Quantum structured light: Non-classical spin texture of twisted single-photon pulses
QQuantum structured light: Non-classical spin texture of twisted single-photon pulses
Li-Ping Yang
1, 2 and Zubin Jacob ∗ Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun 130024, China Birck Nanotechnology Center, School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, U.S.A.
Classical structured light with controlled polarization and orbital angular momentum (OAM) of electromag-netic waves has varied applications in optical trapping, bio-sensing, optical communications and quantum simu-lations. However, the definition of quantum density of spin and OAM at the single-photon level remains elusive.Here, we develop a theoretical framework and put forth the concept of quantum structured light for space-timewavepackets at the single photon level. Our work marks a paradigm shift beyond scalar-field theory as wellas the paraxial approximation and can be utilized to study the quantum properties of the spin and OAM ofall classes of twisted quantum light pulses. Our framework captures the uncertainty in projections of vectorspin and OAM demonstrating their quantum behavior beyond the conventional concept of classical polarization.Specifically, we show that the spin density generates modulated helical texture beyond paraxial limit and ex-hibits distinct statistics for Fock-state vs. coherent-state twisted pulses. We introduce the quantum correlationof the photon spin density to characterize the nonlocal spin noise providing a rigorous parallel with fermionicspin noise operators. Our work paves the way for quantum spin-OAM physics in twisted single photon pulsesand can also opens explorations for new phases of light with long-range spin order.
Structured single-photon pulses are a new frontier for spinand orbital angular momentum (OAM). As a new quantuminformation carrier, single-photon pulses with OAM havebeen achieved in the solid-state system with quantum dots re-cently [1]. They hae been exploited to construct quantum net-work with higher channel capacity [2–6]. The spin and OAMof light have also attracted increasing attention in an emergingresearch field—spin-orbit photonics [7], which studies photonspin-OAM transfer [8–11] and light-matter angular momen-tum exchange in the near-field region [12–14] or transfer ofoptical OAM to a bounded electron [15]. Spin-1 quantizationis also the hallmark of photonic skyrmions and topologicalphotonic phases of matter [16, 17]. However, a fully quan-tum framework to study the quantum properties and to revealthe vector nature of the angular momentum of light is an openproblem.The large body of important work on SAM and OAMphysics are limited to the semi-classical theory of light-matterinteraction [18–20]. Fig. 1 a shows this well-known regimeof twisted laser beams which contain an enormous numberof photons. At the single photon level, these theories breakdown. Even in the conventional state-space description of sin-gle photons or entangled photons ( ψ = | l , s > ), the rich spa-tial texture of SAM and OAM vectors are ignored completely.Specifically, important open questions remain on the full 3Dprojection of photon spin and OAM at the quantum level be-yond the scalar-field theory and paraxial approximation. Fur-thermore, Heisenberg uncertainty relations for photon angularmomenta can a ff ect quantum metrology experiments whichurgently require a new framework. These Heisenberg un-certainty relations between di ff erent photon OAM compo-nents are the canonical quantum characteristics of angularmomentum but have never been investigated. Similarly, forapplications such as secure quantum communication, twistedsingle-photon pulses in the quantum limit with few photons ∗ [email protected] (see Fig. 1 b ) are required. In this technologically importantlimit, quantum statistics of photons will emerge and suppressthe quasi-classical Poisson behavior in traditional OAM laserbeam. All these fundamental as well as technologically rele-vant problems can not be solved with existing semi-classicalapproaches.In this work, we present a new frontier for quantum struc-tured light involving twisted space-time wavepackets of light.We first construct the wave function of a quantum twistedpulse (see Fig. 1 a ), as well as a twisted laser beam (see Fig. 1 b ), from quantum field theory [21] instead of from the single-particle Schr¨odinger equation in the first-quantization pic-ture [22–24]. Combining with our recently discovered quan-tum operators of the angular momenta of light [25], we eval-uate the mean value as well as the quantum uncertainty of thephoton spin operator vector. Apart from the well-establishedglobal properties of polarization, we also investigate the quan-tum properties of the photon spin density vector, i.e., the spintexture, which is a function of space and time. We show thatbeyond the paraxial approximation, the photon spin densityof a Bessel single-photon pulse can exhibit very interestingand rich texture. This marks a significant departure from pre-vious works [18, 19, 26–33]. Our proposed framework pro-vides a powerful and versatile tool to engineer the local photonspin and OAM densities of a quantum structured light pulse,specifically for spatiotemporal optical vortices [34, 35].Non-local spin noise (i.e., spin density correlation) for elec-trons is a fundamental signature of quantum phases of mag-netic condensed matter [36], specifically in new phases ofmatter such as quantum spin liquids without magnetic or-der [37]. However, no such quantum spin noise operator hasbeen defined for photons till date. Our theoretical formalismallows us to overcome this challenge. Here, we introduce thequantum correlation of the photon spin density to character-ize the nonlocal spin noise in light, which can be exploited toexplore new phases of light with long-range spin order. Re-cently, the nitrogen-vacancy centers have been exploited asnanoscale photonic spin density sensors [38]. Thus, the non-local spin density correlation can be measured in compound a r X i v : . [ phy s i c s . op ti c s ] F e b a b Classical limit Quantum limit
Long pulse (beam) withlarge photon number Finite-length pulsewith few photons
OAM OAM
FIG. 1. Schematic of a twisted beam a and a quantum twisted pulse b . measurements with two or multiple NV centers. SPIN AND ORBITAL ANGULAR MOMENTA OF LIGHT
In our recent work [25], we have discovered the full quan-tum operator of the photon spin with the quantum field theory,ˆ S = c (cid:90) d x ˆ π ( r , t ) × ˆ A ( r , t ) . (1)which obeys the canonical commutation relationships[ ˆ S i , ˆ S j ] = i (cid:126) (cid:15) i jk ˆ S k , (2)where (cid:15) i jk is the third-order Levi-Civita symbol. However, wehave shown that only ˆ S obs = ε (cid:82) d r ˆ E ⊥ ( r , t ) × ˆ A ⊥ ( r , t ) andˆ L obs = ε (cid:82) d r ˆ E j ⊥ ( r , t )( r × ∇ ) ˆ A j ⊥ ( r , t ), which are the spin andOAM angular momenta carried by transversely polarized pho-tons, are direct observables. Here, ˆ E ⊥ and ˆ A ⊥ are the trans-verse part of the electric field and the vector potential, respec-tively.Using the circularly polarized plane waves, we can expandthe observable photon spin and OAM operators asˆ S obs = (cid:126) (cid:90) d k (cid:104) ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − (cid:105) e ( k , , (3)ˆ L obs = − i (cid:126) (cid:90) d k (cid:88) λ = ± ˆ a † k ,λ ( k × ∇ k )ˆ a k ,λ , (4)where e ( k , = k / | k | is the unit vector and λ = ± de-note the left circular polarization (LCP) and right circularpolarization (RCP) separately (see Supplementary Material).The ladder operators of the plane wave with wave vector k and polarization λ satisfy the bosonic commutation relation[ˆ a k ,λ , ˆ a † k (cid:48) ,λ (cid:48) ] = δ ( k − k (cid:48) ) δ λλ (cid:48) . The photon helicity is given byˆ Λ = (cid:126) (cid:82) d k (cid:104) ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − (cid:105) .To show the striking symmetry between the angular mo-mentum of photons and electrons, we can define a field oper-ator for light in real space ˆ Ψ ( r ) = [ ˆ Ψ + ( r ) , ˆ Ψ − ( r )] T , whereˆ Ψ λ ( r ) = (cid:112) (2 π ) (cid:90) d k ˆ a k ,λ e i k · r . (5) ( ( $ ( %! ! &" ! = " " FIG. 2. Schematic of the spectral distribution of a Bessel beam.
