Optically tunable bound states in the continuum

We demonstrate the existence of tunable bound states in the continuum (BICs) in a one-dimensional quantum wire with two impurities induced by an intense monochromatic radiation field. We find that there is an interesting type of BIC due to the Fano interference between two optical transition channels, in addition to the ordinary BIC due to geometrical interference between electron wave functions emitted by impurities. In both cases the BIC can be achieved by tuning the frequency of the radiation field. Evidence of the BIC can be obtained by observing the absorption rate of a probe photon.

Figure 1

(a) Energy levels of donor and acceptor atoms. (b) Transitions among energy levels.

Figure 2

Absolute value of the imaginary part of eigenvalues of the Hamiltonian ($p$ sector) as a function of $\Omega +E_l$. The parameters are $T_1=0.2$, $g=0.2$, $E_u=0.1$, and $x_D=2$. Hereafter we use $B/2=1$ as the energy unit. The solid line corresponds to $T_2=0.2$, while the dashed line corresponds to $T_2=0$. The curves on the upper left corner correspond to another solution of Eq. (24). For $T_2=0.2$ (solid line) there is a static BIC at $\Omega +E_l=0.4$ that is independent of the strength of the interaction $g$. In addition there is a dynamic BIC at $\Omega +E_l=-0.2$ that is due to the interaction with the Fano interference. The $\Omega +E_l$ values for which the BIC occurs in the plot are consistent with Eqs. (37) and (38), respectively. When $T_2=0$ (dashed line) the Fano interference is suppressed, so only the first BIC occurs.

Figure 3

Absolute value of the imaginary part of eigenvalues of the Hamiltonian ($p$ sector) as a function of $\Omega +E_l$. The parameters are the same as in Fig. 2 except for $g=0.4$. The solid line corresponds to $T_2=0.2$, while the dashed line corresponds to $T_2=0$. The static BIC that occurs due to the vanishing of the self-energy still occurs at $\Omega +E_l=0.4$, while the dynamic BIC is shifted to $\Omega +E_l=-0.659$ due to a change of $g$.

Figure 4

Absolute value of the imaginary part of eigenvalues of the Hamiltonian ($p$ sector) as a function of $\Omega +E_l$. The parameters are the same as in Fig. 2 except for $x_D=4$. The solid line corresponds to $T_2=0.2$, while the dashed line corresponds to $T_2=0$. There are two BICs at $\Omega +E_l=1.38$ and $0.4$, where the self-energy vanishes. There is another BIC at $\Omega +E_l=-0.2$, for which the self-energy does not vanish.

Figure 5

Absolute value of the imaginary part of eigenvalues of the Hamiltonian ($p$ sector) as a function of $\Omega +E_l$. The parameters are the same as in Fig. 4 except for $g=0.4$. The solid line corresponds to $T_2=0.2$, while the dashed line corresponds to $T_2=0$. There are two static BICs at $\Omega +E_l=1.38$ and $0.4$, while the dynamic BIC is shifted to $\Omega +E_l=-0.659$ due to a change of $g$.

Figure 6

State scheme for pump and probe photons. Depending on the value of $\Omega$ the level $E_l+\Omega$ may be below or above the level $E_u$. The latter case is shown here.

Figure 7

The solid line corresponds to the absorption rate for a pump frequency given by $E_l+\Omega =1.38$, which allows the existence of a static BIC. The dashed line corresponds to the absorption rate for $E_l+\Omega =1.37$, slightly off the static BIC value. When the pump frequency is slightly off the static BIC frequency, the absorption rate (dashed line) takes a sharp profile, which in this case occurs near $\omega -\Omega =0.04$. The sharp profile shape is suppressed when $E_l+\Omega$ is exactly at the BIC value (solid line). For this and the following graph the parameters are $b=2$, $g=0.4$, $x_D=4$, ${\tilde{T}}_2=0.2$, ${\tilde{T}}_1=0.2$, and $E_u=0.1$. We use units where $\hslash =1$. The units of frequency and absorption rate are then the same as the energy unit, $B/2=1$.

Figure 8

The solid line corresponds to the absorption rate for a pump frequency given by $E_l+\Omega =-0.65$, which allows the existence of a dynamic BIC. The absorption rate shows no sharp shape. The dashed line corresponds to the absorption rate for $E_l+\Omega =-0.64$, slightly off the dynamic BIC value. The absorption rate then takes a sharp shape, which in this case occurs near $\omega -\Omega =-0.5$.