Entangled ripples and twists of light: Radial and azimuthal Laguerre-Gaussian mode entanglement 1, 1 Natalia Herrera Valencia, ∗ Vatshal Srivastav, Saroch 1 1 1,2, Leedumrongwatthanakun, Will McCutcheon, and Mehul Malik † 1Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, UK 2 Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Vienna, Austria It is well known that photons can carry a spatial structure akin to a “twisted” or “rippled” wave- front. Such structured light fields have sparked significant interest in both classical and quantum physics, with applications ranging from dense communications to light-matter interaction. Harness- ing the full advantage of transverse spatial photonic encoding using the Laguerre-Gaussian (LG) basis in the quantum domain requires control over both the azimuthal (twisted) and radial (rippled) components of photons. However, precise measurement of the radial photonic degree-of-freedom has proven to be experimentally challenging primarily due to its transverse amplitude structure. Here we demonstrate the generation and certification of full-field Laguerre-Gaussian entanglement between photons pairs generated by spontaneous parametric down conversion in the telecom regime. By precisely tuning the optical system parameters for state generation and collection, and adopting recently developed techniques for precise spatial mode measurement, we are able to certify fidelities up to 85% and entanglement dimensionalities up to 26 in a 43-dimensional radial and azimuthal LG mode space. Furthermore, we study two-photon quantum correlations between 9 LG mode groups, demonstrating a correlation structure related to mode group order and inter-modal cross-talk. In addition, we show how the noise-robustness of high-dimensional entanglement certification can be significantly increased by using measurements in multiple LG mutually unbiased bases. Our work demonstrates the potential offered by the full spatial structure of the two-photon field for enhancing technologies for quantum information processing and communication. I. Introduction schemes [12–14] and controlling the propagation of light through complex media, with applications ranging from The spatial structure of light allows for the study of imaging through multi-mode fibres [15] to the develop- a variety of complex phenomena in the classical and ment of programmable optical circuits [16, 17]. Further- quantum regimes [1, 2]. In this context, the Laguerre- more, access to the complete transverse spatial degree-of- Gaussian (LG) modes of light have emerged as a popular freedom of photons is necessary for harnessing the advan- choice of basis. In addition to their aesthetic proper- tages of high-dimensional encoding for boosting quantum ties, LG modes form a complete and orthonormal ba- communications with higher capacities [18] and enabling sis of optical modes that are solutions of the paraxial noise-resistant entanglement distribution [19, 20]. wave equation in cylindrical coordinates [3]. As such, The capability of performing precise measurements of they are of inherent interest in many areas of physics the radial and azimuthal components of single photons featuring cylindrical or circular symmetries. The appli- (and their coherent superpositions) is key for their use in cations of LG modes extend over research fields as di- quantum information processing [4, 18]. However, us- verse as quantum information [4], matter waves [5, 6], ing the full resource of the transverse field is limited gravitational waves [7], and classical telecommunications by several challenges in accurately measuring full-field [8], where their relation to the quantisation of orbital- Laguerre-Gaussian modes. While multi-outcome mea- arXiv:2104.04506v1 [quant-ph] 9 Apr 2021 angular-momentum (OAM) [9], their unbounded Hilbert surements in both the azimuthal and radial part of spa- space, and their propagation properties have made them tial modes have been demonstrated through interfero- of key interest [10]. metric techniques [21–24], the scalability of such imple- An LG mode is composed of an azimuthal component mentations is difficult. A promising alternative is the given by a twisted helical wavefront of the form eiℓφ, use of multi-plane mode converters [14, 25–27], but loss with the azimuthal index ℓ corresponding to the quan- and cross-talk present additional challenges. On the tised orbital angular momentum (OAM) of photons, and other hand, single-outcome measurements in the spa- a radial component characterised by the radial index p tial degree-of-freedom can be performed through multi- [11]. Control over the radial and azimuthal components plane phase modulations [28], or well-established phase- of light has been key for enhancing classical commu- flattening techniques [29]. In the latter, spatial filters nication protocols based on mode-division-multiplexing composed of spatial light modulators (SLM) and single- mode fibres (SMFs) achieve accurate projective measure- ments