Orbital angular momentum uncertainty relations of entangled two-photon states
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Eur. Phys. J. D (2021) 75 :226
https://doi.org/10.1140/epjd/s10053-021-00243-z
THE EUROPEAN
PHYSICAL JOURNAL D
Regular Article – Quantum Optics
Orbital angular momentum uncertainty relations of
entangled two-photon states
Wei Li1,2,3,a and Shengmei Zhao1,2,b
1
Institute of Signal Processing and Transmission, Nanjing University of Posts and Telecommunications, Nanjing 210003,
Jiangsu, People’s Republic of China
2
Key Lab of Broadband Wireless Communication and Sensor Network, Nanjing University of Posts and
Telecommunications, Nanjing 210003, Jiangsu, People’s Republic of China
3
National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China
Received 22 March 2021 / Accepted 10 August 2021 / Published online 17 August 2021
© The Author(s) 2021
Abstract. The inseparability of quantum correlation requires that the particles in the composite system
be treated as a whole rather than treated separately, a typical example is the Einstein–Podolsky–Rosen
(EPR) paradox. In this paper, we provide a theoretical study on the uncertainty relations of two kinds of
bipartite observables in two-photon orbital angular momentum (OAM) entanglement, that is, the relative
distance and centroid of the two photons at azimuth. We find that the uncertainty relations of the bipartite
observables holds in any two-photon state, and they are separable in two-photon OAM entanglement. In
addition, the entangled state behaves as a single particle in the bipartite representation. Finally, we find
that the uncertainty relations of the bipartite observables can be used to manipulate the degree of the
entanglement of an EPR state.
1 Introduction be totally inferred by the measurement of the other.
Actually, this is, in essence, a quantum behavior, which
There are many remarkable differences between a quan- is caused by the coherence of two-particle state. This
tum particle and a classical particle, like the non- behavior has also been reformulated as either quantum
locality, wave-particle duality, the coherent superposi- steering [14–16] or observer-dependent uncertainty rela-
tion of the wave-function and the collapse of the wave- tion that plays a vital role in quantum witness [17–20].
function caused by measurements. Extending these As we learn from the entangled state, the description
characteristics to multi-particle systems will lead to of the movement behavior of only one particle is inade-
more abundant physical phenomena. One example is quate for a quantum correlated two particle state. In an
the two-photon entangled state which is the coher- isolated two-particle system, its behavior always obey
ent superposition of two-photon product state. The the uncertainty principle of quantum mechanics regard-
long-distance quantum correlation introduced by non- less of the state, like in quantum correlation imaging
locality and coherent superposition, combined with a high spatial resolution can be obtained by a higher
wave-function collapse caused by measurement, two- dimensional momentum entanglement [21,22], the con-
photon entangled state plays an important role in verse correlation between time and frequency domain
the field of quantum computation and quantum infor- for a spontaneous parametric down conversion two-
mation, for example quantum teleportation [1–5] and photon state [23], and the enhancement of the dimen-
entanglement-based quantum key distribution [6–8]. sion and degree of two-photon OAM entanglement by
One of the most important principle for quantum increasing their angular position correlation [24,25].
measurement that is different from classical measure- In this work, we perform a theoretical study on uncer-
ment is the Heisenberg uncertainty relation [9–11], tainty relations of OAM entangled two-photon state.
which states that the incompatible observables in one First, we construct two kinds of conjugate bipartite
quantum particle cannot be determined simultaneously. observables for the composite system which are lin-
While for a two-particle entangled state, the insepara- ear combinations of the operators of the subsystems
bility leads to a counterintuitive uncertainty behavior of in angular position space and OAM space, respectively.
