Optical Kerr nonlinearity in quantum-well microcavities: From polariton to dipolariton

In this paper, we investigate the nonlinear optical properties and photon correlations in a cavity containing quantum wells and interacting with a third-order nonlinear medium. More precisely, we study the interaction of the photonic Kerr nonlinearity with polariton and dipolariton quasiparticles in thermal environments. The optical bistability, the intensity spectrum, and the squeezing spectrum of the transmitted radiation are analyzed. It is shown that the regime of the bistability is reached in the dipolariton cavity faster than attained by the polariton system. Furthermore, the increase of the coupling with the wells reduces the squeezing. More interestingly, it turns out that the additional interactions arising in the dipolariton cavity, between direct and indirect excitons, slow down the variation of the squeezing. As a consequence, dipolaritons offer a greater margin of squeezed light than polaritons, and also than the ${\chi }^{\left(3\right)}$ medium in the absence of quantum wells. Moreover, the dipolariton system is found to be much more robust against the bath temperature in the weak-coupling regime.

Figure 1

Schematic diagram of the polariton cavity. The quantum well is placed in the maximum of the electromagnetic field between a set of Bragg mirrors ensuring high reflectivity. The cavity is driven by a coherent field. A Kerr medium is attached to the cavity and the resulting photons from the nonlinear process are directly injected into the cavity.

Figure 2

Photonic intensities in (a) the strong-coupling regime ($\mathrm{\Omega }=2, \gamma =0.05, \epsilon =100$, and $\kappa =0.1$), and (b) for the weak coupling ($\mathrm{\Omega }=0.008, \gamma =0.05, \epsilon =100$, and $\kappa =0.1$) as a function of the detuning for various nonlinearity coefficients $\alpha$.

Figure 3

(a) Photonic intensity as a function of the coherent pump amplitude $\epsilon$ for various dissipation rates ($\mathrm{\Omega }=0.008, \alpha ={10}^{-6}$, and $\mathrm{\Delta }=0.1$). (b) Photonic intensity as a function of the square of the coherent pump amplitude $\epsilon$ for $\mathrm{\Omega }=0.008, \alpha ={10}^{-6}, \mathrm{\Delta }=0.2, \gamma =0.02$, and $\kappa =0.04$: optical bistability.

Figure 4

Variation of the noise spectrum vs the frequency detuning for various bath temperature values for (a) strong coupling ($\alpha =0, \mathrm{\Omega }=2, \gamma =0.05, \kappa =0.1$, and $\epsilon =100$) and (b) weak coupling ($\alpha =0, \mathrm{\Omega }=0.008, \gamma =0.05, \kappa =0.1$, and $\epsilon =100$).

Figure 5

The zero-temperature dependence of the noise spectrum on the frequency detuning for various nonlinearity coefficients $\alpha$ (a) in the strong-coupling regime ($\mathrm{\Omega }=2$) and (b) in the weak-coupling regime ($\mathrm{\Omega }=0.008$). (c) Density plot of the noise spectrum as a function of the detuning and the coupling constant $\mathrm{\Omega }$ for $\alpha ={10}^{-7}$. In all plots, the other parameters are chosen to be $\gamma =0.05, \epsilon =100$, and $\kappa =0.1$.

Figure 6

Variation of the noise spectrum of the polariton cavity as a function of the photonic nonlinearity $\alpha$ and the temperature of the thermal bath in the (a), (b) strong-coupling regime ($\mathrm{\Omega }=2, \gamma =0.05, \mathrm{\Delta }=1.05, \epsilon =100$, and $\kappa =0.1$) and (c) for weak coupling ($\mathrm{\Omega }=0.008, \gamma =0.05, \mathrm{\Delta }=0.08, \epsilon =100$, and $\kappa =0.1$).

Figure 7

Intensity power spectrum of the output field in the strong-coupling regime as a function of (a) the detuning and (b) the frequency for $\alpha ={10}^{-8}, \mathrm{\Omega }=2, \gamma =0.05, \kappa =0.1, \epsilon =100$, and $n_{\text{th}}=0.5$.

