Double-slit interference of single twisted photons Download: 787次
Recently, an essential quantum feature of photons is the fact that it can carry orbital angular momentum (OAM)[1], which has attracted a lot of attention in classical[2,3] and quantum optics[4]. In classical optics, OAM is related to an optical vortex with a helical phase structure of ( is the topological charge). This helical phase wraps around the axis of propagation times, which results in a phase singularity at the beam center. For the first time, to the best of our knowledge, it has been observed that light with a twisted structure travels slower than the established speed of light[5]. In quantum optics, photons with a twisted or helical phase structure of will carry a quantized OAM of ( is the reduced Planck constant) per photon, providing an infinite-dimensional Hilbert space[4]. Therefore, high-dimensional quantum information can be stored in the OAM of single photons, which holds great potential for quantum computing[6] and communication[7] applications. As a new and important degree of photons, how to understand the wave and particle characters of OAM is a fundamental issue and a basis of the potential application.
The double-slit interference devised by Thomas Young plays an important role in physics and strikingly demonstrates the wave-particle duality of quantum objects. Such an experiment can be performed with particles ranging from individual photons[8,9], electrons[10], neutrons[11], and atoms[12], to large molecules consisting of dozens of atoms[13]. Some counterintuitive features such as entanglement, non-locality, wave-particle duality, and delayed-choice concepts can be demonstrated or tested using a double-slit configuration[14
In this Letter, we have demonstrated the average photon trajectories (APTs) of twisted photons carrying OAM in Young’s double-slit interference. Kocsis et al.[18] measured the APTs of single photons in a double-slit configuration by using the technique of quantum weak measurements[23]. Later, it appears that weak measurements of the local momentum of photons made by Kocsis et al. represent measurements of the Poynting vector in an optical field[24]. This arouses our interest in the study of the APTs of twisted photons. We carry out Young’s double-slit interference with single twisted photons based on a triggered intensified charge-coupled device (ICCD), because the coincidence image can greatly degrade noise. Photons pass one by one through a pair of slits, where single photons from a given slit can be detected (particle-like character). The interference fringes emerge (wave-like character) on a screen behind the slits after many discrete photons impact, as if each particle is passing through both slits and interfering with itself. We find that the distribution of the trajectory end points of the APTs is in agreement with the pattern measured by ICCD. The helical APTs of single twisted photons help us to understand the optical manipulations of OAM and enable new applications in quantum imaging and sensing[25].
We firstly begin with the theoretical analysis, and the equation describing APTs is[26,27]where is the streamlines, is a certain arc length of the corresponding trajectory, and is the speed of light in vacuum. is the real part of the time-averaged Poynting vector carrying electromagnetic energy, and is the energy density of a monochromatic electromagnetic field, where and are the spatial parts of the electric and magnetic fields of light, respectively. ( indicates the unit vector of polarization state); since the polarization state has no influence on APTs, here we only consider the scalar field. Let us consider the simplest example of a twisted light beam—a linearly polarized Laguerre–Gaussian beam propagating along the axis (with a radial mode index )[28]: where are cylindrical coordinates, is the beam waist, is the beam radius, is the curvature radius of the wavefront, and () is the Rayleigh diffraction range.
The distance between the centerlines of the two slits is . Behind the double slit, the superposition of the two twisted beams can be written as where
Substituting Eqs. (
Fig. 1. The three-dimensional and two-dimensional APTs of the twisted light ( ) in the double-slit interference. (a) Three-dimensional structures and their projections at the plane (500 initial points for each slit). (b) Two-dimensional structures in the plane for 60 trajectories.
Spontaneous parametric down-conversion (SPDC) is a well-known nonlinear optical process, in which one photon incident on a nonlinear crystal spontaneously splits into two photons. The experimental setup is shown in Fig.
Fig. 2. Schematic diagram of experimental setup. HWP, half-wave plate; PBS, polarization beam splitter; L, lens; PPKTP, periodically poled KTP; IF, interference filter; QWP, quarter wave plate; FPC, fiber polarization controller; PAD, photon avalanche detector.
In the idler path, a -plate with is used to generate the photons carrying OAM. The -plate can convert a left-handed circularly polarized photon into a right-handed circularly polarized one with an OAM, while it can convert a right-handed circularly polarized photon into a left-handed circularly polarized one with an OAM[31]. Therefore, we use a fiber polarization controller (FPC) to adjust the polarization of the idler photons into left-handed circular polarization. Then, we used a lens with a focal length of 11 mm to collimate the beam coming out of the fiber, and the diameter of the collimated beam is about 2.1 mm. Behind the -plate (), the right-handed circularly polarized idler photons with OAM pass through a quarter wave plate (QWP) to become linearly polarized photons with OAM. Meanwhile, we set the intensifier gate width at a 5 ns coincidence gate time to ensure that only the idler photon events are registered[32]. In order to guarantee that the photons detected at the heralding detector and at the ICCD camera are from the same correlated photon pair, we must introduce a 35 m single-mode fiber (SMF) (corresponding to ) to compensate for the electronic delay of as mentioned above. Clearly, we over compensate for a time delay of (), which will be cancelled by introducing an electronic delay of by the ICCD-provided electronic system. Besides the gate width, there is another timing measure of note in an ICCD: the exposure time, which determines the accumulated time of the signal on the chip before it is read out. At first, we adjust the exposure time to be 100 ms (for each image) to ensure that each acquired frame is photon sparse[33], which leads to only about dozens of single-photon events. When only one image is studied, only a few scattering photons can be found, but no specific pattern can be recorded, because there are not enough detected photons to display the image. However, when summing many images, the pattern and its superposition can be recovered. Figure
Fig. 3. Examples of 100 triggered coincidence images of OAM photons by the ICCD camera and the summing pattern of 100 frame images. Every acquired frame has an exposure time of 100 ms and is photon sparse.
The wave character of twisted photons will be obvious by extending the exposure time, because the number of registered photon events per image grows as the exposure time increases. Figure
Fig. 4. Triggered images by the ICCD for different exposure times, for the OAM twisted photons. (a) The OAM twisted photons themselves and (b) in the double-slit interference. If the photons are detected one by one, the photons appear to be distributed quite randomly. As the total number of accumulated photons increases, the interference fringes appear.
In summary, we provided a classical-like method to describe the microscopic behaviour of photons. We theoretically predicted the APTs in the double-slit interference of single twisted photons, which still go along helical trajectories. Then, we experimentally obtained the low noise diffraction patterns by using the time-gated ICCD camera, which is triggered by a heralded detection. We demonstrated Young’s interference of twisted photons at the single-photon level and the interference of each photon with itself supported by the heralded detection. The interference patterns obtained from averaging many single-photon events correspond to the distributions of the end points of APTs. The average trajectories can also offer a method to explore the microscopic behaviour of other quantum particles.
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Wen-Rong Qi, Rui Liu, Ling-Jun Kong, Zhou-Xiang Wang, Shuang-Yin Huang, Chenghou Tu, Yongnan Li, Hui-Tian Wang. Double-slit interference of single twisted photons[J]. Chinese Optics Letters, 2020, 18(10): 102601.