Density-matrix theory of the optical dynamics and transport in quantum cascade structures: The role of coherence

The impact of coherence on the nonlinear optical response and stationary transport is studied in quantum cascade laser structures. Nonequilibrium effects such as the pump-probe signals, the spatiotemporally resolved electron density evolution, and the subband population dynamics (Rabi flopping) as well as the stationary current characteristics are investigated within a microscopic density-matrix approach. Focusing on the stationary current and the recently observed gain oscillations, it is found that the inclusion of coherence leads to observable coherent effects in opposite parameter regimes regarding the relation between the level broadening and the tunnel coupling across the main injection barrier. This shows that coherence plays a complementary role in stationary transport and nonlinear optical dynamics in the sense that it leads to measurable effects in opposite regimes. For this reason, a fully coherent consideration of such nonequilibrium structures is necessary to describe the combined optical and transport properties.

Figure 1

(Color online) Band structure of the THz QCL from Ref. 37 for the injection barrier width $b=5.5\text{nm}$ under resonance condition. The subbands involved in the laser transition $\left(4,1^{\prime }\right)$ as well as the injector subbands $\left(5,2^{\prime }\right)$ are marked.

Figure 2

(Color online) Gaussian smoothening of the subband occupations $n_{i\mathbf{k}}$ for the two main subbands (injector 5, upper laser state 1) at $T=10\text{K}$: (a) $b=4.5\text{nm}$; (b) $b=11.0\text{nm}$.

Figure 3

(Color online) (a) Injector (5) and upper laser state $\left(1^{\prime }\right)$ showing an anticrossing at the main tunneling barrier at exact resonance of the corresponding Wannier states. Lower laser state (4) is also shown. (b) Tunnel splitting energy $\Delta E=\left|E_5-E_{1^{\prime }}\right|$ for varying bias and different barrier widths $b$ determining the resonance bias at the center of the anticrossing and the tunnel coupling $2\Omega \approx \Delta E$.

Figure 4

(Color online) Population relaxation dynamics using the full set of dynamical equations for a barrier width of $b=5.5\text{nm}$ at $T=10$ and 50 K, determining the stationary nonequilibrium of the system. Inset: Short-time evolution of the subband populations.

Figure 5

(Color online) Tunnel coupling $2\Omega$ and mean broadening $\Gamma =\left({\Gamma }_5+{\Gamma }_{1^{\prime }}\right)/2$ of the injector and the upper laser state for different barrier widths $b$ and at $T=10$ and 50 K (see remark in Ref. 55).

Figure 6

(Color online) Stationary current for both the WSH and the full calculation at resonance for varying barrier widths at $T=10$ and 50 K (see remark in Ref. 55).

Figure 7

(Color online) Dependence of the absorption spectrum on the injection barrier width at $T=10\text{K}$.

Figure 8

(Color online) Pump-probe signals after ultrafast strong pumping and subsequent linear probing at the laser energy for different delay times $t_d$ and barrier widths at $T=10$ and 50 K. Both pump and probe pulses are chosen to be resonant on the laser transition. Insets: Pump-probe signals for small delay times.

Figure 9

(Color online) (a),(c) Spectral function and (b),(d) the imaginary part of the lesser Green’s function for two different barrier widths at $T=10\text{K}$.

Figure 10

(Color online) Spatiotemporal evolution of the electron density of the QCL after ultrafast nonlinear optical excitation at $T=10\text{K}$. The pump pulse parameters are the same as for the pump-probe results.

Figure 11

(Color online) Optically induced population dynamics $n_i\left(t\right)$ for the barrier widths $b=4.5$ and 7.0 nm at $T=10\text{K}$, showing coherent Rabi oscillations as well as subsequent scattering-induced relaxation back into the steady state. The pump pulse parameters are the same as for the pump-probe results (the pump pulse is shown by the dashed line in the figures).