Quantum statistics of light emitted from a pillar microcavity

Abstract

A quantum behavior of the light emitted by exciton polaritons excited in a pillar semiconductor microcavity with embedded quantum well is investigated. Considering the bare excitons and photon modes as coupled quantum oscillators allows for an accurate accounting of the nonlinear and dissipative effects. In particular, using the method of the quantum states representation in a quantum phase space via quasiprobability functions (namely, a P-function and a Wigner function), we study the impact of the laser and the exciton-photon detuning on the second order correlation function of the emitted photons. We determine the conditions under which the phenomena of bunching, giant bunching, and antibunching of the emitted light emerge. In particular, we predict the effect of a giant bunching for the case of a large exciton to photon population ratio. Within the domain of parameters supporting a bistability regime we demonstrate the effect of bunching of photons.

Appendices

Appendix 1: Derivation of basic equations based on the P-representation method

Using the generalized P-representation and using the c-numbers instead of operators we turn from master equation (2) to the Fokker–Planck equation for the P-function [12]:

$$\begin{aligned}&{\frac{\partial P}{\partial t} =\left[ -\frac{\partial }{\partial \phi } \left( -\left( i\varDelta _\mathrm{ph} +\gamma _\mathrm{ph} \right) \phi +\tilde{E}_\mathrm{d} -i\omega _\mathrm{R} \chi \right) -\frac{\partial }{\partial \chi } \left( -\left( -i\varDelta _\mathrm{ex} +\gamma _\mathrm{ex} \right) \chi -i\omega _\mathrm{R} \phi -2i\alpha \chi ^{+} \chi ^{2} \right) +\right. } \nonumber \\&\quad {\left. +\frac{\partial ^{2} }{\partial \chi ^{2} } \left( -i\alpha \chi ^{2} \right) + \mathrm{h.c.}\right] P,} \end{aligned}$$

(8)

where $\chi$ and $\phi$—are c-numbers. A solution of the Fokker–Planck equation (8) is obtained by the method of potentials in the adiabatic limit [12]:

$$\begin{aligned} \begin{array}{l} {P_\mathrm{ss} \left( \chi ,\chi ^{+} \right) =N\chi ^{-2-i\frac{\gamma }{\alpha } } \chi ^{+} {}^{-2+i\frac{\gamma ^{*} }{\alpha } }\exp \left( -\sigma \tilde{E}_\mathrm{d} \frac{1}{\chi } -\sigma ^{*} \tilde{E}_\mathrm{d}^{*} \frac{1}{\chi ^{+} } +2\chi \chi ^{+} \right) }\end{array}, \end{aligned}$$

(9)

where $\gamma =\gamma _\mathrm{ex} +\frac{\omega _\mathrm{R}^{2} \gamma _\mathrm{ph} }{\gamma _\mathrm{ph}^{2} +\varDelta _\mathrm{ph}^{2} } -i\left( \varDelta _\mathrm{ex} +\frac{\varDelta _\mathrm{ph} \omega _\mathrm{R}^{2} }{\gamma _\mathrm{ph}^{2} +\varDelta _\mathrm{ph}^{2} } \right)$, $\sigma =\omega _\mathrm{R} \frac{\gamma _\mathrm{ph} -i\varDelta _\mathrm{ph} }{\alpha \left( \gamma _\mathrm{ph}^{2} +\varDelta _\mathrm{ph}^{2} \right) }$ and N—is a normalization constant.

