Boosting energy-time entanglement using coherent time-delayed feedback

The visibility of the two-photon interference in the Franson interferometer serves as a measure of the energy-time entanglement of the photons. We propose to control the visibility of the interference in the second-order coherence function by implementing a coherent time-delayed feedback mechanism. Simulating the non-Markovian dynamics within the matrix product state framework, we find that the visibility for two photons emitted from a three-level system (3LS) in ladder configuration can be enhanced significantly for a wide range of parameters by decelerating the decay of the upper level of the 3LS.

Figure 1

Franson interferometer with feedback channel. The setup allows to measure the second-order coherence function of two photons emitted from a three-level system (3LS) in ladder configuration. For more pronounced effects, the emitted photons are frequency filtered, which ensures that the first photon is detected by the first detector, $D_1$, and the second photon is detected by the second detector, $D_2$, if a signal is registered. The photons can reach the detectors either via short ($S$) or long paths ($L$) defined by beamsplitters and mirrors. For the $|a\rangle \leftrightarrow |b\rangle$ transition, a feedback mechanism at delay time ${\tau }_{\text{FB}}$ is implemented.

Figure 2

Tensor diagram illustrating the time evolution algorithm. There is a bin describing the three-level system (3LS, blue box) and four types of time bins describing the short path to the first detector (S-1, red box), the short path to the second detector (S-2, orange box), the long path to the first detector (L-1), and the long path to the second detector (L-2, black box). (a) The time bins containing the feedback signal are swapped next to the 3LS bin. (b) The feedback bins, the 3LS bin, and the current time bins are contracted with the stroboscopic time evolution operator (U, purple box). (c) The output bins are placed to the left of the bins in the feedback channel. (d) The procedure is repeated $N$ times. (e) After the time evolution is completed, the bins referring to the short paths are swapped $2N_T$ steps to the left and subjected to a phase shift to account for the delay time between short and long paths. (f) The procedure is repeated $N$ times.

Figure 3

Tensor diagram illustrating the evaluation of the two-time second-order coherence function $G^{\left(2\right)}(t_1,t_2)$, $t_1,t_2\in \{u_1,\cdots ,u_N\}$ where two nested loops are necessary to account for all possible values of $t_1$ and $t_2$. (a) Evaluation for $t_1=t_2=u_1$. (b) The time bins containing the signal registered at detector $D_1$, S-1 and L-1, for time step $u_1$ are exchanged with the corresponding time bins for the next time step $u_2$. (c) Evaluation of $G^{\left(2\right)}(u_1,u_2)$. The procedure is repeated until $G^{\left(2\right)}(u_1,u_k)$ has been evaluated for all $k\in \{1,\cdots ,N\}$. (d) Afterwards, we start the second loop through the MPS by calculating $G^{\left(2\right)}(u_2,u_1)$ and so on until we have completely determined $G^{\left(2\right)}(t_1,t_2)$.

Figure 4

Benchmark of the MPS solution for the second-order coherence function $G^{\left(2\right)}$ without feedback. The MPS solution is compared to the analytical solution (Ana.) for different values of the relative phase ${\varphi }_T$ between the short and the long paths to the detectors. Here ${\mathrm{\Gamma }}_{a}T=2.5$, ${\mathrm{\Gamma }}_{b}T=10$, $\mathrm{\Delta }t=0.05/{\mathrm{\Gamma }}_a$, ${\mathrm{\Gamma }}_a{t}_{\text{max}}=10$. (Inset) Oscillation of the height of the central peak with the delay phase ${\varphi }_T$.

Figure 5

Visibility of the $G^{\left(2\right)}$ interference for the Franson interferometer without feedback as a function of ${\mathrm{\Gamma }}_{a}T$ and ${\mathrm{\Gamma }}_{b}T$ calculated analytically.

Figure 6

Dynamics of the population of the upper level ($\langle {\sigma }_{aa}\rangle$) and the middle level ($\langle {\sigma }_{bb}\rangle$) of a 3LS with ${\mathrm{\Gamma }}_a={\mathrm{\Gamma }}_b$ without feedback (no FB) and with feedback at ${\mathrm{\Gamma }}_a{\tau }_{\text{FB}}=1$ (FB), $\mathrm{\Delta }t=0.05/{\mathrm{\Gamma }}_a$.

Figure 7

Second-order coherence function for a 3LS with ${\mathrm{\Gamma }}_{a}T={\mathrm{\Gamma }}_{b}T=4$ without feedback (no FB) and with feedback at ${\mathrm{\Gamma }}_a{\tau }_{\text{FB}}=1$ (FB), $\mathrm{\Delta }t=0.2/{\mathrm{\Gamma }}_a$, ${\mathrm{\Gamma }}_a{t}_{\text{max}}=20$. (Inset) Oscillation of the height of the central peak with the delay phase ${\varphi }_T$ without feedback (solid line) and with feedback (dashed line).

Figure 8

Comparison of the visibility of the $G^{\left(2\right)}$ interference as a function of ${\mathrm{\Gamma }}_{a}T$ and ${\mathrm{\Gamma }}_{b}T$ without feedback (no FB) and with feedback at ${\tau }_{\text{FB}}=T/4$ (FB), $\mathrm{\Delta }t=0.05T$. (Inset) Enhancement of the visibility of the $G^{\left(2\right)}$ interference with feedback ($V_{\text{FB}}$) in relation to the visibility without feedback ($V_{\text{no}\text{FB}}$) as a function of ${\mathrm{\Gamma }}_{a}T$ for ${\mathrm{\Gamma }}_b={\mathrm{\Gamma }}_a$.