Theory of two-photon processes in quantum dots: Coherent evolution and phonon-induced dephasing

The paper discusses coherent control of charge and spin states of a biexciton system in a quantum dot via coherent two-photon transitions. Rabi oscillations between the ground state of a quantum dot and the biexciton state, as well as oscillations between the two single-exciton states induced by laser pulses with different circular or linear polarizations are studied. The effect of phonon-induced decoherence on these processes is described. System properties and driving conditions that lead to optimal coherent control are identified. It is shown that proper optimization allows one to control the two-qubit biexciton system via two-photon transitions with a high fidelity.

Figure 1

(Color online) The energy-level structure of a confined biexciton and the schematic presentation of the optical couplings between the states for (a) the biexciton Rabi oscillations and for (b) the exciton spin flip.

Figure 2

(Color online) The spectral branches representing the instantaneous eigenstates of the unperturbed Hamiltonian $H_{\text{XX}}+H_{\text{las}}^{\left(\text{lin}\right)}$ as functions of the pulse amplitude $f$ (the adiabatic parameter) for (a) $\Delta =E_{\text{B}}/2$ and for (b) $\Delta =E_{\text{B}}/3$. In both cases, $E_{\text{B}}<0$.

Figure 3

(Color online) (a) Two-photon polarization flip. The occupation of the state $|{\sigma }_{-}⟩$ as a function of the pulse area for $\Delta =E_{\text{B}}/2=-2\text{meV}$ (red solid line), $\Delta =E_{\text{B}}/3$ (green dashed line), and $\Delta =E_{\text{B}}/6$ (blue dotted line). (b) Two-photon Rabi oscillations (Ref. 20). The occupation of the biexciton state as a function of the pulse area for $\Delta =E_{\text{B}}/2=-2$ (red solid line), $-1.9$ (green dashed line), and $-1.7\text{meV}$ (blue dotted line). In both figures, $E_{\text{B}}=-4\text{meV}$ and ${\tau }_0=5\text{ps}$.

Figure 4

(Color online) Two-photon rotation of the linear polarization of an exciton state with a circularly polarized pulse. The occupation of the $|\text{Y}⟩$ state as a function of the pulse area for $E_{\text{B}}=-4\text{meV}$ and two values of $\Delta$: (a) $-3$ and (b) $+2\text{meV}$. Red solid lines: ${\tau }_0=5\text{ps}$; green dashed lines: ${\tau }_0=10\text{ps}$. The initial state is $|\text{X}⟩$.

Figure 5

(Color online) The two-photon Rabi oscillations in the presence of phonon-induced perturbation. (a) The dependence of the damping on temperature for ${\tau }_0=5\text{ps}$: $T=0$ (red solid line), 10 (green dashed line), and 40 K (blue dotted line). The gray dash-dotted line shows the joint occupation of the single-exciton states due to real transitions at $T=40\text{K}$. (b) The dependence on the pulse duration at $T=4\text{K}$: ${\tau }_0=5$ (red solid line), 20 (green dashed line), and 40 ps (blue dotted line).

Figure 6

(Color online) The spectral density of the phonon reservoir at $T=4\text{K}$ (thick gray lines on both plots, arbitrarily scaled) and the spectral functions (a) $S_1$ and (b) $S_2$. Red solid lines: $E_{\text{B}}=-4\text{meV},{\tau }_0=5\text{ps}$; green dashed lines: $E_{\text{B}}=-4\text{meV},{\tau }_0=10\text{ps}$; and blue dotted lines: $E_{\text{B}}=-6\text{meV},{\tau }_0=5\text{ps}$. Here $\Delta =E_{\text{B}}/2$ always.

Figure 7

(Color online) (a) The value of $A_2/\left|\Delta \right|$ for $E_{\text{B}}=-2$ (red solid line), $-6$ (green dashed line), and $-8\text{meV}$ (blue dotted line). Here $\Delta =E_{\text{B}}/2$ always. (b) The first maximum of the biexciton occupation for $\Delta =E_{\text{B}}/2=-2\text{meV}$, $T=4\text{K}$, and pulse durations ${\tau }_0$ as indicated. The gray line is added to guide the eyes through the maxima of the plotted curves.

Figure 8

(Color online) The exciton spin oscillations in the presence of phonon-induced perturbation for $E_{\text{B}}=2\Delta =-4\text{meV}$. (a) The dependence of the damping on temperature for ${\tau }_0=5\text{ps}$: $T=0$ (red solid line), 10 (green dashed line), and 40 K (blue dotted line). The gray dash-dotted line shows the joint occupation of the single-exciton states due to real transitions at $T=0\text{K}$. (b) The dependence on the pulse duration at $T=4\text{K}$: ${\tau }_0=5$ (red solid line), 20 (green dashed line) and 40 ps (blue dotted line).

Figure 9

(Color online) The spectral density of the phonon reservoir at $T=4\text{K}$ (thick gray lines on both plots, arbitrarily scaled) and the spectral functions (a) $S_1$ and (b) $S_2,S_3$ for $\Delta =-2$ (red solid line), $-3$ (green dashed line), and $+2\text{meV}$ (blue dotted line). In all the cases, $E_{\text{B}}=-4\text{meV}$ and ${\tau }_0=5\text{ps}$.

Figure 10

(Color online) (a) The diagram of energy levels and laser frequencies in the case of $\Delta >0$ and $E_{\text{B}}<0$. (b) The two-photon exciton spin oscillations in the presence of phonon-induced perturbation for $\Delta >0$ and $E_{\text{B}}<0$ for ${\tau }_0=5\text{ps}$: $T=0$ (red solid line), 10 (green dashed line), and 40 K (blue dotted line).

Figure 11

(Color online) Dephasing of two-photon processes in a system with $E_{\text{B}}>0$ for ${\tau }_0=5\text{ps}$: (a) two-photon Rabi oscillations, starting from the ground state and (b) two-photon spin flip, starting from the $|{\sigma }_{+}⟩$ state. Red solid line: $T=4\text{K}$; green dashed line: $T=10\text{K}$; and blue dotted line: $T=40\text{K}$.

Figure 12

(Color online) Dephasing of two-photon polarization rotation for ${\tau }_0=5\text{ps}$. (a) $E_{\text{B}}=-4\text{meV}$; red solid line: $\Delta =-3\text{meV},T=4\text{K}$; green dashed line: $\Delta =2\text{meV},T=4\text{K}$; and blue dotted line: $\Delta =2\text{meV},T=40\text{K}$. (b) $E_{\text{B}}=4\text{meV},\Delta =3\text{meV}$; red solid line: $T=4\text{K}$; green dashed line: $T=10\text{K}$; and blue dotted line: $T=40\text{K}$.

Figure 13

(Color online) Rotation of the linear polarization of an exciton state with a circularly polarized pulse with the fine-structure splittings of (a) $\delta =30$ and (b) $100\mu \text{eV}$. The occupation of the $|\text{Y}⟩$ state as a function of the pulse area for $E_{\text{B}}=-4\text{meV}$ and $\Delta =-3\text{meV}$; pulse durations are 10 (red solid lines), 20 (green dashed lines), and 40 ps (blue dotted lines).