Non-classical photonic spin texture of quantum structured light

Classical structured light with controlled polarization and orbital angular momentum (OAM) of electromagnetic waves has varied applications in optical trapping, bio-sensing, optical communications and quantum simulations. However, quantum noise and photon statistics of three-dimensional photonic angular momentum are relatively less explored. Here, we develop a quantum framework and put forth the concept of quantum structured light for space-time wavepackets at the single-photon level. Our work deals with three-dimensional angular momentum observables for twisted quantum pulses beyond scalar-field theory as well as the paraxial approximation. We show that the spin density generates modulated helical texture and exhibits distinct photon statistics for Fock-state vs. coherent-state twisted pulses. We introduce the quantum correlator of photon spin density to characterize nonlocal spin noise providing a rigorous parallel with electronic spin noise. Our work can lead to quantum spin-OAM physics in twisted single-photon pulses and opens explorations for phases of light with long-range spin order. Twisted light beams allow unique control of light-matter interaction, but classical models cannot describe this phenomenon at the single-photon level. Here, the quantum state of structured photons is derived instead from quantum field theory which captures the quantum uncertainty in its angular momentum and the non-local photonic spin density correlation.

S tructured single-photon pulses are an important frontier for spin and orbital angular momentum (OAM) [1][2][3] . As a quantum information carrier, single-photon pulses with OAM have been achieved in the solid-state system with quantum dots recently 4 and have been exploited to construct a quantum network with higher channel capacity [5][6][7][8][9][10] . The spin and OAM of light have also attracted increasing attention in an emerging research field-spin-orbit photonics 11 , which studies photon spin-OAM transfer [12][13][14][15] and light−matter angular momentum exchange in the near-field region [16][17][18] or transfer of optical OAM to bounded electrons 19 or photoelectrons 12,20 . Spin-1 quantization is also the hallmark of photonic skyrmions and topological photonic phases of matter 21,22 . A quantum field theory framework is needed to study the non-classical properties such as 3D noise of the angular momentum of light.
Existing theories of quantum light−matter interaction have advanced over the last two decades to capture a plethora of phenomena related to SAM and OAM of light [23][24][25][26][27][28][29][30][31][32][33][34] . Important outstanding questions remain even within this large body of work which is the focus of this manuscript, namely-photon statistics, 3D quantum spin and OAM vector density, 3D quantum noise in SAM/OAM, and single-photon quantum states. Figure 1a shows the well-known regime of twisted laser beams which contain an enormous number of photons. At the single-photon level, both existing semi-classical 23,24 and approximate quantum theories break down 30,32 . In the widely adopted state-space description of single photons or entangled photons f l; s j ig with Stokes parameters and the Poincaré sphere 28,33 , the rich spatial texture of spin and OAM vectors is ignored completely. Specifically, important open questions remain on the full 3D projection of photon spin and OAM at the quantum level beyond the scalarfield theory and paraxial approximation. Heisenberg uncertainty relations for photon angular momenta can affect quantum metrology experiments which require a quantum theoretic framework. These Heisenberg uncertainty relations between different photon OAM 3D components are the canonical quantum characteristics of angular momentum. Similarly, for applications such as secure quantum communication, twisted single-photon pulses in the quantum limit with few photons (see Fig. 1b) are required. In this technologically relevant limit, quantum statistics of photons will reveal behavior significantly different from the quasi-classical Poisson behavior exhibited by traditional OAM laser beams. These fundamental, as well as technologically relevant problems, require the definition of single-photon quantum state along with OAM/SAM operators.
In this work, we present an important frontier for quantum structured light involving twisted space−time wavepackets of light. We first construct the wave function of a quantum twisted pulse, as well as a twisted laser beam, from quantum field theory 35 instead of from the single-particle Schrödinger equation in the first-quantization picture [36][37][38] . By exploiting the quantum operators of the angular momenta of light 39 , we evaluate the mean value, as well as the quantum uncertainty of the photon spin operator vector. Apart from the well-established global properties of polarization, we also investigate the quantum properties of the photon spin density vector, i.e., the spin texture, which is a function of space and time. We show that beyond the paraxial approximation, the photon spin density of a Bessel single-photon pulse can exhibit rich spatial texture. Our work builds on previous important work in the field 23-34 . Our proposed framework provides a powerful and versatile tool to engineer the local photon spin and OAM densities of a quantum structured light pulse, specifically for spatiotemporal optical vortices 40,41 .
