction with the hole spin will couples
ferromagnetically the atoms. This has been demonstrated in the case of two
manganese atoms (Mn) in both II-VI and III-V semiconductors [7, 8].

We analyze here the case of cobalt (Co) in II-VI semiconductors, a magnetic
element which carries a localized spin with a large sensitivity to local
strain. Co, a 3$d^{7}$ transition element, is usually incorporated in II-VI
semiconductors as a Co2+ ion carrying a spin S=3/2 with an orbital momentum
L=3. All the stable isotopes also carries a nuclear spin I=7/2. The orbital
momentum efficiently connects the Co spin to its strain environment through
the modulation of the crystal field and the spin-orbit coupling making Co an
interesting spin qubit for hybrid spin-mechanical systems [9, 10, 11, 12, 13].

The influence of static local strain on the emission of QDs doped with a
single Co atom has been analyzed [6]. Here we demonstrate that in this diluted
magnetic semiconductor system the exchange coupling is high enough to permit
to spectrally resolve the different magnetic ground states of two Co atoms and
address them independently. We performed the magneto-optic spectroscopy and
resonant optical spectroscopy of CdTe/ZnTe QDs doped with two Co atoms. This
permits first to clearly identify the main parameters controlling the
characteristic emission spectrum of such QDs. We also show that the two Co
spins can be optically oriented by the injection of spin polarized carriers.
The induced correlation between the localized spins is observed in the
photoluminescence (PL) intensity distribution. Under resonant optical
excitation, the absorption of the QDs’ phonon sideband reveals transitions
which strongly depend on a transverse magnetic field. We show that these
characteristic absorptions result from an interplay of the spin states mixing
induced by the presence of an in-plane strain anisotropy at the magnetic atoms
location and the transverse magnetic field.

The rest of the paper is organized as follows: After a short presentation of
the sample and experiments in Sec. II, we discuss in Sec. III the magneto-
optic properties of QDs containing two Co atoms. The fine structure of a
confined exciton in the exchange field of two Co atoms is analyzed and modeled
in detail. In Sec. IV we show that the two Co spins are correlated by there
mutual interaction with confined carriers. We demonstrate that the two Co
spins can be oriented by the optical injection of spin polarized carriers and
measured spin fluctuations under optical excitation in the tens of nanosecond
range. In Sec. V we present and discuss the transverse magnetic field
dependence of the acoustic phonon sideband absorption. The conclusion is drawn
in Sec. VI.

##  II Experimental details

In this study we use self-assembled CdTe/ZnTe QDs grown by molecular beam
epitaxy and doped with Co atoms. The concentration of Co was adjusted to
obtain QDs containing 0, 1, 2 or a few magnetic atoms [6].

Individual QDs are studied by optical micro-spectroscopy at liquid helium
temperature (T= 4.2 K). The sample is mounted on x,y,z piezo actuators and
inserted in a vacuum tube under a low pressure of He exchange gaz. The tube is
immersed in the variable temperature insert of the cryostat filled with liquid
He. The sample temperature is measured thanks to a sensor located in the
copper sample holder. The cryostat is equipped with a vectorial
superconducting coil and a magnetic field up to 9 T along the QD growth axis
and 2T in the QD plane can be applied.

The PL is excited with a tunable dye laser and collected by a high numerical
aperture microscope objective (NA=0.85). For spectral analysis, the PL is
dispersed by a two meters double grating spectrometer (1800 gr/mm) and
detected by a cooled Si charge-coupled-device camera. Power dependence
measurements are performed by using a neutral gradient filter. For the Photo-
Luminescence Excitation (PLE) measurements, the power of the tunable dye laser
is stabilized with an electro-optic variable attenuator when its wavelength is
tuned.

A half wave plate in front of a linear polarizer is used to analyze the linear
polarization. For circular polarization measurement, a quarter-wave plate
oriented at 45° from the linear polarization direction of detection is
inserted in the detection path. When inserted both in the excitation and
detection paths, at 45° from the linear polarization of the excitation laser,
the quarter-wave plate, combined with a half-wave plate and a linear
polarizer, is used to select co-circularly or cross-circularly polarized PL.

For the auto-correlation measurements, a single QD emission line is selected
by the exit slits of the double spectrometer and the PL signal sent to an
Hanbury Brown–Twiss (HBT) setup. The HBT setup is equipped with two Si
avalanche photodiodes in combination with a time-correlated photon-counting
module giving an overall time resolution of around 700 ps.

##  III Individual QDs doped with two Co atoms

In the studied sample, a large fraction of the dots have no magnetic atom
inside, some of them have 1 Co and in some instance 2 Co or more. Dots with a
single Co sufficiently coupled to an exciton give rise to PL with 4 peaks. The
intensity distribution within these four peaks changes from dot to dot with in
general a weaker intensity or even no PL detectable on the inner two lines
[6].

The origin of the 4 peaks can be accounted for by the strongly anisotropic
coupling of the exciton to the four electronic spin states of the Co
$S_{z}$=$\pm 1/2$,$\pm 3/2$. The Co-exciton interaction is dominated by the
coupling with the spin of the hole. In theses self-assembled QDs, the holes
have a heavy-hole character with a spin Jhz=$\pm 3/2$ along the growth axis.
This results in an effective magnetic field on the exciton oriented along the
QDs’ growth direction and given by $g\mu_{B}B_{eff}$=IhCoSz where IhCo is the
hole-Co exchange. I${}_{hCo}\propto\beta|\Psi_{h}(\vec{r}_{Co})|^{2}$ depends
both on the value of the hole wave function at the location $\vec{r}_{Co}$ of
the Co in the QD and on $\beta$ a material dependent quantity [25]. For Co in
CdTe $\beta$ is dominated by the kinetic exchange and is antiferromagnetic
[14].

Figure 1: Low temperature (T=4.2K) photoluminescence spectra of five QDs
containing two Co atoms labeled from QD1 to QD5. The diversity of spectra is
controlled by the ratio of exchange couplings of the two atoms with the
exciton and by the strained induced splitting of the spin of each atom.

The intensity distribution observed in Co-doped QDs arises from the strain
induced fine structure of the magnetic atom. The strain influences the fine
structure of the atom via the deformation of the crystal field and the spin-
orbit coupling. The two inner lines of the exciton-Co system are associated
with the Co spin states Sz=$\pm 1/2$. In self-assembled QDs, the strain at the
magnetic atom position changes from dot to dot [15] but is usually dominated
by negative in-plane bi-axial strain. In this case, the Sz=$\pm 1/2$ spin
states are shifted to a higher energy and their occupancy probability at
cryogenic temperature is weak.

###  III.1 Photoluminescence of QDs containing two Co atoms

For a dot with two Co atoms, similar qualitative arguments suggest that the
dominant effective exchange field felt by the exciton is now
$g\mu_{B}B_{eff}$=I${}_{hCo_{(1)}}$Sz(1)+I${}_{hCo_{(2)}}$Sz(2) which can take
up to 16 values. The observation of dots with two Co strongly coupled to the
exciton should then give rise to more than 4 but less than 16 lines.