Figure 8

Schematic diagram of the dipolariton cavity. The two quantum wells are placed between a highly reflecting set of Bragg mirrors. The wells are coupled via a tunnel effect with a tunneling rate $J$. The cavity is driven by a coherent field. A Kerr medium is attached to the cavity and the resulting photons from the nonlinear process are directly injected into the cavity.

Figure 9

The real parts of the system eigenvalues represented as a function of the decay rate $\gamma$ for $\epsilon =100, \mathrm{\Omega }=J=2, \alpha ={10}^{-6}$, and $\mathrm{\Delta }=0.1$.

Figure 10

Photonic intensity in (a) the strong-coupling regime vs the detuning for $\mathrm{\Omega }=J=2, \gamma =0.05$, and $\kappa =0.1$, and for (b) the weak coupling vs the detuning for $\mathrm{\Omega }=J=0.008, \gamma =0.05$, and $\kappa =0.1$. Both representations are calculated for several nonlinearity coefficients. (c) and (d) are three-dimensional (3D) plots of the photonic intensities vs the tunneling rate $J$ and the detuning in the strong- and weak-coupling regimes, respectively, for $\alpha ={10}^{-8}$ [the other parameters are the same as (a) and (b)].

Figure 11

(a) Photonic intensity as a function of the coherent pump amplitude $\epsilon$ for various damping rates ($\mathrm{\Omega }=0.008, \alpha ={10}^{-6}, \mathrm{\Delta }=0.1$, and $\epsilon =100$). (b) Comparison of the photonic intensities between the polariton and the dipolariton cavity before the regime of the optical bistability for $\gamma =0.06$. The other parameters are the same as (a). (c) Photonic intensity as a function of the square of the coherent pump amplitude $\epsilon$ for $\mathrm{\Omega }=0.008, \alpha ={10}^{-6}, \gamma =0.02, \kappa =0.04$, and three detunings $\mathrm{\Delta }$: optical bistability.

Figure 12

Variation of the spectrum of noise as a function of the frequency detuning for various bath temperature values in the strong-coupling regime. The parameters are $\alpha =0, \mathrm{\Omega }=J=2, \gamma =0.05, \epsilon =100$, and $\kappa =0.1$.

Figure 13

Squeezing spectrum as a function of the detuning and the nonlinearity coefficient for some values of $\alpha$ [(a), (b)], and a continuous variation of $\alpha$ [(c), (d)]. (a) and (c) represent the strong-coupling regime for $\mathrm{\Omega }=J=2$. (b) and (d) represent the weak- coupling regime for $\mathrm{\Omega }=J=0.008$. (e) Density plot of the noise spectrum as a function of the detuning and the coupling constant $\mathrm{\Omega }$ for $J=0.3$ and $\alpha ={10}^{-7}$. In all representations, $\gamma =0.05, \epsilon =100$, and $\kappa =0.1$.

Figure 14

3D and density plots of the squeezing spectrum vs the tunneling rate $J$ in the strong-coupling regime [(a), (b)] for $\mathrm{\Omega }=2$, and for the weak coupling [(c), (d)] for $\mathrm{\Omega }=0.008$. In all representations, the other parameters are $\gamma =0.05, \kappa =0.1$, and $\alpha ={10}^{-8}$.

Figure 15

Variation of the noise spectrum of the dipolariton cavity (and the polariton cavity) as a function of the photonic nonlinearity $\alpha$ and the temperature of the thermal baths in the strong-coupling regime (a) and (b) [$\mathrm{\Omega }=J=2, \mathrm{\Delta }=1.05$ (polariton), and $\mathrm{\Delta }=0.1$ (dipolariton)]. (c) Same conditions as (b) and for higher values of nonlinearity. The three systems are represented together: solid line (dipolariton), dashed line (polariton), and dotted line (no quantum well with $\mathrm{\Delta }=0.1$). (d) The weak-coupling regime ($\mathrm{\Omega }=J=0.008$ and $\mathrm{\Delta }=0.15$). In all representations $\gamma =0.05, \kappa =0.1$, and $\epsilon =100$.

Figure 16

Intensity power spectrum of the output field in the strong-coupling regime plotted against (a) the detuning and (b) the frequency, for $\alpha ={10}^{-8}, \mathrm{\Omega }=J=2, \gamma =0.05, \epsilon =100, \kappa =0.1$, and $n_{dx}=n_{ix}=0.5$.