Entangled Ripples and Twists of Light: Radial and Azimuthal Laguerre-Gaussian Mode Entanglement

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(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”). ℓsℓi division multiplexing [47], and enabling the realisation of The coefficients Cpspi in Eq. (1) can be obtained by very bright entangled sources [48] for multi-photon ex- projecting the measured biphoton state onto LG mode periments. operators for the signal and idler photons. This is cal- culated by taking the overlap integral between the mea- sured state and the corresponding signal and idler mode II. Theory functions over the wave vector space: lsli C = ℓsps ℓipi ΨLG (4) pspi h | i To understand entanglement in the azimuthal and ra- 3 3 ℓs ℓi = d k d k Φ(k , k )[LG (k )]∗[LG (k )]∗, dial components of the transverse spatial field of a pair of s i s i ps s ps i Z Z 3

(a) (b)

(c)

Figure 1. Experimental Setup: a) A 775-nm Ti:Sapphire pulsed laser with a beam waist tailored by a telescope system pumps a non-linear ppKTP crystal and generates pairs of entangled photons at 1550 nm through Type-II spontaneous-parametric- down-conversion (SPDC). Accurate projective measurements in the Laguerre-Gauss basis and any of its mutually unbiased basis are performed with spatial light modulators (SLMs), intensity-flattening telescopes (IFT), and single-mode fibres (SMF). Correlations in the radial and azimuthal LG mode components of the transverse spatial field are obtained by measuring coincidence counts between pairs of photons using two superconducting nanowire detectors (SNSPD) connected to a coincidence 4 counting logic (CC). b) Representation of the complex amplitude describing the full-field mode LG2 and d) the LG MUB mode corresponding to state | 1˜1 i from the second mutually unbiased basis w. r. t. LG basis, and composed of a coherent superposition of 43 LG modes with p = 0, ..., 4 and ℓ = −8, ..., 7. The color scale corresponds to the phase of the mode and the brightness to the absolute value of its amplitude. c) and e) Examples of corresponding computer generated holograms (Type 1 complex amplitude modulation given in [30]) displayed on the SLMs to modulate the phase and amplitude of the LG and LG MUB modes in order to measure their spatial mode content.

