Coherent manipulation of a three-dimensional maximally entangled state

Maximally entangled photon pairs with a spatial degree of freedom is a potential way for realizing high-capacity quantum computing and communication. However, methods to generate such entangled states with high quality, high brightness, and good controllability are needed. Here, a scheme is experimentally demonstrated that generates spatially maximally entangled photon pairs with an orbital angular momentum degree of freedom via spontaneous parametric down-conversion in a nonlinear crystal. Compared with existing methods using postselection, the technique can directly modulate the spatial spectrum of down-converted photon pairs by engineering the input pump beam. In addition, the relative phase between spatially entangled photon pairs can be easily manipulated by preparing the relative phase of input pump states.

Figure 1

Theoretical results of spatial spectrum distributions of down-converted photons and the required intensity profiles of the input beams. [(a)–(d)] The spatial spectrum distributions for the input pump ${|-2\rangle }_p+{|0\rangle }_p+{|2\rangle }_p$, $\sqrt{1.5}{|-2\rangle }_p+{|0\rangle }_p+\sqrt{1.5}{|2\rangle }_p$, $\sqrt{2.5}{|-2\rangle }_p+{|0\rangle }_p+\sqrt{2.5}{|2\rangle }_p$, ${|-2\rangle }_p+{|2\rangle }_p$, respectively, where the beam widths of the pump and detection are $w_p=1.0$, ${\gamma }_s={\gamma }_i=0.5$ [29, 33]. Actually, the coefficients of input state for MES need to change based on the different system of collections. (e) Two-dimnensional (2D) perspective of Fig. 1, where the $x\left(y\right)$ labels represent the modes of signal and idler photons. [(f)–(h)] Corresponding required intensity profiles, where the wavelength of the pump is 780 nm.

Figure 2

Optical layout for generating and detecting maximally entangled states. SLM-V and SLM-I are a spatial visible and infrared light modulator used to modulate and measure the OAM-entangled state. The state ${|-l\rangle }_s{|-l\rangle }_i$ could be converted into ${|-l\rangle }_s{|l\rangle }_i$ by changing the number of reflection mirrors in the idler path.

Figure 3

Spatial spectral distributions and interference results. (a) Coincidences for three-dimensional MES per second. [(b), (d)] Coincidence and interference curve fits for various phase angle of $1/\sqrt{3}({|-2\rangle }_p+e^{i{\theta }_0}{|0\rangle }_p+e^{i{\theta }_2}{|2\rangle }_p)$, where the pentagram and circle data respectively represent two phase parameters $\alpha =2\pi /3$ and $\alpha =\pi /3$ in SLM-I. The error bars were $\pm 2$ standard deviations, assuming the data followed Poisson's distribution. Each coincidence was recorded over $20s$. (c) Theoretical interferences of MES with two phase factors.

Figure 4

Results of Bell inequality and tomographic reconstructions. [(a), (c)] Theoretical Bell-type 3-D surface and measured interference; the period $T$ in panel (c) was used to ensure the spacing of the measurement basis. [(d), (e)] The real and imaginary parts of density matrices of $d=3$, respectively. There were two relatively high (around 0.1) imaginary parts in panel (b), which was the result of a constant phase between low- and high-order states. Each coincidence for quantum-state tomography was recorded over 20 s.

Figure 5

Spatial spectrum distributions. [(a)–(e)] The coincidences for various amplitude-modulated input states as ${|-2\rangle }_p+{|0\rangle }_p+{|2\rangle }_p$, $2{|-2\rangle }_p+{|0\rangle }_p+2{|2\rangle }_p$, $3{|-2\rangle }_p+{|0\rangle }_p+3{|2\rangle }_p$, $4{|-2\rangle }_p+{|0\rangle }_p+4{|2\rangle }_p$, and ${|-2\rangle }_p+{|2\rangle }_p$, respectively, where the normalized coefficients are ignored for simplicity. The coincidence times and windows are 1 s and 1.6 ns in the process of measurements. (f) The phase hologram in the SLM for generating a three-dimensional MES.