Non-local spin noise (i.e., spin density correlation) for electrons is a fundamental signature of quantum phases of magnetic condensed matter 42 , specifically in phases of matter such as quantum spin liquids without magnetic order 43 . However, no such quantum spin noise operator has been defined for photons till date. Our theoretical formalism allows us to overcome this challenge. Here, we introduce the quantum correlator of photonic spin density to characterize the nonlocal spin noise in light. This paves the way to explore exotic phases of light with long-range spin order.
We emphasize that our work is immediately amenable to experimental verification. We predict that for Bessel pulses with large OAM, there will exist large fluctuations in the OAM along orthogonal directions. This additional quantum noise can be verified in metrology experiments even with OAM laser beams. Recently, it was demonstrated that the nitrogen-vacancy (NV) center in diamond can be used as a quantum sensor for detecting the local spinning nature of photons 44 . The spin density of the off-resonant optical beam can induce an effective static magnetic field for the electron spin of the NV center, which itself is an atomic-scale magnetometer working at room temperature. Imaging of our discovered helical spin-density structure in this work can be realized with the same technology in the near future. Furthermore, our proposed non-local spin density correlation can also be measured in compound measurements with two or multiple NV centers.

Results and discussion
Quantum spin and orbital angular momenta of light. The full quantum operator of photon spin is given by 39 and OAML of light in the Lorenz gauge within quantum field Fig. 1 Contrast between a traditional twisted beam and quantum pulse. Schematic of a traditional twisted beam a compared to the quantum twisted pulse b put forth in this paper. The semi-classical theory only captures the mean orbital angular momentum (OAM) of a laser beam with large photon number. However, the quantum effects of photon statistics, vectorial uncertainty relations, and non-local spin noise require a quantum-theoretical framework put forth in this paper.
theory. The operatorsŜ andL obey the canonical commutation relationships where ϵ ijk is the third-order Levi-Civita symbol.  39 . Note,Ê ? andÂ ? are the transverse part of the electric field and the vector potential, respectively.
Using the circularly polarized plane waves, we can expand the observable photon spin and OAM operators as [45][46][47] (please refer to Supplementary Notes 1 and 2) where e(k, 3) = k/|k| is the unit vector and λ = ± denotes the left circular polarization (LCP) and right circular polarization (RCP) separately (see Supplementary Note 1). The ladder operators of the plane wave with wave vector k and polarization λ satisfy the bosonic commutation relation ½â k;λ ;â y k 0 ;λ 0 ¼ δðk À k 0 Þδ λλ 0 . The photon helicity is given byΛ ¼ _ R d 3 kâ y k;þâk;þ Àâ y k;Àâk;À h i . We emphasize that the spin and OAM are separately observable due to the quantum commutation relations 39 To show the striking symmetry between the angular momentum of photons and electrons, we define a field operator for light in real spaceψðrÞ ¼ ½ψ þ ðrÞ;ψ À ðrÞ T , wherê For the source-free case, our defined field operator in the Heisenberg picture satisfies the homogeneous wave equation Now, we can re-express the OAM and helicity operators of light in parallel to their electron counterpartŝ wherep ¼ Ài_∇ is momentum operator andσ z ¼ diag½1; À1 is the Pauli matrix. However, the similar expression for the spin operatorŜ obs can not be obtained in real space. The unit polarization vector e(k, 3) in Eq. (3) for each plane wave is kdependent, i.e., dependent on its spatial momentum.
Quantum wave function of twisted light pulses. In previous sections, we have shown that bothŜ obs andL obs are vector operators. However, in previous studies, usually only their projections on the propagating direction have been fully studied 23,24,34 . Their mean value on the transverse plane and more importantly, their quantum fluctuations have not been investigated. On the other hand, the near-field techniques have now been well developed. This makes it possible to measure and engineer the angular-momentum density of light, which is a vector function of space and time, in experiments. Thus, a fully quantum theory beyond the paraxial approximation to explore all classes of twisted pulses in a united framework is highly desirable.
