Magnetic-field-induced cavity protection for intersubband polaritons

We analyze the effect of a strong perpendicular magnetic field on an intersubband transition in a disordered doped quantum well strongly coupled to an optical cavity. The magnetic field changes the lineshape of the intersubband optical transition due to the interface roughness of the quantum well from a Lorentzian to a Gaussian one. In this regime, a novel form of magnetic-field-induced cavity protection sets in, which strongly reduces the polariton linewidth to the cavity contribution only. Implications of our results for fundamental studies of nonlinear polariton dynamics and for technological applications to polariton lasers are finally highlighted.

Figure 1

(a) Sketch of the system. A semiconductor two-dimensional quantum well is inserted in a metallic cavity. The electric field of the cavity mode oscillates along the $z$ direction, perpendicular to the quantum well plane. The electronic intersubband transitions in the quantum well give rise to an electronic polarization directed along the same direction of the cavity electric field, realising a strong light-matter coupling regime between the cavity and the quantum well. [(b), (c)] Schematic view of the electronic subbands in the disordered quantum well in absence/presence of the external magnetic field perpendicular to the quantum well plane. The lowest optically-active excitations are transitions between the lowest two subbands due to the electrons below the Fermi energy $E_F$. Differently from the clean case, these transitions are not only fully vertical and each electron can jump to different eigenstates of the upper subband.

Figure 2

(a) Spatial profile of a single realization of the disorder potential $\delta U_n\left(\vec{r}\right)$ for the $n=1$ subband. (b) Fourier transform of the same disorder potential $\delta {\tilde{U}}_n\left(\vec{k}\right)$ as a function of the wavenumber $\vec{k}$. Disorder parameters: ${\eta }_{\mathrm{dis}}\approx 0.06$, ${\xi }_c/l_{\mathrm{qw}}=\sqrt{2}$. The numerical methods and the integration parameters are discussed in the Appendix pp3.

Figure 3

(a) Optical density ${\rho }_{\mathrm{qw}}\left(\omega \right)$ (blue-solid line) and Lorentzian fit (black-dashed line). (b) Optical density ${\rho }_{\mathrm{qw}}\left(\omega \right)$ calculated for increasing values of the Fermi energy $E_F$. Disorder parameters: ${\eta }_{\mathrm{dis}}\approx 0.06$, ${\xi }_c/l_{\mathrm{qw}}=\sqrt{2}$. In (a) the Fermi energy is fixed to $E_F/\hslash {\omega }_{\mathrm{qw}}\approx 0.6$, while in (b) a range of values is used as indicated in the legend.

Figure 4

(a) Plot of the FWHM ${\mathrm{\Gamma }}_{\mathrm{qw}}$ of the ISB optical density ${\rho }_{\mathrm{qw}}$ as a function of the disorder strength ${\eta }_{\mathrm{dis}}$ (blue-solid dots). (b) Plot of ${\mathrm{\Gamma }}_{\mathrm{qw}}$ as a function of the disorder correlation strength ${\xi }_c$. Disorder parameters: ${\xi }_c/l_{\mathrm{qw}}=0.5$ [in panel (a)], ${\eta }_{\mathrm{dis}}=0.4$ [in panel (b)]. Fermi energy $E_F/\hslash {\omega }_{\mathrm{qw}}\approx 0.5$. In both panels, the red-dashed line is the Unuma linewidth given by Eq. (23). To numerically implement Eq. (15) we used a finite-width delta function, with small linewidth ${\gamma }_{\delta }/{\omega }_{\mathrm{qw}}\approx 0.002$, which is then subtracted from the numerical data.

Figure 5

(a) Plots of the optical density ${\rho }_{\mathrm{qw}}\left(\omega \right)$ for increasing values of the magnetic field $B/B_0=1,4.5,7$ (red-solid lines). Blue lines show the same quantity in the absence of magnetic field. Disorder parameters: ${\eta }_{\mathrm{dis}}=2/15\approx 0.13$, ${\xi }_c/l_{\mathrm{qw}}=0.5$. Fermi energy $E_F/\hslash {\omega }_{\mathrm{qw}}=0.5$. For this choice, the quality factor of the intersubband transition in the absence of magnetic field is numerically found to be $Q_{\mathrm{qw}}\approx 40$. (b) Plots of the optical density ${\rho }_{\mathrm{qw}}\left(\omega \right)$ with fixed magnetic field $B/B_0=2.5$ for increasing values of the Fermi energy $E_F=0,{\omega }_B,2{\omega }_B$. Here the lowest Landau level is set to be at zero energy, such that the Fermi energy falls in the middle of the disorder-broadened ${\ell }_1=0,1,2$ Landau level. The vertical-dashed line highlights the asymmetry of the central Gaussian. Disorder parameters: ${\eta }_{\mathrm{dis}}=0.08$, ${\xi }_c/l_{\mathrm{qw}}=1$.