For the source-free case, our defined field operator in theHeisenberg picture satisfies the homogeneous wave equation (cid:32) ∇ − c ∂ ∂ t (cid:33) ˆ Ψ ( r , t ) = . (6)Now, we can re-express the OAM and helicity operators oflight in parallel to their electron counterpartsˆ L obs = (cid:90) d r ˆ Ψ † ( r )( r × ˆ p ) ˆ Ψ ( r ) , (7)and ˆ Λ = (cid:90) d r ˆ Ψ † ( r ) ˆ σ z ˆ Ψ ( r ) (8)where ˆ p = − i (cid:126) ∇ is momentum operator and ˆ σ z = diag[1 , − S obs can not be obtained in real space. Di ff erent fromthe electron spin, the polarizations of photons are not intrinsicdegrees of freedom of light because the unit polarization vec-tor e ( k ,
3) in Eq. (59) for each plane wave is k -dependent, i.e.,dependent on its spatial momentum.We emphasize that there still exist significant di ff erencesbetween the spin and OAM of photons. The photon spin an-gular momentum fundamentally originates from the rotationbetween di ff erent polarization modes, which generates the rel-ative phase between the plane-wave modes with the same mo-mentum k . However, the OAM roots in the extrinsic spa-tial rotation fully, which generates the relative phase betweenplane-wave modes with di ff erent momentum k , but with thesame polarization λ . This explains that ˆ S obs and ˆ L obs can stillbe measured independently even though photons do not haveintrinsic spin degrees of freedom. QUANTUM WAVE FUNCTION OF TWISTED LIGHTPULSES
In previous section, we have shown that both ˆ S obs andˆ L obs are vector operators. However, in previous studies, usu-ally only their projections on the propagating direction havebeen fully studied. Their mean value on the transverse planeand more importantly, their quantum fluctuations have beenmarginally investigated. On the other hand, the near-filedtechniques have now been well developed. This makes it pos-sible to measure and engineer the angular-momentum densityof light, which is a vector function of space and time, in ex-periments. Thus, a fully quantum theory beyond paraxial ap-proximation to handle all kinds of twisted pulses uniformlyis highly desirable. We now present this powerful theoreticaltool by generalizing the quantum theory of continuous-modefield [21] to twisted-pulse case.We first define the single-photon wave-packet creation op-erator for a twisted photon pulseˆ a † ξλ = (cid:90) d k ξ λ ( k )ˆ a † k λ , (9)as a coherent superposition of plane-wave modes. The pulseshape and other quantum properties of the pulse are fullydetermined by the spectral amplitude function (SAF) ξ λ ( k ).In the following, we denote the propagating direction of thepulse as the z -axis and work in the cylindrical coordinate in k -space k = k z e z + ρ k e ρ = ρ k cos ϕ k e x + ρ k sin ϕ k e y + k z e z . Here, ρ k is the radial distance from the k z -axis, ϕ k is the azimuthangle, and e denotes the corresponding unit vector. The SAFof a twisted pulse with deterministic OAM can be generallyexpressed as ξ λ ( k ) = √ π η λ ( k z , ρ k ) e im ϕ k . (10)Usually, the amplitude η λ ( k z , ρ k ) is symmetric in the trans-verse plane, i.e, it is independent on the azimuth angle ϕ k .The phase factor exp( im ϕ k ) with an integer m will lead to theOAM of light in z -direction of a single-photon pulse as shownin the following.The SAF is required to satisfy the normalization condition (cid:82) d k | ξ λ ( k ) | =
1. This guarantees that ˆ a † ξλ obeys the bosoniccommutation relation [ˆ a ξλ , ˆ a † ξλ ] = . (11)Then, the wave-packet creation operator ˆ a † ξλ can be treatedas a normal ladder operator of a harmonic oscillator. Using this commutation relation, we can construct the wave func-tion of all kinds of quantum pulses in the standard way, suchas the most common n -photon Fock-state and coherent-statepulses [21, 39] (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = √ n ! (cid:16) ˆ a † ξλ (cid:17) n | (cid:105) , (12)and (cid:12)(cid:12)(cid:12) α ξλ (cid:69) = exp (cid:32) α ˆ a † ξλ −
12 ¯ n (cid:33) | (cid:105) = e − ¯ n / ∞ (cid:88) n = α n √ n ! (cid:12)(cid:12)(cid:12) n ξλ (cid:69) , (13)where ¯ n = | α | is the mean photon number in the coherent-state pulse. The wave function of a squeezed-state pulse, anentangled two-photon pulse [40], or an ultra-short Spatiotem-poral vortex pulse [34, 35] can also be constructed similarly.Here, the polarization of the pulse is fixed as one of the cir-cular polarization. However, linearly or elliptically polarizedquantum pulses can also be constructed with superposition oftwo circular polarization ladder operators ˆ a ξλ ( λ = ± ). Wealso note that a twisted laser beam can be characterized by awave function with a very long pulse length and very largephoton number. Thus, our method can also be used to handlea continuous laser beam.Without loss of generality, we only take the Bessel pulsesas an example to show the quantum properties of the spinand OAM of twisted pulses. Other twisted pulses, such asa Bessel-Gaussian or Laguerre-Gaussian pulse, can be treatedsimilarly. The single-frequency Bessel beam is the superposi-tion 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. 2. Then,the SAF of a Bessel pulse with a Gaussian envelop can beexpress η λ ( k z , ρ k ) as the product of two Gaussian functions η λ ( k z , ρ k ) = (cid:32) σ z π (cid:33) / exp (cid:104) − σ z ( k z − k z , c ) (cid:105) × σ ρ π k ⊥ , c / exp (cid:104) − σ ρ ( ρ k − k ⊥ , c ) (cid:105) . (14)The first Gaussian function with width 1 /σ z and center wavevector k z , c characterize the envelope of the pulse in the prop-agating direction. The pulse length on z -axis in real space isgiven by σ z = c τ p ( τ p the pulse length in time domain). Dif-ferent from previous works [30, 41], we do not add a deltafunction [such as δ ( θ k − θ c )] in the SAF to characterize its dis-tribution property in the xy -plane. This will cause a seriousproblem that the wave functions of the quantum pulses cannotbe well normalized, because (cid:82) d k | ξ λ ( k ) | ∝ δ ( θ k − θ c ). In-stead, we utilize another Gaussian function with width 1 /σ ρ and center value k ⊥ , c = k z , c tan θ c . These two Gaussian func-tions should have the same ratio between center wave-numberand the width, i.e. k z , c σ z = k ⊥ , c σ ρ ≡ C . In the narrowbandwidth limit C (cid:29)
1, our defined SAF is well normal-ized [39]. We also note that in contrast to the Bessel-mode-based method [31], our plane-wave-based framework can bemore easily used to uniformly handle all types of quantumpulses. ! / ! − − ( / ! − − a ) ! = 0.1,- = 0 c ) ! = 0.2,- = 1 b ) ! = 0.2,- = 0 d ) ! = 0.2,- = 2 ! / ! − − ( / ! − − ! / ! − − ( / ! − − ! / ! − − ( / ! − − FIG. 3. Spin texture of a single-photon LCP Bessel pulse on the pulse-center plane with k z , c z = ct . ( a - d ) correspond to di ff erent quantumnumbers m and polar angles θ c . QUANTUM STATISTICS OF THE PHOTON SPIN
Traditionally, the angular momentum carried by eachphoton in a twisted laser beam has been calculated semi-classically via the ratio of angular flux to the energy flux [18,19] and only its projection on the propagating axis has beenstudied. Although the projection of the photon spin and OAMof a non-paraxial beam on the transverse plane has caused at-tention recently [26–29], a systematic and comprehensive in-vestigation of the vector nature of the photon spin and OAM isstill missing. Specifically, the Heisenberg uncertainty relationfor photon OAM has never been investigated. On the otherhand, many researchers have also tried to establish a quan-tum theory of the angular momentum of light in the last twodecades [30–33]. However, a fully quantum framework be-yond the paraxial approximation to handle all kinds of quan-tum pulses has only been established until now.We first calculate the mean value of the spin of a Fock-stateBessel pulse with polarization λ and photon number n , (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ S obs (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = (cid:126) n λ (0 , , cos θ c ) . (15)Here, we see that the magnitude of the spin carried by eachcircularly polarized photon is usually smaller than (cid:126) and ap-proaches to (cid:126) asymptotically in the paraxial limit ( θ c →
0) [27, 30]. This is significantly di ff erent from the helicity,which is exactly (cid:126) . If the SAF of a pulse is symmetric inthe xy -plane, then the mean value of the spin in the xy -planevanishes, i.e., (cid:104) ˆ S obs x (cid:105) = (cid:104) ˆ S obs y (cid:105) =
0. However, we show thatthe quantum fluctuations of photon spin in the xy -plane isnot zero. The standard derivations of the spin of an n -photon Fock-state Bessel pulse are given by ∆ ˆ S obs x = ∆ ˆ S obs y = (cid:126) (cid:112) n / | sin θ c | , ∆ ˆ S obs z = . (16)This is significantly beyond the previous semi-classical the-ory [18, 19], in which the quantum statistics of the photonspin cannot be studied.Similarly, we can evaluate the mean value of the spin ofa coherent-state Bessel pulse with polarization λ and photonnumber ¯ n = | α | , (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ˆ S obs (cid:12)(cid:12)(cid:12) α ξλ (cid:69) = (cid:126) ¯ n λ (0 , , cos θ c ) . (17)Here, we see that the average spin carried by each photon isstill (cid:126) cos θ c and the spin’s projection on xy -plane also van-ishes. However, the quantum statistics of the photon spin fora coherent-state pulse is significantly di ff erent from that of aFock-state pulse, ∆ ˆ S obs x = ∆ ˆ S obs y = (cid:126) (cid:112) ¯ n / | sin θ c | , ∆ ˆ S obs z = ¯ n (cid:126) | cos θ c | . (18)The Poisson statistics of a coherent pulse leads to non-vanishing ∆ ˆ S obs z in contrast to a sub-Poisson Fock-state pulse. QUANTUM STATISTICS OF THE PHOTON OAM
Heisenberg’s uncertainty relation is one of the most impor-tant quantum characteristics for angular momentum. How-ever, this relation for photon OAM has never been addressedtill date. Here, we present a quantitative investigation aboutthe quantum statistics of photon OAM. The mean value ofˆ L obs z for a Fock-state twisted pulse with photon number n , (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L obs z (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = − in (cid:126) π (cid:90) d k η λ ( k ) e − im ϕ k ∂∂ϕ k η λ ( k ) e im ϕ k (19) = mn (cid:126) . (20)This verifies the well-known result obtained from thesemi-classical method that each twisted photon carries m (cid:126) OAM [18, 19]. We see that (cid:104) ˆ L obs z (cid:105) is independent on the pho-ton polarization. It is only determined by the photon num-ber n and integer m in the helical phase factor exp( im ϕ k ) if η λ ( k ) is not a function of ϕ k . We can also verify that, in thiscase, the mean value of photon OAM in xy -plane vanishes,i.e., (cid:104) ˆ L obs x (cid:105) = (cid:104) ˆ L obs y (cid:105) = ∆ ˆ L obs z ) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ( ˆ L obs z ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) − (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L obs z (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = , (21)and( ∆ ˆ L obs x ) = ( ∆ ˆ L obs y ) (22) = n (cid:126) (cid:34)(cid:32) C + (cid:33) x + (cid:32) C + m + (cid:33) x − (cid:35) (23) ≥ n (cid:126) (cid:114) (4 C + C + m +
34 ) − (cid:29) mn (cid:126) , (24)where x = tan θ c ∈ (0 , ∞ ) and we have used the inequalityrelation a x + b / x ≥ | ab | and the narrow-band condition C (cid:29)
1. This automatically give the Heisenberg relation (cid:113) ( ∆ ˆ L x ) ( ∆ ˆ L y ) > (cid:126) (cid:12)(cid:12)(cid:12) (cid:104) ˆ L z (cid:105) (cid:12)(cid:12)(cid:12) = mn (cid:126) . (25)The other two Heisenberg relations for photon OAM are triv-ial due to the vanishing mean values of ˆ L obs x and ˆ L obs y . Similarresults also hold for a coherent-state twisted pulse, but withnon-vanishing ( ∆ ˆ L obs z ) = (cid:126) ¯ nm . QUANTUM SPIN TEXTURE OF A SINGLE-PHOTONPULSE
We show that the spin texture of a single-photon pulse canexhibit a very rich and interesting structure in the case beyondthe paraxial approximation. The photon spin texture is char-acterized by the spin density operatorˆ s ( r , t ) = ε ˆ E ⊥ ( r , t ) × ˆ A ⊥ ( r , t ) . (26)Similar to the electric or magnetic fields, the spin density canbe treated as a vector field and can be measured locally [38].We emphasize that as a vector, the spin density is neitherpurely longitudinal or purely transverse in most case. Inthe single-mode plane-wave limit, the spin density will be aspace-independent constant, i.e., ∇ ×(cid:104) ˆ s ( r , t ) (cid:105) = ∇ ·(cid:104) ˆ s ( r , t ) (cid:105) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ s obs ( r , t ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = λ (cid:16) s ρ e ϕ + s z e z (cid:17) , (27)where s z = n (cid:126) C πσ z σ ρ (cid:26)(cid:2) J m − λ ( k ⊥ , c ρ ) (cid:3) cos θ c − (cid:2) J m + λ ( k ⊥ , c ρ ) (cid:3) sin θ c (cid:27) exp (cid:34) − ( ct − z cos θ c ) σ z cos θ c (cid:35) , (28)and s ρ = n (cid:126) C sin θ c πσ z σ ρ (cid:20) cos θ c J m ( k ⊥ , c ρ ) J m − λ ( k ⊥ , c ρ ) + sin θ c J m + λ ( k ⊥ , c ρ ) J m ( k ⊥ , c ρ ) (cid:21) exp (cid:34) − ( ct − z cos θ c ) σ z cos θ c (cid:35) , (29)with r = ρ e ρ + z e z . The spin density of a coherent pulsecan be evaluated similarly. Here, we can see the followingkey characters of the spin density: (i) its projection in the xy -plane is symmetric around z -axis. This leads to the corre-sponding spacial integral vanishes as shown in previous sec-tion, i.e., (cid:104) ˆ S M , x (cid:105) = (cid:104) ˆ S M , y (cid:105) =
0; (ii) its xy -plane projection isparallel or anti-parallel to the azimuth-angle-dependent unitvector e ϕ . This leads to the helical spin texture as shown inFig. 3. (iii) its xy -plane projection contains the product of twodi ff erent Bessel functions, which can lead to the oscillationbetween clockwise and anti-clockwise structures; (iv) its pro-jection on z is independent on ϕ . For a small angle θ c , the term ∼ cos ( θ c /
2) dominates. Thus, the sign of s z is alwayspositive (negative) for LCP (RCP) pulse. This leads to thenon-vanishing global spin (cid:104) ˆ S M , z (cid:105) .We show the spin texture of an LCP single-photon ( n = k z , c z = ct , at which theGaussian functions in Eqs. (28) and (29) reach their maxima.In this case, the space-dependent spin density is only a func-tion of the radius ρ and the azimuthal angle ϕ contained in e ϕ . For a pulse with small polar angle θ c = . π , almost onlya clockwise structure can be observed in panel a . However,for a pulse with larger polar angle θ c = . π , the oscilla- @/& ! ) ! = 0.2,) ! = 0.1,) ! = 0.05, a (! / = 0/ = ,/6/ = ,/3/ = ,/2/ = 2,/3/ = 5,/6/ = , @/& ! b ! (
10 0 2 4 86 10@/& ! c - = 0 ) ! = 0.1, ×10 " D ' ℏ / & ! - = 0 ) ! = 0.2,- = 1 ) ! = 0.2,- = 2 ) ! = 0.2, Spin density projection D ( E ) Spin density projection D ( E ) FIG. 4.