Emission of different Co-doped QDs measured at T=4.2 K and at zero magnetic
field are presented in Fig. 1. The PL presents in general four dominant lines
with similar intensities. The splitting between the four main lines is
irregular and also changes from dot to dot. The two inner lines can almost
overlap at the center of the structure, as in QD1, or be very close to the two
outer lines, forming two doublets, as in QD5. This contrasts with dots doped
with a single Co where the splitting is regular and the intensity of the inner
and outer lines usually very different [15].

As can be seen from the PL intensity maps of linear polarization in Fig. 2,
the central lines are significantly linearly polarized and their polarization
rate is larger in dots where their energy spacing is smaller. The linear
polarization rate is also generally much lower for the two outer lines.

Weaker intensity optical transitions are in general observed between the main
PL lines. They can also present some linear polarization. The intensity of
these weaker transitions depends on the excitation power. As it can be seen in
Fig. 3, their observations is easier at high excitation power and their
contributions almost disappear at very low excitation. Such weaker intensity
lines between the four main peaks are not observed in QDs containing a single
Co atom.

In the large diversity of observed PL spectrum (see Fig. 1), certain common
feature appear: four dominant PL lines and lower intensity lines with a
relative contribution that depends on the excitation power. This PL structure
is characteristic of dots containing two Co atoms under biaxial in-plane
strain. In the presence of negative biaxial in-plane strain, the spins of each
Co thermalizes on their ground states Sz(i)=$\pm$3/2. There are then four
equivalent spin ground states
$|\pm\frac{3}{2}\rangle_{(1)}|\pm\frac{3}{2}\rangle_{(2)}$ with total spin
Mz=$\pm$3 and $|\pm\frac{3}{2}\rangle_{(1)}|\mp\frac{3}{2}\rangle_{(2)}$ with
Mz=0. These states are degenerate at zero magnetic field and have the same
occupation probability. The other 12 two Co spins configurations contain at
least a component Sz(i)=$\pm$1/2 which is shifted towards high energy by local
biaxial strain. The probability of occupancy of these 12 excited states at
cryogenic temperature is lower than for the 4 ground states.

Figure 2: Top: Linear polarization PL intensity map of QD2, QD3 and QD4.
Bottom: Corresponding linearly polarized PL spectra recorded along two
orthogonal directions. The direction of polarization is measured with respect
to the [100] or [010] axis of the sample (i.e., at 45° from the easy cleavage
axis of the substrate).

When interacting with an exciton, the four 2Co ground states which are
initially degenerate, are split by the exchange interaction with the confined
carriers (see the energy level scheme in Fig. 4). The resulting four
exciton-2Co levels give the main contribution in the PL spectra and four lines
with similar intensities are expected. The lowest and the highest energy lines
corresponds to, respectively, an anti-ferromagnetic and a ferromagnetic
coupling of the hole with the 2Co states
$|\pm\frac{3}{2}\rangle_{(1)}|\pm\frac{3}{2}\rangle_{(2)}$. The two inner
lines correspond to the coupling of the exciton with the 2Co states
$|\pm\frac{3}{2}\rangle_{(1)}|\mp\frac{3}{2}\rangle_{(2)}\rangle$. These
states with total spin Mz=0 generally interact with the exciton as
I${}_{hCo_{(1)}}$ and I${}_{hCo_{(2)}}$, the exchange interaction of the hole
with each Co, are different resulting in a non-zero exchange coupling of the
hole with the two atoms.

If one neglects the exchange interaction with the electron, IeCo(i), usually
much weaker, the relative position of the inner and outer lines depends on the
difference IhCo(1)-IhCo(2). For example, with
I${}_{hCo_{(1)}}$=I${}_{hCo_{(2)}}$, the exchange interaction of the exciton
with the 2 atoms in the configuration Mz=0 cancels out and a single line would
appear in the middle of the structure. As these lines come closer when
IhCo(1)-IhCo(2) decreases, any weak anisotropic coupling term that can mix
$\sigma+$ and $\sigma-$ excitons becomes significant and induces a linear
polarization rate (see QD2, QD3 in Fig. 2). On the contrary, for very
different values of I${}_{hCo_{(1)}}$ and I${}_{hCo_{(2)}}$, the dominant
inner and outer lines come closer together and can even merge when one of the
exchange term tends towards zero. This gives rise to spectrum dominated by two
doublets separated by a large central gap (see QD5 in Fig. 1).

With these simple considerations, in a QD containing two Co atoms, the
splitting of the two outer lines is given by 9/2(IhCo(1)+IhCo(2)) and the
splitting of the two inner lines by 9/2(IhCo(1)-IhCo(2)). The splitting of the
inner lines is also likely to be significantly affected by the exchange
interaction and possible valence band mixing that can couple $\sigma+$ and
$\sigma-$ excitons.

Figure 3: Excitation power dependence of the PL intensity distribution in QD1
at zero magnetic field. (a) PL spectra for two different excitation powers.
(b) Top: Excitation power dependence of the integrated PL intensity of the
lines (i) and (ii). Bottom: Corresponding normalized PL intensity ratio
(I(i)-I(ii))/(I(i)+I(ii)).

The other 12 2Co states also interact with the exciton. As they contain at
least one spin component Sz(i)=$\pm 1/2$, their splitting induced by the
exchange interaction with the exciton are different than for the 4 ground
states (see Fig. 4). The occupation probability of these sates is weaker at
low temperature and they give rise to lower intensity lines. These PL lines
lie between the transition towards the low-energy anti-ferromagnetic states
and the transition towards the high-energy ferromagnetic states.

The contribution in the PL spectra of these higher energy states is expected
to increase with the increase of the effective spin temperature of the two Co
atoms. An increase of the spin temperature can be induced by the injection of
high energy carriers [20]. For Co which presents a large spin to strain
coupling, it can come from an interaction with non-equilibrium phonons
generated during the carriers’ relaxation [17]. Such modification of the
effective spin temperature is consistent with the increase of the number of
observable lines at high excitation power (see Fig.3 in the case of QD1).
Among these weaker intensity lines, the largest contribution in the PL comes
from exciton associated with states containing at least one component
$S_{z(i)}=\pm 3/2$ of one of the atoms Co(i). These states are the first to be
significantly occupied when the spin temperature increases. If both atoms
experience large bi-axial strain, only the four ground states contribution
will be observed at low temperature.

Figure 4: (a) Scheme of the energy levels at zero magnetic field of a QD
containing, in the ground state, two Co atoms with D0,Co(1) $>$ D0,Co(2) and,
in the excited state, a $\sigma+$ exciton with IhCo(1) $>$ IhCo(2). Only the
PL transitions from the four lowest energy states with the largest occupation
probability are presented.

