Phase-sensitive Kerr nonlinearity for closed-loop quantum systems

The third-order susceptibility is investigated in a five-level atomic system in which the laser beams couple the ground state to a four-level closed-loop system. It is found that under the condition of the multiphoton resonance, one can enhance the Kerr nonlinearity of such a medium by properly adjusting the amplitudes and phases of the applied fields. In this case, the linear and nonlinear absorption reduce considerably in a region with a positive group velocity. It is demonstrated that the third-order susceptibility is very sensitive to the relative phase of the applied fields. An analytical model is presented to elucidate such phase control of the Kerr nonlinearity. A comparison is also made between the Kerr-nonlinear indices for the five-, four-, and three-level systems. It is realized that the magnitude of the Kerr nonlinearity for the five-level system is larger than that of the three- and four-level counterparts. Finally, it is shown that effect of Doppler broadening can lead to an enhanced Kerr nonlinearity while maintaining linear and nonlinear absorption.

Figure 1

Schematic diagram of the (a) five-, (b) four-, and (c) three-level quantum systems. (d)–(f) General atom-field states in the new basis.

Figure 2

Linear and nonlinear susceptibility as well as group index versus probe field detuning. (a) Linear absorption (dashed line) and linear dispersion (solid line), (b) nonlinear absorption (dashed line) and Kerr nonlinearity (solid line), and (c) group index. The parameters are ${\gamma }_{14}=0.8\gamma ,{\gamma }_{12}=0.1\gamma ,{\gamma }_{23}=0.1\gamma ,{\gamma }_{43}=0.4\gamma ,{\gamma }_{35}=0.02\gamma ,{\Omega }_{21}={\Omega }_{32}={\Omega }_{43}={\Omega }_{41}=\gamma ,{\Delta }_{43}={\Delta }_{23}={\Delta }_{14}={\Delta }_{12}=0$, and $\varphi =0$. All the parameters are scaled with $\gamma$.

Figure 3

Linear and nonlinear susceptibility as well as group index versus probe field detuning. (a) Linear absorption (dashed line) and linear dispersion (solid line), (b) nonlinear absorption (dashed line) and Kerr nonlinearity (solid line), and (c) group index. The parameters are fields ${\Omega }_{21}=2\gamma ,{\Omega }_{41}=1.1\gamma ,{\Omega }_{32}=\gamma$, and ${\Omega }_{43}=1.9\gamma$. The other parameters are the same as in Fig. 2.

Figure 4

Transmission coefficient versus probe field detuning for (a) ${\Omega }_{21}={\Omega }_{32}={\Omega }_{43}={\Omega }_{41}=\gamma$ and (b) ${\Omega }_{21}=2\gamma ,{\Omega }_{41}=1.1\gamma ,{\Omega }_{32}=\gamma$, and ${\Omega }_{43}=1.9\gamma$. The other parameters are the same as in Fig. 2.

Figure 5

Kerr-nonlinear indices in the case of the five-level KR5 system (solid line), the four-level cascade system (dashed line), and the three-level cascade system (dotted line). The parameters are the same as in Fig. 3, except those for the four-level case where ${\Omega }_{21}={\Omega }_{32}=0$, whereas for the three-level case one has ${\Omega }_{21}={\Omega }_{32}={\Omega }_{41}=0$.

Figure 6

(a) and (c) Linear and (b) and (d) nonlinear susceptibility versus relative phase $\varphi$ for (a) ${\Delta }_{43}={\Delta }_{23}={\Delta }_{14}={\Delta }_{12}=0$ and (b) ${\Delta }_{43}={\Delta }_{23}=2\gamma$ and ${\Delta }_{14}={\Delta }_{12}=0$. The parameters are ${\Omega }_{21}=5\gamma ,{\Omega }_{41}=3\gamma ,{\Omega }_{32}=\gamma$, and ${\Omega }_{43}=2\gamma$. The other parameters are the same as in Fig. 2.

Figure 7

Phase control of (a), (c), and (e) linear and (b), (d), and (f) nonlinear susceptibility for (a) and (b) $\varphi =0$, (c) and (d) $\varphi =\pi /2$, and (e) and (f) $\varphi =\pi$. Here ${\Omega }_{21}={\Omega }_{32}={\Omega }_{43}={\Omega }_{41}=2\gamma$ and the other parameters are the same as in Fig. 2.

Figure 8

Three-dimensional plot of phase control of (a) linear and (b) nonlinear susceptibility Here the parameters are the same as in Fig. 6.

Figure 9

Switching process of the (a) linear absorption, (b) linear dispersion, and (c) Kerr nonlinearity for different values of ${\Delta }_{43}$ and ${\Delta }_{23}$. Here ${\Delta }_p=0.1\gamma ,\varphi =\pi$, and the other parameters are the same as in Fig. 5.

Figure 10

Switching process of the Kerr nonlinearity for different values of $\varphi$. Here ${\Delta }_{43}={\Delta }_{23}=\gamma$ and the other parameters are the same as in Fig. 5.

Figure 11

(a) Linear and (b) nonlinear susceptibility probe field detuning ${\Delta }_p$ including the Doppler broadening $kw=0.1\gamma$. The parameters are the same as in Fig. 2.