Figure 6

(a) Gaussian linewidth (FWHM) ${\mathrm{\Gamma }}_B$ of the central peak of the optical density as a function of the disorder strength ${\eta }_{\mathrm{dis}}$ for a constant value of the magnetic field $B$. The red squares/green dots are the numerical data corresponding to increasing values of $B$, that is $B/{\tilde{B}}_0=0.2$, $B/{\tilde{B}}_0=0.6$. The disorderless reference magnetic field is defined as ${\tilde{B}}_0=\hslash {\omega }_{\mathrm{qw}}/\left(2{\mu }_B^{*}\right)$. The red/green dashed lines are given by the Hikami linewidth ${\mathrm{\Gamma }}_{B,\mathrm{H}}/{\omega }_{\mathrm{qw}}$ while the black-solid line is the Unuma linewidth ${\mathrm{\Gamma }}_{\mathrm{qw},\mathrm{U}}/{\omega }_{\mathrm{qw}}$. Disorder parameters: ${\xi }_c/l_{\mathrm{qw}}=0.5$. (b) Gaussian linewidth ${\mathrm{\Gamma }}_B$ (purple dots) extracted from numerical simulations of the optical density as a function of the magnetic field $B$. The dashed-red line is given by the Hikami linewidth ${\mathrm{\Gamma }}_{B,\mathrm{H}}/{\omega }_{\mathrm{qw}}$, while the dashed-dotted line is given by the small disorder expansion of the Hikami linewidth ${\mathrm{\Gamma }}_{B,\mathrm{H}}^{\left(1\right)}$. Disorder parameters as in Fig. 5: ${\eta }_{\mathrm{dis}}=2/15\approx 0.13$ and ${\xi }_c/l_{\mathrm{qw}}=0.5$. (c) Same plot as in (b) but with different parameters in order to highlight the saturation of the linewidth at high magnetic field. The red-dashed line is given by the Hikami-linewidth ${\mathrm{\Gamma }}_{B,\mathrm{H}}/{\omega }_{\mathrm{qw}}$. The purple-dotted line is the saturation value of the Hikami linewidth given by ${\mathrm{\Gamma }}_{B,\mathrm{H}}^{\infty }/{\omega }_{\mathrm{qw}}\approx 0.094$. Disorder parameters: ${\eta }_{\mathrm{dis}}=1/30\approx 0.03$ and ${\xi }_c/l_{\mathrm{qw}}=4$. In all plots the Fermi energy is $E_F=1.5\hslash {\omega }_B$, in such a way that the central Gaussian is determined by only the two lowest Landau levels. To numerically implement Eq. (15) we used a finite-width delta function, with small linewidth ${\gamma }_{\delta }/{\omega }_{\mathrm{qw}}\approx 0.003$, which is subtracted from the numerical data.

Figure 7

Illustrative examples of the real-space wavefunction of a $n=1$ subband eigenstate in the presence of disorder in the $B=0$ (left) and $B/B_0=0.5$ (right) cases. These eigenstates are obtained via exact diagonalization, as detailed in Appendix pp3, of a square system with lateral size $L_{x,y}/l_{\mathrm{qw}}=14$. Other parameters: $E_F/\left(\hslash {\omega }_{\mathrm{qw}}\right)=0.5$, ${\eta }_{\mathrm{dis}}\approx 0.03$, ${\xi }_c/l_{\mathrm{qw}}\approx 4$.

Figure 8

Logarithmic colorplot of the cavity transmission as a function of the Rabi coupling ${\mathrm{\Omega }}_R$ and the external incident frequency $\omega$ for various values of the magnetic field $B$ in the resonant regime ${\omega }_c={\omega }_{\mathrm{qw}}$. Disorder parameters as in Fig. 5: ${\eta }_{\mathrm{dis}}=2/15\approx 0.13$, ${\xi }_c/l_{\mathrm{qw}}=0.5$. Fermi energy $E_F/\hslash {\omega }_{\mathrm{qw}}=0.5$.The ISB quality factor (without cavity) in the three panels is numerically found to be $Q_{\mathrm{qw}}\approx 40,12,10$, from left to right. An extreme value of the cavity quality factor $Q_c={10}^5$ is taken for illustrative purposes, more realistic values will be considered in the next figures. In physical units, for a QW of thickness $L_z=26$ nm and $m^{*}\approx 0.067m_e$, these adimensional parameters correspond to $\hslash {\omega }_{\mathrm{qw}}\approx 25\mathrm{meV}$ and a reference magnetic field $B_0\approx 1$ T. In the central panel we thus have $B\approx 4.4\mathrm{T}$ while in the right panel $B\approx 6.9\mathrm{T}$.

