Dynamical Magnetic and Nuclear Polarization in Complex Spin Systems: Semi-magnetic II-VI Quantum Dots

Dynamical magnetic and nuclear polarization in complex spin systems is discussed on the example of transfer of spin from exciton to the central spin of magnetic impurity in a quantum dot in the presence of a finite number of nuclear spins. The exciton is described in terms of the electron and heavy hole spins interacting via exchange interaction with magnetic impurity, via hypeprfine interaction with a finite number of nuclear spins and via dipole interaction with photons. The time-evolution of the exciton, magnetic impurity and nuclear spins is calculated exactly between quantum jumps corresponding to exciton radiative recombination. The collapse of the wavefunction and the refilling of the quantum dot with new spin polarized exciton is shown to lead to build up of magnetization of the magnetic impurity as well as nuclear spin polarization. The competition between electron spin transfer to magnetic impurity and to nuclear spins simultaneous with the creation of dark excitons is elucidated. The technique presented here opens up the possibility of studying optically induced Dynamical Magnetic and Nuclear Polarization in Complex Spin Systems.


I. INTRODUCTION
There is currently interest in developing means of localizing and controlling complex spin systems in solid state 1 . This includes electron and/or hole spins in gated 2,3 , selfassembled 4 , nanocrystal 5 and carbon nanotube quantum dots 6 , nitrogen vacancies in diamond 7 and magnetic impurities in II-VI [8][9][10][11][12][13] and III-V 14,15 . The complex spin systems involved include heavy valence holes with spin J = 3/2, nitrogen vacancies with spin M = 1, halffilled shell electrons of mangan M n 2+ impurity atom with spin M = 5/2 or M n 3+ atom with M = 3/2 in II-VI semiconductor quantum dots, or strongly coupled valence hole-Mn atom in InAs/GaAs quantum dots. Extensive theoretical studies have been carried out, predicting rich phase diagram for these systems [17][18][19][20][21] . For NV centers in diamond, carbon nanotube based quantum dots and magnetic impurities in II-VI semiconductor quantum dots the complex spin system interacts with only a finite number of nuclear spins. The controlling of magnetization of complex spin systems is often carried out optically and involves transfer of photon angular momentum into exciton spin, and exciton spin into the spin of the complex spin system 22,23 . This dynamical magnetic polarization (DMP) process is decohered by photon and nuclear spin baths. Recently, first optical experiments on single magnetic impurities in II-VI quantum dots measured the dynamic evolution of the magnetization process 8-10 with theoretical models of DMP based on rate equations 24,25 .
In this work, we develop a microscopic theory of optically driven dynamical magnetic polarization of complex spin systems. The theory describes the transfer of spin from exciton to the central spin of magnetic impurity in a quantum dot in the presence of a finite number of nuclear spins using quantum jump approach [26][27][28] . The exciton is described in terms of the electron and heavy hole spins interacting via exchange interaction with magnetic impurity, via hypeprfine interaction with a finite number of nuclear spins and via dipole interaction with photons. The time-evolution of the exciton, magnetic impurity and nuclear spins is calculated exactly between quantum jumps corresponding to exciton radiative recombination. The collapse of the wavefunction and the refilling of the quantum dot with new spin polarized exciton as in recent experiment by Goryca et al. 8 is shown to lead to build up of magnetization of the magnetic impurity as well as nuclear spin polarization. The competition between electron spin transfer to magnetic impurity and to nuclear spins simultaneous with the creation of dark excitons is elucidated.
The paper is organized as follows. In Section II we describe our model. Section III describes quantum jump approach to time evolution of our system for a system of single MI, single exciton with no nuclear spin. Section IV contains quantum jump approach and the dynamical evolution of MI by train of excitons in the presence of nuclear spins. In sections V and VI we presents numerical results, discussion, conclusion and the summary.