In 2Co-doped QDs, a large diversity of spectrum can then be expected depending
on the position of the two atoms inside the dot and on the strain state at
each magnetic atom location. In addition, as observed in other magnetic
systems, weak intensity lines can also appear due to the recombination of dark
excitons. The presence of anisotropy in the hole-Co exchange interaction and
local deviation from a pure bi-axial strain around the atom could allow a
recombination of the exciton with a change of the spin state of the magnetic
atom and also increase the number of observed lines.

###  III.2 Magneto-optics of the exciton coupled with two Co atoms

Magneto-optics measurements permit to extract information on the carrier /
magnetic atoms exchange interaction. The longitudinal magnetic field
dependence of the PL of four QDs containing two Co atoms are presented in Fig.
5 in two circular polarizations. As for the zero field spectra, a large
diversity in the magnetic field dependence is observed but some common feature
can be extracted.

Figure 5: Intensity map of the longitudinal magnetic field dependence of the
PL of four QDs containing two Co atoms, (a) QD3, (b) QD1, (c) QD2 and (d) QD5.
In (d), low magnetic field spectra showing the evolution of the intensity
distribution among the two doublets of QD5 are presented in the bottom panel.

The four main lines present a Zeeman splitting and a slight diamagnetic shift.
The weaker intensity lines, although sometime presenting anti-crossings,
follow a similar longitudinal magnetic field dependence with comparable Zeeman
splitting and energy shifts.

Figure 6: Detail at low magnetic field of the intensity map of the emission of
two QDs having a very different ratio IhCo(1)/IhCo(2): QD1 with IhCo(1)
$\approx$ IhCo(2) and QD4 with IhCo(1) $\gg$ IhCo(2).

We can also notice in the PL intensity maps of Fig. 5 that the intensity
distribution among the lines is very sensitive to the applied magnetic field.
Whereas the four main lines are clearly observed at zero field, at large field
the PL mainly arises from one line in each circular polarization. A maximum of
PL is observed on the high energy line in $\sigma$\- polarization and on the
low energy line in $\sigma$\+ polarization. This intensity distribution
reflects the efficient thermalisation on the 2Co spins states split by the
Zeeman energy. With a Lande factor of the Co $g_{Co}\approx 2.5$ [15], the two
Co spins thermalizes on the states Sz(i)=-3/2. For an anti-ferromagnetic
coupling with the exciton, this corresponds to a low energy $\sigma$\+ exciton
state
$|\downarrow_{e},\Uparrow_{h}\rangle|-\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$
and to a high energy $\sigma$\- exciton state
$|\uparrow_{e},\Downarrow_{h}\rangle|-\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$,
as observed in the experiment under large magnetic field.

A change in the intensity distribution is also observed for the two inner
lines. In dots with very different IhCo(i) under magnetic field (i.e. PL
spectrum with two doublets separated by a central gap, Fig. 5(d)), most of the
PL comes from the high energy line in $\sigma-$ and low energy line in
$\sigma+$ but a change in the intensity distribution is also observed among
the doublets. The high energy line of the doublet dominates in $\sigma-$
polarization whereas most of the intensity comes from the low energy line of
the doublet in $\sigma+$ polarization (Fig. 5(d)).

To explain this behavior, let’s first consider that I${}_{hCo(1)}\gg$ IhCo(2).
This corresponds to the case of QD5 or QD4. For a $\sigma-$ exciton, the high
energy line is associated with
$|-\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$. The second line of the
high energy doublet corresponds to
$|-\frac{3}{2}\rangle_{(1)}|+\frac{3}{2}\rangle_{(2)}$. Under a positive
magnetic field, the occupation probability of the state Sz(2)=+3/2 decreases
and consequently the PL intensity of the line associated with
$|-\frac{3}{2}\rangle_{(1)}|+\frac{3}{2}\rangle_{(2)}$ becomes weaker than the
high energy line of the doublet. In the opposite circular polarization,
$\sigma+$, the high energy state is now associated with
$|+\frac{3}{2}\rangle_{(1)}|+\frac{3}{2}\rangle_{(2)}$. Under magnetic field
its occupation probability becomes weaker than for
$|+\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$ and the intensity
distribution among the high energy doublet is reversed, as observed in the
experiment (Fig. 5(d)).

When IhCo(1) $\approx$ IhCo(2), the two dominant inner lines are close to the
center of the spectrum. However, the magnetic field dependence of their
intensity distribution is similar. This gives rise to a configuration in which
the distribution of intensity between the two central lines is opposite to
that of the two outer lines (see Fig. 6).

###  III.3 Model of the magneto-optic properties of QDs containing two Co
atoms

Many parameters are necessary to describe the details of the interaction of
confined carriers with two Co spins in a QD. In addition to the confined
carriers spins fine structure, the position of each atom and the detail of the
strain distribution at the magnetic atom’s location can influence the
spectrum. It is hardly possible to independently adjust all these parameters
and we chose here to discuss a simplified model which can explain the main
feature of the observed spectrum.

| IhCo(i) | IeCo(i) | D0(i) | E(i) | gCo(i) |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
|  $\mu$eV |  $\mu$eV | meV | meV |  |  |  |  |  |
Co(1) | 290 | 0 | -1.5 | -0.2 | 2.5 |  |  |  |  |
Co(2) | 210 | 0 | -1.5 | -0.2 | 2.5 |  |  |  |  |
| 2/3$\delta_{0}$ | $\Delta_{lh}$ | $\rho_{s}$ | $\theta_{s}$ | $\delta_{1}$ | $\gamma$ | ge | gh | $\delta_{xz}$ | $\delta_{yz}$
|  $\mu$eV | meV | meV | ∘ |  $\mu$eV |  $\mu$eV$T^{-2}$ |  |  | meV | meV
X | -650 | 25 | 1 | 0 | 100 | 2 | -0.2 | 0.4 | 0 | 0
Table 1: Parameters used in the modeling presented in figure 7 and 8. The
chosen exchange parameters reproduces the overall splitting of QD3. The other
parameters are typical for individual Co [15] or individual CdTe/ZnTe QDs
[19].

The electron-hole-2Co Hamiltonian in a self-assembled QD,
$\mathcal{H}_{X,2Co}$, can be separated into five parts:

$\displaystyle\mathcal{H}_{X,2Co}=\sum_{i}(\mathcal{H}_{s,Co(i)}+\mathcal{H}_{c,Co(i)})+\mathcal{H}_{B}+\mathcal{H}_{eh}+\mathcal{H}_{bd}$
(1)

$\mathcal{H}_{s,Co(i)}$ describes the fine structure of the Co atoms and their
dependence on local strain. For dominant bi-axial in-plane strain, the induced
zero field splitting of the S=3/2 states specific to each Co(i) (i=1,2) reads:

$\displaystyle\mathcal{H}_{s,Co(i)}=\frac{2}{3}D_{0(i)}\left[S_{z_{i}}^{2}-\frac{1}{2}\left(S_{x_{i}}^{2}+S_{y_{i}}^{2}\right)\right]+\frac{E_{i}}{2}\left(S_{x_{i}}^{2}-S_{y_{i}}^{2}\right)$
(2)

where D0(i) splits Sz(i)=$\pm$3/2 and Sz(i)=$\pm$1/2 spin states and E(i) mix
spin states Sz(i)=$\pm$3/2 and Sz(i)=$\mp$1/2 respectively.