Projections of the spin density of a Bessel single-photon pulse on xy -plane (panel a and b) and on z -axis in panel c. Panel a corresponds to the case with fixed azimuthal angle φ =
0, but di ff erent polar angle θ c . Panel b corresponds to the case with fixed polar angle θ c = . π , but di ff erent azimuthal angle φ . Panel c shows the z -component of the spin density. tion between clockwise and counter-clockwise structure canbe observed clearly. This oscillation can only be obtainedbeyond the paraxial approximation. For higher-order Besselpulses with m >
0, the fine structure of the spin density issignificantly di ff erent from the m = c and panel d . We also note that the Bessel pulsewith m = c ), because the spintexture has a vertex instead of a vortex at the center.In Fig. 4, we show more details of the projection of thespin 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 ρ e ϕ with fixed Bessel order index as m = ϕ = x -axis). For apulse with small θ c (see the blue arrows at the bottom), s ρ e ϕ isrelatively small and flat. The amplitude of s ρ decrease with θ c and it vanishes when θ c →
0. For a pulse with larger θ c (seethe yellow arrows at the top), the sign of s ρ oscillates between ± ρ . This explains the oscillation betweenthe clockwise and counter-clockwise structure shown in Fig. 3 b . In panel b , we show the rotation of s ρ e ϕ in xy -plane withfixed m = θ c = . π . In panel c , we show the projectionof the spin density on z -axis as the function of ρ for the fourcases in Fig. 3. Here, we clearly see the oscillation inducedby the Bessel function in Eq. (28). Specifically, the vertex atthe center for m = NONLOCAL SPIN NOISE OF LIGHT
To characterize the nonlocal spin noise of light, we intro-duce the quantum correlation function of the photon spin den-sity. Due to the vector nature of the spin density, the full two-point correlation should be characterized by a 3 × (cid:68) ˆ s obs z ( r , t ) ˆ s obs z ( r (cid:48) , t ) (cid:69) .In the paraxial limit ( θ c ≈ (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ s obs z ( r , t ) ˆ s obs z ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) ≈ (cid:126) (cid:20) δ ( r − r (cid:48) ) n (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) + n ( n − (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ψ m , − λ ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) (cid:21) , (30)and (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ˆ s obs z ( r , t ) ˆ s obs z ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) ≈ (cid:126) (cid:20) δ ( r − r (cid:48) )¯ n (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) + ¯ n (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ψ m , − λ ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) (cid:21) , (31)where ψ λ ( r , t ) = (cid:112) (2 π ) (cid:90) d k ξ λ ( k ) e i [( k · r − ω ) t + λϕ k ] , (32)is the e ff ective wave function of pulse in real space. Thismethod can be easily generalized to higher-order correlations.We note that the delta function δ ( r − r (cid:48) ) in the correlationfunction will not lead to any diverging e ff ect, because a prac-tical probe always measures the averaged photon spin densityover a finite volume instead of the true single-point spin den-sity. On the other hand, this term vanishes in a compositemeasure with r (cid:44) r . In this case, we see that the Poisson andsub-Poisson statistics automatically enter the quantum spin-density correlations. Specifically, the two-point spin densitycorrelation vanishes for a single-photon Fock-state pulse asexpected.Due to the absence of photon-photon interaction, the nonlo-cal spin noise within a light pulse in free space is fully deter-mined by the photon-number statistics. However, we predictthat exotic photonic phases with long-range spin order can ex-ist in a quantum polariton system or a atomic lattice [42, 43]. CONCLUSION
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CONTENTS
Spin and orbital angular momenta of light 2Quantum wave function of twisted light pulses 3Quantum statistics of the photon spin 4Quantum statistics of the photon OAM 4Quantum spin texture of a single-photon pulse 5Nonlocal spin noise of light 6Conclusion 6References 7I. Quantum operator of photon angular momentum and helicity 9A. Commutation relations 10B. Circular polarization representation 11II. Photon OAM operator in reciprocal space 12III. Wavefunction of quantum twisted photon pulse 13A. Fock-state and coherent-state photon pulses 14B. Spectral amplitude function 14IV. Quantum statistics of the angular momenta of twisted photon pulses 15A. Spin angular momentum of twisted photon pulses 15B. Orbital angular momentum of twisted photon pulses 16C. Revisit of commutation relation between the photon spin and OAM 19V. Quantum spin texture of Bessel photon pulse 19VI. Quantum correlation of spin density 21
I. QUANTUM OPERATOR OF PHOTON ANGULAR MOMENTUM AND HELICITY
In this section, we give the quantum operators of the photon angular momentum. First, we give the plane-wave expansion ofthe electromagnetic (EM) field operators in free spaceˆ A ⊥ ( r , t ) = (cid:90) d k (cid:88) λ = (cid:115) (cid:126) ε ω (2 π ) (cid:104) ˆ a k ,λ e ( k , λ ) e i ( k · r − ω t ) + h . c . (cid:105) , (33)ˆ E ⊥ ( r , t ) = i (cid:90) d k (cid:88) λ = (cid:115) (cid:126) ω ε (2 π ) (cid:104) ˆ a k ,λ e ( k , λ ) e i ( k · r − ω t ) − h . c . (cid:105) , (34)ˆ B ( r , t ) = ic (cid:90) d k (cid:115) (cid:126) ω ε (2 π ) (cid:110)(cid:2) ˆ a k , e ( k , − ˆ a k , e ( k , (cid:3) e i ( k · r − ω t ) − h . c . (cid:111) , (35)where ε is the vacuum permittivity, ω = c | k | is the frequency of the mode with wave vector k and e ( k , λ ) ( λ = ,
2) are the unitpolarization vectors of the two transverse modes for each k , i.e., e ( k , · k = e ( k , · k = e ( k , · e ( k , =
0. Here, we worksin the Heisenberg picture, thus all field operators are time-dependent.0Using the bosonic commutation relations for the ladder operators[ˆ a k ,λ , ˆ a † k (cid:48) ,λ (cid:48) ] = δ ( k − k (cid:48) ) δ λ,λ (cid:48) , [ˆ a k ,λ , ˆ a k (cid:48) ,λ (cid:48) ] = [ˆ a † k ,λ , ˆ a † k (cid:48) ,λ (cid:48) ] = , (36)we can verify the following equal-time commutation relations[ ˆ A i ⊥ ( r , t ) , ˆ E j ⊥ ( r (cid:48) , t )] = − i (cid:126) ε δ ( r − r (cid:48) ) δ ⊥ i j , [ ˆ A i ⊥ ( r , t ) , ˆ A j ⊥ ( r (cid:48) , t )] = [ ˆ E i ⊥ ( r , t ) , ˆ E j ⊥ ( r (cid:48) , t )] = , (37)where we have used the identity (cid:80) λ = , e i ( k , λ ) e j ( k , λ ) = δ ⊥ i j .In our previous work [25], we discovered the quantum operator of the full photon spin,ˆ S = c (cid:90) d x ˆ π ( r , t ) × ˆ A ( r , t ) . (38)We have shown that the well-known spin and orbital angular momentum (OAM) of lightˆ S obs = ε (cid:90) d r ˆ E ⊥ ( r , t ) × ˆ A ⊥ ( r , t ) = i (cid:126) (cid:90) d k (cid:104) ˆ a † k , ˆ a k , − ˆ a † k , ˆ a k , (cid:105) e ( k , , (39)ˆ L obs = ε (cid:90) d r ˆ E j ⊥ ( r , t )( r × ∇ ) ˆ A j ⊥ ( r , t ) = − i (cid:126) (cid:90) d k (cid:88) λ = , ˆ a † k ,λ ( k × ∇ k )ˆ a k ,λ , (40)are the directly observable part of photon angular momentum. Here, e ( k , = k / | k | is the unit vector in the propagating directionof each mode. In this paper, we will only study the observable part of photon angular momentum ˆ S obs and ˆ L obs , which both aregauge invariant. The helicity of the photon is defined asˆ Λ = i (cid:126) (cid:90) d k (cid:104) ˆ a † k , ˆ a k , − ˆ a † k , ˆ a k , (cid:105) , (41)which is a Lorentz invariant scalar. A. Commutation relations
