Spatially entangled photon-pair generation using a partial spatially coherent pump beam

We demonstrate experimental generation of spatially entangled photon pairs by spontaneous parametric down-conversion (SPDC) using a partial spatially coherent pump beam. By varying the spatial coherence of the pump, we show its influence on the down-converted photon's spatial correlations and on its degree of entanglement, in excellent agreement with theory. We then exploit this property to produce pairs of photons with a specific degree of entanglement by tailoring of the pump coherence length. This work thus unravels the fundamental transfer of coherence occurring in SPDC processes, and provides a simple experimental scheme to generate photon pairs with a well-defined degree of spatial entanglement, which may be useful for quantum communication and information processing.

Figure 1

(a) Light emitted by a diode laser (${\lambda }_p=405$ nm) is scattered by a static thin diffuser (plastic sleeve) and illuminates a nonlinear crystal of $\beta$-barium borate (BBO) to produce spatially entangled pairs of photons by type-I SPDC. Spectral filters at $810\pm 10$ nm select near-degenerate photons. Lenses $f_1=150$ mm and $f_2=200$ mm image an iris onto the crystal surface. When the diffuser is maintained fixed, the crystal is thus illuminated by a static speckle pattern (b). White scale bar corresponds to 700 $\mu \mathrm{m}$. Momenta of photons are imaged onto an EMCCD camera by imaging the far field via a $f_3=40$-mm lens, and a direct intensity image (c) is recorded by accumulating photons onto an EMCCD camera sensor. Sum-coordinate projection of the joint probability distribution of photon pairs (d) shows a coincidence speckle pattern that reveals the transfer of coherence between the pump and the down-converted fields. Black-red and blue-red colorbars correspond, respectively, to intensity and intensity correlation measurements.

Figure 2

Without diffuser, the direct intensity image (a1) shows a well-defined disk and the $X_{+}$-coordinate projection of $\mathrm{\Gamma }$ (a2) shows a strong antidiagonal. An element $(k{y}_1,k{y}_2)$ of the $X_{+}$-coordinate projection corresponds to the joint probability of detecting one photon at ${\mathbf{k}}_{\mathbf{1}}=(k_{x_1},k_{y_1})$, with no constraints on $k_{x_1}$, together with the second photon at ${\mathbf{k}}_{\mathbf{2}}=(-k_{x_1},k_{y_2})$. The strong antidiagonal is a signature of momentum conservation between photons produced by SPDC with a fully coherent collimated pump beam. When a rotating random diffuser composed by one layer of plastic sleeve is introduced in the apparatus, edges of the direct intensity disk gets blurred (b1) and the width of the antidiagonal on the $X_{+}$-coordinate projection increases (b2). When one photon is detected at $\mathbf{k}$, its twin has now a high probability to arrive on an area that spreads around $-\mathbf{k}$. This area broadens when the coherence length of the pump is decreased by using rougher random diffusers, as shown on direct images (c1) and (d1) and $X_{+}$-coordinate projections (c2) and (d2) measured using, respectively, two layers of plastic sleeve and three layers. Correlation measurements are performed by acquiring a total number of images between $4\times {10}^5$ (no diffuser) and $2\times {10}^7$ (three layers) with an exposure time set between 5 and 30 ms. Black-red and blue-red colorbars correspond, respectively, to intensity and intensity correlation measurements.

Figure 3

(a) Momentum-correlation width ${\sigma }_k$ is represented in the function of coherence length of the pump ${\ell }_c$. Values on the graph correspond to four different measurements performed (0) without diffuser, (1) with one layer of plastic sleeve, (2) two layers, and (3) three layers. Linear regression fits experimental values and returns a slope of $0.8\pm 0.3$ (blue dashed line and shaded area), in accordance with Eq. (2). (b) Schmidt number $K$ is represented in the function of the coherence length of the pump $1/{\ell }_c$ (black circles). The blue dashed curve and the shaded area are a semiempirical model obtained by combining results of the linear regression of ${\sigma }_k$ with the theoretical formula of ${\sigma }_r$ and the $K$ formula [29]. The red dashed curve is a pure theoretical prediction obtained by inserting known experimental parameters (crystal and pump properties) into Eq. (3).

