optical control of initialization of a quantum dot in the ... · key–words: quantum control,...

Optical Control of Initialization of a Quantum Dot in the VoigtGeometry near a Graphene Layer

DIONISIS STEFANATOS, VASILIOS KARANIKOLAS, NIKOS ILIOPOULOS, EMMANUEL PASPALAKISUniversity of Patras

Department of Materials Science265 04 Rio Patras

[email protected], [email protected], [email protected], [email protected]

Abstract: We propose and analyze fast spin initialization for a quantum dot in the Voigt geometry by placing itnear a graphene layer. We show that high levels of fidelity, significantly larger than in the case of the quantumdot without the layer, can be quickly obtained due to the anisotropy of the enhanced spontaneous decay rates ofthe quantum dot near the graphene layer. We initially obtain these results by using a continuous wave optical fieldwith constant control amplitude. We also use state of the art numerical optimal control to find the time-dependentelectric field which maximizes the final fidelity for the same short duration as in the previous case. A better fidelityis obtained with this method.

Key–Words: Quantum control, numerical optimal control, quantum dot, spin initialization, Voigt geometry, opticalpumping, spin dynamics

1 IntroductionSemiconductor quantum dots (QDs) are among themost promising candidates for solid state quantum in-formation processing and for quantum technologiesin general since they allow for engineered energiesand wavefunctions and can be efficiently manipu-lated by electric, magnetic and optical fields [1, 2].They can also be integrated with microphotonic andnanophotonic structures, such as microcavities andnanocavities, photonic crystals, and plasmonic nanos-tructures, allowing for extra control in both the weakand strong light-matter coupling regimes. Spin statesin negatively and positively charged QDs are espe-cially useful for quantum information technologies,with the important problems being the spin initial-ization, coherent manipulation, and readout. A QDstructure that has been studied in detail both theoret-ically and experimentally is based on the spin statesin the Voigt configuration, where a magnetic field isapplied perpendicular to the growth axis of the QD[1, 2, 3, 4, 5, 6, 7, 8, 9, 10].

The level structure for the QD in the Voigt ge-ometry is shown in Fig. 1(a). The natural initial stateof the QD is an incoherent mixture with equal popu-lations in the two electron spin states, states |1〉 and|2〉. A basic process in quantum information proto-cols is the initialization [11], where in this systema specific electron spin state is created. The mostcommon method for initialization relies on the opti-

cal pumping process [3]. Let’s assume that we wishto create state |2〉. We apply an optical field with ax−polarized electric field which couples spin state |1〉to the trion state |4〉. Then, by spontaneous decay thepopulation is pumped into spin state |2〉 after a few de-cay times. In an isotropic photonic environment, forexample in free-space vacuum, the decay rates fromstate |4〉 to states |1〉 and |2〉 are the same. By using ananisotropic photonic environment the decay rate fromstate |4〉 to state |2〉 increases in comparison to the de-cay rate from state |4〉 to state |1〉 and this leads topreferential deexcitation towards state |2〉 in shortertimes. This idea has been explored by embedding theQD in a micropillar [5], placing it in photonic crys-tal nanocavities [6], next to a metallic nanoparticle[7], a metal-dielectric interface [8], and even a bowtienanoantenna [9].

Here, we explore a different nanostructure for ac-celerating the optical pumping process in a QD inthe Voigt geometry. We place the QD next to a sin-gle graphene layer, see Fig. 1(b). Graphene stronglymodifies the spontaneous decay rates of nearby quan-tum emitters [12, 13, 14, 15]. It can also createanisotropic spontaneous decay rates for nearby quan-tum emitters for dipoles parallel and perpendicularto the graphene sheet [12, 14, 15]. In the presentwork we investigate the dynamics of spin initializa-tion for the composite system using optical pumpingwith two different types of optical pulses, using typ-

WSEAS TRANSACTIONS on SYSTEMS and CONTROLDionisis Stefanatos, Vasilios Karanikolas,

Nikos Iliopoulos, Emmanuel Paspalakis

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(a)

(b)

Figure 1: (a) Energy level diagram for a QD in the

Voigt geometry. The magnetic field induces the Zee-

man splitting in the upper and lower levels. The op-

tical field resonant with the |1〉 ↔ |4〉 transition is

x-polarized. (b) The QD is placed a distance z = Rabove a graphene layer, which lies on the x− y plane

at z = 0. The growth axis of the QD is y while the

magnetic field is applied along the x-axis.

ical values of energies for the QD. First, we use acontinuous wave electromagnetic field with constantRabi frequency, and show that, in the presence of thegraphene layer, higher fidelity levels can be achievedin short times compared to the case without graphene.Next, we use state of the art numerical optimal controlto find the time-dependent electric field, leading to atime-dependent Rabi frequency profile, which maxi-mizes the final fidelity for the same short duration asin the previous case. A better fidelity is obtained withthis method.

