Manipulating orbital angular momentum entanglement by using the Heisenberg uncertainty principle

Wei Li; Shengmei Zhao

doi:10.1364/OE.26.021725

1. Introduction

A light beam with a rotational symmetry carries a well-defined orbital angular momentum (OAM), characterized by the winding number l, which ranges from −∞ to ∞ [1]. In the process of spontaneous parametric down conversion (SPDC) in which a high-energy photon is converted into two photons with lower energy, the conservation law is not only fulfilled by energy and momentum, but also fulfilled by angular momentum [2]. If there is no angular momentum exchange between the nonlinear crystal and the incident photons, OAM conservation is fulfilled in SPDC, which gives rise to the generation of OAM entanglement [3]. Since OAM entanglement was first demonstrated experimentally [4], OAM entanglement has served as a promising candidate to accomplish a series of quantum tasks beyond two-dimensional Hilbert-space entanglement [5], for example, dense coding [6], high-dimensional teleportation protocols [7], bit commitment [8], quantum cryptography [9], and high-dimensional entanglement [10–12]. Meanwhile, OAM entanglement has been successfully employed to explore some quantum features in experiments on topics such as the violation of Bell’s inequality for high-dimensional entanglement [12–14], quantum ghost imaging [15] and Einstein-Podolsky-Rosen (EPR) correlation between OAM and angular position [16]. However, experimentally generated OAM entanglement is still a long way from being used directly, as the entanglement spectrum always has a finite bandwidth and the weight distribution for each mode is not uniform. ractical applications of OAM entanglement rely on the technique of entanglement concentration [12,13,17], in which both complex experimental procedures and a good entanglement quality are required. Currently, the scientific community still lacks a thorough understanding of OAM entanglement, and how to manipulate OAM entanglement remains an open question.

Almost all previous research on OAM entanglement was based on mode coupling, i.e., the overlapping of the mode functions between the pump state and the signal and idler states [18–23]. In these studies, the pump and the down-converted states are represented by Laguerre-Gauss (LG) modes, where the mode distribution is determined by the chosen radial modes [20]. This kind of entanglement should be called mode entanglement, it covers azimuthal entanglement and radial entanglement. The total degree of entanglement between the down converted photons is determined by the quantum correlation contributed from these two parts. Nonetheless, the angular position correlation in the azimuthal region has not been discussed yet, even though this is the real cause of the nonuniform distribution of OAM entanglement. Angular position and OAM are conjugate variables connected by Fourier transformation [24], and they form an EPR pair, the quantum correlation of one variable will determine that of the other [25]. The first experimental study of mode distribution of OAM entanglement through angular position correlation was conducted by two-photon interference in the azimuthal domain [26–28]. It has also been shown in experiments that the increase in the dimension of OAM entanglement will lead to a stronger angular position correlation [29]. Further, angular position correlation is closely related to the radial coordinate, when all of the radial modes are contained in this two-photon interference experiment the relationship between angular position correlation and OAM entanglement cannot be determined accurately.

In this paper, we present a theoretical study of OAM entanglement in the azimuthal domain and show that it is an inherent feature of angular-position entangled two-photon states generated by a rotationally symmetric pump beam. We decompose the pump light into a set of cone states characterized by transverse momentum projection. The mode distribution of the OAM entanglement is determined by the radius of the pump cone state, the length of the nonlinear crystal, and the OAM carried by the pump beam. The first two factors influence the OAM entanglement by controlling the quantum angular position correlation between the down-converted two-photon states, while the latter one modulates the OAM entanglement by shifting the diagonals of the OAM correlation spectrum.

2. Theory

We first consider an incident quasi-monochromatic plane-wave-like pump beam with a wave vector of k_p propagating through a uniaxial birefringent nonlinear crystal. The two-photon generation within the scheme of collinear type-II energy-degenerate SPDC is given by [30]

| Ψ 〉 = \iint d k_{s} d k_{i} Φ (k_{p}, k_{s}, k_{i}) | k_{s} 〉 | k_{i} 〉,

where k_s,i are the wave vectors of the signal (s) and idler (i) states. The spectrum function Φ(k_p, k_s, k_i) determines the momentum correlation between the down-converted two photons. Here we omit the discussion of polarization during parametric conversion. Φ(k_p, k_s, k_i) arises from the phase-matching in SPDC, it has the following form [30]

