Amplitude control of a quantum state in a non-Hermitian Rice-Mele model driven by an external field

In the Hermitian regime, the Berry phase is always a real number. It may be imaginary for a non-Hermitian system, which leads to amplitude amplification or attenuation of an evolved quantum state. We study the dynamics of the non-Hermitian Rice-Mele model driven by a time-dependent external field. The exact results show that it can have a full real spectrum for any value of the field. Several rigorous results are presented for the Berry phase with respect to the varying field. We demonstrate that the Berry phase is the same complex constant for any initial state in a single sub-band. Numerical simulation indicates that the amplitude control of a state can be accomplished by a quasiadiabatic process within a short time.

Figure 1

Schematic illustration of the non-Hermitian Rice-Mele model driven by a time-dependent external field. It is $\mathcal{PT}$ symmetric with respect to the ${OO}^{\prime }$ axis in the absence of the field. Constant field $\mathrm{\Phi }$ breaks the $\mathcal{PT}$ symmetry, but keeps the translational symmetry. A time-varying field $\mathrm{\Phi }\left(t\right)$ induces the eddy field in a direction indicated by the red arrow. Together with the distortion, the imaginary potentials can also break the left-right chiral symmetry, inducing a direction of the system indicated by the blue arrow. In the case that two arrows are either the same or the opposite, the dynamics of a state exhibits different behaviors.

Figure 2

Plots of the reduced dynamic phase ${\alpha }_k^{\lambda }\beta$ and the geometric phase ${\gamma }_k^{\lambda }$ as functions of $\varphi$, expressed in Eqs. (21) and (22), for different values of $k$ in a typical (a) Hermitian system with $\delta =-0.15, \mu =0.05$, and $\nu =0$ and (b) non-Hermitian system with $\delta =-0.15, \mu =0$, and $\nu =0.05$. Both systems have a full real spectrum. The real (imaginary) parts of the phases are plotted in red (black). (a) It is shown that both phases are real for the Hermitian system, while the adiabatic phase is complex for the non-Hermitian system. The dynamic phases and the real part of adiabatic phases in both cases have a slight difference. The shapes of all the phases are in agreement with the approximate expressions in Eqs. (33), (34), (37), and (38). Here the reduced dynamic phase ${\alpha }_k^{\lambda }\beta$ and the geometric phase ${\gamma }_k^{\lambda }$ are expressed in units of radian.

Figure 3

Numerical simulations for the time evolutions with different $\beta$. Amplification factor $A_k^{\lambda }\left(\varphi \right)$ and fidelity $f_k^{\lambda }\left(\varphi \right)$ for the initial state with $k=\pi /25$ and $\lambda =-$ are plotted for the system with $\delta =0.15, \mu =0, \nu =-0.2$, and $N=50$. We can see the evolution results are close to that, $A_{\pi /25}^{-}\left(\pi \right)=27.58,f_{\pi /25}^{-}\left(\pi \right)=1,$ obtained in the adiabatic limit. This indicates that the amplitude control could be realized with a high fidelity via a quasiadiabatic process.

Figure 4

The comparison between the dispersion relation (blue dot) and the trajectory of the center of the wave packet (color contour map). The simulation is computed as follows: the wave packet with $k_0=\pi /2, \alpha =0.05$, and $N_{\mathrm{A}}=1900$, in the lower band of the system with $\delta =-0.15, \mu =0, \nu =0.05$, and $N=1000$, subjected in the external field with $\beta =0.001$. This shows that the two are in close proximity owing to the fact that the contribution of the geometric phase to the wave-packet position becomes negligible when compared with the dynamic phase. Only the imaginary part of the geometric phase takes an important role in the dynamics of the wave packet.

Figure 5

The profiles of the time evolution of a wave packet in several typical cases. The initial wave packet is in the form of Eq. (43) with $k_0=1.4\pi , \alpha =0.05$, and $N_{\mathrm{A}}=1000$, and the external field varies with $\beta =0.001$. (a) In the case of $\delta =-0.7, \mu =0$, and $\nu =1.3$, the plot is shown that the probability of the sub-wave-packet in one band increases but the one in another band decreases. In the cases of (b) and (c), we take the same parameters with (a) but opposite $\varphi$ and $\delta$, respectively. Comparing with the profile in (a), we can see the completely opposite behaviors in (b) and (c). This is in agreement with our analytical prediction.

Figure 6

Profiles of the reduced energy defined in Eq. (48) of an evolved wave packet in Eq. (43) controlled by a Gaussian-type time-varying field in Eq. (47). The parameters of the wave packet and system are $k_0=1.5\pi , \delta =-0.7, \mu =0$, and $\nu =1.3$. (a) The shapes of the field as a function of time in Eq. (47) with different $\sigma$. (b) Plots of $E\left(t\right)$ of the field in (a). (c) Same as (b) but with the opposite flux. The simulation result is in agreement with the result $E\left(\infty \right)=-1.14$ for (b) and 1.14 for (c), obtained in the adiabatical limit. This indicates that the amplitude of the wave packet can be well controlled via a quasiadiabatic process in a short time. Here the time $t-\tau$ and the reduced energy $E\left(t\right)$ are expressed in units of $1/J$ and $J$, respectively.