Figure 3: (a) Tripod structure that can be used for holonomic quantum computation. Three degenerate internal states $k=1,2,3$ of an atom, say, are coupled by three laser fields $f_k$ to an excited state $e$. (b) Two of the resulting energy levels form a degenerate pair of dark states that encodes the states of a single qubit. (c) One-qubit holonomies can be obtained by slowly varying the strengths and phases of the laser fields so that the initial and final field configurations coincide. A possible cyclic variation $\text{A}\to \text{B}\to \text{C}\to \text{A}$ in the special case of real-valued $f_k$ is shown. The resulting holonomy is fully determined by the solid angle $\pi /2$ which yields a holonomic gate that takes the logical states $|0\rangle$ and $|1\rangle$ into $|1\rangle$ and $-|0\rangle$, respectively