Φ (k_{p}, k_{s}, k_{i}) = χ (ω_{p}, ω_{s}, ω_{i}) E_{p} (ω_{s} + ω_{i}) \frac{\sin [\frac{1}{2} (k_{p} - k_{s} - k_{i}) \cdot L]}{\frac{1}{2} | k_{p} - k_{s} - k_{i} |} \times \exp [i \frac{1}{2} (k_{p} - k_{s} - k_{i} \cdot L)],

where E_p is the electrical field vector of the pump beam, χ (ω_p, ω_s, ω_i) is the bilinear susceptibility, ω_p,s,i = |k_p,s,i| c/n_p,s,i are the central frequencies of the pump, signal and idler photon states, n_p,s,i are the corresponding refractive indices, L is the propagation vector of the pump beam within the nonlinear crystal, and the exponential term is the phase variances accumulated during the SPDC [30].

It is a common feature that for the generation of OAM entanglement the pump state needs to be rotationally symmetric [4,20]. A schematic illustration of the OAM entanglement generation process is shown in Fig. 1(a), where the plane-wave-like pump beam is focused onto a nonlinear crystal by a focal lens, and the output of the pump state with a spherical phase on its wavefront has a cone structure for its momentum distribution. Just as the momentum originates from the translation symmetry in space, OAM originates from the rotational symmetry, thus it is better to study OAM entanglement in the azimuthal region. In this scenario, we decompose the pump beam into a set of cone states labeled by |p_p|, the transverse wave vector projection. Then the study of OAM entanglement is conducted by fixing the radial coordinate, and is thus only focused on the azimuthal region.

Fig. 1 Schematic illustration of the collinear SPDC in the representation of the cone states. (a) Illustration of the SPDC generated by a pump cone state. The incident rotationally symmetric pump light beam is focused into the nonlinear crystal(NL). During the SPDC, the output of the pump state with a spherical phase on its wavefront can be decomposed into a set of cone states characterized by p_p, the transverse momentum projection within the plane perpendicular to the propagation principal axis. The pump cone state can be visualized as a ring within the transverse plane in the momentum representation, and it has a hollow cone structure in the position space. (b) Transformation of the transverse momentum correlation from cartesian coordinate to polar coordinates. The red and green dashed arrows labeled as q_s and q_i are the transverse momentum components of the signal and idler photons, respectively, and the corresponding solid arrows, labeled as q_s and q_i, represent how they depart from $\frac{p_{p}}{2}$ , which is half the transverse momentum component of the pump photon. Inset: schematic illustration of the down-converted two-photon transverse momentum correlation with respect to the pump momentum.

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We first choose a pump state k_p on a pump cone |p_p|. For a collinear SPDC, the conversion between the pump photon and the signal and idler photons can be divided into two directions, namely parallel and perpendicular to the wave vector of the pump. In this case, the parametric conversion mainly occurs in the direction parallel to k_p, while in the direction perpendicular to k_p there is barely any parametric conversion, otherwise the energy conservation law would be violated. Thus we have

(k_{p} - k_{s} - k_{i}) \cdot L = (k_{p} - k_{s} - k_{i}) L,

where L is the propagation length of the pump state k_p within the nonlinear crystal, while k_p, k_s and k_i are the amplitudes of the longitudinal wave vector components along k_p. In the paraxial approximation, these quantities can be written as follows [31]

\begin{array}{l} k_{p} = | k_{p} |, \\ k_{s (i)} = | k_{s (i)} | (1 - \frac{1}{2} {| \frac{q_{s (i)}}{k_{s (i)}} |}^{2}) . \end{array}

where q_s and q_i are the transverse wave vectors for the signal and idler states, respectively.

By substituting equations (3) and (4) into (2), we obtain the two-photon correlation in the transverse wave vector representation according to

Φ (k_{p}, q_{s}, q_{i}) = E_{p} L \sin c [\frac{1}{2} (\frac{{| q_{s} |}^{2} + {| q_{i} |}^{2}}{| k_{p} |}) L] \exp [i \frac{1}{2} (\frac{{| q_{s} |}^{2} + {| q_{i} |}^{2}}{| k_{p} |}) L] .

Equation (5) describes the cone of the down-converted two photons centered at k_p with a radius Δ_q defined as

\sqrt{\frac{2 π | k_{p} |}{L}}

, as shown in the inset of Fig. 1(b). For the pump state k_p, the down conversion cone is completely determined by the thickness of the nonlinear crystal L.