Now, we check the commutation relations between the angular momenta of light. This can be done much more easily in thek-space. We first check the photon spin[ ˆ S obs i , ˆ S obs j ] = − (cid:126) (cid:90) d k (cid:90) d k (cid:48) (cid:104) ˆ a † k , ˆ a k , − ˆ a † k , ˆ a k , , ˆ a † k (cid:48) , ˆ a k (cid:48) , − ˆ a † k (cid:48) , ˆ a k (cid:48) , (cid:105) e i ( k , e j ( k (cid:48) ,
3) (42) = (cid:126) (cid:90) d k (cid:90) d k (cid:48) (cid:16) ˆ a † k , ˆ a k , − ˆ a † k , ˆ a k , + ˆ a † k , ˆ a k , − ˆ a † k , ˆ a k , (cid:17) δ ( k − k (cid:48) ) e i ( k , e j ( k (cid:48) ,
3) (43) = . (44)This recovers the result obtained by van Enk and Nienhuis in 1994 [44, 45]. For the photon OAM operator, we have[ ˆ L obs i , ˆ L obs j ] = − (cid:126) (cid:90) d k (cid:90) d k (cid:48) (cid:88) λ,λ (cid:48) = , (cid:104) ˆ a † k ,λ ( k × ∇ k ) i ˆ a k ,λ , ˆ a † k (cid:48) ,λ (cid:48) ( k (cid:48) × ∇ k (cid:48) ) j ˆ a k (cid:48) ,λ (cid:48) (cid:105) (45) = − (cid:126) (cid:90) d k (cid:90) d k (cid:48) (cid:88) λ,λ (cid:48) = , (cid:110) ˆ a † k ,λ ( k × ∇ k ) i δ ( k − k (cid:48) ) δ λλ (cid:48) ( k (cid:48) × ∇ k (cid:48) ) j ˆ a k (cid:48) ,λ (cid:48) − ˆ a † k (cid:48) ,λ (cid:48) ( k (cid:48) × ∇ k (cid:48) ) j δ ( k − k (cid:48) ) δ λλ (cid:48) ( k × ∇ k ) i ˆ a k ,λ (cid:111) (46) = − (cid:126) (cid:90) d k (cid:88) λ = , ˆ a † k ,λ (cid:104) ( k × ∇ k ) i ( k × ∇ k ) j − ( k × ∇ k ) j ( k × ∇ k ) i (cid:105) ˆ a k ,λ (47) = (cid:126) (cid:90) d k (cid:88) λ = , ˆ a † k ,λ (cid:15) i jk ( k × ∇ k ) k ˆ a k ,λ = i (cid:126) (cid:15) i jk ˆ L obs k , (48)where we have used the identity ( k × ∇ k ) i ( k × ∇ k ) j − ( k × ∇ k ) j ( k × ∇ k ) i = − (cid:15) i jk ( k × ∇ k ) j . (49)1Di ff erent from previous studies [30, 32, 44, 45], we show that ˆ L obs satisfies the canonical angular momentum commutationrelations. More importantly, we show that the spin and OAM of light commute with each other[ ˆ S obs i , ˆ L obs j ] = (cid:126) (cid:90) d k (cid:90) d k (cid:48) e i ( k , (cid:88) λ (cid:48) = , [ˆ a † k , ˆ a k , − ˆ a † k , ˆ a k , , ˆ a † k (cid:48) ,λ (cid:48) ( k (cid:48) × ∇ k (cid:48) ) j ˆ a k (cid:48) ,λ (cid:48) ] (50) = (cid:126) (cid:90) d ke i ( k , (cid:110) ˆ a † k , ( k × ∇ k ) j ˆ a k , − ˆ a † k , ( k × ∇ k ) j ˆ a k , − ˆ a † k , ( k × ∇ k ) j ˆ a k , + ˆ a † k , ( k × ∇ k ) j ˆ a k , (cid:111) = . (51)This marks a significant departure from previous literature [30, 32, 44, 45]. B. Circular polarization representation
The main task of this work is to explore the quantum properties of the photon spin. Thus, it will much more convenient to usecircular-polarization basis. Now, we define two circularly polarized modes for a given wave vector k ˆ a k , + = √ (cid:0) ˆ a k , − i ˆ a k , (cid:1) , (52)ˆ a k , − = √ (cid:0) ˆ a k , + i ˆ a k , (cid:1) , (53)with corresponding polarization vectors e ( k , + ) = √ e ( k , + i e ( k , , (54) e ( k , − ) = √ e ( k , − i e ( k , . (55)Based on this definition, (cid:15) ( k , + ) corresponds to left-hand polarized mode and (cid:15) ( k , − ) corresponds to right-hand circularly polar-ized light.The corresponding EM field operators are given byˆ A ⊥ ( r , t ) = (cid:90) d k (cid:88) λ = ± (cid:115) (cid:126) ε ω (2 π ) (cid:104) ˆ a k ,λ e ( k , λ ) e i ( k · r − ω t ) + h . c . (cid:105) . (56)ˆ E ⊥ ( r , t ) = i (cid:90) d k (cid:88) λ = ± (cid:115) (cid:126) ω ε (2 π ) (cid:104) ˆ a k ,λ e ( k , λ ) e i ( k · r − ω t ) − h . c . (cid:105) , (57)ˆ B ( r , t ) = c (cid:90) d k (cid:115) (cid:126) ω ε (2 π ) (cid:110)(cid:2) ˆ a k , + e ( k , + ) − ˆ a k , − e ( k , − ) (cid:3) e i ( k · r − ω t ) + h . c . (cid:111) . (58)The photon spin, OAM, and helicity can be re-expressed asˆ S obs = (cid:126) (cid:90) d k (cid:104) ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − (cid:105) e ( k , , (59)ˆ L obs = − i (cid:126) (cid:90) d k (cid:88) λ = ± ˆ a † k ,λ ( k × ∇ k )ˆ a k ,λ , (60)ˆ Λ M = (cid:126) (cid:90) d k (cid:104) ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − (cid:105) (61)For convenience, we will mainly work in the cylindrical coordinate frame in k -space k = k z e z + ρ k e ρ = ρ k cos ϕ k e x + ρ k sin ϕ k e y + k z e z , (62)where ρ k is the radial distance from the k z -axis, ϕ k is the azimuth angle, and e denotes the corresponding unit vector.2 II. PHOTON OAM OPERATOR IN RECIPROCAL SPACE
In this section, we give the quantum operators of photon OAM operators in reciprocal space. In a given coordinate frame, thethree components of L obs are given by ˆ L obs M , x = (cid:90) d k (cid:88) λ = , ˆ a † k ,λ ˆ l x ˆ a k ,λ , (63)ˆ L obs M , y = (cid:90) d k (cid:88) λ = , ˆ a † k ,λ ˆ l y ˆ a k ,λ , (64) L obs M , z = (cid:90) d k (cid:88) λ = , ˆ a † k ,λ ˆ l z ˆ a k ,λ , (65)and their squares are given by( L obs M , x ) = (cid:90) d k (cid:90) d k (cid:48) (cid:88) λ,λ (cid:48) = , ˆ a † k ,λ ˆ a † k (cid:48) ,λ (cid:48) ˆ l x ˆ l (cid:48) x ˆ a k (cid:48) ,λ (cid:48) ˆ a k ,λ + (cid:90) d k (cid:88) λ = , ˆ a † k ,λ ˆ l x ˆ a k ,λ , (66)( L obs M , y ) = (cid:90) d k (cid:90) d k (cid:48) (cid:88) λ,λ (cid:48) = , ˆ a † k ,λ ˆ a † k (cid:48) ,λ (cid:48) ˆ l y ˆ l (cid:48) y ˆ a k (cid:48) ,λ (cid:48) ˆ a k ,λ + (cid:90) d k (cid:88) λ = , ˆ a † k ,λ ˆ l y ˆ a k ,λ , (67)( L obs M , z ) = (cid:90) d k (cid:90) d k (cid:48) (cid:88) λ,λ (cid:48) = , ˆ a † k ,λ ˆ a † k (cid:48) ,λ (cid:48) ˆ l z ˆ l (cid:48) z ˆ a k (cid:48) ,λ (cid:48) ˆ a k ,λ + (cid:90) d k (cid:88) λ = , ˆ a † k ,λ ˆ l z ˆ a k ,λ . (68)where in a Cartesian coordinate frame, the di ff erential operators ˆ l α are given byˆ l x = − i (cid:126) (cid:32) k y ∂∂ k z − k z ∂∂ k y (cid:33) , ˆ l y = − i (cid:126) (cid:32) k z ∂∂ k x − k x ∂∂ k z (cid:33) , ˆ l z = − (cid:126) (cid:32) k x ∂∂ k y − k y ∂∂ k x (cid:33) . (69)For propagating quantum pulses, it is more convenient to evaluate the photon OAM in a spherical or cylindrical coordinateframe usually. Here, we also give the expression of the di ff erential operators in these two coordinate frames.Similar to the real-space counterparts, the reciprocal-space di ff erential operators in a spherical coordinate with k x = k sin θ k cos ϕ k , (70) k y = k sin θ k sin ϕ k , (71) k z = k cos θ k , (72)are given by ˆ l x = i (cid:126) (cid:32) sin ϕ k ∂∂θ k + cot θ k cos ϕ k ∂∂ϕ k (cid:33) , (73)ˆ l y = i (cid:126) (cid:32) − cos ϕ k ∂∂θ k + cot θ k sin ϕ k ∂∂ϕ k (cid:33) , (74)ˆ l z = − i (cid:126) ∂∂ϕ k . (75)We also have the summation of their squaresˆ l x + ˆ l y + ˆ l z = − (cid:126) θ k ∂∂θ k sin θ k ∂∂θ k + θ k ∂ ∂ϕ k . (76)In a cylindrical coordiante with k x = ρ k cos ϕ k , (77) k y = ρ k sin ϕ k , (78) k z = k z , (79)3we have ˆ l x = − i (cid:126) (cid:32) ρ k sin ϕ k ∂∂ k z − k z sin ϕ k ∂∂ρ k − k z ρ k cos ϕ k ∂∂ϕ k (cid:33) , (80)ˆ l y = i (cid:126) (cid:32) ρ k cos ϕ k ∂∂ k z − k z cos ϕ k ∂∂ρ k + k z ρ k sin ϕ k ∂∂ϕ k (cid:33) , (81)ˆ l z = − i (cid:126) ∂∂ϕ k . (82)To evaluate the uncertainty of all components of photon OAM, we also need the the following di ff erential operatorsˆ l x = − (cid:126) ρ k sin ϕ k ∂ ∂ k z + k z sin ϕ k ∂ ∂ρ k + k z ρ k cos ϕ k ∂ ∂ϕ k − k z ρ k sin ϕ k cos ϕ k ∂∂ϕ k − ρ k k z sin ϕ k ∂∂ k z ∂∂ρ k − ρ k sin ϕ k ∂∂ρ k − k z sin ϕ k ∂∂ k z + k z ρ k sin ϕ k cos ϕ k ∂∂ρ k ∂∂ϕ k − k z ρ k sin ϕ k cos ϕ k ∂∂ϕ k + k z ρ k cos ϕ k ∂∂ρ k − k z sin ϕ k cos ϕ k ∂∂ k z ∂∂ϕ k − sin ϕ k cos ϕ k ∂∂ϕ k − k z cos ϕ k ∂∂ k z (cid:35) , (83)ˆ l x = − (cid:126) ρ k cos ϕ k ∂ ∂ k z + k z cos ϕ k ∂ ∂ρ k + k z ρ k sin ϕ k ∂ ∂ϕ k + k z ρ k sin ϕ k cos ϕ k ∂∂ϕ k − ρ k k z cos ϕ k ∂∂ k z ∂∂ρ k − ρ k cos ϕ k ∂∂ρ k − k z cos ϕ k ∂∂ k z − k z ρ k sin ϕ k cos ϕ k ∂∂ρ k ∂∂ϕ k + k z ρ k sin ϕ k cos ϕ k ∂∂ϕ k + k z ρ k sin ϕ k ∂∂ρ k + k z sin ϕ k cos ϕ k ∂∂ k z ∂∂ϕ k + sin ϕ k cos ϕ k ∂∂ϕ k − k z sin ϕ k ∂∂ k z (cid:35) , (84)ˆ l x = − (cid:126) ∂ ∂ϕ k . (85)Their summation gives ˆ l x + ˆ l y + ˆ l z = − (cid:126) (cid:32) k z ∂∂ρ k − ρ k ∂∂ k z (cid:33) − k z ∂∂ k z + k z ρ k ∂∂ρ k + k z ρ k + ∂ ∂ϕ k . (86) III. WAVEFUNCTION OF QUANTUM TWISTED PHOTON PULSE
The single-photon wave-packet creation operator for an arbitrary quantum twisted photon pulse is defined asˆ a † ξλ = (cid:90) d k ξ λ ( k )ˆ a † k λ , (87)which is a coherent superposition of plane waves with fixed circular polarization λ . In this work, we only focus on twistedphoton pulse, thus the spectral amplitude function (SAF) will have a helical wavefront phase. Generally, the SAF of a twistedpulse with determined OAM can be expressed as ξ λ ( k ) = √ π η λ ( k z , ρ k ) e im ϕ k , (88)where η λ ( k z , ρ k ) is determined by the the pulse shape and pulse type as shown in the following. The phase exp( im ϕ k ) with aninteger m finally leads to the OAM in z -direction of a single-photon pulse. To guarantee that ˆ a † ξλ obeys the bosonic commutationrelation [ˆ a ξλ , ˆ a † ξλ ] =
1, the SAF should satisfy the normalization condition (cid:90) d k | ξ λ ( k ) | = . (89)4 A. Fock-state and coherent-state photon pulses
The wave-packet creation operator ˆ a † ξλ can be treated as a normal ladder operator of a harmonic oscillator. Thus, we canconstruct arbitrary photon-pulse state [21]. We first construct the the n -photon Fock-state pulse (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = √ n ! (cid:16) ˆ a † ξλ (cid:17) n | (cid:105) . (90)Using the commutation relation [ˆ a ξλ , ˆ a † ξλ ] =
1, we can verify that this quantum state satisfies the othonormal relation (cid:104) n ξλ | n (cid:48) ξλ (cid:105) = δ nn (cid:48) . (91)Using the relations [ˆ a k (cid:48) ,λ (cid:48) , (cid:16) ˆ a † ξλ (cid:17) n ] = n ξ λ (cid:48) ( k (cid:48) ) δ λλ (cid:48) (cid:16) ˆ a † ξλ (cid:17) n − , (92)[ˆ a † k (cid:48) ,λ (cid:48) , (cid:16) ˆ a ξλ (cid:17) n ] = − n ξ ∗ λ (cid:48) ( k (cid:48) ) δ λλ (cid:48) (cid:16) ˆ a ξλ (cid:17) n − , (93)we can also verify that the photon number in this n -photon Fock-state pulse is exact n , i.e., (cid:104) n ξλ | (cid:90) d k (cid:88) λ (cid:48) = ± ˆ a † k ,λ (cid:48) ˆ a k ,λ (cid:48) | n ξλ (cid:105) = n . (94)Similarly, we can construct a coherent-state pulse (cid:12)(cid:12)(cid:12) α ξλ (cid:69) = exp (cid:32) α ˆ a † ξλ −
12 ¯ n (cid:33) | (cid:105) = e − ¯ n / ∞ (cid:88) n = α n n ! (cid:16) ˆ a † ξλ (cid:17) n | (cid:105) = e − ¯ n / ∞ (cid:88) n = α n √ n ! (cid:12)(cid:12)(cid:12) n ξλ (cid:69) , (95)where ¯ n = | α | is the mean photon number in the coherent-state pulse. We can check that this state is well normalized (cid:68) α ξλ | α ξλ (cid:69) = e − ¯ n ∞ (cid:88) m , n = | α | n √ m ! n ! (cid:68) m ξλ | n ξλ (cid:69) = e − ¯ n ∞ (cid:88) m , n = ¯ n n n ! = . (96)It also has the coherent-state property ˆ a k ,λ (cid:48) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) = αξ λ ( k ) δ λ (cid:48) λ (cid:12)(cid:12)(cid:12) α ξλ (cid:69) , (97) B. Spectral amplitude function
In this subsection, we give the SAF for the Bessel pulse and Laguerre-Gauss pulse, which are the most common twistedphoton pulses used in experiments. In the main text, we only studied the Bessel pulse.We first look at the Bessel pulse. The SAF of a Bessel pulse with a Gaussian envelop can be express η λ ( k z , ρ k ) as the productof two Gaussian functions η λ ( k z , ρ k ) = (cid:32) σ z π (cid:33) / exp (cid:104) − σ z ( k z − k z , c ) (cid:105) × σ ρ π k ⊥ , c / exp (cid:104) − σ ρ ( ρ k − k ⊥ , c ) (cid:105) . (98)The first Gaussian function with width 1 /σ z and center wave vector k z , c characterize the envelope of the pulse in the propagatingaxis. The pulse length on z -axis in real space is given by σ z = c τ p ( τ p the pulse length in time). The second Gaussian functionwith width 1 /σ ρ and center value k ⊥ , c = k z , c tan θ c reflects the fact that a single-frequency Bessel beam is the superposition of allplane waves on a cone in k -space. Thus, the two Gaussian functions should have the same ratio between the center wave-number5and the width, i.e., k z , c σ z = k ⊥ , c σ ρ ≡ C . In the narrow bandwidth limit C (cid:29)
1, our defined SAF is well normalized (cid:90) d k | ξ λ ( k ) | = (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k (cid:90) π d ϕ k π η λ ( k z , ρ k ) = (cid:32) σ z π (cid:33) / (cid:90) ∞−∞ dk z e − σ z ( k z − k z , c ) × σ ρ π / (cid:90) ∞ ρ k k ⊥ , c e − σ ρ ( ρ k − k ⊥ , c ) d ρ k (99) ≈ (cid:32) σ z π (cid:33) / (cid:90) ∞−∞ dk z e − σ z ( k z − k z , c ) × σ ρ π / (cid:90) ∞−∞ e − σ ρ ( ρ k − k ⊥ , c ) d ρ k (100) = . (101)The SAF of a Laguerre-Gaussian (LG) pulse with a Gaussian evelope is given be ξ pm λ ( k ) = N − / (cid:32) σ z π (cid:33) / e − σ z ( k z − k z , c ) × (cid:114) π ρ p + | m | k e im ϕ k exp − w + i ζ ) ρ k , (102)Similarly, the first Gaussian function with width 1 /σ z and center wave vector k z , c characterize the envelope of the pulse in thepropagating axis. The second complicated function with integer numbers m and p characterize the amplitude function of theelegant LG mode in k -space [46]. Here, w is the pulse waist in xy -plane and ζ = z / z R is the reduced coordinate with respect tothe Rayleigh lenght z R = k z w /
2. The normalization factor N is given by N = Γ (2 p + | m | + w / p + | m | + . (103) IV. QUANTUM STATISTICS OF THE ANGULAR MOMENTA OF TWISTED PHOTON PULSES
In this section, we give the details of the evaluation of the mean value and quantum uncertainty of the angular momentum ofFock-state and coherent-state twisted photon pulses.