a single particle. For example in the Einstein–Podolsky– Next, we establish the commutation relation for them
Rosen (EPR) state [12,13], the position and momentum and analyze their uncertainty relations with compu-
which are incompatible observable of one particle can tational simulations. Finally, we studied the mutual
exclusion between two-photon angular position corre-
a lation and OAM correlation for entangled two-photon
e-mail: alfred wl@njupt.edu.cn
b
e-mail: zhaosm@njupt.edu.cn (corresponding author)
123
226 Page 2 of 7 Eur. Phys. J. D (2021) 75 :226
to [24]
Φ (θs , θi , θp ) ≈ A
2 2
(θs − θp ) + (θi − θp )
× sinc
Γ
2 2
(θs − θp ) + (θi − θp )
exp i , (2)
Γ
Fig. 1 Schematic illustration of SPDC two-photon corre- 8|k |
lation. a The purple cone represents a pump cone state, the where A is the normalization constant, Γ = |p |p2 L is
p
red cone represents the down-converted two-photon corre- the radius of down-converted two-photon cone in angu-
lation from the pump photon in the direction of the center lar position representation with |kp | the wave vector
of the red cone, NLC is a nonlinear crystal. b Cross section of the pump beam, L is the propagation distance of
representation of SPDC of the pump cone state in momen- SPDC process within the NLC and |pp | is the radius
tum space of the pump cone state in momentum space. Because a
sinc function varies slightly in the region near around
θp , so Eq. (2) represents a weak correlation between the
state. Modulating the entanglement of the two-photon signal and idler states.
state by using the uncertainty relations of the bipartite According to the Fourier relation between angular
observables has also been discussed in detail. position and OAM [30]
∞
1
|θ = √ exp (−ilθ) |l , (3)
2π l=−∞
2 Uncertainty relation for two-photon
correlation by setting the OAM of the pump beam lp to 0, we have
the two-photon correlation in OAM representation
In a degenerate spontaneous parametric down conver- ∞ ∞
1
sion (SPDC) pumped by a rotationally symmetric light |Ψ (ls , li ) = Φ (ls , li ) δls ,−li |ls |li ,
beam, the down converted two photons will entangle in 2π
ls =−∞ li =−∞
both radial mode and azimuthal mode [26–28]. If only
(4)
the azimuthal modes are concerned, a better way is to
decompose the pump state into a set of cone states,
which is shown as a purple cone in Fig. 1a, and the red where ls and li are the OAM carried by the sig-
cone represents the down-converted two-photon state nal state and idler state, Φ (ls , li ) is the two-photon
in the direction of the center of the red cone. A signif- OAM correlation function Fourier transformed from
icant advantage of this treatment is that the complex Φ (θs − θp , θi − θp ) with respect to θs − θp and θi − θp
radial entanglement can be ignored. In angular position [24], the delta function δls ,−li implies OAM conserva-
representation as shown in Fig. 1b, by considering the tion [29] in the down-conversion process.
rotational symmetry of the pump cone state, the down- As the spaces corresponding to the conjugate observ-
converted two-photon state can be expressed as [24] ables are connected by Fourier transform, now we con-
struct the bipartite observables from the two-photon
correlation function Φ (θs − θp , θi − θp ) and Φ (ls , li ). To
ease the discussion, we temporarily set θp to 0. The
exponential term connecting the two-photon correlation
|Ψ (θs , θi ) = dθs dθi dθp Φ (θp , θs , θi ) |θs |θi , (1) function in conjugate spaces can be recast into
exp (−ils θs − ili θi ) = exp (−iL1 Θ1 − iL2 Θ2 ) , (5)
where the observables ls,i and θs,i for single-photon and
where θp , θs and θi are the angular positions of the the observables L1,2 and Θ1,2 for the joint system are
pump state (p), the signal state (s) and the idler state defined as follows
(i), respectively; Φ (θp , θs , θi ), determined by the phase √ √
matching condition [29], is the angular position corre- L̂1 = 22 ˆls − ˆli , L̂2 = 22 ˆls + ˆli ,
lation function of the down-converted two-photon state √ √ (6)
from the pump state at θp . Under paraxial approxi- Θ̂1 = 22 θ̂s − θ̂i , Θ̂2 = 22 θ̂s + θ̂i .