The conversion from cartesian coordinates to polar coordinates is given in Fig. 1(b). Here p_p, j_s and j_i are the wave vector projections on the cross section of the pump cone for the pump, signal and idler states, respectively, while θ_p,s,i are the corresponding azimuthal angles. In the paraxial approximation, since p_p is far less than k_p, the angle open by the pump cone state is so small that the cross section of the down-conversion cone is approximately parallel to the cross section of the pump cone. According to the trigonometric function calculation formula ${| q_{s,i} |}^{2} = {| \frac{p_{p}}{2} |}^{2} + {| j_{s,i} |}^{2} - | p_{p} | | j_{s,i} | \cos (θ_{s,i} - θ_{p})$ , the spectrum function Φ (k_p, q_s, q_i) can be transformed into:

\begin{array}{l} Φ (θ_{p}, θ_{s}, θ_{i}) = E_{p} L \\ \times \sin c [\frac{\frac{1}{2} {| p_{p} |}^{2} + {| j_{s} |}^{2} + {| j_{i} |}^{2} - | p_{p} | | j_{s} | \cos (θ_{s} - θ_{p}) - | p_{p} | | j_{i} | \cos (θ_{i} - θ_{p})}{2 | k_{p} |} L] \\ \times \exp [i \frac{\frac{1}{2} {| p_{p} |}^{2} + {| j_{s} |}^{2} + {| j_{i} |}^{2} - | p_{p} | | j_{s} | \cos (θ_{s} - θ_{p}) - | p_{p} | | j_{i} | \cos (θ_{i} - θ_{p})}{2 | k_{p} |} L], \end{array}

which represents the radial and azimuthal correlations between the down-converted two photons in polar coordinates. To simplify the discussion, we choose

| j_{s} | = | j_{i} | = \frac{1}{2} | p_{p} |

. Now the down-converted two-photon state in the angular position representation is given by

| Φ (θ_{p}) 〉 = \iint d θ_{s} d θ_{i} Φ (θ_{p}, θ_{s}, θ_{i}) | θ_{s} 〉 | θ_{i} 〉 .

Here |Φ(θ_p)⟩ represents the two-photon state converted from the pump state at angle θ_p.

The Fourier relationship between the angular position and OAM leads to [24]

| θ 〉 = \frac{1}{\sqrt{2 π}} \sum_{l = - \infty}^{\infty} \exp (- i l θ) | l 〉,

where |θ⟩ the angular position state in polar coordinates and |l⟩ is the OAM state. In the angular position representation, the pump cone state carrying an OAM of l_p is expressed as

| Φ_{p} 〉 = \frac{1}{\sqrt{2 π}} d θ_{p} \exp (i l_{p} θ_{p}) | θ_{p} 〉 .

Here, |Φ_p⟩ is the pump cone state and |θ_p⟩ is the angular position state; the amplitude of the pump state is distributed uniformly in the range from 0 to 2π. By substituting equations (7) and (8) into (9), the quantum correlation of the down-converted two-photon state pumped by a cone state in the OAM representation is found to be

\begin{array}{l} | Ψ 〉 = \frac{1}{\sqrt{2 π}} \int d θ_{p} \exp (i l_{p} θ_{p}) | Ψ (θ_{p}) 〉 \\ = \frac{1}{2 π \sqrt{2 π}} \sum_{l = - \infty}^{\infty} \sum_{l^{'} = - \infty}^{\infty} A (l_{s}, l_{i}) | l_{s} 〉 | l_{i} 〉 \int d θ_{p} \exp [i (l_{p} - l_{s} - l_{i}) θ_{p}] \\ = \frac{1}{2 π} \sum_{l_{s} = - \infty}^{\infty} \sum_{l_{i} = - \infty}^{\infty} A (l_{s}, l_{i}) δ_{l_{s}, l_{p} - l_{i}} | l_{s} 〉 | l_{i} 〉 \\ = \frac{1}{2 π} \sum_{l = - \infty}^{\infty} A (l, l_{p} - l) | l 〉 | l_{p} - l 〉 . \end{array}

In equation (10)A (l_s, l_i) is a two dimensional quantum OAM correlation spectra for the signal and idler states with the form of