II. THE MODEL
We consider a semiconductor QD containing a complex spin system M , e.g., magnetic ion (MI), coupled with few nuclear spins of the host material, as in, e.g., CdTe quantum dots. The quantum dot with MI is attached to a smaller quantum dot with no MI where the electrons and valence holes with definite spin polarization are generated arXiv:1202.5352v1 [cond-mat.mes-hall] 24 Feb 2012 optically by circularly polarized light. This is illustrated in Fig.1a where circles describe quantum dots, blue arrow corresponds to electron spin S z = +1/2 and white arrow corresponds to heavy hole spin J z = −3/2 in the small dot. The large dot contains a randomly oriented complex spin M , represented by a magenta arrow, and a number of randomly oriented nuclear spins represented by small arrows. The DMP process starts with transfer of spin polarised exciton from a small QD to a large QD, Fig.1b. As a result of interactions resulting in the flipping of electron, MI and nuclear spins the wavefunction of the large dot evolves in time and becomes a linear combination of bright and dark exciton states, Fig.1c. During this process the small dot is refilled with spin polarized exciton. Simultaneously in time the bright exciton decays due to interaction with the photon field, and at some point in time, it recombines, photon is emitted, and the quantum jump takes place. The states of the magnetic ion and nuclear spins are modified, polarization is increased and the large dot is refilled with spin polarized exciton and the DMP process continues.
We now quantitatively describe the DMP process. We start with the Hamiltonian describing the quantum dot coupled with the photon bath H = H QD + H phQD + H ph . Here H ph is the photon Hamiltonian, H phQD is the Hamiltonian describing coupling of photons with the exciton in a QD and H QD is the QD hamiltonian.
The QD Hamiltonian describes exciton X coupled with the magnetic moment of the complex spin system MI and with nuclear spins I given by Here H x describes the exciton, H m describes the complex spin system MI and the remaining terms in H QD represent X-MI, X-I and MI-I nuclear spins exchange couplings. The exciton Hamiltonian describes the low energy quadruplet |S, J characterized by quantum spin numbers of an electron, S = ±1/2, and a heavy hole, J = ±3/2 in the QD. The complex spin system is described by a total spin M = N i=1 u i , where N is the number of spins u = 1/2 building up the MI system, and H m = i<j J ij u i · u j + DM 2 z with J ij exchange matrix elements building the total spin M . In quantum dots one often includes strain field D leading to splitting of the different M z levels 16 We assume that exchange coupling constants of MI spins with the environment (including excitons and nuclear-spins) are identical. Hence the full QD Hamiltonian can be written as 21 : The exciton Hamiltonian H x = ∆ 0 S z J z + ∆ 1 (S + J − + S − J + ) describes splitting ∆ 0 between the low energy dark exciton doublet | ↑, ⇑ = | + 1/2, +3/2 , | ↓, ⇓ = | − 1/2, −3/2 with total angular momentum j z = ±2 along quantization-axis,ẑ, and higher energy bright exciton doublet | ↓, ⇑ = | − 1/2, +3/2 , | ↑, ⇓ = | − 3/2, +1/2 with j z = ±1. Here ↑ / ↓ and ⇑ / ⇓ represent spin of electron and hole 29 . The bright exciton doublet is split by the anisotropic electron-hole exchange interaction characterized by parameter ∆ 1 which measures the splitting of the two bright exciton states | + 1/2, −3/2 , | − 1/2, +3/2 . ∆ 1 is zero for cylindrical quantum dots and the two bright exciton states correspond to circular photon polarization. The exciton-MI coupling in Eq.1 is given as a sum of the ferromagnetic Heisenberg electron-MI exchange H em = −J em S · M and anti-ferromagnetic Ising exchange interaction H hM = +J hM J z M z . 21 Only electron-MI interaction is responsible for the e-MI spin flip-flop process. The interaction of complex spin MI with nuclear spin associated with the spin complex is denoted here by H M I = A I M · M . This interaction might, for example, describe coupling of manganese d-shell electron spins with manganese ion nuclear spin 16 . With hole spin strongly aligned along the growth z direction the coupling of electron and hole spins to surrounding nuclear spins of isotopes of the QD and barrier material with finite nuclear spin reads Here The population of vacuum can be calculated by ρvacuum = ρ55 + ρ66. The elements of density matrix, not plotted in this figure, are all identical to zero.
is the number of nuclear spins in the QD and the last term describes nuclear spin interaction. We note that for isotropic QD the long range e-h exchange ∆ 1 is zero and the heavy hole spin J z = ±3/2 is preserved.

III. SINGLE MI, SINGLE EXCITON AND NO NUCLEAR SPIN
We start our discussion of DMP by discussing time evolution of magnetization of MI interacting with an exciton in the absence of nuclear spins. To focus on quantum dynamics we consider the simplest complex spin system, M = 1/2, with just two states | ↑ = |M z = 1/2 and | ↓ = |M z = −1/2 and Hamiltonian and H xm,3 = 0 respectively. Here 1 is a 2 × 2 unit matrix.
