Pure helicity vortex and spin-state-dependent quantum coherence in twisted light
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Pure helicity vortex and spin-state-dependent quantum coherence in twisted light
Li-Ping Yang1 and Dazhi Xu2
1
Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun 130024, China
2
Center for Quantum Technology Research and Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement (MOE) and School of Physics,
Beijing Institute of Technology, Beijing 100081, China
The topological charge of a photonic vortex is the essential quantity in singular optics and the critical pa-
rameter to characterize the vorticity of twisted light. However, the definition of the photonic topological charge
remains elusive. Here, we put forth a theoretical formalism to provide a comprehensive treatment of photonic
vortices. We introduce quantum operators for the photon current density and helicity current density based on
the continuity equations from the paraxial Helmholtz equation. Our formalism allows us to introduce flow veloc-
ity and circulation for photonic currents in parallel to their counterparts in superfluids. The quantized circulation
of the photonic currents conserves on propagation and it gives an explicit definition of the photonic topological
arXiv:2110.04705v2 [quant-ph] 3 Nov 2021
charge as the winding number of a photonic vortex. In particular, we predict helicity current generated pure
helicity vortices, in which the photon current vanishes. Finally, we show an interesting effect that the quantum
statistics of twisted photon pairs are essentially determined by their spin states.
I. INTRODUCTION field operator, which allows us to handle photonic quantities
(such as momentum, helicity, OAM, etc.) in real space within
the standard framework of quantum mechanics. Specifi-
Photonic vortices in structured light with helical wave-
cally, building on previous important works in the optical
front have been studied intensively both in theory and experi-
Schrödinger equation for paraxial light [38–40], we define
ments [1–6]. Optical vortex beams have been routinely gener-
quantum operators for the photon current density and helic-
ated in lab [7–11]. Single-photon source for vortex pulses [12]
ity current density. In parallel to superfluids in condensed
and optical vortex lattice [13, 14] have also been achieved.
matter physics, we introduce the flow velocities and the cor-
Recently, the concept of the photonic vortex has also been
responding circulations for these two currents. The conserved
generalized to tempo-spatial pulses [15–17], in which the en-
and quantized circulation automatically connects the photonic
ergy density varies spatially and temporally. Vortex beams
topological charge to the winding number of the photonic vor-
and pulses have been widely applied in optical trapping [18],
tex. We show a particularly interesting result that a pure he-
quantum communication and quantum information [19–21],
licity vortex with vanishing photon (particle) current can be
quantum computation [22], bio-sensing [23], strong-field pho-
obtained via the superposition of left and right circularly po-
toelectron ionization [24], etc. However, the theoretical def-
larized laser beams. On the other hand, the quantum statistics
inition of the photonic topological charge, which is the es-
of the twisted light remain poorly studied both in theory and
sential quantity and critical parameter of photonic vortices,
experiments. Within the well-established theoretical frame-
remains elusive.
work for quantum optical coherence [41], we show a partic-
Existing theories of photonic vortices have advanced over ularly interesting effect that the quantum statistics of twisted
the last two decades to capture a plethora of phenomena re- photon pairs are strongly affected by their spin states. Two
lated to phase singularities of light [25–31]. Based on the he- photons with the symmetric spin state tend to be bunched, and
lical phase of the complex field-amplitude function, M. Berry two photons with the anti-symmetric polarization state behave
defined the total vortex strength (photonic topological charge) more like anti-bunched.
as the signed sum of all the vortices threading a large loop in- This article is structured as follows. In Sec. II, we begin
cluding the propagating axis [28]. This seminal work has trig- by introducing a theoretical formalism for photonic quantities
gered extensive interest in exploring photonic vortices with in real space. In Sec. III, we apply this formalism to inves-
fractional topological charges [31–36]. Akin to the quantum tigate photonic vortices by defining photonic currents, flow
vortices in superfluids, the most basic quantity of a vortex velocities, and the corresponding circulations. In Sec. IV, we
is the corresponding current, which has not got the attention show how to apply our theory to study twisted laser beams
it deserves for photonic vortices. The photon current for a commonly used in experiments. Finally, in Sec. V, we discuss
photonic polarization vortex has even been overlooked com- the quantum coherence and quantum statistics of twisted light.
pletely [29, 30]. Without the associated current, the essential We give a short conclusion in Sec. VI.
link between the photonic topological charge and the vortic-
ity of twisted light is missing. On the other hand, the photon
current defined as the gradient of the phase of the complex II. PHOTONIC FIELD OPERATOR AND WAVE-PACKET
electric field has no clear physical meaning [37], since no con- FUNCTION IN REAL SPACE
tinuity equation exists for this current.
In this paper, we put forth a theoretical formalism to pro- Here, we introduce a quantum formalism to describe the
vide a comprehensive treatment of photonic vortices (see the photonic observables in striking parallel to their electronic
schematics in Fig. 1). We introduce an effective photonic counterparts. In particular, this rather general formalism leads
2
% "
(a) (b)
$
#
!
FIG. 1. Schematic of a twisted beam carrying orbital angular momentum (a) and the in-plane photonic current generated vortex in transverse
plane (b). The subplot in panel (a) shows the cylindrical coordinate r = (ρ cos ϕ, ρ sin ϕ, z).
tonian Ĥ = d3 k λ ~ω k â†k,λ â k,λ and the photonic spin op-
R P
us to define the photon current and helicity current via the con-
tinuity equation of paraxial beams (see Sec. III). The quan- erators, which can be handled more easily in k-space [43].
tized circulation of the current in a paraxial beam automati- However, our new formalism offers significant convenience in
dealing with photonic helicity Λ̂ = ψ̂ σ̂z ψ̂(r) (σ̂z is
R
3 †
cally gives the topological charge (winding number) of a pho- ~ d r (r)
the Puali matrix), OAM L̂ = d rψ̂ (r)(r × p̂)ψ̂(r), vor-
R
tonic vortex [26, 28, 31, 37, 42] (Sec. IV). Moreover, this obs 3 †
formalism enables us to investigate the quantum statistics of a tices, and quantum coherence of light in real space.