Here, we present this powerful theoretical tool by generalizing the quantum theory of continuous-mode field 35,48 to the twistedpulse case.
We first define the single-photon wave-packet creation operator for a twisted photon pulsê as a coherent superposition of plane-wave modes. The pulse shape and other quantum properties of the pulse are fully determined by the spectral amplitude function (SAF) ξ λ (k). In the following, we denote the propagating direction of the pulse as the z-axis and work in the cylindrical coordinate in k-space Here, ρ k is the radial distance from the k z -axis, φ k is the azimuth angle, and e denotes the corresponding unit vector. The SAF of a twisted pulse with deterministic OAM can be generally expressed as Usually, the amplitude η λ (k z , ρ k ) is symmetric in the transverse plane, i.e, it is independent on the azimuth angle φ k . The phase factor expðimφ k Þ with an integer m will lead to the OAM of light in z-direction of a single-photon pulse as shown in the following. The SAF is required to satisfy the normalization condition R d 3 k ξ λ ðkÞ 2 ¼ 1. This guarantees thatâ y ξλ obey the bosonic commutation relation Then, the wave-packet creation operatorâ y ξλ can be treated as a normal ladder operator of a harmonic oscillator. Using this commutation relation, we can construct the wave function of all classes of quantum pulses in the standard way, such as the most common n-photon Fock-state and coherent-state pulses 35 (please refer to Supplementary Note 3) and where n ¼ jαj 2 is the mean photon number in the coherent-state pulse. The wave function of a squeezed-state pulse, an entangled two-photon pulse 49 , or an ultra-short spatiotemporal vortex pulse 40,41 can also be constructed similarly. Here, the polarization of the pulse is fixed as one of the circular polarizations. However, linearly or elliptically polarized quantum pulses can also be constructed with the superposition of two circular polarization ladder operatorsâ ξλ ðλ ¼ ±Þ. We also note that a twisted laser beam can be characterized by a wave function with a very long pulse length and a very large photon number. Thus, our method also captures the cases of continuous OAM laser beams used widely in experiments.
Without loss of generality, we only take the Bessel pulses as an example to show the quantum properties of the spin and OAM of twisted pulses. Other twisted pulses, such as a Bessel−Gaussian or Laguerre−Gaussian pulse, can be treated similarly. The singlefrequency Bessel beam is the superposition of all plane waves on the cone with the same frequency ω = c|k|, k z , and polar angle θ k = θ c as shown in Fig. 2a. Then, the SAF of a Bessel pulse with a Gaussian envelope can be expressed η λ (k z , ρ k ) as the product of two Gaussian functions The first Gaussian function with width 1/σ z and center wave vector k z,c characterizes the envelope of the pulse in the propagating direction. The pulse length on z-axis in real space is given by σ z = cτ p with τ p the pulse length in time domain (please refer to Supplementary Note 3). We show the energy density of a Bessel pulse in Fig. 2b, c. Distinct from previous works 30, 50 , we do not add a delta function [such as δ(θ k − θ c )] in the SAF to characterize its distribution property in the xy-plane. This will cause a serious issue that the wave functions of the quantum pulses cannot be normalized, because ∫d 3 k|ξ λ (k)| 2 ∝ δ(θ k − θ c ). Instead, we utilize another Gaussian function with width 1/σ ρ and center value k ?;c ¼ k z;c tan θ c . These two Gaussian functions should have the same ratio between center wave-number and the width, i.e. k z,c σ z = k ⊥,c σ ρ ≡ C 0 . In the narrow bandwidth limit C 0 ≫ 1, our defined SAF is well normalized (please refer to Supplementary Note 3). We also note that in contrast to the Bessel-modebased method 31 which only applies to Bessel beams, our generalized plane-wave-based framework is amenable to unify the theory of all classes of quantum pulses.
Quantum statistics of the photon spin. Traditionally, the angular momentum carried by each photon in a twisted laser beam has been calculated semi-classically via the ratio of angular flux to the energy flux 23,24 and only its projection on the propagating axis has been studied. Although the projection of the photon spin and OAM of a non-paraxial beam on the transverse plane has caused attention recently [25][26][27]29,34 , a systematic and comprehensive investigation of the vector nature of the photon spin and OAM is still missing. Specifically, the Heisenberg uncertainty relation for photon OAM has never been investigated. On the other hand, many researchers have also tried to establish a quantum theory of the angular momentum of light in the last two decades [30][31][32][33]51 . However, a fully quantum framework to handle arbitrary quantum pulses beyond the paraxial approximation has not been found.
