Abstract
Classical structured light with controlled polarization and orbital angular momentum (OAM) of electromagnetic waves has varied applications in optical trapping, biosensing, optical communications and quantum simulations. However, quantum noise and photon statistics of threedimensional photonic angular momentum are relatively less explored. Here, we develop a quantum framework and put forth the concept of quantum structured light for spacetime wavepackets at the singlephoton level. Our work deals with threedimensional angular momentum observables for twisted quantum pulses beyond scalarfield theory as well as the paraxial approximation. We show that the spin density generates modulated helical texture and exhibits distinct photon statistics for Fockstate vs. coherentstate 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 spinOAM physics in twisted singlephoton pulses and opens explorations for phases of light with longrange spin order.
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Introduction
Structured singlephoton pulses are an important frontier for spin and orbital angular momentum (OAM)^{1,2,3}. As a quantum information carrier, singlephoton pulses with OAM have been achieved in the solidstate 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—spinorbit photonics^{11}, which studies photon spinOAM transfer^{12,13,14,15} and light−matter angular momentum exchange in the nearfield region^{16,17,18} or transfer of optical OAM to bounded electrons^{19} or photoelectrons^{12,20}. Spin1 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 nonclassical 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 singlephoton quantum states. Figure 1a shows the wellknown regime of twisted laser beams which contain an enormous number of photons. At the singlephoton level, both existing semiclassical^{23,24} and approximate quantum theories break down^{30,32}. In the widely adopted statespace description of single photons or entangled photons \(\{\leftl,s\right\rangle \}\) 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 singlephoton 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 quasiclassical Poisson behavior exhibited by traditional OAM laser beams. These fundamental, as well as technologically relevant problems, require the definition of singlephoton 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 singleparticle Schrödinger equation in the firstquantization 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 wellestablished 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 singlephoton pulse can exhibit rich spatial texture. Our work builds on previous important work in the field^{23,24,25,26,27,28,29,30,31,32,33,34}. 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}.
Nonlocal 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 longrange 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 nitrogenvacancy (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 offresonant optical beam can induce an effective static magnetic field for the electron spin of the NV center, which itself is an atomicscale magnetometer working at room temperature. Imaging of our discovered helical spindensity structure in this work can be realized with the same technology in the near future. Furthermore, our proposed nonlocal 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 OAM \(\hat{{{{{{{{\boldsymbol{L}}}}}}}}}\) of light in the Lorenz gauge within quantum field theory. The operators \(\hat{{{{{{{{\boldsymbol{S}}}}}}}}}\) and \(\hat{{{{{{{{\boldsymbol{L}}}}}}}}}\) obey the canonical commutation relationships
where ϵ_{ijk} is the thirdorder LeviCivita symbol. The longitudinal and scalar photons play a significant role in both \(\hat{{{{{{{{\boldsymbol{S}}}}}}}}}\) and \(\hat{{{{{{{{\boldsymbol{L}}}}}}}}}\). However, only the SAM \({\hat{{{{{{{{\boldsymbol{S}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}}={\varepsilon }_{0}\int {{{{{{\mathrm{d}}}}}}}^{3}r{\hat{{{{{{{{\boldsymbol{E}}}}}}}}}}_{\perp }({{{{{{{\boldsymbol{r}}}}}}}},t)\times {\hat{{{{{{{{\boldsymbol{A}}}}}}}}}}_{\perp }({{{{{{{\boldsymbol{r}}}}}}}},t)\) and OAM \({\hat{{{{{{{{\boldsymbol{L}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}}={\varepsilon }_{0}\int {{{{{{\mathrm{d}}}}}}}^{3}r{\hat{E}}_{\perp }^{j}({{{{{{{\boldsymbol{r}}}}}}}},t)({{{{{{{\boldsymbol{r}}}}}}}}\times {{{{{{{\boldsymbol{\nabla }}}}}}}}){\hat{A}}_{\perp }^{j}({{{{{{{\boldsymbol{r}}}}}}}},t)\) carried by transversely polarized photons are directly observable quantities even in the presence of charges^{39}. Note, \({\hat{{{{{{{{\boldsymbol{E}}}}}}}}}}_{\perp }\) and \({\hat{{{{{{{{\boldsymbol{A}}}}}}}}}}_{\perp }\) 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 \([{\hat{a}}_{{{{{{{{\boldsymbol{k}}}}}}}},\lambda },{\hat{a}}_{{{{{{{{\boldsymbol{k}}}}}}}}^{\prime} ,\lambda ^{\prime} }^{{{{\dagger}}} }]=\delta ({{{{{{{\boldsymbol{k}}}}}}}}{{{{{{{\boldsymbol{k}}}}}}}}^{\prime} ){\delta }_{\lambda \lambda ^{\prime} }\). The photon helicity is given by \(\hat{{{\Lambda }}}=\hslash \int {{{{{{\mathrm{d}}}}}}}^{3}k\left[{\hat{a}}_{{{{{{{{\boldsymbol{k}}}}}}}},+}^{{{{\dagger}}} }{\hat{a}}_{{{{{{{{\boldsymbol{k}}}}}}}},+}{\hat{a}}_{{{{{{{{\boldsymbol{k}}}}}}}},}^{{{{\dagger}}} }{\hat{a}}_{{{{{{{{\boldsymbol{k}}}}}}}},}\right]\). 