twisted light (Sec. V). The quantum state for a pulse or a laser beam (an ex-
We start with the effective two-component field operator tremely long pulse) can be constructed with the photon-wave-
ψ̂(r) = [ψ̂+ (r), ψ̂− (r)]T for photons in real space [43] with, packet creation operator âξ = d k λ ξλ (k)â†kλ [43]. The
†
R
3 P
pulse shape is determined P Rby the normalized spectral am-
Z
1
ψ̂λ (r) = p d3 kâ k,λ eik·r , (1) plitude function (SAF) λ d3 k|ξλ (k)|2 = 1. The quantum
(2π)3
states for the most commonly encountered Fock-state √ and
where â k,λ is the annihilation operator for the plane-wave coherent-state pulses are given by |nξ i = (âξ ) |0i/ n! and
† n
mode with wave vector k and circular polarizations λ = ±. |αξ i = D̂ξ (α)|0i, respectively, with
Using the basic equal-time commutation relations of the lad-
der operators [â k,λ , â†k0 ,λ0 ] = δλλ0 δ(k − k0 ), we can verify that D̂ξ (α) ≡ exp αâ†ξ − |α|2 /2 . (5)
our introduced photonic field operator obeys the following
bosonic commutation relations, Utilizing the time evolution operator exp(−iĤt/~), we can ob-
tain a quantum state at time t simply by replacing â†ξ with
[ψ̂λ (r), ψ̂†λ0 (r0 )] = δλλ0 δ(r − r0 ), (2) Z Z X
X
[ψ̂λ (r), ψ̂λ0 (r )] =
0
[ψ̂†λ (r), ψ̂†λ0 (r0 )] = 0. (3) âξ (t) = d k ξλ (k)e
† 3
â kλ = d3 r ξ̃λ (r, t)ψ̂†λ (r). (6)
−iω k t †
λ λ
With this field operator, we can re-express and evaluate
the photonic observables in real space, such as the photon Here, the Fourier transformation of the SAF
= ψ̂ ψ̂(r),
R
3 † Z
number N̂ d r (r) the linear momentum of light 1
ξ̃λ (r, t) = p d3 kξλ (k)ei(k·r−ω k t) . (7)
P̂ = d rψ̂ (r) p̂ψ̂(r) ( p̂ = −i~∇), etc. We note that the field
R
3 †
(2π)3
operator ψ̂(r) satisfies the wave equation, not a Schrödinger-
also satisfies the wave equation. Within our introduced for-
like equation and no probability continuity equation can be
malism, the mean value of a physical quantity can be obtained
constructed [44]. Thus, in most cases, the integral kernel can
via the relations
not be interpreted as the corresponding density operator (see E √ E
Appendix A). However, as shown in Sec. III A, the particle ψ̂λ0 (r0 ) nξ (t) = nξ̃λ0 (r0 , t) (n − 1)ξ (t) , (8)
number density (PND) of a paraxial light can be well charac- E E
terized by ψ̂λ0 (r ) αξ (t) = αξ̃λ0 (r , t) αξ (t) ,
0 0
(9)
X where we have used the identities
n̂(r) ≡ ψ̂† (r)ψ̂(r) = ψ̂†λ (r)ψ̂λ (r). (4) h n i n−1
λ=± ψ̂λ0 (r0 ), â†ξ (t) = nξ̃λ0 (r0 , t) â†ξ (t) ,
We note that not all physical quantities of light can have a sim- h † n i n−1
ple and elegant form in this framework, such as the Hamil- ψ̂λ0 (r0 ), âξ (t) = −nξ̃λ∗0 (r0 , t) âξ (t) .
3
In experiments, the superposition of multiple laser beams perflow leads to quantum vortices with integer winding num-
has been routinely used to obtain various interesting struc- bers [46, 47]. We note that the cornerstone of quantum vor-
tures. Here, we emphasize that the corresponding quantum tices is the directly observable current density of the corre-
state is not a simple superposition of the state of each beam. sponding particle flow. Here, we introduce the photonic coun-
We take the superposition of two beams with strength α and terpart.
α0 and two-component SAFs ξ and ξ0 as an example. The A paraxial laser beam propagating in positive z-direction
corresponding quantum state is given by can be well described by a quasi-single-frequency two-
1 component function Ψ(r, t) = ΨPA (r) exp[i(k0 z − ω0 t)], where
|Ψi = √ D̂ξ (α)D̂ξ0 (α0 ) |0i , (10) ω0 = ck0 is its center frequency. The function ΨPA (r) slowly
N varying in z satisfies the paraxial Helmholtz equation [48, 49],
where N is normalization factor given in Eq. (B5). We can
verify that ψ̂(r) |Ψi = Ψ(r, t) |Ψi (see Appendix B). Here, 1 2
i∂z ΨPA (r) = − ∇ ΨPA (r) (13)
the two-component function Ψ = [Ψ+ , Ψ− ]T with Ψ± (r, t) ≡ 2k0 T
αξ̃± (r, t) + α0 ξ̃±0 (r, t) also obeys the wave equation. In the fol-
where ∇T = e x ∂ x + ey ∂y is the differential operator in xy-
lowing, we only consider quantum pulses or beams. A ran-
plane and e j is the unit vector. Because the parameter t
domly polarized light [27] or thermal light, which can not be
only contributes a phase factor to Ψ(r, t), thus we can fix
described with a pure quantum state, will not be addressed in
the time at t = 0 and take the coordinate z as an effective
this work.
”time” to study the dynamics of the propagating beam. This
paraxial Helmholtz equation can be regarded as the effective
III. PHOTONIC VORTICES IN PARAXIAL BEAMS Schrödinger equation with effective mass k0 [39, 40]. The
two-component function Ψ(r, t), which serves as the many-
body “wave function” of paraxial light, is adequate to charac-
Previously, the scalar electric field has been utilized to terize a uniformly polarized or a “vector” vortex beam [49].
study the photonic vortices [26, 40, 42]. In the work [28], We now introduce an operator to characterize the photon
Berry defined the total vortex strength of a paraxial scalar current density in xy-plane
wave with complex amplitude ψ(r) ∝ exp(imϕ) as
i n †
∂
h i o
ĵN (r) = − ψ̂ (r)∇T ψ̂(r) − ∇T ψ̂† (r) ψ̂(r) .
Z 2π
1 (14)
TC = lim dϕ argψ(r), (11) 2k0
ρ→∞ 2π 0 ∂ϕ
which has been rewritten with the scalar electric field With the help of the paraxial Helmholtz equation (13), we ob-
later [31], tained the continuity equation (see Appendix C)
∂E(r)/∂ϕ ∂
Z 2π
1
TC = lim Im dϕ . (12) hn̂i + ∇T · h ĵN i = 0, (15)
ρ→∞ 2π 0 E(r) ∂z
The strength of the phase singularity has also been called the which reveals the fact that the total particle number within a
topological charge of photonic vortices. We note that these co-moving transverse slice does not change as the light prop-
two definitions are not exactly equivalent to each other. More agates in the z axis. We note that this conservation law is valid
importantly, the physical meaning of the defined topological only under the paraxial approximation. Thus, for photonic
charge in (11) and (12) is unclear. vortices, multiple beams in superposition are required to be
There remain basic questions about photonic vortices hav- parallel to each other [50]. The photon current is a special dis-
ing not been clarified. The obtained topological charge may sipationless flow composed of non-interacting particles. Our
not conserve on propagation [2, 26]. The existence of frac- formalism can also be generalized to vortices in non-linear
tional topological charges in a uniformly polarized light beam media with photon-photon interaction [39, 40].
will cause confusion [28, 31, 45], because the electromagnetic To characterize the dynamics of the local polarization of a
field ∝ exp(iαϕ) with non-integer α is multiple-valued. The paraxial beam, we now introduce the helicity current density,
strength of each vortex cannot be quantitatively analyzed by
(11) and (12). The fundamental link between the photonic i n † h i o
ĵH (r) = − ψ̂ (r)σ̂z ∇T ψ̂(r) − ∇T ψ̂† (r) σ̂z ψ̂(r) . (16)
topological charges and winding numbers of photonic vortices 2k0
is unclear. All these problems will be solved within our pre-
sented formalism conclusively. More importantly, we predict From the paraxial Helmholtz equation, we obtain the corre-
the pure helicity vortex with vanishing photon flow, which can sponding continuity equation,
be measured with circular-polarization-sensitive devices. ∂
hn̂H i + ∇T · h ĵH i = 0. (17)
∂z
A. Photon current density and helicity current density Here, the helicity density in the circular polarization represen-
tation is given by n̂H (r) = ψ̂† (r)σ̂z ψ̂(r). Without loss of gen-
The concept of vortex originates from fluid mechanics. In erality, both the helicity density and the corresponding cur-
superfluids, the quantized circulation of the dissipationless su- rent have been divided by the constant ~. As shown in the4
following, pure helicity current and helicity vortices can be circulations κ± = T v± ·dr. From the hydrodynamic equations
constructed via the superposition of two OAM beams. of v± (see appendix C), we have
Similar to the superfluid in a condensate [46], our defined
∂
" #
PND and helicity density currents fundamentally stem from 1 1 2
κ± = ∇T ∇ |Ψ± |− v± · dr = 0,
2
(24)
the spatial varying phase φ± (r) of the paraxial ∂z 2k2 |Ψ± | T 2
many-photon T
wave-packet function Ψ± (r) = |Ψ± (r)| exp i φ± (r) + k0 z ,
since it is the integral of a perfect differential of single-
1 X valued functions around a closed path. This guarantees that
h ĵN (r)i = |Ψλ (r)|2 ∇T φλ (r), (18) the circulations κN and κH are conserved during propagating.