We first calculate the mean value of the spin of a Fock-state Bessel pulse with polarization λ and photon number n (please refer to Supplementary Note 4), Here, we see that the magnitude of the spin carried by each circularly polarized photon is usually smaller than ℏ and approaches to ℏ asymptotically in the paraxial limit (θ c → 0) 26,30 . This is significantly different from the helicity, which is exactly ℏ.
If the SAF of a pulse is symmetric in the xy-plane, then the mean value of the spin in the xy-plane vanishes, i.e., hŜ obs x i ¼ hŜ obs y i ¼ 0. However, we show that the quantum fluctuations of photon spin in the xy-plane are not zero. The standard derivations of the spin of an n-photon Fock-state Bessel pulse are given by This is significantly beyond the previous semi-classical theory 23,24,34 , in which the quantum statistics of the photon spin cannot be studied. Similarly, we can evaluate the mean value of the spin of a coherent-state Bessel pulse with polarization λ and photon number n ¼ jαj 2 , Here, we see that the average spin carried by each photon is still _ cos θ c and the spin's projection on xy-plane also vanishes. However, the quantum statistics of the photon spin for a coherent-state pulse is significantly different from that of a Fockstate pulse, The Poisson statistics of a coherent pulse leads to non-vanishing ΔŜ obs z in contrast to a sub-Poisson Fock-state pulse.
Quantum statistics of the photon OAM. Heisenberg's uncertainty relation is the canonical quantum characteristics of angular momentum. However, this relation for photon OAM has never been addressed till date. Here, we present a quantitative investigation about the quantum statistics of photon OAM. We discover that for beams with large OAM number, there exist large fluctuations for the OAM operators in the orthogonal directions i.e. in the transverse plane. This quantum effect can be observed in experiment even with traditional OAM laser beams. The mean value ofL obs z for a Fock-state twisted pulse with photon number n is given by, This reduces to the well-known result obtained from the semiclassical method that each twisted photon carries mℏ OAM 23,24 . We see that hL obs z i is independent of the photon polarization. It is only determined by the photon number n and integer m in the helical phase factor expðimφ k Þ if η λ (k) is not a function of φ k . We can also verify that, in this case, the mean value of photon OAM in xy-plane vanishes, i.e., hL obs x i ¼ hL where x ¼ tan θ c 2 ð0; 1Þ and we have used the inequality relation a 2 x 2 + b 2 /x 2 ≥ 2|ab| and the narrow-band condition C 0 ≫ 1. This immediately leads to the Heisenberg relation ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi The other two Heisenberg relations for photon OAM are trivial due to the vanishing mean values ofL obs x andL obs y . Similar results also hold for a coherent-state twisted pulse, but with nonvanishing ðΔL obs z Þ 2 ¼ _ 2 nm 2 . Interesting works have been reported to demonstrate the uncertainty relation between the conjugate variables of angle φ of light and its derivativel z Ài_∂=∂φ in the first-quantization picture 52,53 , i.e., Δφ θ Δl z ≥ ℏ[1 − 2πP(θ)]/2. In contrast, our focus is the Heisenberg uncertainty corresponding to the canonical 3D angular commutation relation of photons. On the other hand, we note that for transverse EM fields, {HamiltonianĤ, momentum P, helicityΛ} has been select as the complete set of commuting observables to specify a photon state usually. Here, we see that a single-photon pulse carrying determinate integer OAM in the propagating direction is not the eigen state of ðL obs Þ 2 . Our predicted large OAM fluctuations in xy-plane can be verified in experiments (see Fig. 3a). The quantum uncertainties ofL obs x andL obs y are linearly proportional to the photon number n in a Bessel pulse as shown in Fig. 3b and proportional to the square of the helical phase index m in Eq. (11) as shown in 3c. From Eq. (23), we see that the OAM fluctuations in the transverse plane are strongly dependent on the polar angle θ c of a Bessel pulse. There exists a minimum-uncertainty angle due to the inequality a 2 x 2 + b 2 /x 2 ≥ 2|ab| (x ¼ tan θ c ) in (24) as shown in Fig. 3d. For a optical pulse, the ratio C 0 between its center wave number and its width is usually very large, e.g., C 0 ≈ 188 for a 50 fs pulse with center wave length λ c = 500 nm. In our numerical simulation, we set C 0 = 100. We note that these large OAM fluctuations in the transverse plane also exist in traditional OAM laser beams, such as the routinely used Laguerre−Gaussian beams in experiments.