The off-diagonal elements of H xm,2 describe mixing of X b and X d via spin-1/2 MI, hence . This state describes a coherent Rabi-oscillations between bright and dark excitons due to MI spin flip-flop.
Note that in |ψ(t) there is no mixing with the vacuum, |0, M z = ±1/2 , unless we take into account the coupling of bright-exciton with radiation field. In the interaction and rotating wave approximation the electronphoton coupling is described by the Hamiltonian that does not directly change the state of MI where b † k and b k are creation and annihilation operators of photon with one specific circular polarization. g k , and ω k are the photon-X coupling constant and photon frequency, and ω b = E b /h. The equation of motion of the QD density matrix, ρ, coupled with thermal bath of photons can be calculated after tracing over photons degrees of freedom. Assuming that photons are in thermal equilibrium and are weakly coupled with excitons in QDs, the equation of motion for exciton density matrix, ρ, can be calculated perturbatively. Up to the second order of perturbation, it is straightforward to show that 28 3hc 3 is the transition rate for the spontaneous emission of photons. ξ is the dipole moment matrix element. Note that in Eq. (3), vacuum can be considered as a shelving-state.
The numerical solutions of Eq.(3) at zero-temperature (n B = 0) are shown in Fig. 2 for a QD with E d = 2 eV and δ = E b − E d = 0, 1, 5 meV. Here we used J em = 1 meV and J hm = 4 meV. The initial state of MI is completely uncorrelated with half of the spins populated in up-direction. As it is shown, because of the coupling with the bath of photons, bright-exciton decays to vacuum without flipping the MI spin and mixing with X d , e.g., |X b , M z → |0, M z . In Fig. 2, we find that ρ 11 = e −Γt /2 and ρ 55 = (1 − e −Γt )/2 fit perfectly the numerical solution of ρ 11 and ρ 55 for all δs. The decay channel of 3: Time evolution of Sz of a train of injected photoelectrons inside QD (circles) and a single MI (stars) with spin S = 1/2 and no nuclear spin. The inset shows the Λ-shape three level optical resonance of bright-and dark-exciton (X b and X d ). The Rabbi-oscillation between X b and X d occurs because of the exchange interaction between the exciton and the system of MI and nuclear spins. The optical selection rule allows decay of X b to vacuum, however, the population of X d decreases indirectly through the conversion of X d to X b .
dark-exciton, is through a transition to bright-exciton and spin-flip of MI. This process is schematically depicted in the inset of Fig. 3. A strong dependence of dark-exciton population on δ is seen in Fig. 2. Consistent with the time-evolution of the density matrix, we propose the exciton wave-function, that fits the density matrix via ρ(t) = |ψ(t) ψ(t)|: with C b↑ = 1/ √ 2. Note that |X b , ↓ and |X d , ↑ coherently oscillate because J em in off-diagonal elements of H xm mix these two states. Also from ρ 66 (Γt >> 1) → 1/2 we deduce |C 0↓ (Γt >> 1)| → 1/ √ 2, and finally C d↓ (t) = 0 because ρ 44 = 0. The rest of coefficients in |ψ(t) can be determined numerically by fitting to the solutions of density matrix that also fulfill the normalization of wave-function ψ|ψ = 1.

