Tunable Landau-Zener transitions using continuous- and chirped-pulse-laser couplings

The laser coupled Landau-Zener avoided crossing has been investigated with an aim towards obtaining the laser source parameters for precise controlling of the state dynamics in a two-level quantum system. The conventional Landau-Zener equation is modified for including the interaction of the system with a laser field during a bias energy sweep and the obtained Hamiltonian is numerically solved for the investigation of the two-state occupation probabilities. We have shown that in the Landau-Zener process, using an additional laser source with controlled amplitude, frequency, and phase, the system dynamics could be arbitrarily engineered. This is while, by synchronous frequency sweeping of a chirped-pulse laser, the system could be guided into a resonance condition, which again gives the remarkable possibility for precise tuning and controlling of the quantum system dynamics.

Figure 1

The probability amplitudes of states 1 and 2 versus time through avoided crossing for ${\widehat{H}}_{12}=0.3$ and $\widehat{\alpha }=1.25$. The inset shows the eigenenergies of the quantum system. The time interval is $-{\widehat{t}}_{\mathrm{end}}\le \widehat{t}\le {\widehat{t}}_{\mathrm{end}}$ where ${\widehat{t}}_{\mathrm{end}}=t_{\mathrm{end}}\tau =20$.

Figure 2

The variation of adiabatic energy levels of $E_1$ and $E_2$ for cases with no field or $\widehat{\mathrm{\Omega }}=0$ (dotted black line), low field intensity of $\widehat{\mathrm{\Omega }}=1/\phantom{12}2{\widehat{H}}_{12}$ with $\theta =\pi$ (dashed black line), and high field intensity of $\widehat{\mathrm{\Omega }}={\widehat{H}}_{12}$ with $\theta =0$ (gray solid line) and $\theta ={cos}^{-1}\left(-{\widehat{H}}_{12}/\phantom{-{\widehat{H}}_{12}\widehat{\mathrm{\Omega }}}\widehat{\mathrm{\Omega }}\right)$ (black solid line). The other parameters are ${\widehat{H}}_{12}=0.3$ and $\widehat{\alpha }=1.25$.

Figure 3

The probability amplitudes of states 1 and 2 versus time through avoided crossing for different field and Rabi frequencies. (a) $\widehat{f}=2.5$; (b) $\widehat{f}=0.75$. The other parameters are ${\widehat{H}}_{12}=0.3$ and $\widehat{\alpha }=1.25$.

Figure 4

The AP of state 1 for sweeping of the Rabi frequency and laser frequency with phase difference of (a) $\theta =0^{\circ }$, (b) $\theta ={45}^{\circ }$, (c) $\theta ={90}^{\circ }$, and (d) $\theta ={180}^{\circ }$. The other parameters are ${\widehat{H}}_{12}=0.3$ and $\widehat{\alpha }=1.25$.

Figure 5

The AP of states 1 and 2 versus Rabi frequency for the laser frequency of (a) $\widehat{f}=0.12$, and (b) $\widehat{f}=0.4$. The other parameters are ${\widehat{H}}_{12}=0.3$, $\theta =0^{\circ }$, and $\widehat{\alpha }=1.25$.

Figure 6

The AP of state 1 for sweeping of the laser frequency and the laser field phase. The other parameters are ${\widehat{H}}_{12}=0.3$, $\widehat{\alpha }=1.25$ and $\widehat{\mathrm{\Omega }}=1$.

Figure 7

The AP of state 1 for sweeping of the Rabi frequency and the laser field phase with (a) ${\widehat{H}}_{12}=0$, and (b) ${\widehat{H}}_{12}=0.3$. The other parameters are $\widehat{f}=1$, and $\widehat{\alpha }=1.25$.

Figure 8

The probability amplitudes of the states (P1 is the solid line and P2 is the dashed line) in the (a) resonance, and (b) 5% off-resonance cases. The considered parameters are ${\widehat{H}}_{12}=0.3$, $\widehat{\alpha }=1.25$, and $\widehat{\mathrm{\Omega }}=0.2$.

Figure 9

The probability amplitudes of the states (P1 is the solid line and P2 is the dashed line) in the resonance case for a limited time interval (a) $\widehat{t}=[0–40]$, and (b) $\widehat{t}=[0–30]$. The insets show the changes of the laser frequency for each case. The considered parameters are ${\widehat{H}}_{12}=0.3$, $\widehat{\alpha }=1.25$, and $\widehat{\mathrm{\Omega }}=0.2$.