mation and assuming that the correlation scale in the
transverse plane is far smaller than the radius of the Here, the bipartite operators L1 and Θ1 can be viewed
pump cone state, Φ (θp , θs , θi ) can be approximated as the relative distance of the two photons in azimuthal
123
Eur. Phys. J. D (2021) 75 :226 Page 3 of 7 226
Fig. 2 Mutual exclusion between two-photon angular position correlation and OAM correlation. Joint probability distri-
bution Φ (Θ1 , Θ2 ) for two-photon angular position correlation for a Γ = 1.59rad2 and b Γ = 0.32rad2 ; Joint probability
distribution Φ (L1 , L2 ) for c for two-photon OAM correlation Γ = 1.59rad2 and Γ = 0.32rad2 . Inset in (b) shows that the
pump light is in a point-like state at θp
region, and L1 and Θ1 are their centroid. Apparently, the uncertainty relation is reformulated as [31–33]
L1,2 and Θ1,2 satisfy the following commutation rela-
tions 1
ΔLi ΔΘj ≥ δi,j |1 − 2πP (Θ)| , (9)
2
L̂i , L̂j = Θ̂i , Θ̂j = 0, L̂i , Θ̂j = iδi,j . (7)
where P (Θ) is the angular probability density at the
boundary of the interval of integration. This inequality
can alternatively be interpreted as mutual exclusion in
From Eqs. (5–8), we can see that Θi and Li are con- determining the values of two incompatible observable.
jugate observables, the two-photon correlation function
Φ (θs , θi ) can be reformulated as Φ (Θ1 , Θ2 ). Accord-
ingly, the two-photon OAM correlation function Φ(L1 ,
L2 ) can be obtained from the two-dimensional Fourier 3 Results and discussion
transformation of Φ (Θ1 , Θ2 ).
According to Heisenberg uncertainty principle, the Let’s first examine the general uncertainty relations for
variances of L̂i and Θ̂j satisfy the bipartite observables in non-entangled two-photon
states. In the following simulation, the down converted
1 1 two-photon state is generated by a point-like pump light
ΔLi ΔΘj ≥ [L̂i , Θj ] = δi,j , (8) at θp , as shown in the inset of Fig. 2b. The amplitude
2 2 distribution of the down converted two-photon cone
state is given by Eq. (2). Because of the rotational sym-
where ΔLi and ΔΘj are their standard deviations. metry of the pump light, varying θp will only shift the
Because of the limited integration range at azimuth, two-photon state along the curve of θs − θi = 0, with-
123
226 Page 4 of 7 Eur. Phys. J. D (2021) 75 :226
out loss of generality, we set θp = 0. The joint proba- ity ΔΘ1 ΔL2 ≥ 12 |1 − 2πP (Θ)|, which arises from the
bility distribution of two-photon states in the angular- breaking of the equalities ΔΦ1 = ΔΦ2 and ΔL1 = ΔL2
position correlation representation and the OAM cor- [18,19]. From Fig. 2, we can see thatP (Θ) = 0 at the
relation representation is shown in Fig. 2. In this fig- boundary of the azimuth integral of Θ1 , which means
ure, we can see that ΔΘ1 = ΔΘ2 and ΔL1 = ΔL2 , that the lower bound value of Heisenberg uncertainty
this is the common phenomenon for non-entangled two- relation is 0.5. Here, ΔΘ1 ΔL2 = 0 in the simula-
particle system [18]. The mutual exclusion between tion gives evidence of the existence of entanglement
Θ1(2) and L1(2) is clearly shown here. When Γ is picked from another perspective. On the contrary, we can also
at 1.59 rad2 in Fig. 2a,c, the joint probability distribu- get ΔΘ2 ΔL1 > ΔΘ1 ΔL1 ≥ 12 |1 − 2πP (Θ)|, which is
2 caused by quantum entanglement.
tion P (Θ1 , Θ2 ) = |Φ (Θ1 , Θ2 )| for two-photon angu-
lar position correlation occupies a large scale, mean- In Fig. 3b, we can see that for the pump cone state,
while the joint probability distribution P (L1 , L2 ) = the value of L2 is strictly equal to 0, which is indepen-
2 dent of the thickness of the NLC as well as the radius of
|Φ (L1 , L2 )| for the two-photon OAM correlation is
the pump cone state. In this case, the two-photon rel-
mainly concentrated near the zero point. While as the
ative distance operator L1 can be separated from the
radius Γ reduces to 0.32 rad2 in Fig. 1b,d, which
centroid operator L2 . The separability is due to the
is 5 times smaller, the joint probability distribution
rotational symmetry of the pump cone state. In angular
P (Θ1 , Θ2 ) occupies a narrower scale, while P (L1 , L2 )
position correlation representation, the down-converted
expands significantly.
two-photon state generated by the pump cone state is
We now extend the trivial uncertainty relationship
between for the two photons in Fig. 2 to the OAM
entangled state, where the pump light is in a cone state |Ψ (Θ1 , Θ2 )
as shown in the inset of Fig. 3b. Inserting Eq. (6) in 2π
Eq. (2), we have the two-photon correlation function in = dΘ1 |Φ1 dΘ2 |Θ2 dθp Φ (Θ1 , Θ2 − θp ) .