k0 λ
This is consistent with Kelvin’s theorem for an ideal classical
1 X fluid [53]. We note that the paraxial approximation is essen-
h ĵH (r)i = λ |Ψλ (r)|2 ∇T φλ (r). (19)
k0 λ tial to the conservation of circulation. Accidental phase sin-
gularity points can be generated by the superposition of two
We emphasize that the continuity equation is essential to non-coaxial beams [37]. These points are not stable and they
define a current. A similar photon current j = Imψ∇ψ (ψ is a will disappear on propagation.
complex scalar function) has been given in [37]. However, its We note that for another case with ∇T φ+ (r) = ∇T φ− (r),
physical meaning is unclear, because no continuity equation we have vN = ∇T φ+ /k0 and conserved path-independent κN .
exists corresponding to this current. In principle, we can also However, the circulation of the helicity density current does
introduce the currents for the photonic spin and OAM den- not necessarily have these good properties. We do not present
sity [51, 52]. However, the corresponding currents are two- a detailed discussion about this case here. In the next subsec-
rank tensors, which are extremely difficult to measure in ex- tion, we only focus on uniformly polarized paraxial beams.
periments. For a paraxial laser beam, the photon current (14) We will show that there exists a direct link between the pho-
and helicity current (16) are enough to characterize its vortic- tonic topological charge and the winding number of a pho-
ity properties. tonic vortex.
B. Circulations of the density currents C. Quantized circulation in uniformly polarized paraxial
beams
To characterize the vorticity of the two currents, we now
introduce two flow velocities, For a uniformly polarized beam, its many-photon wave-
packet function Ψ(r) = αξ̃(r)[c+ , c− ]T can be expressed by
h ĵN (r)i h ĵH (r)i the product of a constant amplitude α, a scalar wave-packet
vN (r) ≡ , vH (r) ≡ . (20)
hn̂(r)i hn̂(r)i function ξ̃(r), and a constant normalized two-component vec-
tor (i.e., |c+ |2 + |c− |2 = 1). The polarization degrees of free-
and the corresponding circulation (vorticity flux) dom does not contribute to the photonic vortices. Thus, the
positive-frequency part of a scalar electric field function E(r)
κN ≡ vN · dr, κH ≡ vH · dr, (21) has been routinely used to study the corresponding photonic
T T vortices [26, 31]. Here, we see that for a uniformly polar-
ized beam, Ψ(r) satisfies both the two conditions mentioned
where the integral is taken along a closed curve in xy-plane in
in previous subsection, i.e., |Ψ+ |/|Ψ− | = |c+ |/|c− | = C and
counter-clockwise direction. Usually, the flow velocity vN(H)
∇T φ+ (r) = ∇T φ− (r). We show that the the circulations of
not only depends on the phase gradient ∇T φλ (r), but also on
the two currents are always quantized in this case and the as-
the density distribution |Ψλ (r)|2 . Thus, the two circulations
sociated photonic vortices can be characterized by an integer
are path-dependent. For a focused beam, the beam waist
winding number.
increases, and the density decreases when leaving the focal
With re-expressed function ξ̃(r) = |ξ̃(r)| exp i φ(r) + k0 z ,
plane. Thus, the circulations κN and κH for a fixed closed-loop
we obtain the flow velocity for the photon current,
in xy-plane are not conserved on propagation.
For a special case with |Ψ+ (r)|/|Ψ− (r)| = C (C is a coor- 1
dinate independent constant), we can obtain conserved path- vN (r) = ∇T φ(r). (25)
k0
independent circulations. In this case, the flow velocities re-
duce to a simplified form, The helicity density flow velocity is obtained by multiplying
vN with a constant (C 2 − 1)/(C 2 + 1). The circulation of the
C2
" #
1 1 photon current is given by
vN = ∇ T φ + (r) + ∇ T φ − (r) , (22)
k0 1 + C 2 1 + C2
1
1
"
C2 1
# κN = vN (r) · dr = δφ(r), (26)
vH = ∇ φ
T + (r) − ∇ φ
T − (r) . (23) T k0
k0 1 + C 2 1 + C2
where δφ(r) is the change in the phase around this closed
To show the conservation of the circulations κN and κH , we curve and the non-vanishing circulation stems from the multi-
introduce two velocities v± = ∇T φ± /k0 and the corresponding valuedness of the phase factor φ(r). The wave-packet function5
$ rents in xy-plane, i.e., ∇T ξ̃(r) = 0. We can easily verify that
# )
* h ĵN i = 0 and vN = 0. However, the helicity density current
&'
and the corresponding flow velocity do not vanish,
hn̂(r)i 1
! ('
" h ĵH (r)i = ∇T φ(r), vH (r) = ∇T φ(r), (27)
k0 k0
where the PND is given by hn̂(r)i = |αξ̃(r)|2 . Thus, pure he-
licity vortices will be obtained from the quantized circulation
%
of the helicity current. This is similar to the net spin current
in condensed matter physics. We also note that the helicity
FIG. 2. Bloch sphere for photonic polarizations. In a circular-
polarization representation, the north and south poles denote the left density vanishes at every point in this case, i.e., h n̂H (r)i = 0.
and right circular polarized states, respectively. The polarization of We can also construct a pure helicity vortices
an arbitrary pure state can be characterized by the eigen state of the with non-vanishing helicity density. With the help
operator σ̂ = {σ̂ x sin θB cos ϕB , σ̂ x sin θB sin ϕB , σ̂z cos θB }. of the Bloch sphere, the polarization of a uni-
formly polarized beam can be characterized a vector
σ̂ = {σ̂ x sin θB cos ϕB , σ̂ x sin θB sin ϕB , σ̂z cos θB } as shown in
of the light Ψ(r) is to be determined uniquely, thus we must Fig. 2. Here, θB and ϕB are the polar angle and azimuthal
have δφ(r) = 2πm, where m is an integer. Therefore, the cir- angle in the Bloch space not in the real space. The two
culation κN is quantized in unit of 2π/k0 = λ0 , i.e., κN = mλ0 . normalized eigen states of σ̂ are given by [4]
This integer number corresponds to the winding number of
the phase φ(r) around the closed loop. Similarly, the the cir- θB −iϕB /2 θB
|↑i = [cos e , sin eiϕB /2 ]T , (28)
culation κH of the helicity current is also quantized in unit of 2 2
λ0 (C 2 − 1)/(C 2 + 1) with the same quantum number m. θB −iϕB /2 θB
|↓i = [− sin e , cos eiϕB /2 ]T . (29)
Since the velocity is irrotational in the transverse plane, 2 2
i.e., ∇T × vN = 0 (but ∇ × vN , 0), thus, to obtain non-
Now, we construct a beam by the superposition of two ellipti-
vanishing circulation, the fluid must contain vortices in xy-
cally polarized light with many-photon wave-packet function
plane, i.e., multi-valued phase factor induced diverging flow
velocity. Thus, this integer number m is also called the topo- h i
Ψ(r) = αξ̃(r) c↑ eiφ(r) |↑i + c↓ e−iφ(r) |↓i . (30)
logical charge of photonic vortices [26]. We also see that for
a uniformly polarized beam, the obtained topological charge √
is always an integer. This is significantly different from pre- For the case |c↑ | = |c↓ | = 1/ 2, we have zero photon current
vious results [28, 31], in which fractional topological charges h ĵN (r)i = 0 but non-vanishing helicity density current
can exist. As shown in Sec. IV B, crossing a zero-amplitude
point, ψ(r) will experience an extra ±π phase jump, which will hn̂(r)i
h ĵH (r)i = cos θB ∇T φ(r). (31)
lead to singularities in the flow velocity. These singularities k0
are essential to obtain quantized integer topological charges
The correponding PND and helicity density are given by
(see Sec. IV C). We emphasize that the diverging flow veloc-
ity can not be detected in experiments. Only the well-behaved hn̂(r)i = |αξ̃(r)|2 , hn̂H (r)i = |αξ̃(r)|2 sin θB cos[2φ(r)+φ0 ], (32)
densities and the corresponding currents are physical observ-
ables. where the constant phase φ0 is determined by the relative
phase between c↑ and c↓ . We can verify that the circulation
of the helicity current is quantized in unit of λ0 cos θB .