We use solution (9) to calculate the correlation functions of any order for excitons:

$$\begin{aligned}&{G^{\left( \mathrm{mn}\right) } =\left\langle \left( \chi ^{+} \right) ^{\mathrm{m}} \chi ^{\mathrm{n}} \right\rangle =\int \chi ^{+} {}^{m} \chi ^{n} P_\mathrm{ss}d\mu } \nonumber \\&\quad {=(-1)^{n+m}\frac{\left( \sigma \tilde{E}_{\mathrm{d}} \right) ^{\mathrm{n}} \left( \sigma ^{*} \tilde{E}_{\mathrm{d}}^{*} \right) ^{\mathrm{m}} \varGamma \left( {\mathrm{i}\gamma ^{*} \big / \alpha } \right) \varGamma \left( -{\mathrm{i}\gamma \big / \alpha } \right) }{\varGamma \left( \mathrm{m}+{\mathrm{i}\gamma ^{*} \big / \alpha } \right) \varGamma \left( \mathrm{n}-{\mathrm{i}\gamma \big / \alpha } \right) }\frac{{}_{0} \mathrm{F}_{2} \left( \mathrm{m}+{\mathrm{i}\gamma ^{*} \big / \alpha } ,\mathrm{n}-{\mathrm{i}\gamma \big / \alpha } ,2 \left| \sigma \right| ^{2} \left| \tilde{E}_{\mathrm{d}} \right| ^{2} \right) }{{}_{0} \mathrm{F}_{2} \left( {\mathrm{i}\gamma ^{*} \big / \alpha } ,-{\mathrm{i}\gamma \big / \alpha } ,2 \left| \sigma \right| ^{2} \left| \tilde{E}_{\mathrm{d}} \right| ^{2} \right) } },d \end{aligned}$$

(10)

where $\varGamma$ is a gamma function and ${}_{0} \mathrm{F}_{2}$ is a hypergeometric function.

The stochastic differential equations can be obtained in the Ito calculus by converting the Fokker–Planck equation (8) into the Ito form [12]:

$$\begin{aligned} \left\{ \begin{array}{l} {\frac{\partial }{\partial t} \phi =-\left( i\varDelta _\mathrm{ph} +\gamma _\mathrm{ph} \right) \phi +\tilde{E}_\mathrm{d} -i\omega _\mathrm{R} \chi ,} \\ {\frac{\partial }{\partial t} \phi ^{+} =-\left( -i\varDelta _\mathrm{ph} +\gamma _\mathrm{ph} \right) \phi ^{+} +\tilde{E}_\mathrm{d}^{*} +i\omega _\mathrm{R} \chi ^{+} ,} \\ {\frac{\partial }{\partial t} \chi =-\left( -i\varDelta _\mathrm{ex} +\gamma _\mathrm{ex} \right) \chi -i\omega _\mathrm{R} \phi -2i\alpha \chi ^{+} \chi ^{2}+\left( 1-i\right) \sqrt{\alpha } \chi \xi \left( t\right) ,} \\ {\frac{\partial }{\partial t} \chi ^{+} =-\left( i\varDelta _\mathrm{ex} +\gamma _\mathrm{ex} \right) \chi ^{+} +i\omega _\mathrm{R} \phi ^{+} +2i\alpha \chi ^{+2} \chi +\left( 1+i\right) \sqrt{\alpha } \chi ^{+} \xi ^{+} \left( t\right) ,} \end{array}\right. \end{aligned}$$

(11)

where $\xi \left( t\right)$ is an independent stochastic function, whose correlation functions satisfy the following relations: $\left\langle \xi \left( t\right) \right\rangle =0$, $\left\langle \xi ^{+} \left( t\right) \right\rangle =0$, $\left\langle \xi \left( t\right) \xi ^{+} \left( t'\right) \right\rangle =\delta \left( t-t'\right)$.

Appendix 2: Derivation of the Wigner function

We use the Wigner function for a visual presentation of the statistical properties of the exciton-polariton system based on the steady-state solution for the P-function (9). The Wigner function one can be expressed in terms of the P-representation as follow [28]:

$$\begin{aligned} \begin{array}{l} {W\left( \chi \right) =\frac{2}{\pi } \mathrm{e}^{-2\left| \chi \right| ^{2} } \int _{C_{x } }\int _{C_{x ^{+} } }P_\mathrm{ss}\left( x ,x ^{+} \right) \exp \left( 2\chi ^{*} x +2\chi x^{+} -2x ^{+} x \right) \mathrm{d} x^{+} \mathrm{d}x}\end{array}. \end{aligned}$$