0
the correlation representation (11)
√
Φ Θ1 , Θ2 − 2θp ≈ A Because of the cyclic symmetry of the integral over θp ,
2π
√ 2 √ 2 we have the integral identity 0 dθp Φ (Θ1 , Θ2 − θp ) =
Θ12 + Θ2 − 2θp Θ12 + Θ2 − 2θp
× sinc exp i . f (Θ1 ), which is independent of Θ2 . Then, we have the
Γ Γ separable state for Θ1 and Θ2
(10)
|Ψ (Θ1 , Θ2 ) = dΘ1 |Θ1 f (Θ1 ) ⊗ dΘ2 |Θ2 . (12)
Actually, Eq. (10) is a variant of Eq. (2), where √ vari-
ables θs − θp , θi − θp are replaced by Θ1 and Θ2 − 2θp
by a unitary transformation. By substituting Eq. (10) Meanwhile, we have the two-photon state in OAM cor-
into Eq. (1), we obtain the joint probability distribution relation representation
of the two-photon state in angular position correlation
representation, as shown in Fig. 3a. It is the extension ∞
of the joint probability in Fig. 2b along Θ1 axis. The |Ψ (L1 , L2 ) = g (L1 ) |L1 ⊗ |L2 = 0 , (13)
joint probability distribution of the corresponding two- L1 =−∞
photon state in the OAM correlation representation is
shown in Fig. 3b, which is derived
√ from the components where g (L1 ) is the Fourier transform of f (Θ1 ). From
along the direction of lp − 2L2 = 0 in the two-photon Eqs. (12, 13), we can see that quantum states dΘ2 |Θ2
OAM correlation spectrum Φ (L1 , L2 ). In this simula- and |L2 = 0 completely inherits the azimuthal modes
tion, the width Γ is chosen as 0.32 rad2 , the value of of the pump state. So conjugate variables Θ2 and L2
Θ2 is evenly distributed in the range of −π to π, while can be viewed as the external degrees of freedom of
the√value of Θ1 is concentrated near 0 with a width the two-photon state. The conjugate variables Θ1 and
of 2Γ . The two-photon joint probability distribution L1 , which describe the relative distances between the
in angular position space is perpendicular to that in two photons in azimuthal region, can be viewed as the
OAM space, this is caused by the opposite correlation internal degrees of freedom of the two-photon state, and
in conjugate spaces [23,34]. the entanglement is determined by the correlation func-
Compared with Fig. 2b,d, we can see at the first tions f (Θ1 ) and g (L1 ). If we only focus on the OAM
glance that in Fig. 3, the values of ΔΘ1 and ΔL1 entanglement of the two-photon state, entanglement
remain unchanged, the value of ΔΘ2 increases and can totally be described by conjugate variables Θ1 and
ΔL2 decreases to 0. The uncertainty relation between L1 , therefore, the two-photon entanglement behaves
L1 and Θ1 , L2 and Θ2 are guaranteed by Eq. (10). like a single particle. The single particle behavior of
For L2 and Θ2 , since the value of P (Θ) is equal to two-photon correlation is totally a quantum mechani-
1/2π in the range of azimuth integral of Θ2 , we have cal phenomenon, and the mutual exclusion between the
ΔL2 · ΔΘ2 = 0. Another equivalent criterion for the standard variances of Θ1 and L1 is the direct result of
existence of entanglement is the violation of inequal- Heisenberg’s uncertainty principle.
