Dynamics of the Kitaev chain model under parametric pumping

Dynamics of the Kitaev chain model under the effect of parametric pumping is studied. Two contributions to dynamical characteristics are considered: from the extended eigenstates and from the edge bound state (zero Majorana modes). It is shown that in the dynamical regime the frequencies of Rabi oscillations for zero Majorana modes are much larger than those related to gapped extended states. In the steady-state regime, the Rabi oscillations are blurred due to relaxation processes, and only oscillations of the characteristics of the model with the pumping frequency exist, producing absorption of the pumping power by extended states. Experimental realizations of the considered effect are discussed.

Figure 1

The contribution from the extended states to the change of the number of fermions in the Kitaev chain under parametric pumping. We use the units where $\hslash =k_B=1$. The parameters are $w=\left|\Delta \right|=1,\left|\mu \right|=1.5w,T=0.01w,\omega =2w,h=0.01w$. (Red) line shows the dynamical contribution. We see that the period of Rabi oscillations is much larger than the one of the pumping frequency $\omega$.

Figure 2

The average in time contribution from extended eigenstates to the change of the number of fermions in the Kitaev chain under parametric pumping as a function of the pumping frequency $\omega$ and chemical potential $\mu$. We use the units where $\hslash =k_B=1$. The parameters are $w=\left|\Delta \right|=1,T=0.01w,h=0.01w$.

Figure 3

The normalized (divided by $h^2)$ absorption of the parametric pumping power $Q_{ex}$ by extended states of the Kitaev chain in the steady-state regime as a function of the pumping frequency $\omega$ and chemical potential $\mu$. Parameters are the same as in Fig. 2.

Figure 4

The contribution (multiplied by $L)$ from the Majorana edge states (zero modes) to the change of the number of fermions in the Kitaev chain under parametric pumping. We use the same units and the same values of parameters as in Fig. 1 and $\gamma =0.2w$. Dashed-dotted (red) line shows the dynamical contribution; dashed (blue) line shows the steady-state contribution, and solid (black) line shows the total contribution. We see that Rabi oscillations due to the Majorana zero mode have much larger frequency than the ones caused by extended states, cf. Fig. 1.

Figure 5

The normalized (divided by ${10}^7{h}^2)$ absorption of the parametric pumping power $Q_b$ by the Majorana zero modes of the Kitaev chain in the steady-state regime as a function of the chemical potential $\mu$. Other parameters are the same as in Fig. 2.