Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities

We study how an oscillating mirror affects the electromagnetically induced transparency (EIT) of an atomic ensemble confined in a gas cell, which is placed inside a microcavity with an oscillating mirror in one end. The cavity field acts as a probe of the EIT system and also produces a light pressure on the mirror. The back-action from the mirror to the cavity field results in several (from one to five) steady-states for this atom-assisted optomechanical cavity, producing a complex structure in its EIT. The multiequilibrium positions of the oscillating mirror have been calculated, and then the susceptibility is studied with respect to the different equilibrium positions. We find that the EIT of the atomic ensemble can be significantly changed by the oscillating mirror, and also the various steady states of the mirror have different effects on the EIT.

Figure 1

Schematic diagram for the atom-assisted optomechanical system considered here. There are three main components: (i) an optical cavity with one fixed mirror; (ii) another oscillating mirror, which is modeled as a quantum-mechanical harmonic oscillator (denoted by a black spring); and (iii) an ensemble of identical three-level atoms, which are confined inside the cavity. Each atom is assumed to have $\Lambda$-type transitions. Here, ${\Delta }_p={\omega }_a-{\omega }_0+{\omega }_0\langle x\rangle /l$ and ${\Delta }_c={\omega }_a-{\omega }_c-\nu$.

Figure 2

Schematic diagram for the functions of the left $Y_{\mathrm{L}}({\Delta }_{p,s})$ and the right $Y_{\mathrm{R}}({\Delta }_{p,s})$ hand sides of Eq. (28) versus ${\Delta }_{p,s}$ in (a) for a large value of ${\Delta }_0$ and (b) for ${\Delta }_0$ in the region of (a) indicated by the red dashed box. In (b), the brown triangle, red crosses, green dot and blue squares denote four different steady state solutions of the self-consistent equation (28). The brown triangle (far right) denotes the solution ${\Delta }_{p,s}^{(1)}$, the four blue squares (top left) denote the solution ${\Delta }_{p,s}^{(2)}$, the red dots on the curve denotes the solution ${\Delta }_{p,s}^{(3)}$, and the green dot denotes the solution ${\Delta }_{p,s}^{(4)}$. The solution ${\Delta }_{p,s}^{(1)}$ is not shown in the regions $c$ and $d$. Here, ${\Delta }_{\mathrm{th}}^{(i)}$ ($i=\mathit{ab},\mathit{bc},\mathit{cd}$) denote the threshold values of ${\Delta }_0$ that separate regions with different number of solutions. Namely, ${\Delta }_{\mathrm{th}}^{(\mathit{ab})}$ is the boundary point of the regions $a$ and $b$, ${\Delta }_{\mathrm{th}}^{(\mathit{bc})}$ is the boundary point of the regions $b$ and $c$, and ${\Delta }_{\mathrm{th}}^{(\mathit{cd})}$ is the boundary point of the regions $c$ and $d$.

Figure 3

The steady-state solutions ${\Delta }_{p,s}^{(i)}$ ($i=1,2,3,4$) versus the atom-cavity detunings ${\Delta }_0$ (when the two mirrors are fixed) for given parameters, e.g., $\kappa ={10}^2$, ${\Delta }_c=0$, ${\omega }_a={10}^6$, ${\gamma }_0={10}^{-6}$, ${\gamma }_2={10}^{-4}$, $g={10}^2$, $\Omega =2$, and $a_{\mathrm{in}}=10$. Hereafter, all the quantities are measured in units of ${\gamma }_1$. Note that here eight parameters determine the system. Recall that ${\Delta }_{p,\mathrm{s}}$ is the steady-state atom-cavity detuning when one mirror is movable. The inset shows the steady-state solution ${\Delta }_{p,s}^{(1)}$, which corresponds to a very large displacement of the mirror. As schematically shown in Fig. 2b, the dotted blue, dashed red, and continuous green lines denote the solutions ${\Delta }_{p,s}^{(2)}$, ${\Delta }_{p,s}^{(3)}$, and ${\Delta }_{p,s}^{(4)}$, respectively. For example, the green cross in Fig. 2b corresponds to a single value of ${\Delta }_0$. Here, ${\Delta }_0$ is swept, and the green cross in Fig. 2b becomes a continuous curve. Same for one red dot and one blue square in Fig. 2b.

