Alloptical Polarization Phase Modulation in Coupled Quantum Dots
 Author: Je Ku Chul, Kyhm Kwangseuk
 Publish: Current Optics and Photonics Volume 1, Issue1, p60~64, 25 Feb 2017

ABSTRACT
We have considered optical nonlinearities of coupled quantum dots theoretically, where an exciton dipoledipole interaction is mediated between the adjacent large and small quantum dots. For increasing a pump pulse area in resonance with the large quantum dot exciton the induced nonlinear refractive index of the small quantum dot exciton has been obtained. As the exciton dipoledipole interaction depends on the relative orientation of two exciton dipoles, the optical nonlinearities for the directions parallel and perpendicular to the coupling axis of the two quantum dots are compared. The directional imbalance of optical nonlinearities in coupled quantum dots can be utilized for a polarization phase modulator by controlling a pump pulse area and propagation length.

KEYWORD
Optical nonlinearities , Exciton , Polarization , Quantum dots

I. INTRODUCTION
During the last decade, coupled quantum dots (CQDs) have been of great interest to study the quantum nature of superposition and entanglement [16]. This can be realized by stacking quantum dots (QDs) vertically with a barrier layer, where the QDs are grown by the StranskiKrastanov growth method. Also, when QDs are formed by the interface fluctuation of a quantum well thickness, laterally neighbored QDs can be found. Likewise, laterally coupled QD structures can also be generated when dense colloidal QDs are dispersed [8, 9]. Additionally, the droplet epitaxy method enables control of the morphology of laterallycoupled QD structures [4, 5].
When the size of the two QDs are different, an energy resonance can be induced by applying an external DCelectric field, resulting in a wavefunction overlap between the QDs via the tunneling effect [6]. In this case, the tunneling effect is limited by a short interdot distance (<10 nm). On the other hand, the exciton dipoledipole interaction can be utilized as an alternative method of interdot coupling [7]. As it is based on the Coulomb interaction, the exciton dipoledipole interaction still works at relatively long distance. As the dipoledipole interaction depends on the dipole orientation, interaction between the two QDs can be controlled selectively by optical polarization direction. The DCelectric control needs extra fabrication processes for electrodes, and the resonance condition also depends on the QD size difference. However, the exciton dipoledipole interaction still works with an energy tolerance among size different QDs, and no extra fabrication is necessary for electrodes in the case of DCelectric control. Therefore, optical control of CQDs is feasible for a practical device and also applicable to inhomogeneous QD ensembles.
In this work, we calculated polarization dependence of optical nonlinearities in lateral CQDs, where the nonlinear refractive indices between the direction parallel and perpendicular to the QD coupled axis are imbalanced as a consequence of the directional dependence of the exciton dipoledipole interaction.
II. METHODS
As shown in Fig. 1, we considered two neighboring spherical QDs, where the surfacetosurface distance (
d = 15Å) and the radii of small (R_{A} = 30Å) and large (R_{B} = 30.5Å) QDs are given. In a practical point of view, this can be easily found in size distributed colloidal QDs. Therefore, we considered CdSe colloidal QDs as a feasible example, but this model can also be utilized for a lateral CQD grown by droplet epitaxy. In the case of a CdSe QD, the exciton ground state is the 1^{L} bright fine state [8, 1013].Although the size difference is small, the energy difference between the two ground exciton states of small (X_{A}) and large (X_{B}) QDs is ~ 14 meV. Nevertheless, the exciton dipoledipole interaction is still effective in the presence of the energy difference between X_{A} and X_{B}. Suppose a single CQD structure is well isolated, this can be analogous to the atomic threelevel system in quantum optics, where electrically induced transparency is studied. However, the origin of optical nonlinearities is different. While an exciting light pulse is resonant with X_{B}, the refractive index of X_{A} also changes through the exciton dipoledipole interaction. For increasing the exciting pulse intensity, the occupancy of both X_{A} and X_{B} are increased. Therefore, the exciton dipoledipole interaction also depends on the exciton occupancy. In this case, the Coulomb interaction needs to be considered carefully by considering the exciton occupancy.
Additionally, it is noticeable that the exciton dipoledipole interaction depends on the relative orientation of two exciton dipoles. For the orientation dependence of a pair of dipoles, two cases were compared, i.e., X_{A} and X_{B} are parallel (Fig. 1(a)) or perpendicular (Fig. 1(b)) to the QD coupled direction, which is denoted by Hpolarized and Vpolarized, respectively.
The Hamiltonian can be given by
H =H _{QD} +V _{C}, whereH _{QD} is the energy of X_{A} and X_{B} and the dipole radiation associated with by a light pulse, andV _{C} describes the Coulomb interaction between X_{A} and X_{B} [7, 14]. More specifically,H _{QD} is given byFor QD_{A}, is the energy of an electron (hole) in the exciton state
λ , which is either Hpolarized or Vpolarized excited exciton state and an unexcited state. and are the creation (annihilation) operators for electrons and holes.E (t ) andd_{λA} are the amplitude of a pump pulse (~ 1 ps) and the interband optical transition matrix element, respectively. Therefore,V _{C} can be described bywhere observable quantities can be defined such as the interband transition〈
β_{λi} α_{λi} 〉 =p_{λi} (t ), the electron (hole) occupancy , and the total polarization .The equations of motion for those observable quantities are described by the semiconductor Bloch equations as [15, 16]