The second term in (1), $\mathcal{H}_{c,Co(i)}$, describes the exchange
coupling of the electron and hole spins with the Co spins. In this model, a
Heisenberg-type exchange interaction is assumed between both Co spins and the
hole and electron spins:

$\displaystyle\mathcal{H}_{c,Co(i)}=I_{hCo(i)}\vec{S_{i}}.\vec{J_{h}}+I_{eCo(i)}\vec{S_{i}}.\vec{\sigma_{e}}$
(3)

with $I_{hCo(i)}$ and $I_{eCo(i)}$ the exchange integrals of the hole
($\vec{J_{h}}$) and the electron ($\vec{\sigma_{e}}$) spins with the Co spins
($\vec{S_{i}}$).

A magnetic field couples via the Zeeman terms to both Co spins and carriers
spins and a diamagnetic shift of the electron-hole pair can also be included
resulting in

$\displaystyle\mathcal{H}_{B}=\sum_{i}g_{Co(i)}\mu_{B}\vec{B}.\vec{S_{i}}+g_{e}\mu_{B}\vec{B}.\vec{\sigma_{e}}+g_{h}\mu_{B}\vec{B}.\vec{J_{h}}+\gamma
B^{2}$ (4)

where ge, gh and gCo(i) are the electron, hole and Co Lande factors and
$\gamma$ a diamagnetic coefficient.

The electron-hole exchange interaction, $\mathcal{H}_{eh}$, contains both
short-range and long-range parts. We consider here QDs with a C2v symmetry
(ellipsoidal flat lenses for instance) where the short-range and the long-
range parts contribute to a splitting $\delta_{0}$ of the bright and dark
excitons and where the long-range part induces a coupling $\delta_{1}$ between
the bright excitons [19].

Figure 7: Calculated zero magnetic field linearly polarized PL spectra of a QD
containing 2 Co atoms with (a) the parameters listed in table 1, (b)
parameters of table 1 and IhCo(2)=75$\mu eV$ and (c) parameters of (b) with
D0(1)=D0(2)=-3.5 meV. For all spectra Teff=20 K. A line broadening with a
Lorentzian of full width at half maximum of 75 $\mu$eV is included in the
spectra. In the PL intensity maps (top panels) the angle is measured from the
[100] axis.

The band Hamiltonian, $\mathcal{H}_{bd}=E_{g}+\mathcal{H}_{vb}$, stands for
the energy of the electron, $E_{g}$, and the heavy-holes and light-holes
energies $\mathcal{H}_{vb}$. We consider here the four lowest energy holes
states $|J_{h},J_{hz}\rangle$ with Jh=3/2. A general form of Hamiltonian
describing the influence of shape or strain anisotropy on the valence band
structure can be written in the basis
($|\frac{3}{2},+\frac{3}{2}\rangle,|\frac{3}{2},+\frac{1}{2}\rangle,|\frac{3}{2},-\frac{1}{2}\rangle,|\frac{3}{2},-\frac{3}{2}\rangle$)
as:

$\mathcal{H}_{vb}=\left(\begin{array}[]{cccc}0&-Q&R&0\\\
-Q^{*}&\Delta_{lh}&0&R\\\ R^{*}&0&\Delta_{lh}&Q\\\ 0&R^{*}&Q^{*}&0\\\
\end{array}\right)$ (5)

with $R=\delta_{xx,yy}-i\delta_{xy}$ and $Q=\delta_{xz}-i\delta_{yz}$. Here,
$\Delta_{lh}$ is the splitting between heavy-hole (hh) and light hole (lh)
which is controlled in QDs both by the in-plane biaxial strain and the
confinement. R describes the heavy-hole/light-hole mixing induced by an
anisotropy in the QD plane $xy$ and Q takes into account an asymmetry in the
plane containing the QD growth axis $z$. The reduction of symmetry can come
from the shape of the QD (Luttinger Hamiltonian) or the strain distribution
(Bir and Pikus Hamiltonian). R is usually written in the form
$R=\rho_{s}/\Delta_{lh}e^{-2i\theta_{s}}$ where $\rho_{s}/\Delta_{lh}$
describes the amplitude of the band mixing and $\theta_{s}$ is the direction
of the main anisotropy responsible for the band mixing measured from the [100]
axis.

Considering only an in-plane anisotropy (Q=0), it follows from (5) that the
band mixing couples the heavy-holes $J_{z}=\pm 3/2$ and the light-holes
$J_{z}=\mp 1/2$ respectively. For such mixing, the short range exchange
interaction, which can be written in the form
$2/3\delta_{0}(\overrightarrow{\sigma}.\overrightarrow{J})$, couples the two
bright excitons. A deformation in a vertical plane (Q term) can couple the
heavy-holes $J_{z}=\pm 3/2$ and the light-holes $J_{z}=\pm 1/2$ respectively.
In this case, the short range electron-hole exchange interaction couples
$|+1\rangle$ and $|+2\rangle$ exciton on one side and $|-1\rangle$ and
$|-2\rangle$ exciton on the other side.

Using the Hamiltonian of the excited state $\mathcal{H}_{X,2Co}$ and the
Hamiltonian of the ground state

$\displaystyle\mathcal{H}_{2Co}=\sum_{i}(\mathcal{H}_{s,Co(i)}+g_{Co(i)}\mu_{B}\vec{B}.\vec{S_{i}})$
(6)

we can compute the PL spectrum of QDs containing two Co atoms. The occupation
probability of the exciton-2Co levels is described by an effective spin
temperature Teff and the optical transition probabilities are obtained
calculating the matrix elements $|\langle S_{z,i}|X,S_{z,i}\rangle|^{2}$.

Figure 8: (a) Calculated magnetic field dependence of the PL of a QD
containing two Co atoms with the parameters of Table 1 and Teff=20K. (b)
Calculated zero field spectra for two effective spin temperatures, Teff=40K
(top) and Teff=10K (bottom). A line broadening with a Lorentzian of full width
at half maximum of 75 $\mu$eV is included in the spectra.

Calculated linearly polarized spectra at zero magnetic field are presented in
Fig. 7. Unless specified, the parameters are those listed in table 1. The
change from dot to dot of the relative position of the inner and outer lines,
the appearance of low intensity lines and the appearance of linear
polarization in the center of the PL spectrum are well explained by the model.

