Multiple transparency windows and Fano interferences induced by dipole-dipole couplings

We investigate the optical properties of a two-level system (TLS) coupled to a one-dimensional array of $N$ other TLSs with dipole-dipole coupling between the first neighbors. The first TLS is probed by a weak field, and we assume that it has a decay rate much greater than the decay rates of the other TLSs. For $N=1$ and in the limit of a Rabi frequency of a probe field much smaller than the dipole-dipole coupling, the optical response of the first TLS, i.e., its absorption and dispersion, is equivalent to that of a three-level atomic system in the configuration which allows one to observe the electromagnetically induced transparency (EIT) phenomenon. Thus, here we investigate an induced transparency phenomenon where the dipole-dipole coupling plays the same role as the control field in EIT in three-level atoms. We describe this physical phenomenon, named a dipole-induced transparency (DIT), and investigate how it scales with the number of coupled TLSs. In particular, we have shown that the number of TLSs coupled to the main TLS is exactly equal to the number of transparency windows. The ideas presented here are very general and can be implemented in different physical systems, such as an array of superconducting qubits, or an array of quantum dots, spin chains, optical lattices, etc.

Figure 1

Pictorial representation of the system: (a) $1+N$ coupled two-level systems (TLSs) and (b) a driven cavity mode coupled to $1+N$ TLSs.

Figure 2

(a) Normalized absorption (black solid line) and dispersion (red dashed line) of the first TLS when coupled to a second one as a function of ${\mathrm{\Delta }}_p/{\gamma }_0$. The black dotted line represents the absorption when $d_0=0$. (b) Imaginary part of the third-order optical susceptibility for two different systems: two TLSs coupled by dipole-dipole interaction (solid line) and a usual three-level system where EIT phenomenon can be observed (dashed line). (c) Real part of the first-order and (d) real part of the third-order optical susceptibility functions for two TLSs coupled by dipole-dipole interaction (solid line) and usual $\mathrm{\Lambda }$ system (dashed line). In (c) the curves overlap perfectly. Parameters used: ${\mathrm{\Omega }}_p=0.03{\gamma }_0,d_0=0.5{\gamma }_0,{\gamma }_1={10}^{-3}{\gamma }_0$.

Figure 3

Normalized absorption $\text{Im}{\langle a\rangle }_{ss}$ (black solid line) and dispersion $\text{Re}{\langle a\rangle }_{ss}$ (red dashed line) of the cavity mode when coupled to two-coupled TLSs as a function of the normalized detuning ${\mathrm{\Delta }}_p/\kappa$. Parameters used: ${\gamma }_0=\kappa ,g=5\kappa ,d=3\kappa ,\left|\epsilon \right|=0.03\kappa$, and ${\gamma }_1={10}^{-3}\kappa$. The black dotted lines represent the absorption when there is no TLS coupled to the cavity mode ($g=0$).

Figure 4

(Left) First eigenenergies as a function of $d/{\gamma }_0$. (Right) Transition rates from the first excited states to the ground state also as a function of $d/{\gamma }_0$. In all these plots we have fixed $d_0=0.5{\gamma }_0$ and $N=2$ in panels (a) and (b), and $N=4$ in panels (c) and (d).

Figure 5

Normalized absorption (black solid line) and dispersion (red dashed line) of the first TLS when coupled to $N=4$ TLSs as a function of ${\mathrm{\Delta }}_p/{\gamma }_0$. The parameters used here are ${\mathrm{\Omega }}_p=0.03{\gamma }_0,{\gamma }_i=\gamma ={10}^{-3}{\gamma }_0$, and $d_0=0.5{\gamma }_0$. The values of the $d$ coupling are (a) $d=d_0/\sqrt{2}$ and (b) $d=2.5{\gamma }_0$. The black dotted lines represent the absorption at $d_0=0$. In (c) and (d) we show the normalized absorption (black solid line) and dispersion (red dashed line) of the cavity mode when coupled to five TLSs (i.e., $N=4$) as a function of ${\mathrm{\Delta }}_p/\kappa$. The parameters used here are $\left|\epsilon \right|=0.03\kappa ,{\gamma }_i=\gamma ={10}^{-3}\kappa$, and (c) $d=0.4\kappa ,g=\sqrt{2}d$, and ${\gamma }_0={10}^{-3}\kappa$; and (d) $d=3.0\kappa ,g=2.0\kappa$, and ${\gamma }_0={10}^{-3}\kappa$. The black dotted lines represent the absorption for $g=0$.

Figure 6

Normalized absorption $\text{Im}{\langle {\sigma }_{+}^0\rangle }_{ss}$ as a function of the normalized detuning ${\mathrm{\Delta }}_p/{\gamma }_0$ for different numbers of TLSs coupled to the main one ($N=7,10,12,$ and 15). The parameters used here were ${\gamma }_0=1,{\mathrm{\Omega }}_p=0.03{\gamma }_0,{\gamma }_i=\gamma ={10}^{-3}{\gamma }_0,d_0=0.5{\gamma }_0$, and $d=d_0/\sqrt{2}$.

Figure 7

Normalized transmission $\text{Im}{\langle a\rangle }_{ss}$ as a function of the normalized detuning ${\mathrm{\Delta }}_p/\kappa$ for $1+N$ TLSs coupled to the cavity mode (for $N=7,10,12,$ and 15). The parameters used here are ${\gamma }_0={\gamma }_i={10}^{-3}\kappa ,\left|\epsilon \right|=0.03\kappa ,d=1.0\kappa$, and $g=\sqrt{2}d$.