Generalized Gross-Pitaevskii model for intersubband polariton lasing

We develop a generalized Gross-Pitaevskii approach to the driven-dissipative dynamics of intersubband polaritons in patterned planar microcavities where the cavity mode is strongly coupled to the intersubband transitions in doped quantum wells. Substantial differences with respect to the case of interband excitonic polaritons are highlighted, in particular, the non-Markovian features of the radiative decay. We show that the method is able to reproduce well the experimental observations, as validated on the linear reflection properties of the cavity. The theoretical framework is then applied in the nonlinear regime to study optical parametric oscillation processes for intersubband polaritons. Our findings open interesting perspectives in view of novel coherent laser sources operating in the mid and far infrared.

Figure 1

Sketch of the microcavity device under consideration. A series of quantum wells (dark blue) is embedded in a planar microcavity device enclosed by a pair of metallic mirrors (yellow). The top mirror is laterally patterned into a grating to ensure radiative coupling of the cavity modes with the external radiation. The device is operated by shining a pump beam of electric field $E_{\mathrm{inc}}$ at an angle to its surface. The reflected field $E_{\mathrm{refl}}$ is dominated by the specular reflection of the pump, but nonlinear phenomena can give rise to other spectral components.

Figure 2

Sketch of the Bragg scattering-mediated coupling of cavity modes to external radiation. An incident photon (full circle) of in-plane momentum $k_p$ and frequency ${\omega }_p$ must lie within the radiative region above the light cone (LC) indicated by the full line. The incident light wavevector changes by multiples of the Bragg wave number $k_B$ as a consequence of the scattering on the grating, thus allowing to resonantly couple to the cavity mode (open circle).

Figure 3

Color plots of the frequency- and momentum-dependent reflectivity $R(\mathbf{k},\omega )$ for empty cavities, i.e., ${\mathrm{\Omega }}_R=0$. The Bragg grating consists of a square wave potential of period $a=4.26\mu \mathrm{m}$ with filling factor 75% (top, comparable to the device used in Ref. [28]) and 25% (bottom). The remaining simulation parameters are reported in Appendix. Roman numerals label the order of each band. The different visibility of the bands (especially evident in the area highlighted by the red rectangles) is the sole consequence of the different filling factor. The diagonal solid line indicates the light-cone condition: the white region below it in the bottom-right direction is inaccessible to reflection experiments.

Figure 4

Reflectivity $R(\mathbf{k},\omega )$ of cavity modes strongly coupled to an intersubband transition. The simulation parameters are reported in Appendix. (Top) Data produced using the form (6) and (7) of radiative loss term, which is exact under monochromatic pumping and can only be used in the linear regime. (Bottom) Data produced using the approximate form (10) of the radiative loss term, which can be used with nonmonochromatic pumps while preserving time locality and is suitable to study nonlinear phenomena. The dots indicate the pump, signal, and idler modes of the OPO discussed in Sec. 4a.

Figure 5

Optical parametric oscillation under continuous-wave pumping at $k_p/(k_B/2)\approx 0.28,{\omega }_p/{\omega }_{\mathrm{ISB}}=1.1$ and the simulation parameters of Appendix. The positions of the pump, signal and idler are marked by dots in Fig. 4. The pump intensity and nonlinear coupling are tuned such that $g|{\psi }_X{|}^2=23.1\mathrm{meV}$. All intensities are normalised to the intensity of the pump mode. (a) Steady state (in practice computed after $250\mathrm{ps}$ of evolution) $k$-space momentum distribution of the emitted radiation. Besides the signal and idler peaks, one sees further replicas due to higher order diffraction from the grating. (b) Momentum-selected (i.e., angle-selected) emitted radiation spectrum in the frequency domain, ${\mathcal{I}}_{\{p,s,i\}}\left(\omega \right)=I(k_{\{p,s,i\}},\omega )$ (c) Intensities of pump, signal and idler emission in real time, $I_{\{p,s,i\}}\left(t\right)\equiv I_{\mathrm{refl}}(k_{\{p,s,i\}},t)$

Figure 6

Emission spectra with continuous wave pump $[\omega /{\omega }_{\mathrm{ISB}}=1.065,k/(k_B/2)\approx 0.28]$ and $150\mathrm{fs}$ (FWHM) Gaussian probe pulses $\left(\omega /{\omega }_{\mathrm{ISB}}=0.89,k=0\right)$. The pump intensity $I_0$ is varied below the OPO threshold at $I_0^{\mathrm{thr}}\approx 0.22$ in these units. The probe intensity is taken very weak at ${10}^{-10}$. (a) Normal $\left(k=0\right)$ reflectivity. The LP absorption dip initially blue-shifts, then develops an amplification peak. The dashed curve is the probe pulse spectrum, normalised to one. For clarity, each curve is offset by 0.5 from the previous one. (b) Idler spectrum at $k/(k_B/2)\approx 0.56$. The idler intensity in the time domain was normalised to the peak probe intensity. For clarity, each curve is offset by 0.25 from the previous one. Notice that the idler vanishes for weak pump intensities.

Figure 7

Emission spectrum with pulsed pump of peak intensity $I_0=0.2$ in the units used in Fig. 5, $\omega /{\omega }_{\mathrm{ISB}}=1.065,2k/k_B\approx 0.28)$ and 150-fs Gaussian pulse probe $\left(\omega /{\omega }_{\mathrm{ISB}}=0.89,k=0,I_0={10}^{-5}\right)$. (a) The pump is a Gaussian packet of increasing duration. The pump and probe pulses are synchronised so that they reach their maximum intensity at the same time. For pump pulses longer than approximately $3\mathrm{ps}$, amplification is observed. (b) Effect of time delay between the probe and the pump pulse using under 5-ps pulsed pump. Curves are offset by 0.5 for clarity.