Figure 4

Steady-state positions $\langle x\rangle {}_s^{(i)}$$(i=1,2,3,4)$ for the movable mirror versus the atom-cavity detuning ${\Delta }_0$, for the same parameters listed in Fig. 3. Here, the continuous black, dotted blue, dashed red, continuous green lines denote $\langle x\rangle {}_s^{(1)}$, $\langle x\rangle {}_s^{(2)}$, $\langle x\rangle {}_s^{(3)}$, and $\langle x\rangle {}_s^{(4)}$, when the atom-cavity detunings ${\Delta }_{p,s}$ take the following steady-state values ${\Delta }_{p,s}^{(1)}$, ${\Delta }_{p,s}^{(2)}$, ${\Delta }_{p,\mathrm{s}}^{(3)}$, and ${\Delta }_{p,s}^{(4)}$, respectively, as shown in Fig. 3.

Figure 5

Steady-state solutions ${\Delta }_{p(s)}^{(i)}$ ($i=1,2,3,4$) schematically shown in Fig. 2b versus ${\Delta }_0$ with $\kappa ={10}^4$. The other parameters here are the same as in Fig. 3. The inset shows the steady state solution ${\Delta }_{p(s)}^{(1)}$ that corresponds to a very large displacement of the mirror. As shown in Fig. 2b, the dotted blue, dashed red, and continuous green lines denote the solutions ${\Delta }_{p(s)}^{(2)}$, ${\Delta }_{p(s)}^{(3)}$, and ${\Delta }_{p(s)}^{(4)}$, respectively.

Figure 6

Steady-positions $\langle x\rangle {}_s^{(i)}$ $(i=1,2,3,4)$ of the movable mirror versus the atom-cavity detuning ${\Delta }_0$, with the same parameters as in Fig. 5. Here, the continuous black, dotted blue, dashed red, continuous green lines denote $\langle x\rangle {}_s^{(1)}$, $\langle x\rangle {}_s^{(2)}$, $\langle x\rangle {}_s^{(3)}$, and $\langle x\rangle {}_s^{(4)}$, when the atom-cavity detunings ${\Delta }_{p(s)}$ (for a moving mirror) takes the following steady-state values: ${\Delta }_{p,s}^{(1)}$, ${\Delta }_{p(s)}^{(2)}$, ${\Delta }_{p,s}^{(3)}$, and ${\Delta }_{p,s}^{(4)}$, respectively, as shown in Fig. 5.

Figure 7

The susceptibilities ${\chi }_1$ and ${\chi }_2$ for the detuning between the atom and the control field ${\Delta }_c=0$ and $\kappa ={10}^2$. We plot only two susceptibilities because there are three stable solutions in this parameter region but two of them are quite similar. Here, the dotted and solid curves correspond to the imaginary and real parts, respectively. The blue and the green colors correspond to ${\chi }_1$ and the dark red and brown colors correspond to ${\chi }_2$, respectively. The EIT-like region is located between the two peaks of Im${\chi }_1$ and Im${\chi }_2$.

Figure 8

The susceptibilities ${\chi }_1$ and ${\chi }_2$ for ${\Delta }_c=2$ and $\kappa ={10}^2$. Here, the dotted and solid curves correspond to the imaginary and real parts, respectively. The blue and the green colors correspond to ${\chi }_1$ and the dark red and brown colors correspond to ${\chi }_2$, respectively. The EIT-like region is located between the two peaks of Im${\chi }_1$ and Im${\chi }_2$.

Figure 9

(a) The susceptibilities ${\chi }_1$ and ${\chi }_2$ for 7${\Delta }_c=0$ and $\kappa ={10}^6$. (b) The magnified figure of (a) for small ${\Delta }_0.$ Here, the dotted and solid curves correspond to the imaginary and real parts, respectively. The blue and the green colors correspond to ${\chi }_1$ and the dark red and brown colors correspond to ${\chi }_2$, respectively. It is shown in these two figures that the oscillating mirror’s influence on EIT can also be observed if the detector’s resolving power is high enough.

Figure 10

The susceptibilities for ${\Delta }_c=0$ and $\kappa =1.6\times {10}^{10}$. Note that this value of the elastic constant $\kappa$ is huge, corresponding to an almost fixed mirror. In this case, the Re${\chi }_1$ and Im${\chi }_1$ occur for mostly positive values of the atom-cavity detuning. When the spring constant becomes softer, as in Fig. 7, the Re${\chi }_1$ and Im${\chi }_1$ have a response that extend over a huge range of values of the atom-cavity detuning ${\Delta }_0$, even for ${\Delta }_c\gtrsim -6$. When $\kappa$ tends to infinity, the susceptibility becomes the same as that in the usual EIT phenomenon. This consistency check is reassuring for our calculations. Here, the dotted and solid curves correspond to the imaginary and real parts, respectively. The blue and the green colors correspond to ${\chi }_1$ and the dark red and brown colors correspond to ${\chi }_2$, respectively. The EIT-like region occurs near zero detuning ${\Delta }_0$.