where Φ(ks, ki) is the measured joint transverse mo- ing the pump and collection mode waists appropriately mentum amplitude (JTMA) of the biphoton state [46]. in order to maximise the correlations between Laguerre- The JTMA is a function determined by the spatial Gaussian modes with non-zero radial components (ps, profile of the pump, the phase matching conditions in pi > 0). SPDC [34, 50, 51], as well as the collection optics. The JTMA describes the measured correlations in the transverse-momentum degree-of-freedom of the gener- III. Experiment ated photons. The result of the overlap integral in Eq. (4) has been As shown in Fig. 1a, we use a Ti:Sapphire femtosec- shown to depend on the real space ratio between the ond pulsed laser (λp = 775nm, 500mW average power) to pump and the down-converted signal and idler mode pump a non-linear ppKTP crystal (1mm 2mm 5mm) × × waists, γ = wp/ws,i [34]. The mode waists wi and ws in order to generate pairs of spatially entangled photons are equivalent to the “collected” mode waists obtained at at 1550 nm through Type-II spontaneous-parametric- the crystal. The azimuthal part of the overlap integral down-conversion (SPDC). A telescope composed of lenses enforces the conservation of OAM, which for the case of L1 and L2 is used to set the 1/e2 radius of the pump to a Gaussian (ℓ =0) pump gives perfect two-photon corre- be wp = 450µm at the crystal (this is the Gaussian pump lations between azimuthal modes with indices ℓs = ℓi. mode waist considered in the ratio γ). After filtering out However, correlations between the radial modes are only− the pump with a dichroic mirror (DM), the signal and perfect in the limit of an infinite pump waist, which idler photons are separated using a polarising beam split- breaks down as the ratio γ 1 [34, 42]. In fact, in anyre- ter (PBS), and sent to spatial light modulators (SLMs) alistic experimental setup with→ finite-sized apertures and that are placed in the Fourier plane of the crystal via a optics, the pump waist is limited to a large extent. Har- 250 mm lens (L3). Note that reflection on the PBS flips nessing the resource of entanglement in the full transverse the sign of the idler mode azimuthal index, converting field thus requires one to increase the ratio γ by optimis- azimuthal anti-correlations into correlations. To perform 4 accurate projective measurements over the complete set (a) of LG modes spanning the field (and any superposition of these), the spatially varying amplitude and phase of the incoming photons are modulated by displaying computer- generated holograms (CGH) on the SLMs [30]. Fig. 1b 4 depicts an example full-field mode (LG2) containing both radial (p = 2) and azimuthal (ℓ = 4) components, with Fig. 1c showing the corresponding CGH used to mea- sure it. Figs. 1d and e depict a full-field mode and its corresponding CGH from the second (r = 1 in Eq.(3)) (b) 43-dimensional LG MUB containing a coherent superpo- sition of modes with p =0, ..., 4 and ℓ = 8, ..., 7. The CGH displayed on an SLM is used− for filtering a particular spatial mode such that it effectively cou- ples to a single-mode fibre (SMF). This method relies on ℓ the orthogonality of the LG modes, where a given LGp is measured by “flattening” its complex amplitude with ℓ a CGH of its complex conjugate [LGp]∗. However, the use of an SMF in this measurement scheme introduces (c) an additional Gaussian factor that results in undesired cross-talk between modes with different indices, which is particularly increased for radial modes [31, 32]. In order to minimise this cross-talk, the orthogonality between the input and projected mode is maintained through the use of a so-called intensity-flattening telescope (IFT, lenses L4 and L5) placed between the SLM and the SMF [32]. The IFTs in our system effectively magnify the back- propagated collection modes at the SLM planes by a fac- tor of 3.3 to mitigate the effects of the SMF Gaussian Figure 2. Two-photon correlations in azimuthal and radial LG modes. Two-photon coincidence counts show- component. In addition, the IFTs allow us to tailor the ing radial (a), azimuthal (b), and full-field (c) correla- collection modes to the size of the quantum spiral band- tions in the standard LG basis of each defined subspace width of the generated state [52], as determined by the {| m i , | n i}m,n (left), and in its second mutually unbiased extent of the JTMA [46]. Note also that increasing the basis {|˜j1 i , | k˜1 i}j,k (right). The inset on each figure illus- size of the collection modes at the SLMs is equivalent to trates the highest order mode measured in that particular reducing the size of the collection mode waists ws,i at the basis. Measuring the correlation matrices for all mutually crystal, which increases the ratio γexp to a value of 5.26, unbiased bases (MUBs) of each high-dimensional space, we in turn reducing radial mode cross-talk between signal obtain fidelities to the maximally entangled state shown in and idler photons[34]. the fourth column of Table I, allowing us to certify entan- glement dimensionalities of dent = 7 in the 11-dimensional radial subspace, dent = 11 in the 13-dimensional azimuthal subspace, and dent = 18 in the 23-dimensional full-field sub- IV. Results space.

We measure two-photon correlations in the azimuthal and radial components of the transverse field with mea- fidelity to fidelity bounds calculated from the Schmidt surements in the LG (standard, or computational) ba- coefficients of the target state Φ+ , we can certify the sis using CGHs for radial modes (ℓ = 0, p 0), az- entanglement dimensionality of| ouri experimental state. ≥ imuthal modes (ℓ Z, p = 0), and full-field modes We construct radial mode bases in dimension d = ∈ (ℓ Z,p 0). With the ability of projecting into any 11 (p =0,..., 10), azimuthal mode bases in d = 13 (ℓ = ∈ ≥ given superposition of spatial modes, we also perform 6,..., 6), and full-field LG mode bases in d = 23 (ℓ = measurements in all mutually unbiased bases with re- −6,..., 5 and p = 0, 1, 2). Since a complete set of d +1 spect to the LG basis (LG MUBs) following the con- −mutually unbiased bases can only be constructed for struction given in [49]. Measurements in the complete prime dimensions d, we use an asymmetrical azimuthal set of LG MUBs allow us to determine the exact fi- bandwidth to build the full-field LG MUBs in a prime + + + delity F (ρ, Φ ) = Tr( Φ Φ ρ) of our experimen- dimension. | i | ih | tally measured state ρ to the maximally entangled state As shown in Fig. 2, our measurements result in strong + 1 Φ = ℓp ℓp , where d is the entangle- diagonal two-photon correlations over the complete set of | i √d ℓ,p | is | ii ment dimensionality and d d is the number of entan- the LG and LG MUBs of the radial, azimuthal and full- P gled modes in the state [53].× By comparing our measured field mode spaces. The off-diagonal correlations in the 5