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 space \(\hat{\psi }({{{{{{{\boldsymbol{r}}}}}}}})={[{\hat{\psi }}_{+}({{{{{{{\boldsymbol{r}}}}}}}}),{\hat{\psi }}_{}({{{{{{{\boldsymbol{r}}}}}}}})]}^{T}\), where
For the sourcefree case, our defined field operator in the Heisenberg picture satisfies the homogeneous wave equation
Now, we can reexpress the OAM and helicity operators of light in parallel to their electron counterparts
and
where \(\hat{{{{{{{{\boldsymbol{p}}}}}}}}}={{{{{\mathrm{i}}}}}}\hslash {{{{{{{\boldsymbol{\nabla }}}}}}}}\) is momentum operator and \({\hat{\sigma }}_{z}={{{{{{{\rm{diag}}}}}}}}[1,1]\) is the Pauli matrix. However, the similar expression for the spin operator \({\hat{{{{{{{{\boldsymbol{S}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}}\) 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 both \({\hat{{{{{{{{\boldsymbol{S}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}}\) and \({\hat{{{{{{{{\boldsymbol{L}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}}\) 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 nearfield techniques have now been well developed. This makes it possible to measure and engineer the angularmomentum 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 continuousmode field^{35,48} to the twistedpulse case.
We first define the singlephoton wavepacket creation operator for a twisted photon pulse
as a coherent superposition of planewave 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 zaxis and work in the cylindrical coordinate in kspace \({{{{{{{\boldsymbol{k}}}}}}}}={k}_{z}{{{{{{{{\boldsymbol{e}}}}}}}}}_{z}+{\rho }_{k}{{{{{{{{\boldsymbol{e}}}}}}}}}_{\rho }={\rho }_{k}\cos {\varphi }_{k}{{{{{{{{\boldsymbol{e}}}}}}}}}_{x}+{\rho }_{k}\sin {\varphi }_{k}{{{{{{{{\boldsymbol{e}}}}}}}}}_{y}+{k}_{z}{{{{{{{{\boldsymbol{e}}}}}}}}}_{z}\). 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 \(\exp ({{{{{\mathrm{i}}}}}}m{\varphi }_{k})\) with an integer m will lead to the OAM of light in zdirection of a singlephoton pulse as shown in the following.
The SAF is required to satisfy the normalization condition \(\int {{{{{{\mathrm{d}}}}}}}^{3}k{\left{\xi }_{\lambda }({{{{{{{\boldsymbol{k}}}}}}}})\right}^{2}=1\). This guarantees that \({\hat{a}}_{\xi \lambda }^{{{{\dagger}}} }\) obey the bosonic commutation relation
Then, the wavepacket creation operator \({\hat{a}}_{\xi \lambda }^{{{{\dagger}}} }\) 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 nphoton Fockstate and coherentstate pulses^{35} (please refer to Supplementary Note 3)
and
where \(\bar{n}= \alpha { }^{2}\) is the mean photon number in the coherentstate pulse. The wave function of a squeezedstate pulse, an entangled twophoton pulse^{49}, or an ultrashort 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 operators \({\hat{a}}_{\xi \lambda }\,(\lambda =\pm \kern1.3pt)\). 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 zaxis 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 xyplane. 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 \({k}_{\perp ,c}={k}_{z,c}\tan {\theta }_{c}\). These two Gaussian functions should have the same ratio between center wavenumber 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 Besselmodebased method^{31} which only applies to Bessel beams, our generalized planewavebased 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 semiclassically 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 nonparaxial 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 Fockstate 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 xyplane, then the mean value of the spin in the xyplane vanishes, i.e., \(\langle {\hat{S}}_{x}^{{{{{{{{\rm{obs}}}}}}}}}\rangle =\langle {\hat{S}}_{y}^{{{{{{{{\rm{obs}}}}}}}}}\rangle =0\). However, we show that the quantum fluctuations of photon spin in the xyplane are not zero. The standard derivations of the spin of an nphoton Fockstate Bessel pulse are given by