A calculated longitudinal magnetic field dependence is also presented in
Fig.8. The exchange interaction terms (see table 1) are chosen to reproduce
the splitting of the four main lines of QD3 and the other parameters are
typical for single Co [15] and individual CdTe/ZnTe QDs [21]. Even if some
details are not described, this simplified spin-effective model reproduces
accurately the main experimental observations under magnetic field (see Fig.
5(a) for QD3). In particular the overall evolution under magnetic field with
its characteristic intensity distribution is well described. The behavior of
the weaker intensity lines and the increase of their contribution with the
increase of an effective spin temperature Teff are also well reproduced (Fig.
5(b)). Additional anticrossings or weak intensity line can appear in the
spectra of some of the dots (see QD2) but the overall behavior remains the
same. Such deviation may result from a further reduction in the symmetry due
to the position of the magnetic atoms or disorientation of the axis of local
strain [5] but these parameters cannot be accurately and independently
adjusted for the two atoms.

###  III.4 Carriers-induced spin-spin correlation

A remarkable feature of 2Co-doped QDs is that it provides a carrier mediated
coupling between both distant Co spins. At zero magnetic field, the exchange
interaction with the exciton, dominated by the coupling with the hole, results
in two ground states
$|\Downarrow_{h},\uparrow_{e}\rangle|\frac{3}{2}\rangle_{(1)}|\frac{3}{2}\rangle_{(2)}$
or
$|\Uparrow_{h},\downarrow_{e}\rangle|-\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$
with a ferromagnetic alignment of the two Co spins.

Figure 9: Correlation function $\chi_{1,2}$ of the two Co spins in a QD. (a)
Temperature dependence for the model presented in the text
$\mathcal{H}_{X,2Co}$ calculated with the parameter of Table 1 and compared
with the spin-spin correlation function for a Heisenberg model for two spins S
= 3/2 with j1,2=-90$\mu$eV. The inset show a calculated PL spectra with the
Teff=15K corresponding to $\chi_{1,2}$=0.032 which can be reproduced with a
Heisenberg coupling j1,2=-90. (b) Longitudinal magnetic field dependence of
the spin-spin correlation function at Teff=10K calculated with
$\mathcal{H}_{X,2Co}$ and parameters of Table 1.

Even in absence of spin polarized carriers, the thermalization of the Co spins
during the lifetime of the exciton induces a positive spin-spin correlation as
would be produced by a ferromagnetic exchange between the Co spins. This
correlation is observed at zero field in the intensity ratio of the high and
low energy lines which reflects the increase of the probability to have
parallel Co spins: $|-\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$ for
a $\sigma+$ exciton and $|+\frac{3}{2}\rangle_{(1)}|+\frac{3}{2}\rangle_{(2)}$
for a $\sigma-$ exciton. The spin correlation function of the two Co atoms is
given by:

$\displaystyle\chi_{1,2}(k_{B}T)=\langle\vec{S_{1}}.\vec{S_{2}}\rangle=\sum_{X}P_{X}(k_{B}T)\langle\psi_{X}|\vec{S_{1}}.\vec{S_{2}}|\psi_{X}\rangle$
(7)

where P${}_{X}=e^{-E_{X}/K_{B}T}/\Sigma_{n}e^{-E_{n}/K_{B}T}$ is the Boltzmann
occupation function of the emitting states. Perfectly aligned
ferromagnetically coupled spins will give $\chi_{1,2}$=9/4.

We can calculate $\chi_{1,2}$ as a function of temperature for the exciton
mediated coupling, controlled by $\mathcal{H}_{X,2Co}$, for a given set of
parameters and for a direct exchange j${}_{1,2}\vec{S_{1}}.\vec{S_{2}}$
between Co spins. These curves are presented in Fig. 9(a). The parameters for
the exciton model are those of Table 1 and for the Heisenberg coupling
$j_{1,2}$=-90$\mu eV$. The two correlations curves present a similar
temperature behavior thought $j_{1,2}$ cannot be adjusted to perfectly match
the exciton induced correlation both at low and high temperature.

To reproduce the intensity ratio which is for instance observed for QD3 in
Fig.2, an effective spin temperature Teff $\approx$ 25 K is used. This gives a
correlation function $\chi_{1,2}$=0.1. This measured spin correlation can be
interpreted as a Zener type ferromagnetism controlled at the nano-scale by
individual carriers [26].

Excitation conditions which minimizes the heating by high energy carriers can
enhance the correlation between the two localized spins. This correlation can
be further improved by the application of a longitudinal magnetic field that
induces a Zeeman splitting of each atom and enhance their alignment (see Fig.
9(b)). The injection of spin polarized carriers that can produce a progressive
orientation of the two Co spins in the exchange field of the exciton can also
significantly enhance their correlation.

##  IV Optical orientation and dynamics of the spin of two Co atoms.

###  IV.1 Optical orientation

Spin polarized excitons can be optically injected in the dots through a
circularly polarized excitation on excited states (see the PLE spectrum of QD3
in the inset of Fig. 10(a)). We can first notice that under these excitation
conditions, the PL is strongly co-circularly polarized with the excitation
(Fig. 10(c)). This shows that the spin polarization of the exciton is well
conserved during its relaxation and during its lifetime in the QD ground
state.

Figure 10: (a) Large band PL spectra of QD3. The inset shows an intensity map
of the PLE of QD3. Arrows point the two excitation energies used in the
experiments discussed in the text. (b) Circularly polarized PL spectra of QD3
for a linearly polarized excitation at 2050 meV. (c) Co and Cross circularly
polarized PL spectra of QD3 for a circularly polarized excitation at 2050 meV.
(d) Excitation power dependence of the PL intensity ratio of the high energy
and low energy lines of QD3 (I(1)-I(5))/(I(1)+I(5)). (e) Transverse magnetic
field dependence of the normalize PL intensity of the high and the low energy
lines of QD3 for a $\sigma_{Co}$ excitation/detection. The inset shows the
transverse magnetic field dependence of the PL intensity map. (f) Magnetic
field dependence of the PL intensity ratio of the high energy and low energy
lines of QD3 (I(1)-I(5))/(I(1)+I(5)).

More interestingly, the relative intensities of the PL lines of the
exciton-2Co complex strongly changes when the polarization of the excitation
is changed from linear to circular. This is presented in Fig. 10(b) and (c)
for an excitation at 2050 meV. For a linearly polarized excitation and a
circular detection the intensity ratio of the high and low energy lines
reflects a thermalisation of the two Co spins in the exchange field of the
exciton.

For a $\sigma_{Co}$ excitation/detection, this intensity ratio is
significantly modified and most of the PL comes from the low energy line. As
each emission line corresponds to a given spin state of the 2 Co, Mz, this
shows that the injection of spin polarized excitons creates a non-equilibrium
distribution of the Co spin states which can be described by a lower effective
spin temperature. In the excitation configuration presented in Fig. 10(c), the
intensity ratio of the high an low energy lines (I(1)-I(5))/(I(1)+I(5))=0.7
can be reproduced with Teff=15K (see inset of Fig. 9(a)) giving an enhanced
correlation function $\chi_{1,2}\approx$ 0.32.

