Ultrafast double-quantum-coherence spectroscopy of excitons with entangled photons

We calculate the four-wave-mixing signal of excitons generated at ${\mathit{k}}_4={\mathit{k}}_1+{\mathit{k}}_2-{\mathit{k}}_3$ by two pulsed entangled photon pairs (${\mathit{k}}_1$, ${\mathit{k}}_2$) and (${\mathit{k}}_3$, ${\mathit{k}}_4$), where all four modes are chronologically ordered. Entangled photons offer an unusual combination of bandwidths and temporal resolution not possible by classical beams. Contributions from different resonances can be selected by varying the parameters of the photon wave function. The signal scales linearly rather than quadratically with the laser field intensity, which allows performance of the measurements at low powers.

Figure 1

(a) Level scheme of the excitonic model system made of three coupled two-level systems. (b) Absorption spectrum. The model parameters are ${\varepsilon }_1=1.518$ eV, ${\varepsilon }_2=1.530$ eV, and ${\varepsilon }_3=1.526$ eV, $J_{12}=10$ meV, $J_{13}=2$ meV, and $J_{23}=3$ meV, and $V_1=V_2=V_3$.

Figure 2

Pulse configuration for the double-quantum-coherence signal.

Figure 3

Pathways contributing to the double-quantum-coherence signal at ${\mathit{k}}_4={\mathit{k}}_1+{\mathit{k}}_2-{\mathit{k}}_3$ (for diagram rules, see [34] and [35]). $s_i$ are the time intervals along the loop, whereas $t$, $t_2$, $t-{\tau }_3$, and $t_2-{\tau }_1$ are real physical time variables.

Figure 4

(a) Entangled photon correlation function $\langle E_1(t_1)E(t_2)\rangle$ for a pulsed entangled photon pair, Eq. (19) (${\tau }_{12}=0$, $T_{12}=40$ fs, $\sigma =20$ fs). (b) Analog plot for two classical impulsive pulses. (c) Fourier transform of (a).

Figure 5

Two-dimensional signal, Eq. (25) (absolute value), showing correlation plots for $T_{12}=T_{34}=20$ fs, ${\Omega }_P=3.0621$ eV, with different pump durations $\sigma$ as indicated. The bottom plot is multiplied by a factor of 6.

Figure 6

Two-dimensional signal, Eq. (26) (absolute value), showing correlation plots for $T_{12}=T_{34}=20$ fs, ${\Omega }_P=3.0621$ eV, with different pump durations $\sigma$. The bottom plot is multiplied by a factor of 100.

Figure 7

Two-dimensional signal, Eq. (25) (absolute value), showing correlation plots for $T_{34}=20$ fs, ${\Omega }_P=3.11$ eV, $\sigma =1/(20\hspace{0.5em}\mathrm{fs})$, and different values of $T_{12}$ as indicated. Plots are multiplied by a factor of 1 (top), 3.5 (middle), and 6.0 (bottom).

Figure 8

Two-dimensional signal, Eq. (26) (absolute value), showing correlation plots for $T_{34}=20$ fs, ${\Omega }_P=3.11$ eV, $\sigma =20\hspace{0.5em}\mathrm{fs}$, and $T_{12}$ values as indicated. Plots are multiplied by a factor of (from top to bottom) 1, 1.3, 2, and 3.0.