Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces

We use spatial light modulators (SLMs) to measure correlations between arbitrary superpositions of orbital angular momentum (OAM) states generated by spontaneous parametric down-conversion. Our technique allows us to fully access a two-dimensional OAM subspace described by a Bloch sphere, within the higher-dimensional OAM Hilbert space. We quantify the entanglement through violations of a Bell-type inequality for pairs of modal superpositions that lie on equatorial, polar, and arbitrary great circles of the Bloch sphere. Our work shows that SLMs can be used to measure arbitrary spatial states with a fidelity sufficient for appropriate quantum information processing systems.

Figure 1

Bloch sphere equivalent for $\pm \ell$ OAM states. The pure state $|\mathbf{a}\rangle$ is defined by its latitude $(0⩽{\theta }_a⩽\pi )$ and longitude $(0⩽{\varphi }_a<2\pi )$ on the sphere. The poles (${\theta }_a=0,\pi$) represent the states $|\ell \rangle ,|-\ell \rangle$, respectively, while around the equator (${\theta }_a=\pi /2$) are the equally weighted superpositions of $|\ell \rangle$ and $|-\ell \rangle$.

Figure 2

For measurements of state $|\mathbf{a}\rangle$ (left), the state measurement of the other photon $|\mathbf{b}\rangle$ will yield maximum coincidence when the state has the same longitude and is reflected about the equator and minimum coincidence when the state has the same latitude and is reflected about the vertical axis (right).

Figure 3

(a) Experimental setup. The down-converted light from the crystal is imaged onto the SLMs, and then reimaged at the plane of the single-mode fibers, where the detection and coincidence measurement takes place. (b) Typical phase masks—with appropriate intensity blazing—as would be displayed on an SLM. Shown are the two OAM eigenstates and four equally weighted superposition states (Laguerre-Gauss and Hermite-Gauss, respectively).

Figure 4

Bell curves for three different great circles around the equator, poles, and an arbitrarily chosen great circle. Each curve corresponds to coincidence measurements between a state on a great circle around one Bloch sphere (right) and one of four different static states (left), shown as points on the spheres. The sinusoidal fringes are indicative of quantum correlations within a purely two-dimensional subspace, and each example shown violates a Bell inequality.

Figure 5

The necessary measurement of both the phase and the intensity of the modes. Pole-to-pole measurements (of the same type as in Fig. 4) are shown, but with only representations of the phase of the desired mode displayed on the SLM (left). The coincidence for the phase and intensity measurement (from Fig. 4) is shown for illustration (right). There is a clear difference between the form of the curves in each case, indicating that the number of modes selected in each case is different.

Figure 6

Two spheres in coincidence, shown with the same projection as Fig. 1. The scale shows the quantum contrast $Q=\frac{C}{S_1{S}_2\Delta t}$, where $C$ is the measured coincidence rate, $S_1,S_2$ are the single count rates for each detector, and $\Delta t$ is the gate time of our coincidence electronics.