Manifestations of quantum holonomy in interferometry

Abelian and non-Abelian geometric phases, known as quantum holonomies, have attracted considerable attention in the past. Here, we show that it is possible to associate nonequivalent holonomies to discrete sequences of subspaces in a Hilbert space. We consider two such holonomies that arise naturally in interferometer settings. For sequences approximating smooth paths in the base (Grassmann) manifold, these holonomies both approach the standard holonomy. In the one-dimensional case the two types of holonomies are Abelian and coincide with Pancharatnam’s geometric phase factor. The theory is illustrated with a model example of projective measurements involving angular momentum coherent states.

Figure 1

(Color online) Direct (upper panel) and iterative (lower panel) holonomy in the interferometer setting. In the direct scenario, the sequence $P_1,\dots ,P_m$ of filtering measurements is applied to the internal state in the 0 arm (upper path). In the 1 arm (lower path), the internal state is exposed to a unitary operation $V$. The intensity for each orthonormal basis vector $\mid 1_k⟩$ of the initial subspace is measured in one of the output beams. Maximum of the total intensity, defined as the sum over all $k$, is obtained when $V$ coincide with the direct holonomy of the sequence. In the iterative scenario, the internal states $\mid a_k⟩$ and $\mid {(a+1)}_k⟩$ are exposed to the unitary operations ${\tilde{V}}_a$ and $V_{a+1}$, respectively, in the two arms. Maximum of the total intensity (sum over $k$) is obtained by varying $V_{a+1}$ and keeping ${\tilde{V}}_a$ fixed. In this way, the unitary operators ${\tilde{V}}_2,\dots ,{\tilde{V}}_m,{\tilde{V}}_1$ are given in an iterative manner, yielding the iterative holonomy as the final unitary ${\tilde{V}}_1$.