2 Dynamics equations for a quantumdot in the Voigt geometry near agraphene monolayer

For the studied system, we consider a singly-chargedself-assembled QD grown along the y-axis. By apply-ing an external magnetic field in the Voigt geometry,

along the x-axis, lifts the degeneracy of electron/holelevels by Zeeman splitting. Then, the ground spin lev-els are |1〉 = | ↓x〉 and |2〉 = | ↑x〉 and the two excitedtrion states are |3〉 = | ↓x↑x⇑x〉 and |4〉 = | ↓x↑x⇓x〉.Here, ⇑ (⇓) and ↑ (↓) denote heavy hole and elec-tron spins, respectively. For the level scheme, see Fig.1(a). Here, the vertical transitions (|1〉 ↔ |4〉 and|2〉 ↔ |3〉) give x-polarized dipole matrix elementsand the cross transitions (|1〉 ↔ |3〉 and |2〉 ↔ |4〉)give z-polarized dipole matrix elements. In this work,we aim to create state |2〉. Thus, we take the QD tointeract with a x-polarized laser field applied at exactresonance with the |1〉 ↔ |4〉 transition, exploring themethod of optical pumping.

In order to study the dynamics of spin initializa-tion, we derive the density matrix equations of thesystem. The Hamiltonian describing the interactionbetween the optical field and the QD system, in thedipole and rotating wave approximations, is

H =4∑

n=1

hωn|n〉〈n|

− h[Ω(t)e−iω0t|4〉〈1|+Ω(t)e−iω0t|3〉〈2|

]+ H.c. (1)

where hωn, n = 1, 2, 3, 4, is the energy of state |n〉and Ω(t) is the generally time-dependent Rabi fre-quency of the applied field. The field is resonant withthe |1〉 ↔ |4〉 transition, thus ω0 = ω4 − ω1 = ω41.Here we use the value hω0 = 1.31 eV. This is a typicalvalue for InAs self-assembled QDs used in relevantstudies [3, 4, 9], and it is also adopted here. We notethat although we did not calculate the electronic struc-ture of the QD numerically here, one can calculate itselectronic structure using several methods, like the ef-fective mass approximation and k · p method [16],empirical tight binding models [17], empirical pseu-dopotential based approaches [18], and others [19].

Using the unitary operator U(t) = e−i∑4

n=1an|n〉〈n|t,

with a1 = ω1, a2 = ω0 + ω1, a3 = ω3 + ω0 − ω41,a4 = ω0 + ω1, Hamiltonian (1) is transformed to theinteraction Hamiltonian

Heff = −h(ω0 − ω21)|2〉〈2| − h(ω0 − ω41)|3〉〈3|−h(ω0 − ω41)|4〉〈4| − h

[Ω(t)|4〉〈1|

+Ω(t)e−i(ω0+ω43)t|3〉〈2|+ H.c.

], (2)

where ωnm = ωn − ωm. Note that hω21, hω43

are the Zeeman splittings of the single-electron spinstates and the heavy-hole spin trion states, respec-tively. Throughout this article we use the valueshω21 = 0.124 meV, hω43 = 0.078 meV [4].



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Using the above Hamiltonian we can easily ob-tain the equations of motion for the density matrix ele-ments of the QD. In the case where the laser frequencyis at exact resonance with the transition |1〉 ↔ |4〉,these are

ρ11 = γ41ρ44 + γ31ρ33 − 2ΩρI41, (3)

ρ22 = γ32ρ33 + γ42ρ44 + 2ΩρI23, (4)

ρ33 = −(γ31 + γ32)ρ33 − 2ΩρI23, (5)

ρ44 = −(γ41 + γ42)ρ44 + 2ΩρI41, (6)

ρI41 = −γ41 + γ422

ρI41 +Ω(ρ11 − ρ44), (7)

˙ρR23 = −γ31 + γ32

2ρR23 + (ω21 + ω43)ρ

I23, (8)

˙ρI23 = −γ31 + γ32

2ρI23 − (ω21 + ω43)ρ

R23

+Ω(ρ33 − ρ22), (9)

where ρ23 = ρ23e−i(ω41+ω43)t and the superscripts

R, I denote real and imaginary parts, respectively.Note that in the above equations we have also incor-porated the spontaneous emission from the upper tothe lower energy levels, with

γ41 = γ32 = γx = Fxγ, γ42 = γ31 = γz = Fzγ(10)

being the radiative decay rates of the correspondingQD transitions modified by the coupling between theQD and the graphene layer.

