Geometry of an adiabatic passage at a level crossing

We discuss adiabatic quantum phenomena at a level crossing. Given a path in the parameter space which passes through a degeneracy point, we find a criterion which determines whether the adiabaticity condition can be satisfied. For paths that can be traversed adiabatically we also derive a differential equation which specifies the time dependence of the system parameters, for which transitions between distinct energy levels can be neglected. We also generalize the well-known geometric connections to the case of adiabatic paths containing arbitrarily many level-crossing points and degenerate levels.

Figure 1

(Color online) Illustration of spin-$1∕2$ evolution generated by variation of external magnetic field. In all cases the angle $\varphi$ grows linearly with time, slowly (adiabatically) as compared to $B$. The variation of the second angle, $\theta$, is different in all cases. Upper plot: $\dot{\theta }=0.005\times B\cos Bt∕2$, middle plot: $\dot{\theta }=0.005\times B\cos Bt$, lower plot: $\dot{\theta }=0.005\times B\cos 2Bt$. Since the diagonal terms in $\tilde{H}$ are dominated by $B$, the middle plot illustrates almost-resonant transition in the rotated frame.

Figure 2

(Color online) Realizations of the path $\theta ={\theta }_0+{\Gamma }^{\beta }$, $\varphi =\mathrm{const}$ for various values of $\beta$. Variation along the solid paths can be adiabatic at each point, down to the degeneracy, while adiabaticity for the dashed paths would require infinite time of the process.

Figure 3

(Color online) Schematic structure of the control manifold $\mathcal{M}$. The path $\Gamma$ is formally embedded in the subspace $\mathcal{T}$ tangential at ${\mathbf{R}}_0$ to the degeneracy domain $\mathcal{E}$.

Figure 4

Quantum optical STIRAP process. (a) The level configuration, (b) applied pulses, (c) the energy of the eigenstates. The system which initially was in the atomic state ∣1⟩ follows the eigenstate $\mid E_0⟩$ and during the process is transformed into ∣3⟩.

Figure 5

Geometry of the states (a) $\mid E_0⟩$ and (b) $\mid E_{-}⟩$. The loops denote the traversed path during the passage. Dashed lines are the directions of constant states in the subspace {∣1⟩,∣3⟩}. In (b) the amplitude $\epsilon$ changes along these lines, however it does not contribute in the first order to the mixing of $\mid E_0⟩$ and $\mid E_{-}⟩$.

Figure 6

Solution of the equation of motion for $\Gamma$, parametrizing the path of the three-level STIRAP. (a) $\dot{\Gamma }$ as a function of $\Gamma$, shows already that near the level crossing, the rate of variation decreases to zero. (b) The solution for already a long duration of adiabatic evolution. (c) The time dependence of $\ln \left(\Gamma \right)$ shows that the parameter decreases in time slower than exponentially. In these plots the ratio $\Delta ∕{\Omega }_0=1$, and $\delta =0.01$.