A. quantum jump Algorithm
As noted in Ref. 27, detection of photons requires spontaneous emission due to vacuum fluctuations, i.e., the photo-emission is a stochastic process, described by quantum jump approach 26,27 . The time evolution of the density matrix interacting with photons is given by Eq. (3). At zero temperature in which n B = 0, it is described by a standard Lindblad master equation (ME) 26 The formulation of quantum jump starts from Eq.(5). In the absence of MI, a recipe for quantum jump algorithm can be found in Ref. 26. For completeness of our presentation first we review this algorithm and then generalize it for a system in the presence of MI and nuclear spins. We consider optical transition in a two-level system consisting of bright-exciton and vacuum without considering an intermediate transition to dark-exciton. This condition is fulfilled if we disregard presence of any MI and nuclear spin. Here the quantum jump operators areP = |0 X b |,P † = |X b 0|, henceP †P = |X b X b |. Starting at t = 0 with the initial condition |ψ(t = 0) = |X b , we calculate the time evolution of the system in a discrete time-steps δt. In each time-step we evaluate the quantum jump probability by calculating δq 0 = Γ(δt) ψ|P †P |ψ = Γδt and drawing a random number r. If r < δq 0 a quantum jump occurs and |ψ collapses to |0 , otherwise |ψ(t + δt) = e −Γδt(P †P )/2 |ψ(t) . Here the generator for the time-evolution operator is a non-Hermitian Hamiltonian H eff = −ihΓ(P †P )/2. So at t = 0 + δt we have |ψ(0 . The last term keeps the norm of |ψ constant (if we use the norm of wave-function as a constraint in our calculation). At this time δq 1 = Γ(δt)e −Γδt . We draw r and if r < δq 0 + δq 1 then |X b → |0 and a photon is detected and calculation is terminated. Otherwise, |ψ(δt In nth-step δq n = Γ(δt)e −nΓδt thus we calculate an accumulative quantum jump probability δp n = n k=0 Γ(δt)e −kΓδt , hence and if r < δp n quantum jump occurs. As time advances, the chance for a quantum jump becomes more likely, however, the probability amplitude for X b in |ψ vanishes with the same rate simultaneously.
In nth-step if there is still no quantum jump, then |ψ(nδt) = e −Γδt(|X b X b |)/2 |ψ([n − 1]δt) = e −Γ(nδt)/2 |X b + √ 1 − e −Γ(nδt) |0 . The quantum jump algorithm in the presence of MI is similar to the absence of MI, with a difference that the time-evolution of wave-function is generated by effective Hamiltonian H eff = H QD − ihΓδt(P †P )/2 that allows an intermediate transition to the dark-exciton due to spinexchange with MI. Therefore the description of quantum jump process in the presence of MI is based on a three level system depicted in the inset of Fig. 4, and consist of |0 , |X b , |X d and MI.

IV. DYNAMICAL EVOLUTION OF MI BY TRAIN OF EXCITONS IN THE PRESENCE OF NUCLEAR SPINS
As illustrated in Fig.1 a small quantum dot is continuoulsy refilled by a non-resonant circularly polarized CW laser. The spin polarized excitons transfer into the QD containing the complex spin system MI. We assume therefore a train of incoming bright excitons |X b ≡ | ⇓, ↑ interacting with MI in the quantum dot. Each electron in the exciton transfers spin to MI and creates a superposition of dark and bright excitons entangled with nuclear spins. At the bright exciton recombination time, t r , photon is detected, quantum jump takes place, dark exciton wavefunction is erased and spin is transferred from bright exciton into a MI and nuclear spin complex and a new exciton is created. The exciton removal is performed by using the quantum jump projector method 26,27 described below and yields the wavefunction of the MI and nuclear spins.
The basis for combined exciton-spin system is composed of three groups of basis states: vacuum |0, M z , I z1 , . . . , I zN b , bright exciton |X b , M z , I z1 , . . . , I zN b and dark exciton |X d , M z , I z1 , . . . , I zN b .
Only the vacuum and bright exciton group of states are coupled to the photon field via projectors P λ = |0, λ X b , λ| with states |λ = |M z , I z1 , . . . , I zN b describing a total of N S = (2M z + 1)(2I z + 1) N b complex spin MI and nuclear spin states. In the following we use symbols |λ and |µ to represent |M z , I z1 , . . . , I zN b .
The time evolution of the density matrix ρ = |Ψ Ψ| in ME, Eq.(5), can be generalized by M z → λ. As described in section III A, the wave-function |Ψ subjected to stochastic "birth-death" process of recombination and photo-excitation can be used for the time propagation of the system coupled with radiation-field and undergoing quantum jump process. At t = 0 we start with the initial state |Ψ n=0 (t = 0) = |0 |χ 0 of MI and the nuclear spinbath. The index n counts the number of quantum jump events. The state |χ 0 = λ C n=0 λ |λ is a random linear combination of all possible configurations with the coefficients C (0) λ being uniformly distributed random complex numbers. We note that if we were to compute expectation value M z for this random state we would obtain a finite value. However, averaging over many sets of C (0) λ yields no initial magnetization.