123Eur. Phys. J. D (2021) 75 :226 Page 5 of 7 226
Fig. 3 Two-photon correlation spectrum in the representation of correlation operators. By the consideration of the rota-
tional symmetry of the pump cone state, we obtain a the joint probability distribution band P (Θ1 , Θ2 ) in angular position
space, b the corresponding joint probability distribution band P (L1 , L2 ) in OAM space. In this simulation, the width of
two-photon angular position function Γ is set to 0.32 rad2 . Inset in (b) shows that the pump light is in a cone state
Due to the single particle behavior of two-photon entanglement
OAM entanglement, the mutual exclusion between ΔΘ1
and ΔL1 is relevant to the degree of entanglement of
the system. Now we study the uncertainty relation for
two-photon correlation operators, and discuss its influ- 1
ence on the degree of entanglement of the system. Fig- K= , (16)
l P2(l, −l)
ure 4 shows the joint probability distributions of two-
photon state in azimuthal region for different values
of Γ . In the upper row of Fig. 4, the joint probabil-
ity P (θs , θi ) rotates π4 in the original azimuth repre-
sentation compared with P (Θ1 , Θ2 ). The bottom row which is an estimation of the mean number of modes
of Fig. 4 shows the corresponding two-photon OAM that participate in the entanglement. In addition,
entangled spectrum. From this graph we can see that we use H as the Heisenberg uncertainty function
as Γ reduces from 1.59 rad2 to 0.16 rad2 , the relative Δ θs√−θ 2
i
Δ ls√
−li
2
. When Γ = 1.59 rad2 in Fig. 4a,b,
distance between the two photons in angular position we have the average distance between two photons in
space is closer and, on the contrary, their relative dis- θs√
−θi
angular position space Δ ≈ 1.1 rad, the aver-
tance in OAM space and the degree of entanglement 2
become larger. ls√
−li
age distance in OAM space Δ 2
≈ 0.45 /rad, the
Next, we investigate the uncertainty relations for the
two-photon relative distance operators. The standard Heisenberg uncertainty function H ≈ 0.5, the Schmidt
deviation of two-photon angular position correlation is number K ≈ 1.2. In this case, the mode distribution
of two-photon OAM entanglement mainly concentrates
at l = 0. As Γ decreases to 0.32 rad2 in Fig. 4c, d, we
2 2 θs√
−θi ls√
−li
θs − θi dθs dθi |Φ (θs , θi )| (θs − θi ) have Δ ≈ 0.313 rad, Δ ≈ 3.82 /rad,
Δ √ = 2 . 2 2
2 2 dθs dθi |Φ (θs , θi )| H ≈ 1.2, the Schmidt number K ≈ 8.3. Now the two-
photon OAM entanglement spectrum occupies a larger
(14)
scale, and the average dimension reaches to 8. When
Γ continues
todecreases to 0.16 rad2 in Fig. 4e, f, we
Correspondingly, the standard deviation of two-photon θs√−θi ls√−li
have Δ 2
≈ 0.17 rad, Δ 2
≈ 7.1 /rad,
OAM correlation is
H ≈ 1.21, the Schmidt number K ≈ 16.4. At this time,
the dimension of two-photon OAM entanglement spec-
ls − li ∞ trum continues to expand, and the average dimension
Δ √ = 2l2 P (l, −l). (15) reaches to 16.4. We can see that the Heisenberg uncer-
2 l=−∞ tainty relations for quantum correlation for all values
of Γ are satisfied and the degree and dimension of two-
photon OAM entanglement can be enlarged by increas-
2
where P (l, −l) = |Φ (l, −l)| . Here, we use the Schmidt ing the strength of two-photon angular position corre-
number K to represent the dimension and degree of lation.
123226 Page 6 of 7 Eur. Phys. J. D (2021) 75 :226
Fig. 4 Uncertainty relation for two-photon quantum correlation in azimuthal region. Joint probability distribution
P (θs , θi ) in angular position space for a Γ =1.59 rad2 , c Γ =0.32 rad2 , e Γ =0.16 rad2 . The corresponding joint probability
distribution P (ls , li ) in OAM space for (b) Γ =1.59 rad2 , d Γ =0.32 rad2 , f Γ =0.16 rad2
4 Conclusion comment: All data can be obtained from the formulae in the
article.]
In this paper, we have studied the uncertainty relations
of bipartite observables in two-photon OAM entangled Open Access This article is licensed under a Creative Com-
state. Two kinds of conjugate bipartite observables are mons Attribution 4.0 International License, which permits
defined, which can be viewed as the centroid and rela- use, sharing, adaptation, distribution and reproduction in
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correlation, improvement of the resolution of quantum
correlation imaging by increasing the correlation size
in the momentum space, and inferring the correlation
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