The change in the intensity distribution between the $\pi$ and the $\sigma$
polarization configurations reflects an optical pumping of the two Co spins
for a $\sigma$ excitation i.e. a progressive orientation of the two Co spins
in the exchange field of successively injected spin polarized excitons. This
cumulative effect is confirmed by the excitation power dependence of the high
energy/low energy lines intensity ratio (I(1)-I(5))/(I(1)+I(5)) presented in
Fig. 10(d). The increase of the intensity ratio with the increase of the
excitation power results from an enhancement of the efficiency of pumping at
high power. A saturation and a slight decrease of this intensity ratio is
however observed at high excitation power. It can result from a heating of the
spins by (i) the non-resonantly carriers created in the wetting layer and/or
(ii) the interaction with non-equilibrium phonons generated during the
relaxation of carriers [20, 17].

The high energy/low energy lines intensity ratio in $\sigma_{Co}$
configuration also strongly depends on a transverse magnetic field. This is
presented in Fig. 10(e) for a circularly polarized excitation of QD3 at 2050
meV (see Fig. 10(a)). As the transverse field is applied, a decrease of the
intensity of the low energy line and an increase of the intensity of the high
energy line are simultaneously observed. The corresponding normalized
intensity ratio is presented in Fig. 10(f). The reduction of this intensity
ratio corresponds to a decrease of the spin polarization of the two atoms. The
depolarization curve in transverse magnetic field presents a slight asymmetry.

The opposite evolution of the intensity of the low- and high-energy lines
shows that the transverse magnetic field dependence is not simply the result
of a reduction in exciton spin polarization. Such a reduction would lead to a
change in the polarization rate of all the exciton-2Co lines, but not to a
change in the intensity distribution between the lines. The transverse
magnetic field dependence results indeed from a reduction in the optical
pumping efficiency. As observed in Fig. 10(f), a transverse field of about 0.4
T is required to significantly erase the optical pumping.

The effect of the weak magnetic field can be neglected in the excited state
where the Zeeman energy of the atoms remains much weaker than the exciton
exchange field oriented along the growth axis of the QD. Depolarization should
then be controlled by the precession of the two Co spins in the transverse
magnetic field. In the QD ground state, this precession is limited by the fine
structure of Co atoms induced by bi-axial strain. However, as all the 2Co
spins ground states with Mz=$\pm$3 and Mz=0 are degenerate, a weak mixing term
can induce a coherent coupling between them.

The asymmetry in the depolarization can come from the presence of in-plane
anisotropy in the magnetic atoms fine-structure (E term). Depending on the in-
plane orientation of this anisotropy, the response of the energy levels of the
atoms to the transverse field can depend on its orientation. This has been
observed in the time domain in the case of individual Mn atoms [22]. Probing
in detail this anisotropy would require further magneto-optic measurements
with a change of the direction of the transverse magnetic field in the plane
of the QD which could not be performed in the available set-up. The influence
of the E term on 2Co dots will be discussed more in details in Sec. V.

The observation of an optical orientation shows that, even in the presence of
the strain induced magnetic anisotropy D0, some spin-flip of the magnetic
atoms can occur during the lifetime of an exciton. However this experiment
does not permit to identify the main channel of spin-flip responsible of the
pumping and their time-scale.

###  IV.2 Spin dynamics revealed in auto-correlation measurements

Observing the time evolution of the PL intensity of a magnetic dot can be used
to probe the spin fluctuations of the Co atoms in the presence of the exciton.
This spin dynamics is responsible for the observed optical orientation. PL
intensity fluctuations and their characteristic time can be revealed by the
auto-correlation signal of the emission of a given line of the dot.

Figure 11: Auto-correlation of the $\sigma_{Co}$ PL intensity of the low
energy line (5) of QD3 for two different energies of excitation, (a) 2050 meV
and (b) 2020.8 meV and in each case for two excitation powers, P0 and 3P0. (c)
Transverse magnetic field dependence of the auto-correlation signal for an
excitation at 2050 meV and an excitation power P0.

To estimate the auto-correlation signal we used a HBT setup with a time
resolution of about 700 ns [19]. In this experiment, the detection of a first
circularly polarized photon from a given line indicates that the system is in
a particular spin state of the two atoms, Mz. The probability of delayed
detection of a second photon with the same energy and polarization depends on
the probability of conservation of the spin state and on the excitation rate.
This photons’ coincidence measurement is a good approximation of the auto-
correlation of the PL intensity as far as the considered time delay remains
smaller than the inverse of the photon collection rate limited in these
experiments to a few kHz by the collection efficiency of a single QD emission.

An auto-correlation signal is presented in Fig. 11 for a detection of the
intensity of the low energy line of QD3, under excitation at 2050 meV and at
zero magnetic field. Because of the limited time resolution of the
experimental setup and the short lifetime of the exciton (around 250 ps in
CdTe/ZnTe QDs), the antibunching characteristic of a single-photon emitter is
not observed. However, a photon bunching with an amplitude of about 6 and a
characteristic width in the tens of nanoseconds range is obtained. This
clearly shows intermittent QD emission with the amplitude of the bunching
proportional to the OFF/ON ratio of the selected PL line.

As presented in Fig. 11, the amplitude of the bunching depends on the
excitation energy and slightly increases for a resonant laser on a lower
energy and sharper excited state (Fig. 11(b)). The width of the bunching
reveals the timescale of the intensity fluctuations. For both excitation
energies, the speed of fluctuations increases with the increase of the
excitation intensity.

Charge fluctuations are unlikely in the studied magnetic QDs as only the
neutral exciton is observed in the PL spectra (see Fig. 10(a)). The measured
fluctuations timescale is controlled both by the fluctuations in the excited
state, when the QD is occupied by an exciton and by the spin dynamics in the
ground state when the QD is empty. The reduction of the linewidth of the
bunching with the increase of the excitation intensity could traduce a faster
spin dynamics in the excited state when an exciton is present in the dot than
in the empty dot. It is also possible that the high-energy and high-intensity
optical excitation generates non-equilibrium acoustic phonons that can induce
spin flips of the Co spins [17]. The two processes probably contribute to the
acceleration of the spin dynamics and this experiment does not permit to
discriminate the main channels of spin relaxation.

As it was observed in the optical orientation experiment, a weak transverse
magnetic field also significantly affects the auto-correlation signal (Fig.
11(c)) with both a decrease of the amplitude and of the width of the photon
bunching peak. As the exciton-2Co exchange interaction is much larger than the
Zeeman energy, the transverse field dependence of the measured dynamics is a
consequence of the field induced mixing of the Co spins states in the empty
dot.