A. Spin angular momentum of twisted photon pulses
We first study the photon spin for a Fock-state Bessel pulse. It mean value is given by (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ S obs (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = (cid:126) (cid:90) d k (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) e ( k ,
3) (104) = (cid:126) n λ (cid:90) d k | ξ λ ( k ) | e ( k ,
3) (105) ≈ (cid:126) n λ π (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k (cid:90) π d ϕ k | η λ ( k z , ρ k ) | (sin θ c cos ϕ k , sin θ c sin ϕ k , cos θ c ) (106) = (cid:126) n λ (0 , , cos θ c ) , (107)where in the thirst step, we have used the approximation the unit vector e ( k , = k / | k | ≈ (sin θ c cos ϕ k , sin θ c sin ϕ k , cos θ c ).Now, we calculate the mean values of ( ˆ S obs x ) , ( ˆ S obs y ) , and ( ˆ S obs z ) separately, (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ( ˆ S obs z ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) ≈ (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (cid:34) (cid:126) (cid:90) d k (ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − ) cos θ c (cid:35) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (108) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (cid:126) cos θ c (cid:90) d k (cid:90) d k (cid:48) (cid:104) ˆ a † k ,λ ˆ a † k (cid:48) ,λ ˆ a k (cid:48) ,λ ˆ a k ,λ + δ ( k − k (cid:48) )ˆ a † k ,λ ˆ a k ,λ (cid:105) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (109) = (cid:126) cos θ c (cid:34) n ( n − (cid:90) d k | ξ λ ( k ) | (cid:90) d k (cid:48) (cid:12)(cid:12)(cid:12) ξ λ ( k (cid:48) ) (cid:12)(cid:12)(cid:12) + n (cid:90) d k | ξ λ ( k ) | (cid:35) = n (cid:126) cos θ c . (110)6 (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ( ˆ S obs x ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) ≈ (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (cid:34) (cid:126) (cid:90) d k (ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − ) sin θ c cos ϕ k (cid:35) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (111) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (cid:126) sin θ c (cid:90) cos ϕ k d k (cid:90) cos ϕ (cid:48) k d k (cid:48) (cid:104) ˆ a † k ,λ ˆ a † k (cid:48) ,λ ˆ a k (cid:48) ,λ ˆ a k ,λ + δ ( k − k (cid:48) )ˆ a † k ,λ ˆ a k ,λ (cid:105) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (112) = n (cid:126) sin θ c . (113) (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ( ˆ S obs y ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) ≈ (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (cid:34) (cid:126) (cid:90) d k (ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − ) sin θ c sin ϕ k (cid:35) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (114) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (cid:126) sin θ c (cid:90) sin ϕ k d k (cid:90) sin ϕ (cid:48) k d k (cid:48) (cid:104) ˆ a † k ,λ ˆ a † k (cid:48) ,λ ˆ a k (cid:48) ,λ ˆ a k ,λ + δ ( k − k (cid:48) )ˆ a † k ,λ ˆ a k ,λ (cid:105) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (115) = n (cid:126) sin θ c . (116)Thus, the standard derivations of the spin of an n -photon Fock-state Bessel pulse are given by ∆ ˆ S obs x = ∆ ˆ S obs y = (cid:126) (cid:112) n / | sin θ c | , ∆ ˆ S obs z = . (117)Similarly, we can calculate the mean value and the quantum uncertainty of the spin of a coherent-state Bessel pulse: (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ˆ S obs (cid:12)(cid:12)(cid:12) α ξλ (cid:69) = (cid:126) (cid:90) d k (cid:68) α ξλ (cid:12)(cid:12)(cid:12) (ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) e ( k , = (cid:126) ¯ n λ (0 , , cos θ c ) , (118)where ¯ n = | α | is the mean photon number of the pulse. The mean values of ˆ S M , x , ˆ S M , y , and ˆ S M , z are given by (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ( ˆ S obs z ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) ≈ (cid:68) α ξλ (cid:12)(cid:12)(cid:12) (cid:34) (cid:126) (cid:90) d k (ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − ) cos θ c (cid:35) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) (119) = (cid:68) α ξλ (cid:12)(cid:12)(cid:12) (cid:126) cos θ c (cid:90) d k (cid:90) d k (cid:48) (cid:104) ˆ a † k ,λ ˆ a † k (cid:48) ,λ ˆ a k (cid:48) ,λ ˆ a k ,λ + δ ( k − k (cid:48) )ˆ a † k ,λ ˆ a k ,λ (cid:105) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) (120) = ¯ n (¯ n + (cid:126) cos θ c . (121) (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ( ˆ S obs x ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) ≈ (cid:68) α ξλ (cid:12)(cid:12)(cid:12) (cid:34) (cid:126) (cid:90) d k (ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − ) sin θ c cos ϕ k (cid:35) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) (122) = (cid:68) α ξλ (cid:12)(cid:12)(cid:12) (cid:126) sin θ c (cid:90) cos ϕ k d k (cid:90) cos ϕ (cid:48) k d k (cid:48) (cid:104) ˆ a † k ,λ ˆ a † k (cid:48) ,λ ˆ a k (cid:48) ,λ ˆ a k ,λ + δ ( k − k (cid:48) )ˆ a † k ,λ ˆ a k ,λ (cid:105) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) (123) =
12 ¯ n (cid:126) sin θ c . (124) (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ( ˆ S obs y ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) ≈ (cid:68) α ξλ (cid:12)(cid:12)(cid:12) (cid:34) (cid:126) (cid:90) d k (ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − ) sin θ c sin ϕ k (cid:35) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) (125) = (cid:68) α ξλ (cid:12)(cid:12)(cid:12) (cid:126) sin θ c (cid:90) sin ϕ k d k (cid:90) sin ϕ (cid:48) k d k (cid:48) (cid:104) ˆ a † k ,λ ˆ a † k (cid:48) ,λ ˆ a k (cid:48) ,λ ˆ a k ,λ + δ ( k − k (cid:48) )ˆ a † k ,λ ˆ a k ,λ (cid:105) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) (126) = n (cid:126) sin θ c . (127)Thus, the standard derivations of the spin of an n -photon coherent-state Bessel pulse are given by ∆ ˆ S obs x = ∆ ˆ S obs y = (cid:126) (cid:112) ¯ n / | sin θ c | , ∆ ˆ S obs z = ¯ n (cid:126) | cos θ c | . (128) B. Orbital angular momentum of twisted photon pulses
In this subsection, we give the details of the photon OAM evalution. The mean value of ˆ L obs z for a Fock-state twisted pulse isgiven by (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L obs z (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = − in (cid:126) π (cid:90) d k η λ ( k ) e − im ϕ k ∂∂ϕ k η λ ( k ) e im ϕ k (129) = nm (cid:126) (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k (cid:90) π d ϕ k π η λ ( k z , ρ k ) = mn (cid:126) , (130)7which is only determined by the photon number n and integer m in the helical phase factor exp( im ϕ k ). The mean values ofphoton OAM in xy -plane are given by (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L obs x (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = − in (cid:126) π (cid:90) d k η λ ( k ) e − im ϕ k (cid:32) ρ k sin ϕ k ∂∂ k z − k z sin ϕ k ∂∂ρ k − k z ρ k cos ϕ k ∂∂ϕ k (cid:33) η λ ( k ) e im ϕ k (131) = − in (cid:126) π (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k (cid:34) η λ (cid:32) ρ k ∂∂ k z − k z ∂∂ρ k (cid:33) η λ (cid:90) π sin ϕ k d ϕ k − im k z ρ k η λ (cid:90) π cos ϕ k d ϕ k (cid:35) (132) = , (133) (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L obs y (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = in (cid:126) π (cid:90) d k η λ ( k ) e − im ϕ k (cid:32) ρ k cos ϕ k ∂∂ k z − k z cos ϕ k ∂∂ρ k + k z ρ k sin ϕ k ∂∂ϕ k (cid:33) η λ ( k ) e im ϕ k (134) = in (cid:126) π (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k (cid:34) η λ (cid:32) ρ k ∂∂ k z − k z ∂∂ρ k (cid:33) η λ (cid:90) π cos ϕ k d ϕ k + im k z ρ k η λ (cid:90) π sin ϕ k d ϕ k (cid:35) (135) = . (136)To obtain the Heisenberg uncertainty relations of photon OAM, we first evaluate the mean values of ( ˆ L obs x ) , ( ˆ L obs y ) , and( ˆ L obs z ) . The value of ( ˆ L obs z ) can be obtained straightforwardly, (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ( ˆ L obs z ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (cid:90) d k (cid:88) λ (cid:48) ˆ a † k ,λ (cid:48) ˆ l z ˆ a k ,λ (cid:48) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (137) = − (cid:126) n ( n − (cid:34)(cid:90) d k ξ ∗ λ ( k ) ∂∂ϕ k ξ λ ( k ) (cid:35) + n (cid:90) d k ξ ∗ λ ( k ) ∂ ∂ϕ k ξ λ ( k ) (138) = (cid:126) m n (139)However, the evaluation of (cid:104) ( ˆ L obs x ) (cid:105) and (cid:104) ( ˆ L obs y ) (cid:105) is much more complicated. We first make some simplification, (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ( ˆ L obs x ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) (cid:90) d k (cid:88) λ (cid:48) ˆ a † k ,λ (cid:48) ˆ l x ˆ a k ,λ (cid:48) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (140) = n ( n − (cid:34)(cid:90) d k ξ ∗ λ ( k )ˆ l x ξ λ ( k ) (cid:35) + n (cid:90) d k ξ ∗ λ ( k )ˆ l x ξ λ ( k ) (141)where the first term vanishes and the second term gives (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ( ˆ L obs x ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = n (cid:90) d k ξ ∗ λ ( k )ˆ l x ξ λ ( k ) = − n (cid:126) (cid:90) d k ξ ∗ λ ( k ) cos ϕ k k z ρ k ∂ ∂ϕ k + k z ρ k ∂∂ρ k (142) + sin ϕ k ρ k ∂ ∂ k z + k z ∂ ∂ρ k − ρ k k z ∂∂ρ z ∂∂ k z − ρ k ∂∂ρ k − k z ∂∂ k z ξ λ ( k ) (143) = − n (cid:126) (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k η λ (cid:32) k z ∂∂ρ k − ρ k ∂∂ k z (cid:33) − m k z ρ k + k z ρ k ∂∂ρ k − k z ∂∂ k z η λ , (144)where other terms containing the product sin ϕ k cos ϕ k in ˆ l x have been dropped because they will vanish after integral over ϕ k .Similarly, the mean value of ˆ L obs y ) can be simplified as (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L y (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = − n (cid:126) (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k η λ (cid:32) k z ∂∂ρ k − ρ k ∂∂ k z (cid:33) − m k z ρ k + k z ρ k ∂∂ρ k − k z ∂∂ k z η λ . (145)8Using the SAF of Bessel pulses, we have the following relations, − (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k η λ (cid:32) k z ∂∂ρ k − ρ k ∂∂ k z (cid:33) η λ = − (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k η λ ρ k ∂ ∂ k z + k z ∂ ∂ρ k − ρ k k z ∂∂ρ z ∂∂ k z − k z ∂∂ k z − ρ k ∂∂ρ k η λ (146) = − (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k η λ (cid:110)(cid:104) σ z ( k z − k z , c ) − σ z (cid:105) ρ k + k z (cid:104) σ ρ ( ρ k − k ⊥ , c ) − σ ρ (cid:105) − σ z σ ρ k z ( k z − k z , c ) ρ k ( ρ k − k ⊥ , c ) + σ z k z ( k z − k z , c ) + σ ρ ρ k ( ρ k − k ⊥ , c ) (cid:105) η λ (147) ≈ σ z k ⊥ , c + σ z σ ρ + σ ρ k z , c + σ ρ σ z + − − = (cid:34) C (tan θ c + cot θ c ) +