3 Fast initialization with optical con-trol

As we have already mentioned above, the natural ini-tial state of the system is an equal incoherent mixtureof states |1〉 and |2〉, thus ρ11(0) = ρ22(0) = 1/2while the populations of the rest levels and the coher-ence terms are all zero. Starting from this initial state,the goal is to apply the appropriate optical field in or-der to quickly prepare the spin state |2〉 with high fi-delity. In the following we separately investigate howthis can be efficiently achieved with constant controlΩ and with time-dependent control Ω(t).

3.1 Constant Rabi frequencyWe consider a specific case where the QD is placed adistance R = 5 nm above the graphene layer, whilethe chemical potential of the layer is set to the valueμ = 1 eV. We also consider that the layer is placed ina dielectric environment with ε1 = ε2 = 4. Note thatdielectric environments with permittivity value 4 havebeen considered in other works that study the sponta-neous decay of quantum emitters near graphene layers

0 5 10 15 200.5

0.6

0.7

0.8

0.9

1

Ω/γ

ρ 22(T

)

(a)

0 5 10 15 200.5

0.6

0.7

0.8

0.9

1

Ω/γ

ρ 22(T

)

(b)

Figure 2: Final population of level |2〉, ρ22(T ), ver-

sus the constant control Ω for four durations T =1/γ, 2/γ, 3/γ, 4/γ (from bottom to top) and: (a) in

the presence of a graphene layer placed a distance

R = 5 nm from the QD with chemical potential μ = 1eV, (b) in the absence of the graphene layer. A realistic

dielectric environment with ε1 = ε2 = 4 is considered

in both cases.

[13], as they model very well the permittivity valuesof several materials that are used as hosts to graphene,like hBN, GaN and SiO2. With the above parame-ter values one can calculate the corresponding Pur-cell factors using electromagnetic theory, and specifi-cally the method of scattering superposition, see Ref.[14] for the details of the method. The resulting val-ues are Fx = 3.99307 and Fz = 6.00242. Usingthese values we solve numerically Eqs. (3)-(9) andsimulate the response of the quantum dot system. InFig. 2(a) we plot the final population of level |2〉,ρ22(T ), versus the constant control Ω for four dura-tions T = 1/γ, 2/γ, 3/γ, 4/γ (from bottom to top).For comparison, in Fig. 2(b) we plot the final pop-ulations obtained in the absence of graphene, whereFx = Fz = 2, for the same durations (again with in-



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0 1 2 3 40.5

0.6

0.7

0.8

0.9

1

γ t

ρ 22(t)

Figure 3: Time evolution of state |2〉 population ρ22(t)when a constant Rabi frequency Ω = 5γ is applied,

for a realistic dielectric environment with ε1 = ε2 = 4and (a) in the presence of a graphene layer of chemi-

cal potential μ = 1 eV placed a distance R = 5 nm

from the QD (red solid line), (b) in the absence of a

graphene layer (blue dashed line). Observe that the

fidelity obtained with the layer is larger than the one

without it.

creasing duration from bottom to top). Obviously, theanisotropy in the Purcell factors induced by the pres-ence of the graphene layer results in higher initializa-tion fidelities which are obtained at earlier times. Thisis further illustrated in Fig. 3, where we display thetime evolution of the population ρ22(t) in the presence(red solid line) and in the absence (blue dashed line) ofgraphene, for a constant control Ω = 5γ (a value closeto optimal as shown in Fig. 2) and duration T = 4/γ.For example, at t = 3/γ the value ρ22(3/γ) = 0.9962is obtained with graphene, while the correspondingvalue without graphene is just 0.9691. This kind ofimprovement is quite important for quantum informa-tion processing applications, where large values of fi-delity close to unity are necessary.

3.2 Time-dependent Rabi frequencyHere we recourse to numerical optimization to obtaintime-dependent controls Ω(t) which are zero at theinitial and final times

Ω(0) = Ω(T ) = 0, (11)

while maximize the final population of level |2〉. Thiskind of controls are more appropriate for experimentalimplementation, since they avoid the initial and finaljumps of the square pulse.