This is significantly beyond the previous semiclassical 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 coherentstate Bessel pulse with polarization λ and photon number \(\bar{n}= \alpha { }^{2}\),
Here, we see that the average spin carried by each photon is still \(\hslash \cos {\theta }_{c}\) and the spin’s projection on xyplane also vanishes. However, the quantum statistics of the photon spin for a coherentstate pulse is significantly different from that of a Fockstate pulse,
The Poisson statistics of a coherent pulse leads to nonvanishing \({{\Delta }}{\hat{S}}_{z}^{{{{{{{{\rm{obs}}}}}}}}}\) in contrast to a subPoisson Fockstate 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 of \({\hat{L}}_{z}^{{{{{{{{\rm{obs}}}}}}}}}\) for a Fockstate twisted pulse with photon number n is given by,
This reduces to the wellknown result obtained from the semiclassical method that each twisted photon carries mℏ OAM^{23,24}. We see that \(\langle {\hat{L}}_{z}^{{{{{{{{\rm{obs}}}}}}}}}\rangle\) is independent of the photon polarization. It is only determined by the photon number n and integer m in the helical phase factor \(\exp ({{{{{\mathrm{i}}}}}}m{\varphi }_{k})\) 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 xyplane vanishes, i.e., \(\langle {\hat{L}}_{x}^{{{{{{{{\rm{obs}}}}}}}}}\rangle =\langle {\hat{L}}_{y}^{{{{{{{{\rm{obs}}}}}}}}}\rangle =0\) (please refer to Supplementary Note 4).
The quantum variances of the three components of photon OAM for a Fockstate Bessel pulse are given by
and
where \(x=\tan {\theta }_{c}\in (0,\infty )\) and we have used the inequality relation a^{2}x^{2} + b^{2}/x^{2} ≥ 2∣ab∣ and the narrowband condition C_{0} ≫ 1. This immediately leads to the Heisenberg relation
The other two Heisenberg relations for photon OAM are trivial due to the vanishing mean values of \({\hat{L}}_{x}^{{{{{{{{\rm{obs}}}}}}}}}\) and \({\hat{L}}_{y}^{{{{{{{{\rm{obs}}}}}}}}}\). Similar results also hold for a coherentstate twisted pulse, but with nonvanishing \({({{\Delta }}{\hat{L}}_{z}^{{{{{{{{\rm{obs}}}}}}}}})}^{2}={\hslash }^{2}\bar{n}{m}^{2}\).
Interesting works have been reported to demonstrate the uncertainty relation between the conjugate variables of angle φ of light and its derivative \({\hat{l}}_{z}\equiv {{{{{\mathrm{i}}}}}}\hslash \partial /\partial \varphi\) in the firstquantization 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, {Hamiltonian \(\hat{H}\), momentum \(\hat{{{{{{{{\boldsymbol{P}}}}}}}}}\), helicity \(\hat{{{\Lambda }}}\)} has been select as the complete set of commuting observables to specify a photon state usually. Here, we see that a singlephoton pulse carrying determinate integer OAM in the propagating direction is not the eigen state of \({({\hat{{{{{{{{\boldsymbol{L}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}})}^{2}\).
Our predicted large OAM fluctuations in xyplane can be verified in experiments (see Fig. 3a). The quantum uncertainties of \({\hat{L}}_{x}^{{{{{{{{\rm{obs}}}}}}}}}\) and \({\hat{L}}_{y}^{{{{{{{{\rm{obs}}}}}}}}}\) 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 minimumuncertainty angle due to the inequality a^{2}x^{2} + b^{2}/x^{2} ≥ 2∣ab∣ (\(x=\tan {\theta }_{c}\)) 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 singlephoton pulse
We show that the spin texture of a singlephoton 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
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 planewave limit, the spin density will be a spaceindependent constant, i.e., \({{{{{{{\boldsymbol{\nabla }}}}}}}}\times \langle {\hat{{{{{{{{\boldsymbol{s}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}},t)\rangle ={{{{{{{\boldsymbol{\nabla }}}}}}}}\cdot \langle {\hat{{{{{{{{\boldsymbol{s}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}},t)\rangle =0\).
The mean value of the spin density of a Fockstate Bessel pulse is given (please refer to Supplementary Note 5)
where
and
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 xyplane is symmetric around zaxis. This causes the corresponding spatial integral to vanish as shown in the previous section, i.e., \(\langle {\hat{S}}_{x}^{{{{{{{{\rm{obs}}}}}}}}}\rangle =\langle {\hat{S}}_{y}^{{{{{{{{\rm{obs}}}}}}}}}\rangle =0\); (ii) its xyplane projection is parallel or antiparallel to the azimuthangledependent 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 xyplane 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 anticlockwise structures in the spin texture; (iv) its projection on z is independent on φ. For a small angle θ_{c}, the term \(\sim \! {\cos }^{4}({\theta }_{c}/2)\) dominates. Thus, the sign of s_{z} is always positive (negative) for LCP (RCP) pulse. This leads to the nonvanishing global spin \(\langle {\hat{S}}_{M,z}\rangle\).
