Implementation of stimulated Raman adiabatic passage in degenerate systems by dimensionality reduction

We consider the problem of the implementation of stimulated Raman adiabatic passage (STIRAP) processes in degenerate systems, with a view to be able to steer the system wave function from an arbitrary initial superposition to an arbitrary target superposition. We examine the case of an $N$-level atomic system consisting of $N-1$ ground states coupled to a common excited state by laser pulses. We analyze the general case of initial and final superpositions belonging to the same manifold of states, and we cover also the case in which they are nonorthogonal. We demonstrate that for a given initial and target superposition, it is always possible to choose the laser pulses so that in a transformed basis the system is reduced to an effective three-level $\Lambda$ system, and standard STIRAP processes can be implemented. Our treatment leads to a simple strategy, with minimal computational complexity, which allows us to determine the laser-pulse shape required for the wanted adiabatic steering.

Figure 1

$N$-level system consisting of $N-1$ ground states coupled to a common excited state. (a) Interaction scheme in the basis $\left\{\right|i\rangle \}$ with couplings ${\Omega }_i$, as in Eq. (1). (b) Interaction scheme in the basis $\left\{\right|{\varphi }_i\rangle \}$ with couplings ${\tilde{\Omega }}_i$, as in Eqs. (23).

Figure 2

Numerical solutions for the time evolution of the five-level system starting from the initial state $|{\psi }_i\rangle =\left(\right|1\rangle +|2\rangle )/\sqrt{2}$. The pulses are determined, following the procedure outlined in the text, to transfer the system into the state $|{\psi }_f\rangle =\sqrt{2/10}|1\rangle -\sqrt{2/10}|2\rangle -\sqrt{5/10}|3\rangle +\sqrt{1/10}|4\rangle$. (a) Pulse shapes for the Rabi frequencies ${\tilde{\Omega }}_1$, ${\tilde{\Omega }}_2$ in the transformed basis $\left\{\right|{\varphi }_i\rangle \}$. (b) Pulse shapes for the Rabi frequencies ${\Omega }_i$ in the atomic basis $\left\{\right|i\rangle \}$. (c) Population of the ground and excited states in the atomic basis, and fidelity $F$ of preparation of the target state. The parameters of the simulation are ${\Omega }_0=30$, $\tau =160$, $w=150$, $t_0=500$.

Figure 3

Numerical solutions for the time evolution of the five-level system starting from the initial state $|{\psi }_i\rangle =-\sqrt{3/5}|1\rangle +\sqrt{1/5}|2\rangle -\sqrt{1/10}|3\rangle +\sqrt{1/10}|4\rangle$. The pulses are determined, following the procedure outlined in the text, to transfer the system into the state $|{\psi }_f\rangle =\sqrt{2/5}|1\rangle +\sqrt{3/10}|2\rangle -\sqrt{1/5}|3\rangle +\sqrt{1/10}|4\rangle$. (a) Pulse shapes for the Rabi frequencies ${\tilde{\Omega }}_1$, ${\tilde{\Omega }}_2$ in the transformed basis $\left\{\right|{\varphi }_i\rangle \}$. (b) Pulse shapes for the Rabi frequencies ${\Omega }_i$ in the bare atomic basis $\left\{\right|i\rangle \}$. (c) Population of the ground and excite states, and fidelity $F$ of preparation of the target state. The parameters of the simulation are ${\Omega }_0=30$, $\tau =160$, $w=150$, $t_0=500$.