In order to find Ω(t), we use the freely avail-able optimal control solver BOCOP [20], which can

0 1 2 30

5

10

15

γ t

Ω/γ

(a)

0 1 2 30.5

0.6

0.7

0.8

0.9

1

γ t

ρ 22(t)

(b)

Figure 4: For a realistic dielectric environment with

ε1 = ε2 = 4, a graphene layer with chemical poten-

tial μ = 1 eV placed a distance R = 5 nm from the

QD, and duration T = 3/γ: (a) Optimal control Ω(t)using a polynomial of order p = 6 in Eq. (12), (b) cor-

responding ρ22(t). The final value ρ22(T ) = 0.9979is larger than the value 0.9962 obtained with constant

control and for the same duration in Fig. 3.

easily incorporate boundary conditions like (11). Wealso exploit a feature provided by this program, whichallows to express the control as a polynomial func-tion of time. Specifically, we consider Ω as an extrastate variable, on which we impose boundary condi-tions (11), while we place the polynomial control inits derivative. The corresponding equation in normal-ized time τ = γt is

d

dτ

(Ω

γ

)=

p∑k=0

akτk, (12)

where we have used a p-th order polynomial for thederivative, so Ω is of order p+1, and ak are the dimen-sionless unknown coefficients to be determined by theoptimization procedure. We additionally impose the



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Table 1: Optimal coefficients ak, k = 0, 1, . . . p, for

the polynomial control (14) with p = 6, when μ = 1eV, R = 5 nm, ε1 = ε2 = 4, and the duration is set to

T = 3/γ.

ak, p = 6

146.2137

-907.9316

1812.3362

-1688.2531

806.7791

-191.4565

17.8462

constraint0 ≤ Ω(t)/γ ≤ 20, (13)

in order to restrict the field amplitude within experi-mentally feasible values. Now we can state the op-timal control problem. For system equations (3)-(9),which can easily be expressed using normalized timeand dimensionless variables, augmented by Eq. (12)for Ω, we would like to find the coefficients ak whichmaximize the final value ρ22(T ) for a specific dura-tion T , while satisfy the boundary condition (11) andthe constraint (13). Having found ak we can integrateEq. (12) and obtain the field

Ω(t)

γ=

p+1∑m=1

am−1

m(γt)m (14)

expressed in the original time t.As in the case with constant control, we consider

a specific example with μ = 1 eV, R = 5 nm, ε1 =ε2 = 4, while the duration is set to T = 3/γ. Theoptimal coefficients found using BOCOP for polyno-mial of order p = 6 in Eq. (12) are listed in Table 1.The resulting control field Ω(t) and the correspond-ing evolution ρ22(t) are displayed in Fig. 4. The finalvalue ρ22(T ) is 0.9979, larger than the value 0.9962obtained with constant control Ω = 5γ for the sameduration in Fig. 3, as expected from an optimizationprocedure. Note that if we increase the polynomial or-der in the control, for example to p = 7, a marginalimprovement in the fidelity at the fifth decimal digit isobserved, and this is why we use the value p = 6 inEq. (12).

The suggested methodology is advantageouscompared to other recent works where control fieldspolynomial in time with coefficients optimized withrespect to an objective are used, see for example Refs.[21, 22, 23]. In these studies, some of the coefficientare fixed in order to satisfy the boundary conditions atinitial and final times. For each value of the remaining

free coefficients, the system equations are simulatedand the resulting objective value is recorded. The op-timal coefficients are those optimizing the objective.Usually, one or two free coefficients are employed,since the number of simulations and thus the compu-tational time grows exponentially with the number offree parameters. With the optimal control solver BO-COP that we exploit here, we can use a large num-ber of coefficients in the optimization, as we did forp = 6, while the optimal solution is obtained muchfaster than with the above described practice. Finally,note that instead of polynomial control functions wecould have used trigonometric functions, as did for adifferent problem in a recent publication [24].

4 ConclusionWe studied the problem of fast spin initialization fora QD in the Voigt geometry placed near a graphenelayer. We showed that high levels of fidelity, signif-icantly larger than in the case of the QD without thelayer, can be quickly obtained due to the anisotropyof the enhanced spontaneous decay rates of the QDnear the graphene layer. We obtained this improventby first using a continuous wave electromagnetic fieldwith constant control amplitude. Next, we used stateof the art numerical optimal control to find the time-dependent electric field which maximized the final fi-delity for the same short duration as in the previouscase. A better fidelity was obtained with this method.Closing, we would like to point out that a full para-metric study of the initialization dynamics for variousdistances between the QD and the graphene layer, theconstant Rabi frequency and the chemical potential ofthe graphene is needed and we have started workingon it.

Acknowledgements: The research is implementedthrough the Operational Program “Human ResourcesDevelopment, Education and Lifelong Learning”and is co-financed by the European Union (Euro-pean Social Fund) and Greek national funds (projectEΔBM34, code MIS 5005825).

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