We show the spin texture of an LCP singlephoton (n = 1) Fockstate 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 spacedependent 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 scalarfield theory and the paraxial approximation. For higherorder 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 counterclockwise 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 xyplane and zaxis, respectively. In panel a, we look at the projection of the spin density on xyplane s_{φ}e_{φ} with fixed Bessel order index as m = 0 and the azimuthal angle as φ = 0 (i.e., along the xaxis). For a pulse with small θ_{c} (see the blue arrows at the bottom), s_{φ}e_{φ} is relatively small and flat. The amplitude of s_{φ} decreases with θ_{c} and it vanishes when θ_{c} → 0. For a pulse with larger θ_{c} (see the yellow arrows at the top), the sign of s_{φ} oscillates between ±1 with increasing ρ. This explains the oscillation between the clockwise and counterclockwise structures shown in Fig. 4b. In panel b, we show the rotation of s_{φ}e_{φ} in xyplane with fixed m = 0 and θ_{c} = 0.2π. In panel c, we show the projection of the spin density on zaxis as the function of ρ for the four cases in 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 twopoint correlation should be characterized by a 3 × 3 correlation matrix as shown in the Supplementary Note 6. Here, we only describe the equaltime correlator \(\left\langle {\hat{s}}_{z}^{{{{{{{{\rm{obs}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}},t){\hat{s}}_{z}^{{{{{{{{\rm{obs}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}}^{\prime} ,t)\right\rangle\).
In the paraxial limit (θ_{c} ≈ 0), the twopoint correlation functions for a Fockstate and a coherentstate pulse are given by (please refer to Supplementary Note 6)
and
where
is the effective wave function of a pulse in real space. This method can be easily generalized to higherorder correlations.
We note that the delta function \(\delta ({{{{{{{\boldsymbol{r}}}}}}}}{{{{{{{\boldsymbol{r}}}}}}}}^{\prime} )\) 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 singlepoint spin density. On the other hand, this term vanishes in a composite measurement with \({{{{{{{\boldsymbol{r}}}}}}}}\neq {{{{{{{\boldsymbol{r}}}}}}}}^{\prime}\). In this case, we see that the Poisson and subPoisson statistics automatically enter the quantum spindensity correlations. Specifically, the twopoint spin density correlation vanishes for a singlephoton Fockstate pulse as expected.
We now propose to detect the nonlocal spin density correlation via compound measurements of two NV centers, which have been exploited as nanoscale quantum sensors for photonic spin density measurements recently^{44}. As shown in Fig. 6a, we fixed one quantum sensor on the zaxis 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 Fockstate and coherentstate Bessel pules in Fig. 6b−d. Here, we see that in the fewphoton limit, there exist significant differences between Fockstate and coherent pulses. This difference fundamentally roots in the quantum statistics of photons and it will disappear in the largephoton limit.
The electronic groundstate of a negatively charged NV center is a spin1 system, which has been routinely used as a highly sensitive nanoscale 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., \({{{{{{{{\boldsymbol{B}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}\propto \langle {\hat{{{{{{{{\boldsymbol{s}}}}}}}}}}^{{{{{{{{\rm{obs}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}})\rangle\). Thus, an NV center can be exploited as a nanoscale photonic spin sensor.
Currently, imaging of singlephoton level spin density and the corresponding correlation is extremely challenging in experiments. However, our discovered interesting texture of spin density and nonlocal 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 photonphoton interaction, the nonlocal spin noise within a light pulse in free space is fully determined by the photonnumber statistics. However, we predict that exotic photonic phases with longrange 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 nonlocal 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 nanoscale spin sensors, which have been proposed and demonstrated in our previous experiment^{44}. The photonic OAM density and the corresponding nonlocal 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
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Acknowledgements
This work is supported by the funding from DARPA Nascent Light−Matter Interactions. L.P.Y. is also supported by the funding from the Ministry of Science and Technology of China (No. 2021YFE0193500).
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L.P.Y. and Z.J. conceived the idea and wrote the paper. L.P.Y. performed the calculation under the supervision of Z.J.
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Yang, LP., Jacob, Z. Nonclassical photonic spin texture of quantum structured light. Commun Phys 4, 221 (2021). https://doi.org/10.1038/s4200502100726w
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DOI: https://doi.org/10.1038/s4200502100726w
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