14 tan θ c +
34 cot θ c − (cid:35) , (148) − (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k η λ k z ρ k ∂∂ρ k η λ = σ ρ (cid:90) ∞−∞ dk z (cid:90) ∞ d ρ k k z ( ρ k − k ⊥ , c ) η λ ≈ , (149) (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k η λ k z ∂∂ k z η λ = − σ z (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k k z ( k z − k z , c ) η λ ≈ − . (150)We have used a Gaussian function to replace the square of the delta function δ ( ρ k − k ⊥ , c ) in the SAF of Bessel pulses. This leadsto the divergence of the integral of the term η λ k z /ρ k . Here, we use the relation k z /ρ k = cot θ c , to obtain (cid:90) ∞−∞ dk z (cid:90) ∞ ρ k d ρ k η λ k z ρ k η λ ≈ cot θ c . (151)Then, we finally obtain (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L x (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L y (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (152) ≈ n (cid:126) (cid:34) C (tan θ c + cot θ c ) +
14 tan θ c + ( 34 + m ) cot θ c − (cid:35) (153)Now, we verify the Heisenberg uncertainty relations for photon OAM. The polar angle θ c ∈ (0 , π/
2) for a Bessel pulse is verysmall usually. Let tan θ c = x ∈ (0 , ∞ ), then we have (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L x (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ L y (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (154) = n (cid:126) (cid:34)(cid:32) C + (cid:33) x + (cid:32) C + m + (cid:33) x − (cid:35) (155) ≥ n (cid:126) (cid:114) (4 C + C + m +
34 ) − (156) (cid:29) mn (cid:126) , (157)where we have used the relation a x + b / x ≥ | ab | and the fact C (cid:29)
1. We can easily verify the Heisenberg uncertaintyrelation for all Bessel pulses (cid:113) ( ∆ ˆ L x ) ( ∆ ˆ L y ) > (cid:126) (cid:12)(cid:12)(cid:12) (cid:104) ˆ L z (cid:105) (cid:12)(cid:12)(cid:12) = mn (cid:126) (158)On the eigenstates of ˆ L obs z , the mean values of ˆ L obs x and ˆ L obs y vanish. Thus, the other two Heisenberg uncertainty relations aretrivial. Similar results for coherent-state twisted Bessel pulses can be obtained in the same way.9 C. Revisit of commutation relation between the photon spin and OAM
We now verify the commutation relations between ˆ S obs and ˆ L obs via evaluating the matrix element of the product of theoperators ˆ S obs i and ˆ L obs j . Here, we take the commutator [ ˆ S obs i , ˆ L obs z ] as an example. The matrix element of the operators betweentwo arbitrary Fock states | n ξλ (cid:105) and | ˜ n ˜ ξ ˜ m ˜ λ (cid:105) is given by (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) ˆ S obs i ˆ L obs z (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (159) = − i (cid:126) (cid:90) d k (cid:90) d k (cid:48) (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) (cid:16) ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − (cid:17) (cid:88) λ (cid:48) ˆ a † k (cid:48) ,λ (cid:48) ∂∂ϕ k (cid:48) ˆ a k (cid:48) ,λ (cid:48) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) e i ( k ,
3) (160) = − i λ (cid:126) (cid:90) d k (cid:90) d k (cid:48) (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) ˆ a † k ,λ (cid:34) ˆ a † k (cid:48) ,λ ∂∂ϕ k (cid:48) ˆ a k ,λ + δ ( k − k (cid:48) ) ∂∂ϕ k (cid:48) (cid:35) ˆ a k (cid:48) ,λ (cid:12)(cid:12)(cid:12) n ξλ (cid:69) e i ( k ,
3) (161) = − i λ (cid:126) δ λ ˜ λ (cid:90) e i ( k , d k (cid:90) d k (cid:48) (cid:34) (cid:112) n ( n − n (˜ n −
1) ˜ ξ ∗ ˜ m ˜ λ ( k ) ˜ ξ ∗ ˜ m ˜ λ ( k (cid:48) ) ∂∂ϕ k (cid:48) ξ λ ( k ) ξ λ ( k (cid:48) ) (cid:68) (˜ n − ˜ ξ ˜ m ˜ λ | ( n − ξλ (cid:69) + √ n ˜ n ˜ ξ ∗ ˜ m ˜ λ ( k ) δ ( k − k (cid:48) ) ∂∂ϕ k (cid:48) ξ λ ( k (cid:48) ) (cid:68) (˜ n − ˜ ξ ˜ m ˜ λ | ( n − ξλ (cid:69)(cid:35) (162) (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) ˆ L obs z ˆ S obs i (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (163) = − i (cid:126) (cid:90) d k (cid:90) d k (cid:48) (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) (cid:88) λ (cid:48) ˆ a † k (cid:48) ,λ (cid:48) ∂∂ϕ k (cid:48) ˆ a k (cid:48) ,λ (cid:48) (cid:16) ˆ a † k , + ˆ a k , + − ˆ a † k , − ˆ a k , − (cid:17) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) e i ( k ,
3) (164) = − i λ (cid:126) (cid:90) d k (cid:90) d k (cid:48) (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) ˆ a † k (cid:48) ,λ (cid:34) ˆ a † k ,λ ∂∂ϕ k (cid:48) ˆ a k (cid:48) ,λ + ∂∂ϕ k (cid:48) δ ( k − k (cid:48) ) (cid:35) ˆ a k ,λ (cid:12)(cid:12)(cid:12) n ξλ (cid:69) e i ( k ,
3) (165) = − i λ (cid:126) δ λ ˜ λ (cid:90) e i ( k , d k (cid:90) d k (cid:48) (cid:34) (cid:112) n ( n − n (˜ n −
1) ˜ ξ ∗ ˜ m ˜ λ ( k (cid:48) ) ˜ ξ ∗ ˜ m ˜ λ ( k ) ∂∂ϕ k (cid:48) ξ λ ( k (cid:48) ) ξ λ ( k ) (cid:68) (˜ n − ˜ ξ ˜ m ˜ λ | ( n − ξλ (cid:69) + √ n ˜ n ˜ ξ ∗ ˜ m ˜ λ ( k ) ∂∂ϕ k (cid:48) δ ( k − k (cid:48) ) ξ λ ( k (cid:48) ) (cid:68) (˜ n − ˜ ξ ˜ m ˜ λ | ( n − ξλ (cid:69)(cid:35) , (166)where we note the fact that the inner product (cid:104) ˜ n ˜ ξ ˜ m ˜ λ | n ξλ (cid:105) is independent on both k and k (cid:48) . It can be easily verified that (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) (cid:104) ˆ S obs i , ˆ L obs z (cid:105) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) ˆ S obs i ˆ L obs z (cid:12)(cid:12)(cid:12) n ξλ (cid:69) − (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) ˆ L obs z ˆ S obs i (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = , (167)because the di ff erential operator ∂/∂ϕ k (cid:48) does not act on the function ξ ∗ λ ( k ) and the order between ∂/∂ϕ k (cid:48) and the delta function δ ( k − k (cid:48) ) does not change the value of the double integration.Similarly, we can verify that (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) (cid:104) ˆ S obs i , ˆ L obs x (cid:105) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = (cid:68) ˜ n ˜ ξ ˜ m ˜ λ (cid:12)(cid:12)(cid:12) (cid:104) ˆ S obs i , ˆ L obs y (cid:105) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = . (168)On the other hand, we can also evaluate the matrix element of the commutator [ ˆ S obs i , ˆ L obs j ] between two arbitrary coherent statesand obtain the same result (cid:104) ˜ α ˜ ξ ˜ m ˜ λ | [ ˆ S obs i , ˆ L obs j ] | α ξλ (cid:105) =
0. The Fock-state set {| n ξλ (cid:105)} or the coherent-state set {| α ξλ (cid:105)} can form acomplete basis. Thus, the zero matrix elements of the commutator leads to the operator identity [ ˆ S obs i , ˆ L obs j ] = V. QUANTUM SPIN TEXTURE OF BESSEL PHOTON PULSE
In this section, we give the details about the quantum properties of the photon spin texture, i.e., the photon spin density,ˆ s obs ( r , t ) = ε ˆ E ⊥ ( r , t ) × ˆ A ⊥ ( r , t ) (169) = − i (cid:126) π ) (cid:88) λ (cid:48) λ (cid:48)(cid:48) (cid:90) d k (cid:48) (cid:90) d k (cid:48)(cid:48) (cid:114) ω (cid:48) ω (cid:48)(cid:48) + (cid:114) ω (cid:48)(cid:48) ω (cid:48) ˆ a † k (cid:48) ,λ (cid:48) ˆ a k (cid:48)(cid:48) ,λ (cid:48)(cid:48) e − i [( k (cid:48) − k (cid:48)(cid:48) ) · r − ( ω (cid:48) − ω (cid:48)(cid:48) ) t ] e ∗ ( k (cid:48) , λ (cid:48) ) × e ( k (cid:48)(cid:48) , λ (cid:48)(cid:48) ) . (170)In the narrow bandwidth limit, we have ω (cid:48) ≈ ω (cid:48)(cid:48) . The spin densities of an n -photon Fock-state and a coherent-state pulses aregiven by (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ s obs ( r , t ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) ≈ − in (cid:126) (2 π ) (cid:90) d k (cid:48) (cid:90) d k (cid:48)(cid:48) ξ ∗ λ ( k (cid:48) ) ξ λ ( k (cid:48)(cid:48) ) e − i [( k (cid:48) − k (cid:48)(cid:48) ) · r − ( ω (cid:48) − ω (cid:48)(cid:48) ) t ] e ∗ ( k (cid:48) , λ ) × e ( k (cid:48)(cid:48) , λ ) , (171)0 ! " ! ! " FIG. 5. Relation between the wave-vector-dependent coordinate frame (the green arrows) and the fixed lab coordinate (the blue arrows). and (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ˆ s obs ( r , t ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) ≈ − i ¯ n (cid:126) (2 π ) (cid:90) d k (cid:48) (cid:90) d k (cid:48)(cid:48) ξ ∗ λ ( k (cid:48) ) ξ λ ( k (cid:48)(cid:48) ) e − i [( k (cid:48) − k (cid:48)(cid:48) ) · r − ( ω (cid:48) − ω (cid:48)(cid:48) ) t ] e ∗ ( k (cid:48) , λ ) × e ( k (cid:48)(cid:48) , λ ) , (172)respectively.Using r = ( ρ cos ϕ, ρ sin ϕ, z ) and k = ( ρ k cos ϕ k , ρ k sin ϕ k , k z ), we can expand the phase factor as e − i [( k (cid:48) − k (cid:48)(cid:48) ) · r − ( ω (cid:48) − ω (cid:48)(cid:48) ) t ] = exp (cid:110) i (cid:104) ( ω (cid:48) − ω (cid:48)(cid:48) ) t − ( k (cid:48) z − k (cid:48)(cid:48) z ) z − ρρ (cid:48) k cos( ϕ − ϕ (cid:48) k ) + ρρ (cid:48)(cid:48) k cos( ϕ − ϕ (cid:48)(cid:48) k ) (cid:105)(cid:111) . (173)We can also expand k -dependent polarization unit vectors e ( k , λ ) in the fixed lab coordinate frame (see Fig. 5), e ( k , λ ) ≈ e − i λϕ k cos θ c e λ − e i λϕ k sin θ c e − λ − √ θ c e z , (174)where e λ = (cid:16) e x + i λ e y (cid:17) / √