At t = 0 + a bright exciton created in neighboring QD enters the central QD. The creation of |X b and annihi-lation of |0 are described by operator |X b 0|, hence the wave-function of the system with one exciton is given by where C n=1 λ (t = 0 + ) = C n=0 λ (t = 0). The initial wavefunction of the injected bright exciton, MI and nuclearspins is an uncorrelated state. However, the Hamiltonian H QD that accounts for the exchange coupling, creates quantum correlation in the exciton-MI-nuclei complex and |Ψ evolves into a linear combination of all configurations, including an entangled state between bright and dark excitons. As function of time, the bright-exciton decays to vacuum because of coupling with quantized electromagnetic-field.
To be consistent with the quantum jump algorithm we discretize the time t into small steps. Note that because of the small eh-and MI-nuclear-spin couplings (J ne , J nh and J nh ) the excitons and MI evolve in a frozenfluctuating field of nuclear-spins [31][32][33] . The eh recombination time, t r , is the smallest time-scale in our model, hence δt << t r .
In Eq.
Because H eff is timeindependent, we employ the method based on Bessel-Chebyshev polynomial expansion 32 to calculate the time evolution of the wave-function Note that H QD is Hermitian and thus exp(−iH QD δt/h) is a unitary operator that conserves the norm of wavefunction, hence |C n=1 X b ,µ (δt)| 2 + |C n=1 X d ,µ (δt)| 2 = 1. This is in contrast with the operator exp(−Γδt/2 λ P † λ P λ ) that is non-unitary and does not preserve norm of wavefunction, however, because it describes the decay of X b to vacuum, we build a norm-conserving wave-function by adding vacuum. After applying exp(−Γδt/2 λ P † λ P λ ) in Eq. 9 we find In the limit of Γ = 0 the coefficients with and without tilde used in Eqs. 9-10 are identical. Matching the initial conditions between Eq.(7) and Eq.(10) implies C n=1 X b ,λ (δt = 0) = C n=1 λ (t = 0 + ) and C n=1 X d ,λ (δt = 0) = 0. Using an iterative procedure to propagate the time we find where the coefficients C n=1 X b ,λ (t), C n=1 0,λ (t), and C n=1 X d ,λ (t) are determined numerically. This wave-function describes a correlated state of bright and dark excitons as well as vacuum. Because of the spin flip-flop process of electron with MI and nuclear-spins the initially formed bright exciton |X b mixes with the dark-exciton |X d .
From the Lindblad ME, the quantum jump transition rate is given by Γ jump = Γρ b with ρ b representing the population of the bright exciton obtained from the full QD density matrix after tracing over MI and nuclear spin degrees of freedom. The quantum jump probability δp jump = Γ jump δt is then calculated and compared with a random number r generated between zero and one. If r < tr 0 dtδp jump a quantum jump takes place, photon is recorded and the quantum dot is in the ground state. The elapsed time t r recorded for this quantum jump is the eh-recombination time. At t = t r we allow exciton to annihilate by spontaneous emission of a photon. The operator that allows annihilation of bright exciton and creation of vacuum is |0 X b |. Hence where C n=2 λ (t = t r ) = C n=1 X b ,λ (t = t r ). Immediately after annihilation of exciton, a new spin polarized exciton tunnels into the quantum dot from the neighboring dots. The spin polarized exciton interacts with the MI spin M and nuclear spins I in a state modified by the previous exciton. At t = t r + 0 + , second bright exciton X b created in the neighboring QD tunnels into the central QD with matching the initial conditions that requires C n=3 λ (t = t r + 0 + ) = C n=2 λ (t = t r ). Note that this state is not correlated. The quantum correlation appears from the time evolution of wave-function generated by exchange couplings in H eff right after t = t r where C n=3 X b ,λ (t r + 0 + ) = C n=3 λ (t r + 0 + ) and C n=3 X d ,λ (t r + 0 + ) = 0 are matching conditions. As we see, there is no a type of linear combination between |0 and {|X b , |X d }, because there is no Rabi-oscillations between vacuum and excitons.