Figure 4

Sum-coordinate projections of $\mathrm{\Gamma }$ measured with (a) no diffuser, (b) one-layer rotating diffuser, (c) two layers, and (d) three layers. Values of ${\sigma }_k$ are estimated in each case by measuring the width of the central spot using a Gaussian model [37, 38]. White scale bar corresponds to $0.05\mu {\mathrm{m}}^{-1}$.

Figure 5

(a) Light emitted by a diode laser (${\lambda }_p=405\mathrm{nm}$) is scattered by a static thin diffuser (plastic sleeve) and illuminates a nonlinear crystal of $\beta$-barium borate (BBO) to produce spatially entangled pairs of photons by type-I SPDC. Spectral filters at $810\pm 10\mathrm{nm}$ select near-degenerate photons. Lenses $f_1=150\mathrm{mm}$ and $f_2=200\mathrm{mm}$ image an iris onto the crystal surface. When the diffuser is maintained fixed, the crystal is thus illuminated by a static speckle pattern. White scale bar corresponds to $700\mu \mathrm{m}$. Positions of photons at the output surface of the crystal are imaged onto an EMCCD camera via a single-lens imaging system $f_3/2=20\mathrm{mm}$. The direct intensity image (b) recorded by accumulating photons onto an EMCCD camera sensor is a speckle pattern. Minus-coordinate projection of the joint probability distribution of photon pairs (c) shows a strong peak at its center that reveals the strong correlations between positions of the pairs.

Figure 6

Direct intensity of the down-converted field at the crystal plane is imaged onto the EMCCD camera using the experimental configuration described in Fig. 5 without diffuser (a1), with a rotating diffuser composed by one layer of plastic sleeve (b1), two layers (c1), and three layers (d1). All intensity patterns are homogeneous and identical. When the camera is used to measure the joint probability distribution $\mathrm{\Gamma }$, the $X_{-}$-coordinate projection of $\mathrm{\Gamma }$ (see Sec. pp6-s1) camera shows a very strong diagonal in all four cases: without diffuser (a2), with one layer of plastic sleeve (b2), two layers (c2), and three layers (d2). These projections show that position correlations do not depend on the coherence properties of the pump. Minus-coordinate projections of $\mathrm{\Gamma }$ taken without diffuser (a3), with one layer (b3), two layers (c3), and three layers (d3) show the same peak at their center, which highlights the strong correlations between positions of the pairs. Fitting these images with the model of Eq. (C2) allows determining values of ${\sigma }_r$ reported in Table 2.

Figure 7

(a) Apparatus used to Fourier image the pump field at the crystal plane onto the camera. It is similar to the one shown in Fig. 1 without the crystal and all the filters. (b) Apparatus used to image the pump field at the crystal plane onto the camera. It is similar to the one shown in Fig. 5 without the crystal and all the filters. Using configuration (b), intensity distribution of the pump beam at the crystal plane is imaged on the camera without diffuser (c), with a rotating diffuser composed of one layer of plastic sleeve (d), two layers (e), and three layers (f). All intensity patterns are homogeneous and identical. Using configuration (b), intensity distribution of the pump beam in the momentum space is imaged on the camera without diffuser (g), with a rotating diffuser composed of one layer of plastic sleeve (h), two layers (i), and three layers (j). Without diffuser, the pump beam is focused onto the camera and the width of the peak ${\sigma }_{p_0}$ is used to estimate the beam waist $\omega =1/{\sigma }_{p_0}\approx 125\mu \mathrm{m}$. When rotating diffusers are inserted, the peak broadens and its width ${\sigma }_p$ provides an estimation of the correlation length ${\ell }_c$ using the formula ${\ell }_c=2/\sqrt{{\sigma }_p^2-{\sigma }_{p_0}^2}$. Values of ${\ell }_c$ are reported in Table 3.