D. Pure helicity vortices The helicity density can be measured with a pixelated
polarization filter array, which has been routinely used in
Previously, the normalized Stokes parameters have been polarization-sensitive imaging [57, 58]. On the other hand,
utilized to characterized polarization singularities in “vector” atoms can have asymmetric circular polarized light-induced
beams [29, 30, 54, 55]. However, for the same beam, there transitions. Thus, our predicted pure helicity current and pure
exist three types of polarization vortices depending on the se- helicity vortex can be detected via imaging the created atomic
lected polarization basis [56]. Our introduced helicity cur- vortex via 2-photon Raman processes [59].
rent and the associated helicity vortices are uniquely defined.
More importantly, we show that there exist pure helicity-
vortex beams, in which the photon current and the correspond- IV. PHOTONIC VORTICES IN PARAXIAL BEAMS
ing vortices vanish.
We first consider a beam by superposition of two circu- Recently, the quantized topological charge of photonic vor-
larly polarized beams. The corresponding many-photon wave- √ tices in paraxial beams or pulses have attracted increasing in-
packet function is given by Ψ(r) = αξ̃(r)[eiφ(r) , e−iφ(r) ]T / 2, terest [49, 60, 61]. We now apply the theory presented in
where the scalar function ξ̃(r) does not contribute to the cur- the previous section to investigate the properties of vortices6
(a) particle-number density (b) PND current (c) flow velocity
12
0 0.2 0.4 0.6 0.8 0 1 2 3 ×10 0 0.5 1
−1 −1 −
0 10 0 10 10 10
−5 −5 −5
5 5 5
)/
0 0 $ )/ 0 0 $ )/ 0 0 #$
# #* # #
#* !/ 5 !/ 5 !/
5
−5 −5 −5
*
10 0 10 0 10 0
−1 −1 −1
FIG. 3. Photonic vortex in a paraxial Laguerre-Gaussian beam. (a) Rescaled particle number density (PND) hn̂(ρ, ϕ, 0)i/|αN pm |2 . (b) Rescaled
photon current density h ĵN (ρ, ϕ, 0)i/|αN pm |2 . (c) Flow velocity vN (ρ, ϕ, 0). The diverging velocity at the center leads to the photonic vortex.
Here, the parameters in the simulation are taken as p = 1, m = 1, and w0 = 10λ0 . The length is in the unit of λ0 (center wave length of the
beam).
in a paraxial Laguerre-Gaussian (LG) beam and a Bessel- z = 0 plane is given by
Gaussian (BG) beam, which are widely used in experiments.
√ |m| 2
We explicitly calculate the associated PND, current density,
2
2ρ 2ρ −ρ2 /w2
and specifically the flow velocity. We then analyze the flow- hn̂(ρ, ϕ, 0)i = αN pm |m|
L p 2 e 0 . (34)
w0 w0
velocity singularities due to the ill-defined phase at the zeros
of the many-photon wave-packet function and the contribution For m , 0, the mean PND vanishes on the z-axis with scal-
of these singularities to the winding number of a photonic vor- ing ∼ ρ2|m| . We show the rescaled PND hn̂(ρ, ϕ, 0)i/|αN pm |2 in
tex. In particular, we exemplify that no fractional topological Fig. 3 (a). We see that there is a hole at the center.
charge exists in a uniformly polarized beam. Finally, we show The corresponding in-plane photon current density is given
the pure helicity current and helicity vortex in a superposition by
of two BG beams.
mλ0
h ĵN (ρ, ϕ, 0)i = hn̂(ρ, ϕ, 0)i × eϕ , (35)
2πρ
A. Laguerre-Gaussian beam which only has a tangent component. We see that this current
density is proportional to the PND and the integer m. It is
We now apply the theory we presented in the previ- also modulated by the function 1/ρ. We note that the photon
ous section to study the properties of photonic vortices in current is well-defined on the whole xy-plane and it vanishes
a paraxial linearly polarized LG beam. The correspond- on the z-axis due to the vanishing PND (limρ→0 hn̂i ∝ ρ2|m| ).
ing many-photon wave-packet function is given by Ψ(r) = The rescaled photon current h ĵT (ρ, ϕ, 0)i/|αN pm |2 is shown in
√ Fig. 3 (b). For m > 0, it flows in a counter-clockwise direction.
αξ̃LG,pm (ρ, ϕ, z)[1, 1]T / 2, where the complex constant α de-
notes the strength of the beam. The pulse profile is character- For a linearly polarized beam, both the helicity density and
ized by the wave-packet function [62] helicity current density are zero (not shown).
The flow velocity of the photon current is given by,
!2p+|m|+1 √ |m| mλ0 X λ0
1 2ρ vN = eϕ + δ(ρ − ρZA, j )eρ . (36)
ξ̃LG,pm (ρ, ϕ, z) =N pm 2p
|q(z)| 2πρ 2
q(z) w0 j
ρ2
2ρ2
× L|m| +imϕ , The Laguerre polynomial L|m|
p 2
2
exp ik
0 z− 2
(33) p (x) has p zeros [63], which lead
w0 |q(z)| w0 q(z) to p zero-PND circles with radius ρZA, j in xy-plane (not shown
p (2ρZA, j /w0 ) = 0. We can see that there are
in Fig. 3), i.e., L|m| 2 2
where Lmp (x) is the associated Laguerre polynomials with a two types of singularities in the flow velocity. The first type
non-negative integer p and an integer m, q(z) = 1 + iz/zR , zR = is a singularity point lying at the center of a vortex as shown
kw20 /2 = πw20 /λ0 is the Rayleigh length, and w0 is the Gaus- in Fig. 3 (c) and the second type (not shown) comes from the
sian beam waist radius. The constant factor N pm is determined π-phase jump when the wave-packet function Ψ cross a zero-
by the normalization condition V d3 r|ξ̃LG,pm (ρ, ϕ, z)|2 = 1 and
R
value curve. A closed velocity singularity curve does not con-
V is the effective volume of the laser beam. tribute to the winding number of a vortex, because any integral
In previous section, we have shown that both the current loop in xy-plane will always cross the singularity curve even
and flow velocity are in the transverse plane perpendicular to times. The accumulated phases cancel out with each other.
the propagating direction. Without loss of generality, we only Thus, the topological winding number of the vortex in this
consider the focal plane (z = 0) in the following. The PND in LG beam is m.7
We note that the diverging flow velocity is unphysical and where the diverging radial velocity comes from the π-phase
cannot be observed in experiments. These singularities are jump at zeros of the Bessel function and ρZA, j denotes the ra-
fully due to the ill-defined phase of the complex wave-packet dius of the jth zero-amplitude circle. As explained in previous
function Ψ(r) at the zero-value points. On the other hand, subsection, only the first term in vN will contribute to the vor-
no singularity exists in the truly observable PND and photon tex. Thus, the topological winding number of the vortex in
current density. this BG beam is m.