Figure 9

(a) Plot of the electronic contribution to the polariton linewidth estimated using Eq. (30) as a function of Rabi coupling ${\mathrm{\Omega }}_R$, left (right) parts corresponding to the upper (lower) polariton. The blue and red lines refer to the $B=0$ and $B=4.5B_0$ cases, respectively. (b) Three examples of the cavity transmission $T_c\left(\omega \right)$ given by Eq. (13) as a function of the incident frequency $\omega$ in the presence/absence of magnetic field (blue and red lines, respectively) for increasing values of the Rabi coupling ${\mathrm{\Omega }}_R$ (from left to right) as indicated by the dashed, dotted, and dashed-dotted lines in panel (a). (c) The quality factor of the lower (light) and upper (dark) polaritons $Q_{\pm }={\omega }_{\pm }/{\mathrm{\Gamma }}_{\pm }$ as a function of the Rabi frequency ${\mathrm{\Omega }}_R$. Blue and red lines refer to the cases in the absence/presence of magnetic field $B/B_0=0,4.5$. The black dotted and black dashed-dotted lines indicate the usual upper/lower polariton quality factors obtained averaging between cavity and ISB linewidths, $Q_{\pm ,\mathrm{std}}=2({\omega }_c\pm {\mathrm{\Omega }}_R/2)/({\gamma }_c+{\mathrm{\Gamma }}_{\mathrm{qw}})$. The light/dark red dashed lines indicate the upper bound to the polariton quality factor set by the cavity losses as defined in Eq. (32). Same system parameters as in Fig. 8 except for the realistic value $Q_c=150$ of the cavity quality factor.

Figure 10

[(a), (b)] Logscale color plot of the cavity transmission $T_c\left(\omega \right)$ as a function of the magnetic field $B/B_0$ and the external incident frequency $\omega /{\omega }_{\mathrm{qw}}$. In (a) ${\mathrm{\Omega }}_R/{\omega }_{\mathrm{qw}}=0.15$ and in (b) ${\mathrm{\Omega }}_R/{\omega }_{\mathrm{qw}}=0.4$. (c) Quality factor of the upper polariton as a function of the magnetic field $B/B_0$. The green and red solid lines refer to the $\mathrm{\Omega }/{\omega }_{\mathrm{qw}}=0.15$ and 0.4 values used in panels (a) and (b), respectively. Parameters for all plots: ${\omega }_c={\omega }_{\mathrm{qw}}$, ${\eta }_{\mathrm{dis}}=0.1$, ${\xi }_c/l_{\mathrm{qw}}=1$, Fermi energy $E_F/\hslash {\omega }_{\mathrm{qw}}=0.5$. For these parameters, the ISB quality factor (without cavity) is found to be $Q_{\mathrm{qw}}\approx 22$, while the cavity quality factor is set to the realistic value $Q_c=200$.

Figure 11

Logscale contour-plot of the cavity-protection magnetic field $B_{\mathrm{cp}}$ as defined in (37) as a function of the intersubband frequency ${\omega }_{\mathrm{qw}}$ and the disorder correlation length ${\xi }_c$. Here, $B_{\mathrm{cp}}$ is given in units of Tesla, $\hslash {\omega }_{\mathrm{qw}}$ is given in meV, and ${\xi }_c$ in nm. The disorder amplitude is kept constant to ${\xi }_0=0.3\mathrm{nm}$. The effective Bohr magneton corresponding to the effective mass of electrons in GaAs is used, ${\mu }_B^{*}\approx 0.86\mathrm{meV}/\mathrm{T}$. The purple star indicates the position of the setup inspired by [18] described in the text.

Figure 12

Logscale contour plot of the cavity-protection magnetic field $B_{\mathrm{cp}}^{\left(1\right)}$ as predicted by Eq. (38) as a function of the intersubband frequency $\hslash {\omega }_{\mathrm{qw}}$ and the QW quality factor $Q_{\mathrm{qw}}$ (in logscale). Here, $B_{\mathrm{cp}}^{\left(1\right)}$ is given in units of Tesla and $\hslash {\omega }_{\mathrm{qw}}$ in meV. The red hexagon and green circle mark the position of, respectively, typical THz [7, 52, 53] and MIR [11] devices.