##  V Resonant photo-luminescence of QDs doped with two Co atoms

To avoid heating of spins by the injection of high-energy carriers or carrier
induced spectral diffusion, resonant optical excitation is generally preferred
for optical control of individual spins or generation of high quality
individual photons with QDs [27]. Resonant excitation combined with the
detection of quasi-resonant PL has been successfully used for the
initialization of the spin of individual Mn2+, Cr2+ atoms and hole-Cr+ complex
[16, 17, 18].

We use here resonant optical excitation of the neutral exciton lines to
analyze the spin transfer mechanism among the exciton-2Co complex. This
technique also enables to study the extent to which the two Co spins can be
optically initialized by resonant optical pumping.

For this experiment we select a dot, QD3, that is mainly neutral (i.e. no
significant contribution of the charged excitons is observed in the PL, see
Fig. 10). The resonant PL and the spin dynamics in QD2 are unlikely to be
affected by any charge fluctuations in the dot. This dot also presents a large
splitting which permits to independently address each emission line with a
resonant optical excitation. Quasi-resonant PL is studied in QD3 by scanning a
tunable laser on the high energy side of the exciton-2Co PL spectra while
detecting the emission of the low energy line (1) or the central line (3) (see
Fig. 10 for the labeling of the lines).

###  V.1 Absorption measured in the acoustic-phonon side-band

A PLE spectra obtained in $\pi_{cross}$ excitation/detection configuration for
a detection on the low energy line (1) of QD3 is presented in Fig. 12 (a) and
the corresponding quasi-resonant PL intensity map in Fig. 12 (b). A quasi-
resonant PL is observed for excitation very close to the emission line and up
to a few milli-electronvolts above it. A progressive decrease in PL intensity
occurs with increasing the excitation energy. This side-band is characteristic
of the absorption in the acoustic phonon band of the observed emission line
[23, 24].

Figure 12: (a) PLE intensities detected on line (1) of QD3 as a function
excitation energy at zero magnetic field and under a transverse field
Bx=-0.25T. The inset shows a detail of the PL intensity map of line (1) at
Bx=0T for an excitation around (5). (b) PL intensity map of the low energy
line (1) of QD3 for a tunable quasi-resonant excitation at zero magnetic field
(left) and under a transverse field Bx=-0.25T (right) for a cross-linear
excitation/detection configuration. (c) and (d) PLE intensity detected on line
(1) of QD3 under Bx=-0.25T for a cross-circular excitation/detection and for a
co-circular excitation/detection configuration respectively. (e) Integrated PL
intensity of line (3) of QD3 as a function excitation energy at zero magnetic
field and under a transverse magnetic field Bx=-0.25T. (f) Transverse magnetic
field dependence of the quasi-resonant $\pi_{cross}$ PL intensity for a
detection on line (1) and an excitation on line (4). (g) Corresponding PL
intensity map of line (1).

Surprisingly, at zero magnetic field, the quasi-resonant PL spectra does not
reveal any excitation transfer. Excitation transfer usually appears as an
increase of the PL which becomes larger than the acoustic phonon background
for resonant excitation on some specific excitonic line. This was for instance
observed in the case of Mn2+, Cr2+ or hole-Cr+ in similar QDs and permit to
identify the dominant spin relaxation channels of the exciton / magnetic atoms
complex [16, 17, 18].

The absence of any additional PL signal superimposed on the acoustic phonon
background shows that in the case of the 2Co system there is no efficient
excitation transfer during the exciton lifetime. This can either come from (i)
an efficient resonant optical pumping which prevents the resonant absorption
or (ii) from an absence of significant spin relaxation for a resonantly
created exciton.

At zero magnetic field, the only feature observed in the PLE is a dip that
appears in the absorption background when the excitation laser is scanned
around the high-energy line (5). The exact energy position of the dip follow
the excitation laser as it crosses the exciton line broadened by spectral
diffusion (inset of Fig. 12(a)). Dips are not observed for an excitation on
the other lines. This shows that this reduction in PL intensity for an
excitation on (5) does not result from saturation of absorption by a direct
resonant excitation of a transition. Rather, such saturation would produce a
dip for excitation on each of the excitonic lines.

As the low- and the high-energy lines correspond to the same spin states of
the two Co atoms, i.e. $|\frac{3}{2}\rangle_{(1)}|\frac{3}{2}\rangle_{(2)}$ or
$|-\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$ for a $\pi_{cross}$
excitation/detection, the reduction in PL intensity reflects a decrease of
occupancy probability of these two spin states. This reduction is induced by a
resonant optical pumping: resonant absorption followed by a spin-flip in the
exciton-2Co system empties the states that are resonantly excited.

###  V.2 Transverse magnetic field dependence of the absorption of the
acoustic phonon side-band

The behavior of the absorption in the acoustic phonon band significantly
changes when a weak transverse magnetic field is applied. This is illustrated
in Fig. 12(a) and (b) for a detection on the low energy line (1) of QD3: under
a transverse field, additional dips appear in the side-band for resonant
excitation of some of the higher energy excitonic lines. For a transverse
field Bx=-0.25T, dips are clearly observed for an excitation on the higher
energy lines (3) and (4) and also for an excitation of the weaker intensity PL
line (2).

Quasi-resonant PL with a similar behavior in transverse magnetic field is also
observed for a circularly polarized excitation in the acoustic phonon band. In
this case, the PL is mainly co-circularly polarized with the excitation (Fig.
12(c) and (d)). This confirms the good stability of the spin of the carriers
coupled with the spins of the two magnetic atoms.

For a detection on the central line (3) (see Fig. 12(e)), at zero magnetic
field, an unperturbed acoustic phonon absorption background is observed except
for an excitation on (4). When a weak transverse magnetic field is applied, a
dip appears in the phonon side-band for excitation around the high energy line
(5) showing that a transverse field creates a link between the outer and the
inner lines.

The amplitude of the dips in the acoustic phonon bands are controlled by the
transverse magnetic field. This is illustrated in Fig. 12(f) and (g) for a
detection on (1) and a resonant excitation on (4). When the transverse
magnetic field is increased, a progressive decrease of the resonant PL is
observed. A minimum of resonant PL intensity is reached around Bx=0.5 T. As it
was observed for the optical orientation under injection of spin polarized
carriers, a slight asymmetry in the transverse magnetic field dependence is
observed. It likely arises from the orientation of the in-plane strain
anisotropy at the origin of the Co atoms fine structure. A full width at half
maximum on the dependence of the transverse field can nevertheless be defined
with a value around 0.16 T.