B. Bessel-Gaussian beam C. Fractional topological charge controversy
In this subsection, we study the vortex in a paraxial linearly In previous studies, photonic vortices with fractional topo-
polarized BG beam. The corresponding many-photon wave- √ logical charge have been found both in theory [28, 31] and
packet function is given by Ψ(r) = αξ̃BG,pm (ρ, ϕ, z)[1, 1]T / 2, experiments [32, 33]. In Sec. III, we show that the circulation
where the complex constant α denotes the strength of the of a paraxial beam is not quantized and even not conserved
beam. The pulse profile is characterized by the wave packet in most cases. However, for a uniformly polarized beam, the
function [64] circulated is conserved on propagation and the corresponding
βpρ
! winding number must be an integer. Now we show that half-
1
ξ̃BG,pm (ρ, ϕ, z) =N p Jp integer topological charges obtained in a uniformly polarized
q(z) q(z) beam is due to the improper evaluation of the circulation.
sin2 θ p ρ2
To resolve the controversy, we look at a simple example
+imϕ , (37)
× exp ik z 1− −
0
2q(z)
2
w0 q(z)
of the superposition of two co-axial BG beams with the same
polarization, strength, and beam profile, but with different he-
where J p (x) is the Bessel function of the first kind with a non- lical phase factors. The many-photon wave-packet function
negative integer p, β p = k0 sin θ p is the scaling factor of the can be expressed in the form Ψ(r) = [c+ , c− ]T αξ̃(r) with nor-
Bessel function, and θ p is the half-angle of a conical wave that malized constants |c+ |2 + |c− |2 = 1. In the z = 0-plane, the
forms the Bessel beam. The constant factor N p is determined wave-packet function is given by,
by the normalization condition V d3 r|ξ̃BG,pm (ρ, ϕ, z)|2 = 1.
R
ξ̃(ρ, ϕ, 0) = N p J p β p ρ e−ρ /w0 eimϕ + einϕ
2
The PND of a BG beam in z = 0 plane is given by (41)
m − n i(m+n)ϕ/2
= 2N p J p β p ρ e−ρ /w0 cos
2 22 2
hn̂(ρ, ϕ, 0)i = αN p J p β p ρ e−ρ /w0 . (38) ϕe . (42)
2
We show the rescaled PND hn̂(ρ, ϕ, 0)i/|αN p |2 in Fig. 4 (a). In Fig. 5 (a), we plot the rescaled PND for two mixed beams
For p , 0, there is a hole with vanishing PND on the z-axis. with m = 1 and n = 4,
On the other hand, a Bessel function J p (x) has an infinite num-
m−n 2
ber of real zeros, thus a Bessel beam will have an infinite num- hn̂(ρ, ϕ, 0)i/|αN p |2 = 2J p β p ρ e−ρ /w0 cos
2
ϕ . (43)
ber of zero-amplitude circles in the transverse plane (see the 2
red-dashed curves). Here, ρZA, j denotes the radius of the jth Here, we see that except the zero-amplitude circles (not
circle. shown) due to the Bessel function J p (x), there are |m − n| = 3
The corresponding photon current density is given by zero-amplitude cut lines from the center to infinity (the red-
mλ0 dashed lines). The photon current density is given by [see
h ĵN (ρ, ϕ, 0)i = hn̂(ρ, ϕ, 0)i × eϕ . (39)
2πρ Fig. 5 (b)],
The rescaled photon current h ĵT (ρ, ϕ, 0)i/|αN p |2 is shown in (m + n) λ
Fig. 4 (b). We see that this current is modulated by the PND h ĵN (ρ, ϕ, 0)i = hn̂(ρ, ϕ, 0)i × × eϕ . (44)
2 2πρ
and it flows in a counter-clockwise direction for m > 0. We
note that the photon current also has a hole on the z-axial for Different from Eqs. (35), a half-integer enters the photon cur-
p > 0 due to the vanishing PND (hn̂i ∝ ρ2p → 0), but it di- rent here.
verges when ρ → 0 if p = 0 and m , 0. However, in this case, We note that the winding number of the photonic vortex is
the transverse component of the wave vector diverges [50], still an integer in this case. The flow velocity now contains
i.e., −iΨ∇T Ψ → ∞, thus the paraxial approximation loses its the contributions from three parts,
validity on z-axis. On the other hand, the ”kinetic energy” di- m+n λ0 X λ0
verges in this case [47] (i.e., − d3 rΨ∇2T Ψ/2k0 → ∞), thus vN = eϕ + δ(ρ − ρZA, j ) eρ
R
×
infinite large laser power is required to prepare this vortex 2 2πρ j
2
beam with p = 0 but non-zero m. |m−n|
λ0
!
X 2 j−1
Similar to the LG beam, the flow velocity for a BG light − (−1) j δ ϕ− π eϕ . (45)
beam also has a radial component j=1
2 |m−n|
mλ0 X λ0
vN = eϕ + δ(ρ − ρZA, j )eρ . (40) As explained in previous subsections, the first term determines
2πρ j
2 the position of the vortex core, the second term results from8
(a) particle-number density (b) PND current (c) flow velocity
0 0.1 0.2 0.3 0 1 2 3 ×10/0 0 0.5 1
diverging
zero- velocity
amplitude (not shown)
circles
−1 −1 −
0 10 0 10 10 10
−5 −5 −5
5 5 5
)/
0 0 $ )/ 0 0 $ )/ 0 0 #$
# #* # #
#* !/ 5 !/ 5 !/
5
−5 −5 −5
*
10 0 10 0 10 0
−1 −1 −1
FIG. 4. Photonic vortex in a Bessel-Gaussian beam. (a) Re-scaled particle number density (PND) hn̂(ρ, ϕ, 0)i/|αN p |2 . Red-dashed curves
denote the zero-amplitude circles at ρZA, j resulting from J p (β p ρZA, j ) = 0. (b) Re-scaled photon current density h ĵN (ρ, ϕ, 0)i/|αN p |2 . (c) Flow
velocity vN (ρ, ϕ, 0). The diverging radial component of the flow velocity ∝ eρ δ(ρ − ρZA, j ) on the zero-amplitude circles has not been shown.
Here, the parameters in the simulation are taken as p = 1, m = 1, w0 = 10λ0 , θ p = 0.05π.
(a) particle-number density (b) PND current (c) flow velocity
0 0.5 1 0 0.1 0.2 0 0.5 1
−10 −10 −10
zero-
amplitude
−5 cut lines −5 −5
diverging
'/)*
'/)*
'/)*
0 0 0 velocity
(not shown)
5 5 5
10 10 10
−10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10
+/), +/), +/),
FIG. 5. Photonic vortex in a superposition of two Bessel-Gaussian beams. (a) Re-scaled particle number density (PND) hn̂(ρ, ϕ, 0)i/|αN p |2 .
Red-dashed lines denote the zero-amplitude cut lines at φ = π(2 j − 1)π/|m − n| ( j = 1, 2, ..., |m − n|) [see Eq. (43)]. (b) Re-scaled photon current
density h ĵN (ρ, ϕ, 0)i/|αN p |2 . (c) Flow velocity vN (ρ, ϕ, 0). The diverging terms due to crossing the zero-amplitude curves have not been shown.
Here, the parameters in the simulation are taken as p = 1, m = 1, n = 4, w0 = 10λ0 , θ p = 0.05π.
π-phase jump when crossing the zero-amplitude circles, and D. Helicity vortex
the third term comes from the π-phase jump crossing the cut
lines. In this subsection, we investigate the pure helicity current
and pure helicity vortex in superposition of two BG beams.
