Nonlinear Landau-Zener-Stückelberg-Majorana interferometry

We investigate Landau-Zener-Stückelberg-Majorana (LZSM) interferometry in a nonlinear two-level system subjected to a sinusoidal driving field. A comprehensive analysis for the interference patterns is shown from which we can analytically obtain the condition of constructive and destructive interferences in some cases. In the presence of nonlinear interaction, the level structure is changed, which can lead to an asymmetric deformation of the interference patterns and a significant position shift of the interference fringes. Our findings suggest an application of the nonlinear LZSM interferometry in accurately calibrating the parameters characterizing a nonlinear two-level system, as well as its coupling with the driven fields.

Figure 1

Time evolution of the energy levels for different offsets: (a) ${\epsilon }_0=0$ and (b) ${\epsilon }_0=5$. The time-dependent adiabatic energy levels (i.e., $\mathrm{\Delta }=1$) are plotted by the thick dashed-dotted or dashed lines, while the diabatic energy levels (i.e., $\mathrm{\Delta }=0$) are shown by the thin solid lines. The parameters $A=10$ and $\omega =1$ are used.

Figure 2

Population probability ${\left|a\left(t\right)\right|}^2$ at time $t=50/\mathrm{\Delta }$ for an initial preparation of the system in the state $(a,b)=(0,1)$ at $t_0=0$. For numerical implementation, we have considered $A/\mathrm{\Delta }=2.5$ with different nonlinearities: (a) $c/\mathrm{\Delta }=0$ and (b) $c/\mathrm{\Delta }=1.05$.

Figure 3

Time evolution of the population probability ${\left|a\left(t\right)\right|}^2$ for weak coupling with different nonlinearities: (a) $c/\omega =0$, (b) $c/\omega =0.5$, (c) $c/\omega =1.0$, (d) $c/\omega =1.5$, (e) $c/\omega =2.0$, and (f) $c/\omega =2.5$. Olive solid lines indicate exact numerical results, while red dashed lines are approximate analytical results. For implementation, we have used $A/\omega =10.5$, $\mathrm{\Delta }/\omega =0.05$, and ${\epsilon }_0/\omega =3$.

Figure 4

LZSM interference patterns for weak driving with different nonlinearities: (a) $c/\omega =0$, (b) $c/\omega =3$, (c) $c/\omega =5$, (d) $c/\omega =7$, (e) $c/\omega =9$, and (f) $c/\omega =11$. Population probability ${\left|a\left(t\right)\right|}^2$ viewed as a function of $\mathrm{\Delta }/\omega$ and ${\epsilon }_0/\omega$ for $A/\omega =0.05$ from the initial time $t_0=0$ to $t=2\pi /\omega$.

Figure 5

Nonlinear LZSM interference patterns for weak driving at ${\epsilon }_0=0$. Population probability ${\left|a\left(t\right)\right|}^2$ viewed as a function of $\mathrm{\Delta }/\omega$ and $c/\omega$ for $A/\omega =0.05$ from the initial time $t_0=0$ to $t=2\pi /\omega$. Black dashed-dotted line (with slope 1/2) is plotted to denote the boundary between strong and weak oscillations.

Figure 6

LZSM interference patterns for strong driving with different nonlinearities: (a) $c/\omega =0$, (b) $c/\omega =0.2$, (c) $c/\omega =0.4$, (d) $c/\omega =0.6$, (e) $c/\omega =0.8$, and (f) $c/\omega =1$. Time-averaged occupation probability $\frac{1}{t-t_0}{\int }_{t_0}^t{\left|a\left(t^{\prime }\right)\right|}^{2}d{t}^{\prime }$ (with $t_0=0$ and $t=50\pi /\omega$) viewed as a function of $\mathrm{\Delta }/\omega$ and ${\epsilon }_0/\omega$ for $A/\omega =50$.

Figure 7

(a) Nonlinear LZSM interference patterns. Final occupation probability ${\left|a\left(t\right)\right|}^2$ (with $t=8.5\pi /\omega$) viewed as a function of $c/\omega$ and ${\epsilon }_0/\omega$ for $A/\omega =10.5$ and $\mathrm{\Delta }/\omega =0.05$. We also show the profiles of (a) with different ${\epsilon }_0$: (b) ${\epsilon }_0/\omega =0$ and (c) ${\epsilon }_0/\omega =6$. Olive solid lines indicate exact numerical results, while red dashed lines are approximate analytical results.