The position-momentum (EPR) entanglement of two particles is of special importance for fundamental research in quantum physics and quantum information processing. The existing methods for preparing position-momentum entanglement are mainly based on nonlinear crystal and atomic systems. Devices in atomic systems are complex and difficult to adjust, and the efficiency of entangled photons generated by β-BaB₂O₄ (BBO) crystals is low. In this study, we first use ghost imaging and ghost interference techniques to achieve efficient generation of position-momentum entangled photons in the periodically polarized potassium titanium phosphate (PPKTP) crystal optical path, and verify the entanglement properties. The sampling time for a single pixel is 10 s, and the experimental setup design is relatively simple. The entangled photon source analyzed in this study can provide assistance for quantum imaging and preparation of super-entangled states. It also demonstrates broad application potential in fields such as quantum information processing and quantum communication protocols.

The preparation of EPR entanglement is mainly based on nonlinear effects of the medium. The atoms in crystals and cold atomic clusters are almost stationary, making them ideal media for generating position and momentum entangled photons, given that the mass of photon entanglement does not decrease owing to atomic motion. The existing EPR entanglement preparation methods are mainly based on nonlinear crystal and atomic systems, including spontaneous parametric down-conversion (SPDC) effects based on BBO crystals, spontaneous four-wave mixing (SFWM) effects based on cold atomic systems, and spontaneous Raman scattering (SRS) effects. Experiments have found that high-quality entanglement sources can also be prepared based on the SFWM and SRS effects of thermal atomic systems. The characterization of EPR entanglement can be achieved through ghost imaging and ghost interference. The positional uncertainty can be calculated using ghost imaging data whereas the uncertainty of the displacement can be calculated using ghost interference data. Whether the EPR entanglement inequality is met or not must be verified to evaluate the entanglement degree. The EPR preparation scheme based on atomic systems has a narrow bandwidth of entangled photons at the order of 1 MHz. However, the experimental setup is relatively complex. Among such atomic systems, cold atom systems require multiple lasers for atomic cooling, while the power, frequency, timing, and other parameters of the cooling laser have strict requirements. Thermal atomic systems require a heating device to accurately control the temperature, and the SFWM and SRS effects based on atomic systems require a pump beam and a coupling beam, which have inconsistent wavelengths and high requirements for spatial optical path calibration. Therefore, atomic systems feature a large volume, multiple devices, and complex operations. Nonlinear crystals can operate at room temperature, but the SPDC process of BBO crystals has a lower efficiency in generating entangled photons, which requires a higher pump power, usually exceeding 30 mW. Collecting entangled photons requires a long time to accumulate. The efficiency of PPKTP crystals at room temperature exceeds 10 times that of BBO, requiring low pump power, and making it a better choice for preparing entanglement sources. This study is based on the spontaneous parametric down-conversion effect of PPKTP crystals to prepare an efficient EPR entanglement source. The experimental results show that the uncertainty of the position and momentum of the entangled photon pairs calculated through ghost imaging and ghost interference images satisfies the entanglement paradox inequality, thereby verifying the entanglement characteristics.

The preparation and characterization of EPR entanglement based on the SPDC effect of PPKTP crystals include the following processes. First, the SPDC effect of the pump light source in the PPKTP crystal is exploited to generate entangled photon pairs. Then, the entangled photon pairs along the path are separated, the imaging object with the signal photon is irradiated, a signal photon is collected using a multimode fiber, and a single photon detector is employed for photon detection. Subsequently, the idle photon is divided into two paths, the power of each path is adjusted using a half-wave plate and a polarizing beam splitter prism, and either imaging or interferometric measurements are taken. After passing through a slit installed on a translation platform, the idle photon enters the fiber coupling head and undergoes position scanning. Photons are collected using a single photon counter and measured in accordance with the signal photons for ghost imaging. The other possibility for the idle photon is to collect photons using a short focal length lens and perform position scanning for ghost interference and measurement in accordance with the signal photon. Finally, the positional and momentum uncertainties are calculated using ghost imaging and ghost interference data, respectively, to verify whether they meet the EPR entanglement inequality and perform entanglement characterization.

The experimental results of ghost imaging and ghost interference are shown in Fig. 3. We established the calibrated theoretical models by fitting the data. Regarding ghost imaging, we fitted the data by performing a convolution of the double slit with a Gaussian function that takes into account the finite size. We compared the resulting fitting curve with the real transfer function of the metal bar to evaluate Δx. For ghost interference, we first fitted the experimental data by applying a Fourier transformation to the double slits and a tunable parameter for visibility, and compared the fitted curve with the Fourier transformation of the real double slits, thereby obtaining Δp. The values obtained from experimental measurements are listed in Table 1. We obtained the uncertainties of position and momentum by fitting the ghost images and ghost interference and compared them with the ideal curves, thereby verifying the EPR paradox inequality. The data exhibit clearly both ghost imaging and ghost interference and satisfy the EPR paradox inequality.

In this study, we implement a two-photon position-momentum entanglement preparation based on the spontaneous parametric down-conversion effect of PPKTP crystals, and use ghost imaging and ghost interference methods for entanglement characterization. By fitting experimental data with theory, we demonstrate that the prepared EPR entangled photon pairs satisfy the entanglement inequality. The principle of this method is reliable, achieving efficient preparation and characterization at room temperature. Only one pump light is needed for preparation, and the device is simple and easy to implement. It can effectively reduce interference from factors such as random jitter of optical components, and has low energy consumption. Our EPR entanglement source may have potential application in fields such as quantum imaging, quantum teleportation based on continuous variable entanglement, quantum key distribution, and quantum detection. This efficient method for preparing entangled sources provides a new approach for future applications. In this regard, the Gaussian characteristics of pump light sources may be changed and modulated to further improve the entanglement quality.