The many-photon wave-packet function in the focal plane is
given by
Ψ(ρ, ϕ, 0) = αN p J p β p ρ e−ρ /w0 c↑ eimϕ |↑i+c↓ e−imϕ |↓i , (46)
2
Half-integer topological charge has been obtained [31] in
a linear combination of optical vortices with the definition in
with |c↑ |2 = |c↓ |2 = 1/2. The PND in this plane is the same as
Eq. (12). However, we note that this fractional topological
a BG beam as given in Eq. (38) [see Fig. 4 (a)]. Different from
charge is because the definition in Eq. (12) has not taken the
the PND, the helicity in this plane has also been modulated by
contribution from the third term in Eq. (45). Different from the
the azimuth angle
closed zero-amplitude curve, the open zero-amplitude lines
can contribute to the winding number of the vortex. If m + n is hn̂H (ρ, ϕ, 0)i = hn̂(ρ, ϕ, 0)i sin θB cos(2mϕ + φ0 ). (47)
an even integer, then |m − n| must also be an even integer and
the third term vanishes. The topological winding number of We plot the helicity density in the focal plane in Fig. 6 (a) with
the vortex is (m + n)/2. If m + n is an odd integer, then |m − n| θB = π/4 and φ0 = 0. The zero-amplitude circles due to the
must also be an odd integer and the third term will contribute zeros of the Bessel function also exist (not shown).
an extra π-phase. The topological winding number of the vor- The photon current vanishes in a pure helicity vortex beam.
tex is (m + n + 1)/2. Similarly, a cut line connecting two vor- The helicity current is given by
tices [65] will also contribute to the circulation of each vortex.
Here, we see fractional topological charge can not be obtained mλ0
by superposition of uniformly polarized OAM beams. h ĵH (ρ, ϕ, 0)i = hn̂(ρ, ϕ, 0)i cos θB × eϕ . (48)
2πρ9
(a) helicity density (b) helicity current (c) helicity flow velocity
−0.1 0 0.1 0 1 2 ×10./ 0 0.5 1
−1 −1 −
0 10 0 10 10 10
−5 −5 −5
5 5 5
)/
0 0 $ )/ 0 0 $ )/ 0 0 #$
# #* # #
#* !/ 5 !/ 5 !/
5
−5 −5 −5
*
10 0 10 0 10 0
−1 −1 −1
FIG. 6. Pure helicity vortex in a superposition of two Bessel-Gaussian beams. (a) Re-scaled helicity density hn̂H (ρ, ϕ, 0)i/|αN p |2 . (b)
Re-scaled helicity current density h ĵH (ρ, ϕ, 0)i/|αN p |2 . (c) Helicity flow velocity vH (ρ, ϕ, 0). The diverging terms due to crossing the zero-
amplitude curves have not been shown. Here, the parameters in the simulation are taken as p = 1, m = 1, w0 = 10λ0 , θ p = 0.05π.
The helicity flow velocity vH is obtained by multiplying the of photon-number-density correlation addressed here. On the
velocity vN in Eq. (40) by a factor cos θB [see Fig. 6 (C)]. other hand, we only consider equal-time correlation and co-
Similar to the vortex of the photon current in a BG beam, the herence functions in the following. The time-dependence
zero-amplitude circles do not contribute to the winding num- can be easily recovered with the field operator ψ̂λ (r, t) in the
ber of the vortex. The topological charge of this pure helicity Heisenberg picture or with time-varying quantum states in the
vortex is also m. Schrödinger picture.
We now propose a new correlation function corresponding
to the photonic helicity density,
V. QUANTUM COHERENCE OF TWISTED LIGHT
X
H (r, r ) =
G(2) 0
λλ0 hψ̂†λ (r)ψ̂†λ0 (r0 )ψ̂λ0 (r0 )ψ̂λ (r)i. (51)
In his seminal work [41], Glauber introduced a succes- λλ0
sion of quantum correlation functions and quantum coherence
functions for optical light, laying the foundation for modern which can be exploited to reveal the correlation information
quantum optics. We now re-express the quantum coherence about the polarization degrees of freedom of light. Here,
functions of light with our introduced field operator ψ̂(r). We we do not consider the correlation functions for the pho-
also propose a quantum correlation function for the photonic tonic spin and OAM densities. A 3 × 3 correlation matrix is
helicity density, which can be directly measured in experi- needed to fully characterize the corresponding quantum cor-
ments. When applying to twisted photon pairs, we find an relations [43]. On the other hand, it is extremely challenging
interesting phenomenon that two photons with a symmetrical to measure the spin or OAM density correlation at few-photon
spin state tend to be bunched, and two photons with an anti- level currently.
symmetric polarization state behave more like anti-bunched. We first check the quantum coherence of the pure-helicity
The quantum statistics of a twisted photon pair are strongly vortex laser beam with many-photon wave-packet function
dependent on its spin (polarization) state. (30). We can verify that a coherent-state light (even with
Quantum statistics of light is essentially characterized by highly sophisticated structure) has a constant quantum coher-
its high-order correlations, which can be measured via high- ence function, i.e., g(2) (r, r0 ) = 1. Its helicity correlation func-
order quantum interference experiments [66]. Without loss tion is simply the product of the helicity density at each point,
of generality, we only take the most commonly used second- H (r), r ) = hn̂H (r)ihn̂H (r )i. Non-trivial quantum co-
i.e., G(2) 0 0
order correlation in coincidence measurements as an example herence only exists in truly quantum light, such as Fock-state
X photon pulses, squeezed light pulses, etc. In the following,
G(2) (r, r0 ) = hψ̂†λ (r)ψ̂†λ0 (r0 )ψ̂λ0 (r0 )ψ̂λ (r)i. (49) we focus on entangled photon pairs [67–69], which has been
λλ0 extensively explored for Bell’s inequalities testing [70], quan-
The corresponding second-order coherence function in real tum key distribution [71], ghost imaging [72, 73], ect.
space is given by The quantum state of a photon pair can be written as [74],
G(2) (r, r0 ) E 1 XZ Z
g(2) (r, r0 ) = . (50) Pξ = √ d3 k1 d3 k2 ξλ1 ,λ2 (k1 , k2 )â†k2 ,λ1 â†k1 ,λ1 |0i . (52)
hn̂(r)ihn̂(r0 )i 2 λ1 λ2
We emphasize that paraxial approximation is also required
here, otherwise the physical meaning of n̂(r) is unclear. In The normalization requirement of the state |Pξ i requires
2
d3 k1 d3 k2 ξλ1 ,λ2 (k1 , k2 ) = 1. On the other hand,
P R R
Glauber’s original work, the electric field operator has been λ1 λ2
used to defined the quantum coherence function. It is actu- photons are bosons, thus the two-photon SAF should also be
ally based on the energy-density correlation function instead symmetric, i.e., ξλ1 ,λ2 (k1 , k2 ) = ξλ2 ,λ1 (k2 , k1 ). Via the Fourier10
(=1 (=2 (=3
light, we have g(2) (r, r) = 2. Here, we show an exotic phe-
nomenon for entangled twisted photon pairs. The spatial dis-
symmetrized
spin state
tribution of the quantum coherence function g2 (r, r0 ) is con-
1
trolled by the photonic spin state. The g(2) -function is reversed
0.8
when the photonic spin state changes from symmetric to anti-
0.6 symmetric.
