Solitons in cavity-QED arrays containing interacting qubits

We reveal the existence of polariton soliton solutions in the array of weakly coupled optical cavities, each containing an ensemble of interacting qubits. An effective, complex Ginzburg-Landau equation is derived in the continuum limit, taking into account the effects of cavity field dissipation and qubit dephasing. We have shown that an enhancement of the induced nonlinearity can be achieved by two orders of magnitude with negative interaction strength, which implies a large negative qubit-field detuning as well. Bright solitons are found to be supported under perturbations only in the upper (optical) branch of polaritons, for which the corresponding group velocity is controlled by tuning the interacting strength. With the help of perturbation theory for solitons, we also demonstrate that the group velocity of these polariton solitons is suppressed by the diffusion process.

Figure 1

Schematic for our proposed 1D qubit-cavity QED arrays, in which each cavity contains an ensemble of qubits with the interaction Hamiltonian among them described by the LMG model, ${\widehat{H}}_{\mathrm{LMG}}\propto {\widehat{J}}_z^2$.

Figure 2

(a) Two branches in the dispersion relation, labeled as upper ${\omega }_{+}$ and lower ${\omega }_{-}$ branches, are shown as a function of the wave number $k$ at the resonance condition $\Delta =0$, but for different qubit interaction strengths, $\eta =-1$, 0, and 1, respectively. (b) The carrier frequencies ${\omega }_{\pm }$ and (c) the group velocity $v_{\pm }$ for the wave packet are shown as a function of interaction strength $\eta$ under the condition $k=0.1$, $\gamma =0$, but with different frequency detunings, i.e., $\Delta =-2$, 0, and $+2$. Other parameters used are $\alpha =d=1$, $g=10$, and $n_{\mathrm{pol}}=0.01$.

Figure 3

Nonlinear coefficient $C_{\pm }$ of the reduced NLSE in Eq. (17) is shown for two frequency branches ${\omega }_{\pm }$, as a function of the qubit interaction strength $\eta$ for different frequency detunings $\Delta$. Inset shows a enlargement close to $\eta =0$. The parameters used are $k=0.1$, $\alpha =d=1$, $n_{\mathrm{pol}}=0.01,$ and $g=10$, respectively.

Figure 4

Polariton number density $n_{\mathrm{pol}}$ vs qubit interaction strength $\eta$ required for soliton formation taken for $\gamma =0.1$, $\eta =1$. Other parameters are the same as in Fig. 3.

Figure 5

Snapshots of bright soliton profile ${\left|\Psi \right|}^2$ as a function of effective coordinate $x_{\mathrm{eff}}=x-\zeta \left(\tau \right)$ at different times of propagation $\tau$. In the inset temporal behavior of soliton group velocity $v$ is plotted. The parameters are (cf. Refs. [23, 44]) $g=2\pi \times 1.7\mathrm{THz}$, $\Gamma =2\pi \times 12.1\mathrm{GHz}$, ${\gamma }_c=2\pi \times 50\mathrm{GHz}$, $\alpha =2\pi \times 0.75\mathrm{THz}$, $\eta n_{\mathrm{pol}}=2\pi \times 24.3\mathrm{GHz}$, $d=400\mathrm{nm}$, $k={10}^4{\mathrm{m}}^{-1}$, and $\delta =0$, respectively.