This transverse magnetic field dependence of the dips confirm that they do not
arise from a saturation of the absorption of another line of the dot as the
oscillator strength of a transition is not expected to be significantly
affected by such weak magnetic field. However, these experiments demonstrate
that a transverse magnetic field creates a link between the optical
transitions that do not share the same 2Co spins ground states. For instance,
transition (1) is associated with
$|+\frac{3}{2}\rangle_{(1)}|+\frac{3}{2}\rangle_{(2)}$ or
$|-\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$ and transitions (3) and
(4) with $|-\frac{3}{2}\rangle_{(1)}|+\frac{3}{2}\rangle_{(2)}$ or
$|+\frac{3}{2}\rangle_{(1)}|-\frac{3}{2}\rangle_{(2)}$. The corresponding
optical transitions are independent at zero field.

Whereas a large effect of the transverse field is observed for a resonant
excitation on the intermediate lines (2), (3), (4), the dip on the high energy
line is not significantly affected in the investigated field range. This can
come from the fact that to observe a signal for an excitation in the acoustic
phonon band at the energy of the high energy line, the excitation intensity is
large. It likely saturates the absorption of the high energy line when on
resonance and prevents the effect of the weak transverse field which only act
on the empty dot.

###  V.3 Discussion on the transverse magnetic field dependence of the
resonant PL

For the quasi-resonant PL experiments discussed here, the low-energy line is
continuously excited by absorption in the acoustic phonon band. This
absorption is not spin-selective and all the optical transitions of the dots
are simultaneously excited, though with slightly different probabilities,
depending on their distance to the resonant laser. The excitation in the
acoustic-phonon band does not contribute to an optical pumping of the two Co
spins. In the case of a linearly polarized excitation, the excitons are not
spin-polarized, nor can they induce optical Co spins orientation. The weak
transverse field used in this experiment does not significantly affect the
energy levels of the exciton-2Co complex.

In the ground state the four spin states with the largest occupation
probability $|\pm 3/2\rangle_{(i)}$ are initially degenerate at zero magnetic
field and they can be coherently coupled by any weak mixing terms. The spin
states $|\pm 1/2\rangle_{(i)}$ are pushed to high energy by the bi-axial
strain term D0. The presence of in-plane strain anisotropy induces a fine
structure term $E$ which couples spin states $|+3/2\rangle_{(i)}$ and
$|-1/2\rangle_{(i)}$ on one side and $|-3/2\rangle_{(i)}$ and
$|+1/2\rangle_{(i)}$ on the other side. The coupling is controlled by the
ratio $E/D_{0}$. A magnetic field applied in the plane of the QD induces a
coupling of the spin states $|\pm 3/2\rangle_{(i)}$ and $|\pm
1/2\rangle_{(i)}$ respectively. Combined with the presence of a $E$ term,
these two mixing induces a coupling between the $|\pm 3/2\rangle_{(i)}$ spin
states of each atom which are initially degenerate. The resulting mixing of
the four ground states depends on the amplitude of the applied transverse
magnetic field.

A dip in the acoustic phonon band of the low energy line (1) for a resonant
excitation of a higher energy line in $\pi_{cross}$ excitation-detection
configuration means a decrease of the population of the spin state
corresponding to the low energy line, $|\pm 3/2\rangle_{(1)}|\pm
3/2\rangle_{(2)}$. This shows that under a transverse magnetic field, a
resonant excitation of the lines associated with $|\pm 3/2\rangle_{(1)}|\mp
3/2\rangle_{(2)}$ (lines (3) or (4)) introduces a channel of pumping for the
spin states $|\pm 3/2\rangle_{(1)}|\pm 3/2\rangle_{(2)}$.

This channel of pumping is induced by the combined action of the E term and
the transverse magnetic field. For instance in the presence of a transverse
field, the ground state $|+3/2\rangle_{(1)}$+$|-3/2\rangle_{(2)}$
corresponding to line (3) of QD3, becomes
($|+3/2\rangle_{(1)}$+$\epsilon_{(1)}$
$|-3/2\rangle_{(1)}$)+($|-3/2\rangle_{(2)}$+$\epsilon_{(2)}$
$|+3/2\rangle_{(2)}$). With this mixing, a resonant optical excitation on line
(3) can also excite the dot if the spins are initially in the state
$|+3/2\rangle_{(1)}$+$|+3/2\rangle_{(2)}$. This introduces an efficient
channel of optical pumping as an absorption on the central line and a
recombination projects the two Co spins on the states $|\pm 3/2_{(1)};\mp
3/2_{(2)}\rangle$ different from the initial 2Co spins ground state.

A similar scenario explains that the resonant excitation of some transitions
with weak PL signal efficiently decrease the intensity of the low energy line.
Let’s consider for instance the optical transition associated with the 2Co
spins state $|-3/2\rangle_{(1)}$)+$|-1/2\rangle_{(2)}$. It corresponds to the
lowest energy level containing a $S_{z}=\pm 1/2$ component (see Fig. 4). If we
take into account the mixing induced by the fine structure term $E$ it becomes
($|-3/2\rangle_{(1)}$)+($|-1/2\rangle_{(2)}$+$\xi_{(2)}$
$|+3/2\rangle_{(2)}$). Under a transverse field this state becomes
($|-3/2\rangle_{(1)}$+$\epsilon_{(1)}$
$|+3/2\rangle_{(1)}$)+($|-1/2\rangle_{(2)}$+$\xi_{(2)}$ $|+3/2\rangle_{(2)}$).
A resonant optical excitation of the corresponding line can also excite the
dot if the spins are initially in the state
$|+3/2\rangle_{(1)}$+$|+3/2\rangle_{(2)}$. This introduces a channel of
pumping controlled by the transverse field and the initial strain induced
coupling.

##  VI Conclusion

To conclude, we demonstrated the possibility to optically access the spin
states of two Co atoms in a semiconductor host. Magneto-optic measurements and
modeling of QDs containing two Co atoms show that a large diversity of spectra
can be obtained. PL spectra depend on the relative position of the two atoms
inside the dot and on the strain state at each magnetic atom location. In most
of the dots, the emission is dominated by four main lines resulting from the
contribution of the ground states of the two Co spins with M${}_{z}=\pm$3 or
Mz=0.

The presence of an exciton in the dot induces a spin-spin correlation with an
enhancement of the occupation probability of the ferromagnetic ground states
M${}_{z}=\pm$3\. The injection of spin polarized excitons permit an optical
orientation of the two Co spins showing that some spin-flips among the
exciton-2Co system are possible. Spin fluctuations under optical excitation in
the tens of nanosecond range are revealed with auto-correlation measurements.

The four 2Co ground states can be addressed by resonant optical excitation of
the exciton. The resonant excitation of the phonon side-band reveals
absorption which strongly depends on a transverse magnetic field. These
characteristic absorption feature arises from an interplay of the spin states
mixing induced by the presence of an in-plane strain anisotropy and the
transverse magnetic field. This tunable channel of pumping which strongly
depends on the magnetic atoms fine structure $E/D_{0}$ could be used for a
probing of the modulation of local strain in the vicinity of the magnetic
atoms [28].

###### Acknowledgements.

We acknowledge the financial support from NationalScienceCenter Poland,
project number 2021/41/B/ST3/04183.

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