Optically mediated coherent population trapping in asymmetric semiconductor quantum wells

A quantum well (QW), with only three transitions, which are all dipole allowed because of the QW asymmetry, is investigated when driven coherently by two infrared optical fields. One field couples the $\mid 1⟩-\mid 2⟩$ and $\mid 2⟩-\mid 3⟩$ transitions simultaneously, while a second field couples the $\mid 1⟩-\mid 3⟩$ transition. The absorption linear response is computed allowing the illustration of some features due to quantum coherence. Without the second field, the absorption of a weak probe is that of a strongly saturated system. As the coupling strength of the second field is increased the absorption strength of a weak probe increases, by over an order of magnitude factor in absorption strength compared to the case in the absence of the second field. The results are shown to be due to an optically mediated, coherent population trapping effect.

Figure 1

Three level energy schemes for interaction with one or more resonant coupling fields (derived from Ref. 10). (a) Cascade (ladder) (b) lambda, and (c) V schemes.

Figure 2

Schematic of the energy level arrangement for the asymmetric quantum wells considered in this paper. Subband levels are labeled in ascending energy: ∣1⟩, ∣2⟩, and ∣3⟩. There are three possible optical transitions (frequencies): $\mid 1⟩-\mid 2⟩$ $\left({\omega }_{21}\right)$, $\mid 2⟩-\mid 3⟩$ $\left({\omega }_{32}\right)$, and $\mid 1⟩-\mid 3⟩$ $\left({\omega }_{31}\right)$. Three coupling fields with optical frequencies ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ couple separately to the transitions $\mid 1⟩-\mid 2⟩$, $\mid 2⟩-\mid 3⟩$, and $\mid 1⟩-\mid 3⟩$, respectively. The coupling field detunings from the transition frequencies are defined as ${\Delta }_1={\omega }_{21}-{\omega }_1$, ${\Delta }_2={\omega }_{32}-{\omega }_2$, and ${\Delta }_3={\omega }_{31}-{\omega }_3$. (b) Schematic of the asymmetric conduction band barrier arrangement to create an asymmetric quantum well in which all transitions are dipole allowed.

Figure 3

Time evolution of the QW electron populations for all three levels ∣1⟩ (solid line), ∣2⟩ (dotted line), and ∣3⟩ (dashed line) in the presence of three driving optical fields, each possessing Rabi frequencies ${\Omega }_1={\Omega }_2={\Omega }_3=5\mathrm{meV}$ from $t=0$ (when the fields were switched on). (a) Under resonant driving conditions, ${\Delta }_1={\Delta }_2={\Delta }_3=0$ and (b) under a detuned driving configuration, ${\Delta }_1={\Delta }_2=0$ and ${\Delta }_3=10\mathrm{meV}$.

Figure 4

Linear absorption response of an asymmetric quantum well (transition frequencies: ${\omega }_{21}={\omega }_{32}=125\mathrm{mev}$ and ${\omega }_{31}=250\mathrm{meV}$) in the frequency range of both the $\mid 1⟩-\mid 2⟩$ and $\mid 2⟩-\mid 3⟩$ transitions under the resonant coupling field configuration of ${\omega }_1={\omega }_2=125\mathrm{meV}$ and ${\omega }_3=250\mathrm{meV}$ (the detunings are all zero). (a) ${\Omega }_1={\Omega }_2={\Omega }_3=0$; Lorentzian linear absorption response of the transitions $\mid 1⟩-\mid 2⟩$ and $\mid 2⟩-\mid 3⟩$. (b) Dotted line: ${\Omega }_1={\Omega }_2=5\mathrm{meV}$ and ${\Omega }_3=0$. Solid line ${\Omega }_1={\Omega }_2={\Omega }_3=5\mathrm{meV}$. In all cases ${\gamma }_i=1\mathrm{meV}$ $(i=1,2,3)$ and ${\gamma }_{12}={\gamma }_{23}={\gamma }_{13}=5\mathrm{meV}$.

Figure 5

(Color online) Three-dimensional plot [for the parameters in Fig. 4b solid line] depicting the evolution of the linear absorption spectrum of both the $\mid 1⟩-\mid 2⟩$ and $\mid 2⟩-\mid 3⟩$ transitions as a function of ${\Omega }_3$ (${\Omega }_1$ and ${\Omega }_2$ are held fixed at $5\mathrm{meV}$).

Figure 6

Steady state populations ${\rho }_{ii}\left(\infty \right)$ of levels ∣1⟩, ∣2⟩, and ∣3⟩ (${\rho }_{11}\left(\infty \right)$, ${\rho }_{22}\left(\infty \right)$, and ${\rho }_{33}\left(\infty \right)$, respectively, as a function of coupling field Rabi frequency ${\Omega }_3$. The parameters for the asymmetric quantum well transition frequencies and coupling field configuration is given in the caption of Fig. 2.

Figure 7

Comparison of the linear absorption response for the driven system of Fig. 4 for two different regimes of the level ∣3⟩ population decay rate. Solid line: ${\gamma }_1={\gamma }_3=1\mathrm{meV}$. Dotted line: ${\gamma }_1={\gamma }_3=2.5\mathrm{meV}$.

Figure 8

Schematics of the dressed state pictures for a triply driven three level QW by three strong driving fields. (a) ${\Omega }_1={\Omega }_2=5\mathrm{meV}$ and ${\Omega }_3=0$. We obtain symmetric splitting of the three original eigenstates, ∣1⟩, ∣2⟩, and ∣3⟩. Each level has now developed into a triplet of dressed energy eigenstates. The energy separation of these triplet states is equal: $\mid i,a⟩\to \mid i,c⟩=5\sqrt{2}\mathrm{meV}$ and $\mid i,c⟩\to \mid i,b⟩=5\sqrt{2}\mathrm{meV}$, where $i$ is the original eigenstate label, 1, 2, or 3. There are five spectral contributions to the absorption spectrum, denoted by the absorption routes (solid arrows) and gain routes (dotted arrows) (i)–(v) (for $i=1,2$). (b) ${\Omega }_1={\Omega }_2={\Omega }_3=5\mathrm{meV}$. Now the triplet reduces to an asymmetric doublet (as the two dressed states are degenerate). The energy separations of the states are now: $\mid i,a⟩\to \mid i,b⟩=0$ and $\mid i,a⟩\to \mid i,c⟩$ and $\mid i,b⟩\to \mid i,c⟩=15\mathrm{meV}$. There are three spectral contributions to the absorption spectrum, denoted by the absorption routes (solid arrows) and gain routes (dotted arrows) (i)–(iii) (for $i=1,2$). There is an identical set of absorption/gain contributions for $i=2,3$.