Quantum state engineering with twisted photons via adaptive shaping of the pump beam

High-dimensional entanglement is a valuable resource for quantum communication, and photon pairs entangled in orbital angular momentum (OAM) are commonly used for encoding high-dimensional quantum states. However, methods for the preparation of maximally entangled states of arbitrary dimensionality are still lacking, and currently used approaches essentially rely on filtering and entanglement concentration. Here, we experimentally realize a method for the generation of high-dimensional maximally entangled OAM states of photon pairs which does not require any of these procedures. Moreover, the prepared state is restricted to the subspace of the specified dimensionality, thus requiring minimal spatial postselection.

Figure 1

Experimental setup. $\text{L}1=200$ mm, $\text{L}2=100$ mm, $\text{L}3=\text{L}4=11$ mm. O1 and O2: $20\times$ and $10\times$ microscopic objectives, respectively. IF: $810\pm 5$-nm interference filter. SPCM: Single-photon-counting modules.

Figure 2

Normalized (divided by the maximal value) coincidence count rates as a function of detection mode azimuthal numbers $l_s$ and $l_i$ for the cases of the pump beam with (a) $l=-2$, (b) $l=0$, and (c) $l=2$ (experiment).

Figure 3

Spiral spectra of maximally entangled (a) qutrits, (b) ququarts, and (c) ququints. Associated intensity and phase profiles of the pump beams calculated from the experimentally obtained coefficients ${\alpha }_l$ in (2) are shown in the bottom row. The corresponding nonzero ${\alpha }_l$ are (a) ${\alpha }_{-2}=0.76-0.11i$, ${\alpha }_0=-0.12+0.15i$, ${\alpha }_2=0.30-0.53i$, (b) ${\alpha }_0=0.09-0.02i$, ${\alpha }_2=-0.02-0.19i$, ${\alpha }_4=0.57-0.01i$, ${\alpha }_6=0.77-0.21i$, (c) ${\alpha }_{-4}=-0.25-0.73i$, ${\alpha }_{-2}=0.19-0.10i$, ${\alpha }_0=-0.07+0.11i$, ${\alpha }_2=0.14-0.14i$, ${\alpha }_4=-0.54+0.09i$.

Figure 4

Experimental phases of qutrit terms $\left|-1,-1\right.〉$ and $\left|1,1\right.〉$ for different rotation angles (a) $\theta =0$, (b) $\theta =\pi /8$, (c) $\theta =\pi /4$, and (d) $\theta =3\pi /8$, followed by the corresponding intensity profiles of the pump beam. The phase of the $\left|0,0\right.〉$ term is taken to be zero as a reference. Phase errors are calculated from Monte Carlo simulations of Poissonian counting statistics.

Figure 5

Density matrices $\rho$ of the reconstructed qutrit states. (a) A maximally entangled qutrit state, and (b) a nonmaximally entangled state from Eq. (5).