Quantum spin texture of a single-photon pulse. We show that the spin texture of a single-photon pulse can exhibit a very rich and interesting structure in the case beyond the paraxial approximation. The photon spin texture is characterized by the spin density operator s obs ðr; tÞ ¼ ε 0Ê? ðr; tÞ Â ? ðr; tÞ: ð26Þ Similar to the electric or magnetic fields, the spin density can be treated as a vector field and can be measured locally 44 . We emphasize that as a vector, the spin density is neither purely longitudinal or purely transverse in most cases. In the singlemode plane-wave limit, the spin density will be a spaceindependent constant, i.e., ∇ hŝ obs ðr; tÞi ¼ ∇ Á hŝ obs ðr; tÞi ¼ 0.
The mean value of the spin density of a Fock-state Bessel pulse is given (please refer to Supplementary Note 5) where with r = ρe ρ + ze z . The spin density of a coherent pulse can be evaluated similarly. Here, we can see the following key characters of the spin density: (i) its projection in the xy-plane is symmetric around z-axis. This causes the corresponding spatial integral to vanish as shown in the previous section, i.e., hŜ obs x i ¼ hŜ obs y i ¼ 0; (ii) its xy-plane projection is parallel or anti-parallel to the azimuth-angle-dependent unit vector e φ and it does not have a radial component. This leads to the helical spin texture as shown in Fig. 4; (iii) its xy-plane projection contains the product of two different Bessel functions. The sign of a Bessel functions flips when crossing its zeros. This leads to the oscillation between clockwise and anti-clockwise structures in the spin texture; (iv) its projection on z is independent on φ. For a small angle θ c , the term $ cos 4 ðθ c =2Þ dominates. Thus, the sign of s z is always positive (negative) for LCP (RCP) pulse. This leads to the non-vanishing global spin hŜ M;z i.
We show the spin texture of an LCP single-photon (n = 1) Fock-state Bessel pulse in Fig. 4. Here, we only look at the spin density vector field on the plane k z,c z = ct, at which the Gaussian functions in Eqs. (28) and (29) reach their maxima. In this case, the space-dependent spin density is only a function of the radius ρ and the azimuthal angle φ contained in e φ . For a pulse with small polar angle θ c = 0.1π, almost only a clockwise structure can be observed in panel a. However, for a pulse with a larger polar angle θ c = 0.2π, the oscillation between clockwise and counterclockwise structure can be observed clearly. This oscillation can only be obtained beyond the scalar-field theory and the paraxial approximation. For higher-order Bessel pulses with m > 0, the fine structure of the spin density is significantly different from the m = 0 case. The innermost ring changes from clockwise to counter-clockwise as shown in panels c and d. We also note that the Bessel pulse with m = 1 is very special (see panel c), because the spin texture has a peak instead of a hole at the center.
In Fig. 5, we show more details of the projection of the spin density vector field on xy-plane and z-axis, respectively. In panel a, we look at the projection of the spin density on xy-plane s φ e φ   Fig. 4. Here, we clearly see the oscillation induced by the Bessel function in Eq. (28). Specifically, the vertex at the center for m = 1.
Nonlocal spin noise of light. To characterize the nonlocal spin noise of light, we introduce the quantum correlation function of the photon spin density. Due to the vector nature of the spin density, the full two-point correlation should be characterized by a 3 × 3 correlation matrix as shown in the Supplementary Note 6.