0.4 We now apply the our theory to investigate the quantum
anti-symmetrized
0.2 correlations of eliptically polarized photon pairs with a sym-
spin state
0 metric spin state ∼ |↑i ⊗ |↓i + |↓i ⊗ |↑i. For simplicity, we
consider a photon pair with two-photon SAF,
h i
(=1 (=2 (=3 ξλ1 ,λ2 (k1 , k2 ) = NΘλ1 λ2 η(k1 )η(k2 ) eim(ϕk1 −ϕk2 ) + c.c , (57)
FIG. 7. Quantum coherence of twisted photon pairs. Each disk Here, the function η(k), which determines the pulse shape and
shows the g(2) as a function of azimuthal angle difference of the two transverse-plane distribution, is independent on ϕk [43]. The
photons ϕ − ϕ0 ∈ [0, 2π) and the red dashed line denotes the case polarization of the photon pair is described by a symmetric
ϕ − ϕ0 = 0. The upper and lower rows denote the g(2) -function of 2 × 2 matrix
twisted photon pairs with symmetric and anti-symmetric spin states,
Θ++ Θ+− − sin θB e−iϕB cos θB
" # " #
respectively.
Θ= = . (58)
Θ−+ Θ−− cos θB sin θB eiϕB
The normalization of state |Pξ i requires d3 k |η(k)|2 = 1 and
R
transformation, we can also re-express |Pξ i with the field op-
erators, N = 4(1 + δm,0 ) −1/2 . This type of photon pairs can be gen-
E 1 XZ Z erated by a two-atom light source [75]. Deterministic photon
Pξ = √ d3 r1 d3 r2 ξ̃λ1 ,λ2 (r1 , r2 )ψ̂†λ2(r2 )ψ̂†λ1(r1 )|0i , (53) pairs can be generated via bundle-emission processes [76, 77].
2 λ1 λ2 The two-photon wave-packet in real space can be expressed
in the form of (see Appendix D)
where the two-photon wave packet in real space
0
h i
Z Z ξ̃λλ0 (r, r0 ) = N η̃(r)η̃(r0 )Θλλ0 eim(ϕ−ϕ ) + c.c . (59)
1
ξ̃λ1 ,λ2 (r1 , r2 ) = d k1 d3 k2 ξλ1 ,λ2 (k1 , k2 )ei(k1 ·r1 +k2 ·r2 ) ,
3
(2π)3 With this wave-packet function, we obtain the PND and helic-
2
(54) ity density as hn̂(r)i = 2 |η̃(r)|2 and hn̂H (r)i = 0, respectively.
is also normalized λ1 λ2 d3 r1 d3 r2 ξ̃λ1 ,λ2 (r1 , r2 ) = 1 and
P R R
This vanishing net helicity density results from the fact that
symmetric. we have required the two photons in the pulse to have op-
We now give the quantum correlation and coherence func- posite helicity. The corresponding correlation functions are
tions for an arbitrary two-photon pulse. The PND and helicity given by,
density correlations are given by
2
G(2) (r, r0 ) = 8N 2 |η̃(r)|2 η̃(r0 ) 1 + cos[2m(ϕ − ϕ0 )] , (60)
2
X
G(2) (r, r0 ) = 2 ξ̃λλ0 (r, r0 ) , (55) G(2)
H (r, r ) = −G (r, r ) cos 2θ B .
0 (2) 0
(61)
λλ0
and We obtain a simple relation between G(2) (r, r0 ) and G(2) 0
H (r, r )
for non-interacting photons in free space. More sophisticated
2
X
H (r, r ) = 2
G(2) 0
λλ0 ξ̃λλ0 (r, r0 ) , (56) and interesting photon helicity correlation structure can be ob-
λλ0 tained in a nonlinear medium, such as the Rydberg-atom ar-
rays induced photon-photon interaction [78, 79]. In experi-
respectively. Here, the factor 2 comes from the fact that there ments, the helicity correlation can be measured via nano-scale
are two photons in the pulse. The quantum coherence func- quantum sensors [80].
tion function can be easily obtained with the help of the PND The striking property of the twisted photon pair is that its
2
at each point hn̂(r)i = 2 λλ0 dr0 ξ̃λλ0 (r, r0 ) . The helicity
P R
quantum coherence function is now modulated by the az-
2 imuthal angle difference of the two photons,
density is given by hn̂H (r)i = 2 λλ0 λ dr0 ξ̃λλ0 (r, r0 ) .
P R
1
g(2) (r, r0 ) = 1 + cos[2m(ϕ − ϕ0 )] .
(62)
2(1 + δm,0 )
Spin-state dependent quantum statistics
For m = 0 case, the quantum coherence function reduces to a
It is well-known that statistics of identical particles are constant g(2) = 1/2 for an un-twisted two-photon Fock state.
strongly dependent on their spin property. For fermions with For m , 0 case, the g(2) -function reaches it maximum 1 when
the same spin state, the two-particle correlation vanishes when ϕ − ϕ0 = 0. Thus, the two photons in a twisted photon pair
|r − r0 | → 0, i.e., g(2) (r, r) = 0. For photons in a thermal prefer to be bunched compared to a regular un-twisted one.11
For a twisted photon pair with symmetric spin state |↑i ⊗ |↑i of a photonic vortex—the winding number of the photonic
or |↓i ⊗ |↓i, its quantum coherence function is the same as currents. We predict the pure helicity vortex, which can be
Eq. (62). However, the helicity density is not zero for these measured in experiments.
two cases, hn̂H (r)i = ±2 |η̃(r)|2 cos θB . We also study the quantum statistics of twisted light with
Another interesting phenomenon is that the quantum statis- our proposed theoretical formalism. We show an interest-
tics of twisted photon pairs are reversed if their spin state is ing effect: the statistical behaviors are essentially different for
anti-symmetric ∼ |↑i ⊗ |↓i − |↓i ⊗ |↑i. The corresponding two- twisted photon pairs with symmetric and anti-symmetric spin
photon SAF is given by states. Entangled twisted photons, which possess spatially
h i varying quantum coherence function in the transverse plane,
ξλ1 ,λ2 (k1 , k2 ) = NΘλ1 λ2 η(k1 )η(k2 ) eim(ϕk1 −ϕk2 ) − c.c. , (63) can be utilized to enhance resolution in quantum imaging [83]
and detect the texture of a target in the quantum-illumination-
where m , 0. The polarization of the photon pair is now based radar system [84, 85].
described by an anti-symmetric constant matrix L.P.Y is supported by the funding from Ministry of Sci-
ence and Technology of China (No.2021YFE0193500). D.X.
Θ++ Θ+−
" # " #
0 1 is supported by NSFC Grant No.12075025.
Θ= = . (64)
Θ−+ Θ−− −1 0
The quantum coherence of photon this type of twisted photon Appendix A: Photonic OAM operators in real space
pairs is given by
1 Historically, the discovery of the optical OAM has boosted
g(2) (r, r0 ) = 1 − cos[2m(ϕ − ϕ0 )] . (65) the development of optical phase singularities researches [37].
2 Here, we show how to evaluate the OAM of light in real
In contrast to Eq. (62), the g(2) -function now reaches it mini- space within our proposed theoretical framework in this work.
mum 0 when ϕ − ϕ0 = 0. Thus, the two photons in a twisted The directly observable part of the photonic OAM is given
by L̂obs = ε0 d3 r Ê⊥j (r, t)(r × ∇)Â⊥j (r, t) [86], which can be
R
photon pair prefer to be anti-bunched compared to a regular
un-twisted one. rewritten with the photonic field operator as [43],
In Fig. 7, we contrast the g(2) -function for twisted photon Z Z
pairs with a symmetric spin state (the upper row) and with an L̂obs = d3 rψ̂† (r)(r × p̂)ψ̂(r) ≡ d3 rψ̂† (r)l̂ψ̂(r). (A1)
anti-symmetric spin state (the low row). We find that the two
rows are precisely complementary to each other. Thus, the To obtain the quantum uncertainties of the photonic OAM, we
quantum statistics of twisted photon pairs are reversed when need the square of its three components,
their spin state changes from symmetric to anti-symmetric. Z Z Z
Here, we use two simple examples to show the influence of (Lobs
j )2
= d 3
r d r ψ̂
3 0 †
(r) ψ̂ † 0
(r ) l̂ l̂
j j
0
ψ̂(r) ψ̂(r 0
)+ d3 rψ̂† (r)l̂2j ψ̂(r).
the photon spin on the quantum statistics of light. The SAF
of entangled photon pairs generated via an SPDC process will (A2)
be more complicated [81, 82]. However, we have revealed the In a Cartesian coordinate, the three components of the dif-
essential feature of spin-dependent statistics in twisted photon ferential operator l̂ = (l̂ x , l̂y , l̂z ) are given by
pairs. ∂ ∂
!