To summarize the above procedure and make connection across tunneling of exciton and photo-emission we formally introduce a projector Q n→n+1 = |X n+1 b X n b | in nth step of quantum jump. The superscripts refer to annihilated nth and created n + 1th exciton. Note that Q n→n+1 |X n b = |X n+1 b and Q n→n+1 |X d = Q n→n+1 |0 = 0, hence the quantum jump operator projects out any correlated state composed of superposition of bright and dark exciton to a new born bright exciton. The new wavefunction in the QD then can be constructed as |Ψ(t = t + r ) = Q n→n+1 |Ψ(t = t − r ) where t ± r = t r ± η and η → 0. In this state, |X n+1 b is initially uncorrelated from MI and nuclear-spins. At t = t + r it can be expressed as We observe that after quantum jump the new injected exciton X n+1 b starts with the normalized (factor A N ) state of MI and nuclear spins λ C n X b ,λ (t r )|λ which was left over by the previous bright exciton X n b at the time of radiative recombination. Detecting a photon erased the dark exciton wave-function and modified the state of both MI and nuclear spins. This is the DMP mechanism discussed here. With initial condition established, the time evolution of the entangled state of photo-carriers with MI and spin-bath then can be calculated after updating the coefficients C's. At the end one needs to average over initial conditions. Although the procedure discussed here describe the immediate refilling of central QD right after annihilation of the exciton, we can implement a waiting time between the recombination and refilling process.

V. NUMERICAL RESULTS AND DISCUSSION
Our approach to DMP is illustrated for parameters based on (Cd,Mn)Te QDs withJ em = 15 meV nm 3 , J hm = 60 meV nm 3 corresponding to the exchange coupling in the bulk materials, hence J em =J em |φ e (R m )| 2 and J hm =J hm |φ h (R m )| 2 . The symmetrical quantum dot is assumed with ∆ 1 = 0. Here φ e/h ( R m ) is the e/h envelop-wavefunction in the central dot at R m , the position of MI. J eh = 0.6 meV 29 , and J ne , J nh , J nm , and J nn are initialized as random numbers with a mean value in the order of 1 µeV, however, the realistic value for nuclear hyperfine interaction is reported within 1 neV 34 , three orders of magnitude smaller than the energy scales used in our finite size calculation.
Here we discuss numerical results with nuclear spins, immediately after refilling of QD by bright exciton. Figs. 4, 5 and 6 illustrate DMP/DNP and the quantum jump trajectories for exciton, MI and nuclear-spins. In Fig. 4 a single quantum jump trajectory for spin-1/2 MI is plotted. In Figs. 5 and 6 the ensemble average of twenty quantum jump trajectories for DMP/DNP by a series of the injected photo-carriers for spin-1/2 (Fig. 5) and spin-5/2 Mn (Fig. 6) are plotted. Each curve consists of thousands of time-steps and points. To assign the legends, every hundred points, symbols like circle, star and triangle are superimposed on each curve. As shown the MI and the average polarization of N b = 15 nuclear spins gradually builds-up by a train of injected bright excitons. At t = t r , one pair of eh collapses to vacuum with S e,z (t) < 1/2 as part of the e-spin is transferred to MI. An empty dot instantaneously absorbs the second photo- generated eh pair with total angular momentum j z = −1 that transfers to spin of MI and nuclei before its removal. We repeat this procedure until the spin polarization in MI and nuclear-spins builds-up. The method presented here is limited to a finite number of nuclear spins because of exponentially increasing with the number of nuclear spins computational effort. However, a systematic study of the convergence of the numerical results by increasing N b shows satisfactory outcomes around N b = 15. For Mn, each of five Mn electrons interact with the exciton and nuclear spins with the same exchange coupling because of a symmetry that Mn electrons follow to interact with the environment. As it is shown in Fig. 6, after approximately ten injection and annihilation of excitons in the central QD, S z approaches to 5/2 for Mn, reaching the maximum polarization.

VI. SUMMARY
In conclusion, dynamical magnetic and/or nuclear polarization in complex spin systems is discussed on the example of transfer of spin from exciton to the central spin of magnetic impurity in a quantum dot in the presence of a finite number of nuclear spins. The exciton is described in terms of the electron and heavy hole spins interacting via exchange interaction with magnetic impurity, via hyperfine interaction with a finite number of nuclear spins and via dipole interaction with photons. The time-evolution of the exciton, magnetic impurity and nuclear spins is calculated exactly between quantum jumps corresponding to exciton radiative recombination. The collapse of the wave-function and the refilling of the quantum dot with new spin polarized exciton is shown to lead to build up of magnetization of the magnetic impurity as well as nuclear spin polarization. The competition between electron spin transfer to magnetic impurity and to nuclear spins simultaneous with the creation of dark excitons is elucidated. The technique presented here opens up the possibility of studying optically induced Dynamical Magnetic Polarization in Complex Spin Systems.

VII. ACKNOWLEDGEMENT
RMA thanks Steve Girvin for useful discussion and the support from Texas Advanced Computing Center (TACC) for computer resources.