of spatial modes. In this case, limitations arise from the loss associated with the use of SLMs, where it ∗ Email address: [email protected] is necessary to implement intensity masking in order to † Email address: [email protected] modulate the amplitude of the field [30]. In addition, the 2 issue of mode-dependent detection efficiencies presents photons created via SPDC, let us write the two-photon additional hurdles, especially when coherent superposi- state in the LG spatial mode basis: tions of modes are being measured [31, 32]. ℓsℓi ΨLG = C ℓsps ℓipi . (1) While the precise description of the full transverse | i pspi | i | i ℓi,pi,ℓs,ps structure of light requires both indices (ℓ and p), the X generation and detection of LG modes in the quan- Here, the subscripts s and i refer to the signal and idler tum domain initially focused on their azimuthal com- photon respectively. For simplicity, we use the expres- ponent [10, 29], with early experiments demonstrating sions ℓsps and ℓipi in Eq. (1) to refer to a single their use in quantum communication and entanglement | i | ℓs i ℓi photon in modes LGps and LGpi , respectively. The co- (please see [4] for a review of these). The “forgotten” ra- efficient Cℓsℓi 2 indicates the joint probability of finding dial quantum number p [33] has emerged in recent years | pspi | a signal photon in mode LGℓs and an idler photon in as the subject of several theoretical investigations [34– ps ℓi 37], with various experimental demonstrations of quan- mode LGpi . While these coefficients are primarily deter- tum radial correlations in the transverse field of pho- mined by the SPDC process, they also depend on the tons produced through spontaneous parametric down- collection optics involved in the measurement scheme. conversion (SPDC) [38–41]. These studies have shown An ideal LG entangled state exhibits perfect two-photon anti-correlations in azimuthal modes (ℓs = ℓi) and per- that although perfect azimuthal correlations can be easily − observed due to the conservation of OAM, entanglement fect correlations in radial modes (ps = pi). However, in the radial part is only available when the the pump and due a variety of reasons that can be attributed to mea- detected mode waists are finely tuned [34, 42]. Other ex- surement imperfections and the optics used, cross-talk periments have implemented spatial-mode measurements between modes can appear that reduces the quality of that only modulated the phase of the LG modes [42– the measured state. Below, we briefly discuss the origin 44], or used phase-only discretisation of the radial space of radial mode cross-talk. through Walsh functions [45]. However, access to the full An LG mode in the transverse momentum space can modal bandwidth of spatially entangled photons through be explicitly written in cylindrical coordinates as [34]: accurate measurements of radial modes requires both am- 2 ℓ ℓ w p! ρw | | plitude and phase-sensitive detection. This becomes par- LGp(ρ, φ)= (2) 2π(p + ℓ )! √2 ticularly clear when implementing projections onto co- s | | 2 2 2 2 herent superposition states, which are of key importance ρ w ℓ ρ w exp − L| | exp [iℓ (φ)] , when studying entanglement. × 4 p 2 Here we demonstrate the generation and measure- where w is the beam waist in the transverse position ment of the full transverse spatial field of a two-photon ℓ state generated through spontaneous parametric down- space (we have assumed z = 0), and Lp| |( ) is the as- · conversion (SPDC) in the telecom regime, certifying sociated Laguerre polynomial. In order to understand up to 26-dimensional radial and azimuthal Laguerre- the entanglement certification method used here, it is Gaussian mode entanglement with a dimensionality of also important to define bases that are mutually unbi- 43. Accurate spatial state projections onto states of the ased with respect to the LG basis (LG MUB). For prime LG basis (and any superposition of these) are performed dimensions d, these can be calculated by following the with an “intensity-flattening” technique that we have re- construction [49]: cently demonstrated in both the classical and quantum d 1 regime [32, 46]. Our measurement scheme optimises cor- 1 − jm+rm2 ˜jr = ε m (3) | i √ | i relations in the radial component through the fine adjust- d m=0 ment of the mode sizes in both the generation and detec- X tion systems, allowing us to certify entanglement between where m m denotes the standard LG basis, ε = 2πi{| i} transverse spatial modes of light spanning over 21 LG exp d is the principal complex d-th root of unity, mode groups. In addition, our high-dimensional entan- r 0,...,d 1 labels the chosen LG MUB, and j ∈ 0 {,...,d −1 labels} the basis elements. Figs. 1.b,d gled states are generated at 1550 nm, making it possible ∈ { − } to interface them with optical fibers at extremely low show examples of LG and LG MUB modes carrying both loss for quantum information schemes based on space- azimuthal and radial components (“twists and ripples”).