Role of chiral symmetry in a kicked Jaynes-Cummings model

We have studied the role of chiral symmetry in a periodically kicked Jaynes–Cummings (KJC) model by freezing an initial phase. We show that commensurate kicks ($2\pi k$ periodicity with integer $k$) conserve the chiral symmetry in the KJC model under the resonant condition, while incommensurate kicks break the symmetry. The chiral symmetry preserves the phase of an initial state against phase fluctuations during the dynamical evolution, but broken chiral symmetry erases the initial phase. The frozen phase is preserved within a finite evolution time for slight deviations of the kick period from an integer multiple of $2\pi$ and small variations of detuning from the resonant condition. The chiral symmetry-protected phase information is noteworthy as it provides various uses in quantum computation and information.

Figure 1

Schematic diagrams of driven Jaynes–Cummings models. The upper panel shows a coupled system with a two-level atom and an optical cavity, and the lower panel depicts a coupled system with a circuit QED and transmission line. The external driving potential $K\left(t\right)$ can be sinusoidal oscillation or discrete kicks.

Figure 2

Eigenphases ${\varepsilon }_{\alpha }$ of the Floquet operator $U(T_{-},0_{-})$ vs the atom–cavity coupling strength $g$ with the strength of the driven potential $\xi =1.5$ and the period of the kicks, (a) $T=6$ and (b) $T=2\pi$.

Figure 3

(a) Absolute value and (b) phase of amplitude distribution coefficients $C_n^{\eta }$ and $D_n^{\eta }$ with respect to the basis $|n,\eta \rangle$ with eigenstate index $\alpha =6$, atom–cavity coupling strength $g=1.7$, driven potential strength $\xi =1.5$, and period $T=2\pi$. In both panels, the blue circle line and red star line indicate the absolute value and phase of $C_n^{\eta }$ and $D_n^{\eta }$, respectively. In (b), the yellow dotted line indicates the summation of the phases of $C_n^{\eta }$ and $D_n^{\eta }$, which gives the stable global phase ${\theta }_g=-0.2605$.

Figure 4

(a) and (b) Amplitudes and (c) and (d) phases of the distribution function ${\mathcal{P}}_n^{\eta }\left(M\right)$ of the final state with respect to evolution time $MT$ and the basis $|n,\eta \rangle$ with initial phase ${\theta }_0=0.5$ and initial state $n_0=12$. Other parameters are atom–cavity coupling strength $g=1.7$ and driven potential strength $\xi =1.5$. The periods of the kicks are $T=6$ and $T=2\pi$ in (a), (c) and in (b), (d), respectively. In (d), the blue color indicates the phase of the even state $\theta ={\theta }_0$, while the yellow color indicates the phase of the odd state $\theta ={\theta }_0+\pi /2=2.071$.

Figure 5

Log-log plots of phase deviation $\sigma \left(\zeta \right)$ for the initial state as a function of perturbation $\zeta$. (a) Plot with a deviation of kick period, $\zeta =\delta$, with $\mathrm{\Delta }=0$. (b) Plot with a variation of detuning parameter, $\zeta =\mathrm{\Delta }$, with $\delta =0$. Other parameters are initial phase ${\theta }_0=0.5$, atom–cavity coupling strength $g=1.7$, and driven potential strength $\xi =1.5$. The blue dotted line takes the evolution period $M=100$, and the red circle line takes the evolution period $M=500$. The dark dashed line is given as (a) ${log}_{10}\sigma \left(\delta \right)={log}_{10}\delta +3.6$ and (b) ${log}_{10}\sigma \left(\mathrm{\Delta }\right)={log}_{10}\mathrm{\Delta }+1.5$.