Fidelity with multiple bases (a) 1 (b) (c) 1500 (f) 2 0.6 10 3 B 1000 25 4 20 0 B 0.55 24 5 30 500 B 40 23 6 0 B 0 0.5 22 (d) (e) 1500

7 Fidelity B 21 10 1000 8 20 0.45 Mode Group (idler) 30 500 9 B 40 18 0 12 3 4 5 6 7 8 9 10 20 30 40 10 20 30 40 5 10 15 20 Mode Group (signal) Number of MUBs

Figure 3. Full LG-mode entanglement in a 43-dimensional subspace. (a) Two-photon coincidence counts showing correlations in the standard LG basis of radial and azimuthal modes belonging to 9 different mode groups (indicated by the pink lines). (b-e) Two-photon coincidence counts showing correlations in the first 4 mutually unbiased bases (LG MUBs) with respect to the standard basis. Correlations in 21 mutually unbiased bases allow us to lower bound the fidelity of our state to a maximally entangled state, and certify an entanglement dimensionality of dent = 26. The advantage of using measurements in more LG MUBs is shown in (f), where the estimated fidelity allows us to violate higher dimensionality bounds (Bdent−1), thus allowing us to certify higher entanglement dimensionality as the number of MUBs used increases. standard radial basis (Fig. 2a) correspond to the expected the coincidence counts of the LG basis, as demonstrated cross-talk originating from the overlap integral in Eq. 4 in [53]. + with our γexp = 5.26, with additional cross-talk due to Fidelities to the maximally entangled state ( Φ ) and experimental imperfections such as alignment. The cross- a non-maximally entangled target state ( Φ| ) ini each | i talk between adjacent azimuthal modes, i.e. ℓs = ℓi 1 mode space are shown in Table I. Errors in the fidelities ± observed in Fig. 2b can be attributed to experimental are calculated via Monte-Carlo simulation of the experi- imperfections such as misalignment and finite SLM reso- ment that propagates statistical Poisson error associated lution. These two effects can also be observed in the full- to the photon count rates. With an appropriate choice of field correlations shown in Fig. 2c. The cross-talk in the target state, is possible to obtain higher fidelities F (ρ, Φ) LG MUBs (second column of Fig. 2) arises from the un- compared to the ones calculated with respect to the max- even distribution of the diagonal correlations in the stan- imally entangled state F (ρ, Φ+). Nevertheless, the fi- dard basis. This probability distribution is determined delity bounds for certifying entanglement dimensionality by the quantum spiral bandwidth of the SPDC state, become harder to violate for the tilted-witness, result- which dictates a higher coincidence probability for lower ing in the same or lower certified dimensionality. The order LG modes [34], and a mode-dependent coupling entanglement dimensionalities shown in Table I are thus efficiency that originates from the use of SMFs [31, 32]. calculated using the fidelity F (ρ, Φ+) of our measured Optimising the parameter γ increases the available en- state to a maximally entangled target state [53]. tanglement by flattening the quantum spiral bandwidth, To demonstrate the potential of our technique for ac- allowing us to access higher-order correlations and reduc- cessing very high-dimensional LG entanglement, we use ing the cross-talk in the LG MUB measurements. azimuthal and radial modes to construct a full-field stan- Even though we can optimise the correlations over the dard LG basis with dimension d = 43, composed of LG transverse spatial field of the biphoton state through the modes with ℓ = 8,..., 7 and p = 0,..., 4. This basis manipulation of the pump and collection mode waists, contains states with− mode group order (MG =2p+ l +1) a “flat” spiral bandwidth can only be achieved with an up to 9. The LG mode groups consist of modes that| | ex- infinite pump waist. Since the SPDC state is a non- perience the same Gouy phase, and are of immense in- maximally entangled state, one can also characterise its terest in the study of multi-mode waveguides. As shown spatial correlations through a tilted-basis witness that in Fig. 3a, the cross-talk between modes in the standard uses the fidelity to a general, non-maximally entangled basis has a particular structure. When examining them target state F (ρ, Φ) [53]. The performance of this tilted with respect to their respective mode groups (pink lines), witness improves with an appropriate choice of the tar- we can see that very little cross-talk is present within get state, with fidelity ideally approaching unity. An in- the same mode group. However, alignment imperfections formed guess for a good target state can be made from the leading to correlations between azimuthal modes with measured correlations in the standard basis, which then ℓs = ℓi for a given p result in cross-talk between adjacent 6 informs the construction of appropriate tilted-bases, in mode groups MGi = MGs 1. On the other hand, radial ± manner similar to LG MUBs. In this process, it is impor- mode correlations between modes with ps = pi manifest 6 tant to correct for the effects of mode-dependent loss in as cross-talk that skips a mode group MGi = MGs 2. ± 6