In the upper two examples, the spin and spatial degrees l̂ x = −i~ y − z , (A3)
∂z ∂y
freedom of the photon pairs are highly entangled. Thus, the
∂ ∂
!
photon-number-density and helicity-density correlations ex- l̂y = −i~ z − x , (A4)
hibit almost the same correlation behavior. However, the ∂x ∂z
properties of these two correlations could be significantly ∂ ∂
!
different in more general cases, such as randomly polarized l̂z = −i~ x − y . (A5)
∂y ∂x
Fock-state photon pairs. Quantum correlation can still exist in
photon-number density, but the helicity density will be com- It is more convenient to evaluate the OAM of a paraxial pulse
pletely uncorrelated. or beam in a cylindrical coordinate as shown in Fig. 1a. Then,
the three differential operators are given by
∂ ∂ ∂
!
z
VI. CONCLUSION l̂ x = −i~ ρ sin ϕ − z sin ϕ − cos ϕ , (A6)
∂z ∂ρ ρ ∂ϕ
∂ ∂ ∂
!
We have put forth a quantum framework for structured z
l̂y = i~ ρ cos ϕ − z cos ϕ + sin ϕ , (A7)
quantum light in real space by introducing the effective field ∂z ∂ρ ρ ∂ϕ
operator of photons. We exploit our presented theoretical for- ∂
malism to study the photonic vortices. We show the analogy l̂z = −i~ . (A8)
∂ϕ
between the photonic vortex and its counterpart in superfluids
by defining quantum operators for the photonic currents. We For a laser pulse or a beam, the mean value of the OAM
also give an unambiguous definition of the topological charge is obtained by simply replacing the field operators ψ̂(r) and12
ψ̂† (r) with the functions Ψ(r) and Ψ∗ (r), respectively. This is commute with each other, i.e., [D̂ξ1 (α1 ), D̂ξ2 (α2 )] = 0, because
also the reason that OAM of light has usually been handled only creation operators â k are involved in D̂ξ .
classically [87–89]. We emphasize that no paraxial approx- The superposition of multiple laser beams can be described
imation [87, 88] is required for the global quantities. Our by the quantum state
theory provides a powerful and versatile tool to handle the
OAM of all classes of quantum structured light pulses in real 1 Y
|Ψi = √ D̂ξ j (α j )|0i (B2)
space, specifically for the one with spatiotemporal optical vor- N j
tices [15–17]. !
We note that the integral kernal in Eq. (A1) can not be in- 1 X 1 2
= √ exp α j âξ j − |α j | |0i.
†
(B3)
terpreted as the photonic OAM density, which by-definition is N j
2
given by
j j
Using the relation,
M (r) = ε0 Ê ⊥ (r)(r × ∇) Â⊥ (r).
l̂obs (A9)
α j â†ξ α j â†ξ α∗j α j0 âξ j ,â†ξ 0
P
α∗j âξ j α∗j âξ j
P P
j, j0
P P
With the the plane-wave expansion of Ê⊥ and Â⊥ [43], we e j
e j j =e j j e j
e j , (B4)
have
we obtain the normalization factor
ωk ∗
Z Z r
−i~ X
l̂obs (r) ≈ d 3
k d 3 0
k e (k, λ) · e(k0 , λ0 ) X
M
(2π)3 λλ0 ω k0 N = exp α j α j0 [âξ j , âξ j0 ] ,
∗ †
(B5)
0 j, j0
× â†k,λ e−ik·r (r × ∇)â k0 ,λ0 eik ·r , (A10)
with the overlap of the SAFs of the two pulses
where we have neglected the counter-rotating wave
terms a†k,λ â†k0 ,λ and â k,λ â k0 ,λ . In the paraxial limit, we
XZ
[âξ j , â†ξ j0 ] = d3 kξ∗j,λ (k)ξ j0 ,λ (k) (B6)
have e∗ (k, λ) · e(k0 , λ0 ) ≈ δλλ0 exp iλ(ϕk − ϕk0 ) , where we
λ
have used the relations XZ
= d3 rξ̃∗j,λ (r, t)ξ̃ j0 ,λ (r, t). (B7)
2 θk 2 θk 1
e(k, λ) = e−iλϕk
cos iλϕk
eλ −e sin e−λ − √ sin θk ez , (A11) λ
2 2 2
Utilizing the relation
√
and eλ = (e x + iλey )/ 2 [43]. Thus, even for a quasi-single-
√
h i
frequency paraxial beam with ω k /ω k0 ≈ 1, the OAM den- a k , D̂ξ (α) = αξ(k)D̂ξ (α),
sity only reduces to
we show that ψ̂(r) |Ψi = Ψ(r, t) |Ψi with Ψ(r, t) = j α j ξ̃ j (r, t)
P
satisfying the wave equation ∇2 − c12 ∂t∂ 2 Ψ(r, t) = 0. We
2
l̂obs (r, t) ≈ ψ̃ˆ † (r)(r × p̂)ψ̃ˆ (r), (A12)
note that our presented formalism is significantly different
where from the previous one based a six-component photon wave
1
Z function [90, 91]. Our introduced two-component field op-
ψ̃ˆ λ (r) ≡ p d3 kâ k,λ ei(k·r−λϕk ) (A13) erator ψ̂(r) satisfies the wave equation [43] and is compati-
(2π)3 ble to light-matter interaction. However, the six-component
wave function satisfies a Schrödinger-like equation and is in-
It is more convenient to evaluate both the photonic spin and
compatible to light-matter interaction.
OAM densities in k-space. The angular momentum density
In the circularly-polarization representation, a linearly po-
of light will be of increasing interest for experiments in the
larized laser pulse can be described by a state
near future.
|α|2
" #
E 1
αξ,λ=1 = exp √ α(â†ξ,+ + â†ξ,− ) − |0i, (B8)
Appendix B: Quantum state of the superposition of multiple 2 2
|α|2
" #
laser beams E i
αξ,λ=2 = exp √ α(âξ,+ − âξ,− ) −
† †
|0i, (B9)
2 2
A laser beam can be regarded as a quantum light pulse with
extremely long pulse length and enormous number of photons. with
The corresponding quantum state is given by E 1 E
ψ̂(r) αξ,λ=1 = √ [αξ(r, t), αξ(r, t)]T αξ,λ=1 , (B10)
∞ 2
X αn † n
= D̂ξ (α)|0i = e−|α| /2
2
E
αξ â |0i . (B1) i
n! ξ
E E
n=0 ψ̂(r) αξ,λ=2 = √ [αξ(r, t), −αξ(r, t)]T αξ,λ=2 . (B11)
2
where the operator D̂ξ is given in Eq. (5) and â†ξ can be the. It Elliptically polarized quantum pulses can be constructed in a
can be easily verified that two operators D̂ξ1 (α1 ) and D̂ξ2 (α2 ) similar way.You can also read