whereby the nonlinear refractive index spectrum
ñ (ω ,N (Θ)) =n (ω ,N (Θ)) + ίκ(ω ,N (Θ)) can be obtained.III. RESULTS AND DISCUSSION
As shown in Fig. 1(d), the occupancy of X_{B} is increased gradually with increasing a pump pulse area Θ, which becomes maximized at Θ = π. For an isolated QD, we have already calculated the refractive index change, where the real and imaginary parts of the refractive index spectrum were shown for pump intensity (~ Θ2) [15]. Although the exciton dipoledipole interaction has been well known in terms of resonance energy transfer (FRET), its nonlinearity is barely considered. However, as shown schematically in Fig. 1(c), the optical nonlinearity of X_{A} can appear through the exciton dipoledipole interaction as
N _{B} increases. Since it depends on a frequency difference with respect to its resonance as shown schematically in Fig. 1(c), we calculated the real and imaginary parts of the nonlinear refractive index spectrum near the X_{A} energy under resonant excitation of X_{B}.Figure 2(a) and (b) show the real (n(ω, Θ)) and imaginary (κ(ω, Θ)) parts of the nonlinear refractive index spectrum for the Hpolarized excitation. As Θ is increased, a spectral splitting is seen in the n(ω, Θ) and κ(ω, Θ) spectrum, which is a distinguishing feature of the dipoledipole interaction compared to the results of an isolated QD. This result reminds us of the spectral splitting of dressed states as a consequence of lightmatter interaction. For example, a spectral splitting of the emitter in a cavity appears as the cavity field becomes strongly coupled with the emitter, and recently the cavity field can be replaced with the strong local surface plasmon of a metal nanostructure. In this sense, the nonlinear exciton dipoledipole interaction is quite similar, and the spectral splitting indicates a degree of the nonlinear coupling between two exciton dipoles.
From a practical point of view, a pulse laser is necessary to induce transient optical nonlinearities, and the preferential direction of the exciton dipoledipole interaction can be selected by the pump pulse polarization. However, the energy difference (a few meV) between the H and Vpolarized states of an exciton pair is vulnerable to thermal transition. Therefore, both the H and Vpolarized states contribute to the optical nonlinearities. As shown in Fig. 2(c) and (d), the nonlinear spectrum for the Vpolarized excitation are different from those for the Hpolarized exciton pair due to the directional dependence of the exciton dipoledipole interaction. Suppose the transient optical nonlinearities are utilized for alloptical phase modulation, the imbalanced nonlinear refractive index between the H and Vpolarized should be considered.
In Fig. 3, the changes of the nonlinear refractive index with respect to the linear refractive index, i.e. Δn(ω, Θ) = n(ω, Θ > 0) − n(ω, Θ = 0) and Δκ(ω, Θ) = κ(ω, Θ > 0) − κ (ω, Θ= 0), are shown at negativedetuned (ΔE = −0.1 meV) (a), resonant (ΔE = 0) (b), and positivedetuned (ΔE = +0.1 meV) (c) probe light for the H and Vpolarized exciton pairs, respectively. When the probe light is resonant (ΔE = 0), both the H and Vpolarized excitons are similar in the real part change of the nonlinear refractive index Δn_{H}K≃Δn_{V}. On the other hand, a significant difference is shown in the imaginary part change. Therefore, the incident linear polarization of a probe light barely rotates, but the ellipticity becomes enhanced. For the positive and negative detuned probes, the pump pulse area should be large enough (Θ > 0.4π to observe a significant polarization modulation. In experimental point of view, this implies more than 16% of the exciton saturation intensity is necessary.
When the imbalanced optical nonlinearities of the H and Vpolarized exciton are utilized for a practical alloptical polarization modulation, a precise phase modulation needs to be predicted. As an example, we have considered the modulated polarization of the negative detuned probe (ΔE = −0.1 meV), the modulated polarization can be represented in terms of rotational (
θ ) and elliptical (ε) angles as shown in Fig. 4(b). Given the refractive index change of X_{A} induced by pumping X_{B} with a pulse area (Θ), the modulated polarization Jones vector (, ) can be described aswhere the amplitude of an incident probe light and the induced phase components are denoted by
E _{0} andφ _{x, y}, respectively. Regarding the density of QDs in a feasible device, we assume ∼100. This implies the propagation length z ~ 10μm with a multiple QD layers. Elliptical polarization states can also be characterized by using Stokes parameters aswhereby the elliptical (
ϵ ) and rotational (θ ) angles can be obtained asIn Fig. 4(a), the polarization state for increasing excitation pulse area (Θ) is mapped in the normalized Poincar´e sphere, which enables to overview the degree of linearity and ellipticity. Fig. 4(b) also shows
ϵ (Θ) andθ (Θ). Interestingly, remains near zero whilst increases. Therefore, this low pulse area range (Θ < 0.5π) can be utilized for rotational polarization control similar to the Faraday rotator. On the other hand, the ellipticity becomes significant when 0.5π < Θ < π. This range is useful as a waveplate by extending the propagation length. Consequently, a precise phase modulation can be realized by excitation and propagation length.IV. CONCLUSION
In conclusion, we have proposed that coupled quantum dots can be utilized as an alloptical polarization phase modulator. When an exciton dipoledipole interaction is present between the size different two quantum dots, resonant excitation to the large quantum dot gives rise to optical nonlinearities of the small quantum dot. As the induced optical nonlinearities depend on the orientation of an exciton pair, a modulated polarization phase can be controlled in terms of excitation and propagation length.

