Path identity as a source of high-dimensional entanglement

Abstract

We present an experimental demonstration of a general entanglement-generation framework, where the form of the entangled state is independent of the physical process used to produce the particles. It is the indistinguishability of multiple generation processes and the geometry of the setup that give rise to the entanglement. Such a framework, termed entanglement by path identity, exhibits a high degree of customizability. We employ one class of such geometries to build a modular source of photon pairs that are high-dimensionally entangled in their orbital angular momentum. We demonstrate the creation of three-dimensionally entangled states and show how to incrementally increase the dimensionality of entanglement. The generated states retain their quality even in higher dimensions. In addition, the design of our source allows for its generalization to various degrees of freedom and even for the implementation in integrated compact devices. The concept of entanglement by path identity itself is a general scheme and allows for construction of sources producing also customized states of multiple photons. We therefore expect that future quantum technologies and fundamental tests of nature in higher dimensions will benefit from this approach.

Keywords: entanglement by path identity; high-dimensional entanglement; orbital angular momentum; path indistinguishability.

Fig. 1.

Basic concept. Gray boxes labeled with underlined uppercase letters represent nonlinear crystals, each pumped coherently and each generating with a small probability a pair of photons via an SPDC process. Each generated photon pair is in OAM state $\left|\mathrm{0,0}\right.⟩$ with a small contribution of higher-order terms. The two down-converted photons then propagate along their paths in the direction indicated by the arrows and acquire phase shifts ${\phi }_i$ as well as additional quanta of OAM due to phase and mode shifters. (A) The pump beam, represented by an arrow, gives rise to an SPDC process in crystals A and B. Photons generated in crystal A are reflected into crystal B such that their paths are overlapped with paths of photons generated in crystal B. As a consequence, the two coherent SPDC processes in crystals A and B are indistinguishable and the generated photon pairs leave the setup in a two-dimensionally entangled Bell state $1/\sqrt{2}\hspace{0.17em}\left(\left|\mathrm{0,0}\right.⟩+\exp \left(i\phi \right)\left|\mathrm{1,1}\right.⟩\right)$. The quantum of OAM is imparted to the photon by a spiral phase plate shown in Inset. (B) A schematic picture of the setup in A, where the pump beam is not shown. (C) The addition of the third crystal to the setup increases the entanglement dimension by one. The resulting state is thus $1/\sqrt{3}\hspace{0.17em}\left(\left|\mathrm{0,0}\right.⟩+\exp \left(i{\overline{\phi }}_1\right)\left|\mathrm{1,1}\right.⟩+\exp \left(i{\overline{\phi }}_2\right)\left|\mathrm{2,2}\right.⟩\right)$, where ${\overline{\phi }}_1={\phi }_2$ and ${\overline{\phi }}_2={\phi }_1+{\phi }_2$. (D) One can stack multiple setups from A to acquire a series of $d$ crystals that produces a $d$-dimensionally entangled state $1/\sqrt{d}\left(\left|\mathrm{0,0}\right.⟩+\exp \left(i{\overline{\phi }}_1\right)\left|\mathrm{1,1}\right.⟩+\cdots +\exp \left(i{\overline{\phi }}_{d-1}\right)\left|d-1,d-1\right.⟩\right)$, where the relative phases ${\overline{\phi }}_i={\sum }_{j=d-i}^{d-1}{\phi }_j$ are adjusted by an appropriate choice of phase shifters ${\phi }_j$. The magnitudes of the individual modes are modified by varying the power with which the respective crystals are pumped.

Fig. 2.

Experimental setup. Three-dimensional states are created by elements in boxes labeled 1st dim, 2nd dim, and 3rd dim. Three periodically poled KTP crystals A, B, and C are pumped with a continuous-wave laser beam at the central wavelength of 405 nm. Frequency-degenerate down-converted photons created by type II collinear SPDC propagate along identical paths into the detection system shown in box Detection. Photons originating in crystal B are created in $\left|\mathrm{2,2}\right.⟩$ OAM mode because of a spiral phase plate (SPP $+4$) inserted after the first beam splitter (BS). In addition, photons originating in crystal C are created in $\left|-2,-2\right.⟩$ mode due to an extra mirror that effectively works as a $-8$ mode shifter as is explained in the main text. The pump beam is separated from the down-conversion beam by dichroic mirrors (DM) and a band-pass filter (BPF). Before detection, the two down-converted photons are separated on a polarizing beam-splitter (PBS). The state tomography in the OAM degree of freedom is done by projective measurements (27) where specific holograms are projected on two spatial light modulators (SLMs). The reflected photons are subsequently coupled into single-mode fibers and detected by single-photon detectors (Det). The resulting signals are postprocessed by a coincidence counting module (&). The relative phases ${\phi }_1$ and ${\phi }_2$ can be adjusted by phase shifters implemented with trombone systems (TS). The magnitudes of individual terms in the quantum state are controlled by setting the splitting ratio of the beam splitters. For the detailed diagram of the experimental setup see SI Appendix.

Fig. 3.

Examples of the three-dimensionally entangled states $\left|{\psi }_1\right.⟩$, $\left|{\psi }_4\right.⟩$, and $\left|{\psi }_5\right.⟩$ produced (Table 1). With our method we can control the relative phases, as demonstrated in A and B, as well as relative magnitudes, as shown in C. Only real parts are shown; imaginary parts lie in the range $\left(-0.12,0.12\right)$ for all cases. Background noise contributions lie in the range $\left(-0.04,0.04\right)$ for all cases and are explicitly shown only in A. This background is omitted in B and C to improve readability. Green (solid) and orange (hatched) bars represent positive and negative values of reconstructed density matrices, respectively. Gray translucent bars represent the theoretical expectation. Fidelities of the measured states with their reference states are $87.0\pm 0.5\%$, $89.0\pm 0.4\%$, and $84.8\pm 0.8\%$, respectively.