Here, we only describe the equal-time correlator s obs z ðr; tÞŝ obs z ðr 0 ; tÞ . In the paraxial limit (θ c ≈ 0), the two-point correlation functions for a Fock-state and a coherent-state pulse are given by (please refer to Supplementary Note 6) n ξλ ŝ obs z ðr; tÞŝ obs z ðr 0 ; tÞ n ξλ % _ 2 δðr À r 0 Þn ψ Àλ ðr; tÞ 2 h þ nðn À 1Þ ψ Àλ ðr; tÞ 2 ψ Àλ ðr 0 ; tÞ 2 i ; ð30Þ and α ξλ ŝ obs z ðr; tÞŝ obs z ðr 0 ; tÞ α ξλ % _ 2 δðr À r 0 Þ n ψ Àλ ðr; tÞ 2 h þ n 2 ψ Àλ ðr; tÞ 2 ψ Àλ ðr 0 ; where ψ λ ðr; tÞ is the effective wave function of a pulse in real space. This method can be easily generalized to higher-order correlations. We note that the delta function δðr À r 0 Þ in the correlation function will not lead to any diverging effect, because a practical probe always measures the averaged photon spin density over a finite volume instead of the true single-point spin density. On the other hand, this term vanishes in a composite measurement with r 6 ¼ r 0 . In this case, we see that the Poisson and sub-Poisson statistics automatically enter the quantum spin-density correlations. Specifically, the two-point spin density correlation vanishes for a single-photon Fock-state pulse as expected.
We now propose to detect the non-local spin density correlation via compound measurements of two NV centers, which have been exploited as nano-scale quantum sensors for photonic spin density measurements recently 44 . As shown in Fig. 6a, we fixed one quantum sensor on the z-axis and move the other one to image the distribution of the spin density correlation in the transverse plane. We contrast the spin density correlations in Fock-state and coherent-state Bessel pules in Fig. 6b−d. Here, we see that in the few-photon limit, there exist significant differences between Fock-state and coherent pulses. This difference fundamentally roots in the quantum statistics of photons and it will disappear in the large-photon limit.
The electronic ground-state of a negatively charged NV center is a spin-1 system, which has been routinely used as a highly sensitive nano-scale magnetometer at room temperature 54 . A laser with wavelength shorter than 637 nm is required to excite the NV to its electronic excited states. A red circularly polarized laser pulse (target pulse) with wavelength around 800 nm will not excite the NV, but only induce energy shifts in the three ground spin states. Recent work has shown that these energy shifts function as a static magnetic field for the NV spin 44 , which is linearly proportional to the local spin density of the target beam, i.e., B eff / hŝ obs ðrÞi. Thus, an NV center can be exploited as a nano-scale photonic spin sensor.
Currently, imaging of single-photon level spin density and the corresponding correlation is extremely challenging in experiments. However, our discovered interesting texture of spin density and non-local spin noise also exists in traditional OAM beam, which can be demonstrated in the near future. On the other hand, due to the absence of photon-photon interaction, the nonlocal spin noise within a light pulse in free space is fully determined by the photon-number statistics. However, we predict that exotic photonic phases with long-range spin order can exist in a quantum polariton system or an atomic lattice 55,56 .

Conclusion
We have established the fully quantum framework for photonic angular momenta of quantum structured pulses, as well as the corresponding quantum texture. Our approach presents a paradigm shift for the photonics community as it can be exploited to study the quantum properties and to reveal the vector nature of the angular momentum of light. We have shown that the spin texture of a Bessel pulse can exhibit a very interesting structure beyond the paraxial limit. Our proposed non-local spin noise will open a frontier for studying exotic phases of photons with longrange spin order. This spin noise can be measured in compound measurements with multiple nano-scale spin sensors, which have been proposed and demonstrated in our previous experiment 44 . The photonic OAM density and the corresponding non-local OAM density noise can also be handled within our proposed theoretical framework, which will be addressed in our future work.

Data availability
The data that support this study are available at https://github.com/yanglp091/ PhontonicSpinTexture.

Code availability
The code that supports this study is available at https://github.com/yanglp091/ PhontonicSpinTexture Received: 8 May 2021; Accepted: 14 September 2021; Fig. 6 Non-local spin noise. a Our proposed spin density correlation measurement with two nitrogen-vacancy (NV) centers (NV1 and NV2) at r and r 0 , respectively. b−d Contrast of spin density correlation between Fock-state and coherent-state pulses. Here, n is the mean photon number in each pulse and we show the results for the case with m = 1 and θ c = 0.2π. One of the quantum sensor is fixed on the z-axis and the other sensor can move in xy-plane.