2. Thus e ∗ ( k (cid:48) , λ ) × e ( k (cid:48)(cid:48) , λ ) = i λ √ (cid:20) e − i λϕ (cid:48)(cid:48) k sin θ c cos θ c + e − i λϕ (cid:48) k sin θ c sin θ c (cid:21) e λ (175) + i λ √ (cid:20) e i λϕ (cid:48)(cid:48) k sin θ c sin θ c + e i λϕ (cid:48) k cos θ c θ c (cid:21) e − λ (176) + i λ (cid:20) e i λ ( ϕ (cid:48) k − ϕ (cid:48)(cid:48) k ) cos θ c − e − i λ ( ϕ (cid:48) k − ϕ (cid:48)(cid:48) k ) sin θ c (cid:21) e z , (177)where we have used relations e λ × e ∗ λ = − i λ e z , e z × e ∗ λ = i λ e − λ , and e z × e λ = − i λ e λ .For convenience, we define the following amplitude function ψ λ ( r , t ) = (cid:112) (2 π ) (cid:90) d k ξ λ ( k ) e i [( k · r − ω ) t + λϕ k ] , (178)which is the e ff ective wave function of the photon pulse in real space. For a Bessel pulse, we have ψ λ ( r , t ) = (cid:115) k ⊥ , c πσ z σ ρ i m + λ J m + λ ( k ⊥ , c ρ ) e i ( m + λ ) ϕ exp (cid:34) − ( ct − z cos θ c ) σ z cos θ c − ik z , c ( z − t / cos θ c ) (cid:35) (179) = (cid:115) C π V p i m + λ J m + λ ( k ⊥ , c ρ ) e i ( m + λ ) ϕ exp (cid:34) − ( ct − z cos θ c ) σ z cos θ c − ik z , c ( z − t / cos θ c ) (cid:35) , (180)1where the constant C = k z , c σ z = k ⊥ , c σ ρ (cid:29) V p = σ z σ ρ characterizes the e ff ective volumeof the photon pulse, and we have used the Jacobi-Anger expansion e iz cos θ = ∞ (cid:88) n = −∞ i n J n ( z ) e in θ , (181) e − iz cos θ = e iz cos( θ + π ) = ∞ (cid:88) n = −∞ i n J n ( z ) e in ( θ + π ) = ∞ (cid:88) n = −∞ ( − i ) n J n ( z ) e in θ , (182)with the n th Bessel function of the first kind J n ( z ) = π (cid:90) π e i ( n ϕ − z sin ϕ ) d ϕ. (183)Then, we have (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ s obs (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = s λ e λ + s − λ e − λ + λ s z e z (184)where s − λ = s ∗ λ , s λ = − in (cid:126) × i λ √ θ c (cid:20) cos θ c ψ ∗ m , ( r , t ) ψ m , − λ ( r , t ) + sin θ c ψ ∗ λ ( r , t ) ψ m , ( r , t ) (cid:21) (185) = − in (cid:126) C sin θ c √ πσ z σ ρ e − i λϕ (cid:20) cos θ c J m ( k ⊥ , c ρ ) J m − λ ( k ⊥ , c ρ ) + sin θ c J m + λ ( k ⊥ , c ρ ) J m ( k ⊥ , c ρ ) (cid:21) exp (cid:34) − ( ct − z cos θ c ) σ z cos θ c (cid:35) , (186)and s z = − in (cid:126) × i (cid:20)(cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) cos θ c − | ψ λ ( r , t ) | sin θ c (cid:21) (187) = n (cid:126) C πσ z σ ρ (cid:26)(cid:2) J m − λ ( k ⊥ , c ρ ) (cid:3) cos θ c − (cid:2) J m + λ ( k ⊥ , c ρ ) (cid:3) sin θ c (cid:27) exp (cid:34) − ( ct − z cos θ c ) σ z cos θ c (cid:35) . (188)Using the relation e ± λ = ( e x ± i λ e y ) / √
2, we can express the spin density vector as (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ s obs ( r , t ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) = − s ρ e x sin λϕ + λ s ρ e y cos λϕ + λ s z e z = λ (cid:16) s ρ e ϕ + s z e z (cid:17) , (189)where s ρ = n (cid:126) C sin θ c πσ z σ ρ (cid:20) cos θ c J m ( k ⊥ , c ρ ) J m − λ ( k ⊥ , c ρ ) + sin θ c J m + λ ( k ⊥ , c ρ ) J m ( k ⊥ , c ρ ) (cid:21) exp (cid:34) − ( ct − z cos θ c ) σ z cos θ c (cid:35) (190)The spin density of a coherent-state Bessel pulse can be evaluated similarly. VI. QUANTUM CORRELATION OF SPIN DENSITY
In this section, we show how to evaluate the quantum correlations of the photon spin density. Due to the vector nature of thespin density, the full two-point correlation should be characterized by a 3 × G ( r , t ; r (cid:48) , t (cid:48) ) = (cid:68) ˆ s obs x ( r , t ) ˆ s obs x ( r (cid:48) , t (cid:48) ) (cid:69) (cid:68) ˆ s obs x ( r , t ) ˆ s obs y ( r (cid:48) , t (cid:48) ) (cid:69) (cid:68) ˆ s obs x ( r , t ) ˆ s obs z ( r (cid:48) , t (cid:48) ) (cid:69)(cid:68) ˆ s obs y ( r , t ) ˆ s obs x ( r (cid:48) , t (cid:48) ) (cid:69) (cid:68) ˆ s obs y ( r , t ) ˆ s obs y ( r (cid:48) , t (cid:48) ) (cid:69) (cid:68) ˆ s obs y ( r , t ) ˆ s obs z ( r (cid:48) , t (cid:48) ) (cid:69)(cid:68) ˆ s obs z ( r , t ) ˆ s obs x ( r (cid:48) , t (cid:48) ) (cid:69) (cid:68) ˆ s obs z ( r , t ) ˆ s obs y ( r (cid:48) , t (cid:48) ) (cid:69) (cid:68) ˆ s obs z ( r , t ) ˆ s obs z ( r (cid:48) , t (cid:48) ) (cid:69) . (191)Here, we only concern the equal-time correlator (cid:68) ˆ s obs z ( r , t ) ˆ s obs z ( r (cid:48) , t ) (cid:69) . To simplify the calculation, we take the paraxial approxi-mation, i.e., θ c ≈
0. Then, the two-point spin-density correlations for a Fock-state and coherent-state pulse are given by (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ s obs z ( r , t ) ˆ s obs z ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) ≈ (cid:126) (2 π ) (cid:90) d k (cid:90) d k (cid:90) d k (cid:90) d k e − i [( k − k ) · r + ( k − k ) · r (cid:48) − ( ω − ω + ω − ω ) t ] e i λ ( ϕ k − ϕ k + ϕ k − ϕ k ) × (cid:68) n ξλ (cid:12)(cid:12)(cid:12) ˆ a † k ,λ ˆ a † k ,λ ˆ a k ,λ ˆ a k ,λ + δ ( k − k )ˆ a † k ,λ ˆ a k ,λ (cid:12)(cid:12)(cid:12) n ξλ (cid:69) (192) = (cid:126) (cid:20) n ( n − (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ψ m , − λ ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) + δ ( r − r (cid:48) ) n (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) (cid:21) , (193)2and (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ˆ s obs z ( r , t ) ˆ s obs z ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) ≈ (cid:126) (2 π ) (cid:90) d k (cid:90) d k (cid:90) d k (cid:90) d k e − i [( k − k ) · r + ( k − k ) · r (cid:48) − ( ω − ω + ω − ω ) t ] e i λ ( ϕ k − ϕ k + ϕ k − ϕ k ) × (cid:68) α ξλ (cid:12)(cid:12)(cid:12) ˆ a † k ,λ ˆ a † k ,λ ˆ a k ,λ ˆ a k ,λ + δ ( k − k )ˆ a † k ,λ ˆ a k ,λ (cid:12)(cid:12)(cid:12) α ξλ (cid:69) (194) = (cid:126) (cid:20) ¯ n (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ψ m , − λ ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) + δ ( r − r (cid:48) )¯ n (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) (cid:21) (195)Here, we see that the Poisson and sub-Poisson statistics automatically enter the quantum spin-density correlations.We note that the delta function δ ( r − r (cid:48) ) in the correlator will not lead to any diverging e ff ect, because a practical probe alwaysmeasures the averaged photon spin density over a finite volume instead of the true single-point spin density. Now we considerthe e ff ective volume of the detector is V d (cid:28) λ c . For a Fock-state Bessel pulse, the averaged photon spin density detected by thisprobe is given by (cid:68) n ξλ (cid:12)(cid:12)(cid:12) V d (cid:90) V d d r ˆ s obs z ( r , t ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) ≈ n (cid:126) (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) . (196)The corresponding spin correlation is given by (cid:68) n ξλ (cid:12)(cid:12)(cid:12) V d (cid:90) d r ˆ s obs z ( r , t ) (cid:90) d r (cid:48) ˆ s obs z ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) n ξλ (cid:69) ≈ (cid:126) n ( n − (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ψ m , − λ ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) , for r (cid:44) r (cid:48) (cid:126) n ( n − (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) + n (cid:126) V d (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) , for r = r (cid:48) Then, the uncertainty of the spin density in z -direction is given by ∆ ˆ s obs z ≈ (cid:126) √ n (cid:114) V d (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) . (197)Usually, the second term will be very small, i.e., the ratio of these two terms can be approximated as (cid:112) V p / V d (cid:29) (cid:112) λ c / V d (cid:29) (cid:68) α ξλ (cid:12)(cid:12)(cid:12) V d (cid:90) V d d r ˆ s obs z ( r , t ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) ≈ ¯ n (cid:126) (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) . (198)The corresponding spin correlation is given by (cid:68) α ξλ (cid:12)(cid:12)(cid:12) V d (cid:90) d r ˆ s obs z ( r , t ) (cid:90) d r (cid:48) ˆ s obs z ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) α ξλ (cid:69) ≈ (cid:126) ¯ n (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ψ m , − λ ( r (cid:48) , t ) (cid:12)(cid:12)(cid:12) , for r (cid:44) r (cid:48) (cid:126) ¯ n (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) + ¯ n (cid:126) V d (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) , for r = r (cid:48) Then, the corresponding uncertainty of the spin density in z -direction is given by ∆ ˆ s obs z ≈ (cid:126) (cid:114) ¯ nV d (cid:12)(cid:12)(cid:12) ψ m , − λ ( r , t ) (cid:12)(cid:12)(cid:12) ..