Table I: Measured fidelities F (ρ, Φ+ ) and F (ρ, Φ ) of experimental LG-entangled states (ρ) to the maximally ( Φ+ )| andi non-maximally| i ( Φ ) entangled target states | i | i + Type ℓ p ddent F (ρ, Φ ) F (ρ, Φ) Radial 0 0,..., 10 11 7 61.8 0.5 % 69.0 1.0 % ± ± Azimuthal 6,..., 6 0 13 11 83.3 0.3 % 85.7 0.6 % − ± ± Full Field 6,..., 5 0, 1, 2 23 18 75.1 0.1 % 75.5 0.1 % − ± ± Full Field† 8,..., 7 0,..., 4 43 26 59.5 0.9 %– − ±

The third column lists the entanglement dimensionality dent certified using the fidelity to the maximally entangled state F (ρ, Φ+). † This fidelity value corresponds to a lower bound to the fidelity obtained from measurements in 21 of the 44 MUBs of the 43-dimensional space.

This structure can be understood as a natural conse- glement with measurement settings tailored to the spa- quence of the way mode groups are defined, with a factor tial distribution of the measured two-photon state, and of two appearing before the index p. an intensity-flattening technique that ensures accurate Instead of estimating the exact fidelity through mea- state projections onto any given spatial mode, while min- surements in all LG MUBs for a given dimension, the imising radial mode cross-talk. Careful control over the certification of high-dimensional entanglement is possi- mode waists of the pump and collected photons increases ble by lower bounding the fidelity to a given target en- the correlation strength within the radial and azimuthal tangled state with measurements in at least two MUBs. components of the two-photon field and allows for the It has been theoretically shown that as the number of certification of entanglement dimensionalities up to 26. bases used for lower bounding the fidelity is increased, By measuring correlations in a 43-dimensional set of LG a higher tolerance to noise is obtained [53]. This is modes and its mutually unbiased bases (LG MUBs), we especially advantageous in scenarios where there is a are able to characterise high-dimensional entanglement in lot of noise present to begin with. Here we experi- modes spanning nine LG mode groups, and observe how mentally demonstrate this noise advantage, as can be two-photon inter-modal cross-talk follows a structure re- clearly seen in Fig. 3f. Increasing the number of LG lated to the mode group orders. In addition, we demon- MUBs used for lower bounding the fidelity from 2 to strate clearly how measurements in additional LG MUBs 21 results in an improvement in the fidelity bound from enables one to certify high-dimensional entanglement in F˜(ρ, Φ+) = 44.6 0.9% to a value of 59.5 0.9%. This a noise-robust manner. Our techniques are significant for improvement in fidelity± corresponds to an increased± cer- the emerging field of high-dimensional quantum informa- Full tified entanglement dimensionality from dent = 20 to 26. tion and will prove beneficial for quantum technologies The results show that the use of additional MUBs offers harnessing Laguerre-Gaussian modes of light. a significant advantage for entanglement certification in Note: We have recently become aware of a related noisy regimes. work [41] that characterised LG radial mode entangle- ment through full quantum state tomography in state spaces with local dimension 4.

V. Conclusion Acknowledgments We have demonstrated high-dimensional entanglement in the telecom regime between two photons in their full- This work was made possible by financial support from field Laguerre-Gaussian (LG) spatial mode basis consist- the QuantERA ERA-NET Co-fund (FWF Project I3773- ing of radial and azimuthal components. We are able N36) and the UK Engineering and Physical Sciences Re- to harness the complete resource of LG mode entan- search Council (EPSRC) (EP/P024114/1).

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