12. Norris D. J., Efros Al. L., Rosen M., Bawendi M. G. (1996) [Phys, Rev. B] Vol.53 P.16347

[FIG. 1.] Schematic diagram of coupled quantum dots when the excitons of large (QDA) and small (QDB) quantum dots are parallel (a) or perpendicular (b) to the coupling direction. With increasing a pump pulse area, which is in resonance with QDA energy (c,e), the enhanced exciton population at QDA (d) gives rise to a transient optical nonlinearity of the detuned (Δ E < 0 and ΔE > 0) and resonant (ΔE = 0) probe.

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[FIG. 2.] The real (n) (a, c) and imaginary (κ) (b, d) refractive index spectrum of XA for increasing the pump pulse area (Θ) resonant to the QDB exciton energy, where the polarization of a pump pulse is either parallel (Hpolarized) or perpendicular (Vpolarized) to the QD coupling direction.

[FIG. 3.] Refractive index change in real (Δn) and imaginary (Δκ) parts for increasing either H or Vpolarized pulse area (Θ) resonant with the QDB exciton when the probe energy is below (a), resonant (b), and above (c) the QDA exciton.

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[FIG. 4.] For increasing the excitation of a pump pulse area (Θ), the linear polarization of a probe, which is offresonant (ΔE = ？0.1 meV) to XA exciton, becomes elliptically polarized due to the phase retardation arising from the imbalanced transient optical nonlinearity between the H (denoted by x) and Vpolarized (y) direction. The elliptical polarization for excitation is mapped into the Poincar´e sphere (a) and characterized in terms of rotational (θ) and elliptical angles (ε) (b).