Quantum Dot Lattice as Nano-Antenna for Collective Spontaneous Emission

Abstract

We present a theory for the collective spontaneous emission of timed Dicke states in a periodic 2D-array of quantum dots (QDs) coupled by dipole-dipole (d-d) interactions. The master equation is first reformulated with respect to the timed Dicke basis. As a result, we obtain simple analytical relations for the spontaneous decay rate, collective Lamb shift and radiative pattern. The collective spontaneous emission in QD-array manifests itself in strong directivity, whereby the radiative pattern consists of a set of strong radiative lobes. The direction of the first lobe is dictated by the pumping direction, while the other lobes correspond to diffractive rays due to the periodicity. The influence of d-d interactions on the radiation decay of timed Dicke states in QD arrays is identical to the influence of an environment to single-particle excited states similar to the action of a structured photonic reservoir. For a rectangular 2D-array, the equivalent structured photonic reservoir has a form of a hollow rectangular waveguide with perfectly conductive walls. For lattice periods comparable to the radiation wavelength the decay rate shows sharp peaks due to Van-Hove singularities in the photonic density of states (PDOS) similar to the Purcell effect in photonic crystals. The optical nanoantenna under study allows tuning of the radiation pattern by varying the timing.

Appendices

Appendices

4.1.1 Appendix A: Scalar Green Function for Electromagnetic Field in Square 2D-Lattice

Let us obtain the presentation for scalar Green function (4.14), which is convenient for numerical calculations. We start with the presentation

$$ \frac{e^{i{k}_0r}}{4\pi r}=\frac{1}{{\left(2\pi \right)}^3}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{e^{i\left({k}_xx+{k}_yy+{k}_zz\right)}}{k_x^2+{k}_y^2+{k}_z^2-{k}_0^2}d{k}_x}}}d{k}_yd{k}_z. $$

(4.21)

Using the integral relation

$$ {\displaystyle {\int}_{-\infty}^{\infty}\frac{e^{i\xi \left(z-{z}^{\prime}\right)}}{\xi^2-{\kappa}^2}d\xi }=\frac{i\pi }{\kappa }{e}^{i\kappa \left|z-{z}^{\prime}\right|}, $$

and integrating over k _z , we transform (4.21) to

$$ \frac{e^{i{k}_0r}}{4\pi r}=\frac{i\pi }{{\left(2\pi \right)}^3}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{e^{i\left({k}_xx+{k}_yy\right)}}{\kappa }{e}^{i\kappa \left|z\right|}}}d{k}_xd{k}_y $$

(4.22)

with $\kappa =\sqrt{k_0^2-{k}_x^2-{k}_y^2}$. Based on Eq. (4.14) and using (4.22), we obtain the periodic Green’s function

$$ \begin{aligned} G\left(x,y,z;{\omega}_0\right)&=\frac{i\pi }{{\left(2\pi \right)}^3}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{e^{i\kappa \left|z\right|}}{\kappa }d{k}_xd{k}_y}}\\ &\quad \times {\displaystyle \sum_{p,q=-\infty}^{\infty } \exp \left[i{k}_x\left(x-pa\right)+i{k}_y\left(y-qa\right)+iqa{k}_0 \cos {\theta}_{\mathrm{inc}}\right]}\end{aligned} $$

Using the well-know Poisson’s summation formula

$$ {\displaystyle {\sum}_{m=-\infty}^{\infty } \exp (imx)}=2\pi {\displaystyle {\sum}_{m=-\infty}^{\infty}\delta \left(x-2\pi m\right)}, $$

(4.23)

and integrating over k _x,y , we finally obtain the scalar Green’s function in the form (4.15).

4.1.2 Appendix B: Spontaneous Emission Rate of QD-Lattice in the Limit of Infinitely Large Period

When the lattice period is infinitely large, the spontaneous emission rate of the QD lattice can be obtained from Eq. (4.17), as

$$ \underset{a\to \infty }{ \lim }{\tilde{\Gamma}}_{\mathbf{K}}=\underset{a\to \infty }{ \lim}\frac{\mu^2}{\hslash {\varepsilon}_0{a}^2}{\displaystyle \sum_m\left[{k}_0^2-{\left(\frac{2\pi m}{a}\right)}^2\right]{\displaystyle \sum_n\frac{1}{k_{mn}}}}, $$

(4.24)

where $m,n{=}0,\pm 1,\pm 2,\cdots$, with the restriction ${\left(2\pi m/a\right)}^2{+}{\left[\left(2\pi n/a\right){+}{k}_0 \cos {\theta}_{\mathrm{inc}}\right]}^2\linebreak \le {k}_0$. By use of the mathematical identity,

$$ \underset{l\to \infty }{ \lim}\frac{2\pi }{l}{\displaystyle \sum_{p=-\infty}^{\infty }f\left(\frac{2\pi p}{l}\right)}\to {\displaystyle {\int}_{-\infty}^{\infty }f\left(\xi \right)}d\xi, $$

(4.25)

the inner sum in Eq. (4.24) transformed to the integral

$$ {\displaystyle \underset{-\sqrt{k_0^2-{\left(2\pi m/a\right)}^2}}{\overset{\sqrt{k_0^2-{\left(2\pi m/a\right)}^2}}{\int }}\frac{d\xi }{\sqrt{k_0^2-{\left(2\pi m/a\right)}^2-{\xi}^2}}}=\pi, $$

(4.26)

where $\left|{k}_0\right|>2\pi \left|m\right|/a.$ Then Eq. (4.24) will be simplified to

$$ \underset{a\to \infty }{ \lim }{\tilde{\Gamma}}_{\mathbf{K}}=\frac{\mu^2}{4\pi \hslash {\varepsilon}_0}\underset{a\to \infty }{ \lim}\left(\frac{2\pi }{a}\right){\displaystyle \sum_m\left[{k}_0^2-{\left(\frac{2\pi m}{a}\right)}^2\right]}. $$

(4.27)

Again, by employing Eq. (4.25), the sum over m can also be exchanged to integration which gives rise to

$$ \underset{a\to \infty }{ \lim }{\tilde{\Gamma}}_{\mathbf{K}}=\frac{\mu^2}{4\pi \hslash {\varepsilon}_0}{\displaystyle {\int}_{-{k}_0}^{k_0}\left({k}_0^2-{\zeta}^2\right)d\zeta }=\frac{\mu^2{k}_0^3}{3\pi \hslash {\varepsilon}_0}={\Gamma}_0, $$

(4.28)

which is equal to the spontaneous emission rate of a single QD in free space.