Three-level Landau-Zener dynamics

We compute Landau-Zener probabilities for three-level systems with a linear sweep of the uncoupled energy levels of the $3\times 3$ Hamiltonian matrix $H\left(t\right)$. Two symmetry classes of Hamiltonians are studied: For $H\left(t\right)\in$ su(2) (expressible as a linear combination of the three spin-1 matrices), an analytic solution to the dynamical problem is obtained in terms of the parabolic cylinder $D$ functions. For $H\left(t\right)\in$ su(3) (expressible as a linear combination of the eight Gell-Mann matrices), numerical solutions are calculated. In the adiabatic regime, full population transfer is obtained asymptotically at large times, but at intermediate times, all three levels are populated, and Stückelberg oscillations can manifest from the occurrence of two avoided crossings. For the open system, (wherein interaction with a reservoir occurs), we numerically solve a Markovian quantum master equation for the density matrix with Lindblad operators that models interaction with isotropic white Gaussian noise. We find that Stückelberg oscillations are suppressed and that the decoherence cannot be modeled in terms of a simple exponential.

Figure 1

(a) Energy eigenvalues of the three-level Hamiltonian versus time for $a=-1$, $\mathrm{\Omega }=\mathrm{\Delta }=1$ shown as solid curves, and the eigenvalues for $\mathrm{\Omega }=\mathrm{\Delta }=2$ shown as dashed curves (level 1 eigenvalue is the blue curve on top, 2 is the red curve in the middle, and 3 is the green curve on bottom). (b) Probability of levels 1 (blue curve on bottom right), 2 (orange curve in the middle right), and 3 (green curve on top right) versus time for $a=-1$, $\mathrm{\Omega }=\mathrm{\Delta }=1$. The blue curve ${\rho }_{11}\left(t\right)$ is initially unity at the initial time $t_0$, ${\rho }_{11}\left(t_0\right)=1$. (c) Probabilities ${\rho }_{11}$, ${\rho }_{22}$, and ${\rho }_{33}$ versus time for $a=1$ and $\mathrm{\Omega }=\mathrm{\Delta }=2$. This case is nearly fully adiabatic, with population staying on the eigenvector with the highest eigenvalue, hence ${\rho }_{22}\to 1$ at large time. The straight red line on top in (b) and (c) is the sum of the probabilities, $\mathrm{Tr}\left[\rho \right]$, which equals unity throughout the dynamics, as does the purity, $\mathrm{Tr}\left[{\rho }^2\right]$.

Figure 2

(a) Probability of levels 1 and 2 in the two-level system versus time for $a=-1$ and $\mathrm{\Delta }=1$. The initial state at early times is level 1 (blue curve). (b) Probability of the levels versus time for $a=-1$ and $\mathrm{\Delta }=2$. (c) Probability with $a=-1$ and $\mathrm{\Delta }=2e^{-{(t/2\sigma )}^2}$ with $\sigma =5$.

Figure 3

(a) Eigenvalues of the three-level Hamiltonian versus time for $a=-1$, $\mathrm{\Delta }=1$, and $\mathrm{\Omega }=5$. The level 1 eigenvalue is the blue curve on top, 2 is the green curve in the middle, and 3 is the red curve on bottom. (b) Probability of the levels 1, 2, and 3 versus time. The straight red line on top is the sum of the probabilities, $\mathrm{Tr}\left[\rho \right]$, which remains unity during the dynamics, as does the purity $\mathrm{Tr}\left[{\rho }^2\right]$.

Figure 4

Probabilities ${\rho }_{11}$ (blue curve on top left), ${\rho }_{22}$ (orange curve), and ${\rho }_{33}$ (green curve on top right) versus time for $a=1$ and $\mathrm{\Omega }=\mathrm{\Delta }=2$ [same parameters as in Fig. 1] with isotropic decay ($p_0=0.1$). The red dashed line shows the sum of the probabilities, $\mathrm{Tr}\left[\rho \right]$, which remains equal to unity throughout the dynamics, and the purple dotted curve shows the purity, $\mathrm{Tr}\left[{\rho }^2\right]$, which uniformly decreases from unity as a function of time and goes asymptotically to 1/3 at large times. Compare with Fig. 1 for the unitary (no decay) case.