(12)

Then substituting (9) in (12), changing variables in the integral as $x\rightarrow -E_\mathrm{d}\sigma /t$ and using the Schläfli’s integral [29]

$$\begin{aligned} J_{\nu } \left( z\right) =\frac{\left( z/2\right) ^{\nu } }{2\pi i} \int _{C}t^{-\nu -1} \exp \left[ t-\frac{z^{2} }{4t} \right] \mathrm{d}t, \end{aligned}$$

(13)

we obtain the following relation for the excitonic Wigner function:

$$\begin{aligned} W\left( \chi \right) =N'e^{-2\left| \chi \right| ^{2} } \left| \frac{J_{-i\frac{\gamma }{\alpha } -1} \left( \sqrt{8 \tilde{E}_\mathrm{d} \sigma \chi ^{*}} \right) }{\left( \chi ^{*}\right) ^{-\left( i\frac{\gamma }{\alpha } +1\right) /2} } \right| ^{2}, \end{aligned}$$

(14)

where $N'$ is a normalization constant, defined by the following expression:

$$\begin{aligned} N'=\frac{2}{\pi \left| \left( 2 \tilde{E}_\mathrm{d} \sigma \right) ^{-i\frac{\gamma }{\alpha } -1} \right| } \frac{\varGamma \left( -i\frac{\gamma }{\alpha } \right) \varGamma \left( i\frac{\gamma ^{*} }{\alpha } \right) }{{}_{0} \mathrm{F}_{2} \left( {\mathrm{i}\gamma ^{*} \big / \alpha } ,-{\mathrm{i}\gamma \big / \alpha } ,2\left| \sigma \tilde{E}_\mathrm{d} \right| ^{2} \right) }. \end{aligned}$$

(15)

Here we use the following expansion of the power series of the Bessel function [30]

$$\begin{aligned} \begin{array}{l} J_{\nu }(\sqrt{t})=\left( \frac{t}{4}\right) ^{\nu /2}\sum \limits _{k=0}^{\infty }\frac{(-1)^{k}(t/4)^{k}}{k!\varGamma (k+\nu +1)}\end{array}. \end{aligned}$$

(16)

According to (14) the Wigner function is always positive and equals the squared modulus of the Bessel function with the complex index $-i\gamma /\alpha -1$. The Wigner function of photons is obtained from the governing principle (3):

$$\begin{aligned} \begin{array}{l} {W\left( \phi \right) =N''\exp \left( {-2\frac{\left( \gamma _\mathrm{ph} \mathrm{Im}(\phi ) +\varDelta _\mathrm{ph} \mathrm{Re}(\phi ) \right) ^{2} +\left( \gamma _\mathrm{ph} \mathrm{Re}(\phi ) -\varDelta _\mathrm{ph} \mathrm{Im}(\phi ) -\tilde{E}_\mathrm{d} \right) ^{2} }{\omega _\mathrm{R}^{2} } }\right) } {\left| \frac{J_{-i\frac{\gamma }{\alpha } -1} \left( \sqrt{8\tilde{E}_\mathrm{d} \sigma \frac{\left( i\tilde{E}_\mathrm{d} +\phi ^*\left( \varDelta _\mathrm{ph} -i\gamma _\mathrm{ph} \right) \right) }{\omega _\mathrm{R} } } \right) }{\left( \frac{\left( i\tilde{E}_\mathrm{d} +\phi ^*\left( \varDelta _\mathrm{ph} -i\gamma _\mathrm{ph} \right) \right) }{\omega _\mathrm{R} } \right) ^{\left( -i\frac{\gamma }{\alpha } -1\right) /2} } \right| ^{2}}\end{array}, \end{aligned}$$

(17)

where we introduced new normalized constant $N''=\frac{\omega _\mathrm{R}^{2} }{\varDelta _\mathrm{ph}^{2} +\gamma _\mathrm{ph}^{2} } N'.$