A (l_{s}, l_{i}) = \iint d θ_{s} d θ_{i} ϕ (θ_{p}, θ_{s}, θ_{i}) \exp [- i l_{s} (θ_{s} - θ_{p}) - i l_{i} (θ_{i} - θ_{p})],

which is a two-dimensional Fourier transformation of ϕ (θ_p, θ_s, θ_i). From equation (10), we can see that OAM entanglement in SPDC arises from the continuous rotational symmetry of the pump state, where the entanglement spectrum A (l, l_p − l) is one of the diagonals of A (l_s, l_i) shifted by l_p from the zero point. Thus, pumped by a rotationally symmetric light beam, the two-photon OAM entanglement is completely determined by A (l_s, l_i), which is a conjugate part of ϕ (θ_p, θ_s θ_i). In practical SPDC experiments conducted on a nonlinear crystal with a specified cut angle, and in which perfect phase-matching is guaranteed within a small angle range, the tunable parameters are the radius of the pump cone state p_p, the length of the nonlinear crystal L, and the OAM of the pump state l_p. In the following, we will present the results of a detailed study of how these parameters influence OAM entanglement.

3. Results and Discssion

Figure 2 shows the simulation results of the evolution of the OAM correlation spectra A (l_s, l_i) with different values of |p_p| and L. The pump wavelength is 400 nm and the wavelengths of the signal and idler states are both equal to 800 nm. In Fig. 2(a–c), the nonlinear crystal length L is 1 mm, and in Fig. 2(d–f), L is 6 mm. A (l_s, l_i) is a weak two-dimensional sinc-like function of l_s and l_i peaking at l_s,i = 0. According to Fig. 2, the bandwidth of A (l_s, l_i) increases with |p_p| and L. This phenomenon can be explained by the Heisenberg uncertainty principle. In a collinear SPDC with a pump state of k_p, both |q_s| and |q_i| are confined within a small down conversion cone (see equation (5)). Here we choose an appropriate pump cone state in which |q_s(i)| ≪ |p_p|/2 then, we will have $\cos (θ_{s, (i)} - θ_{p}) \approx 1 - \frac{1}{2} {(θ_{s, (i)} - θ_{p})}^{2}$ . Furthermore, by choosing |j_s(i)| = |p_p|/2, equation (6) can be re-written as

\begin{array}{l} Φ (θ_{p}, θ_{s}, θ_{i}) \approx E_{p} L \sin c [\frac{{| p_{p} |}^{2} ({(θ_{s} - θ_{p})}^{2} + {(θ_{i} - θ_{p})}^{2})}{8 | k_{p} |} L] \\ \times \exp [i \frac{{| p_{p} |}^{2} ({(θ_{s} - θ_{p})}^{2} + {(θ_{i} - θ_{p})}^{2})}{8 | k_{p} |} L] . \end{array}

By choosing θ_p = 0, the angular distribution of the down-converted two-photon state is confined within a small angle range with a width of

Δ (θ_{s}^{2} + θ_{i}^{2}) = \frac{8 π | k_{p} |}{{| p_{p} |}^{2} L} .

Here,

Δ (θ_{s}^{2} + θ_{i}^{2})

can be viewed as the uncertainty in the relative angle distribution of the signal and idler states on the pump cone; the expression is inversely proportional to L and the square of |p_p| As a result, the larger the pump cone radius |p_p| is (and the longer the crystal length L is), the more certain we can be about knowing the relative angular position distribution of the down-converted two-photon states. According to Heisenberg uncertainty principle, for two conjugate variables of one particle, an increase in the certainty of one variable will lead to a decrease of the certainty in the other. Here, we found that the uncertainty principle is also applicable to two-part correlated systems. Therefore, by increasing the relative angular position correlation of the down-converted two-photon state, the bandwidth of A (l_s, l_i) will increase.

Fig. 2 Evolution of the two-photon OAM quantum correlation spectrum A (l_s, l_i) with respect to the radius of the pump cone state, |p_p|, at L = 1 mm (a–c) and L = 6 mm (d–f).

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Figure 3(a) shows the dependence of the mode probability distribution P (l, −l) of the two-photon OAM entanglement on the radius of the pump cone state |p_p|, in which l ranges from −15 to 15 and P (l, −l) is equal to

P (l, - l) = \frac{{| A (l, - l) |}^{2}}{\sum_{l^{'}} {| A (l^{'}, - l^{'}) |}^{2}} .

Here, we set l_p = 0 and L = 1 mm. Now the radius of the two-photon down-conversion cone Δ_q is 2π/0.02 rad/mm, and A (l, −l) is a diagonal that crosses the zero point of A (l_s, l_i). From this, it can be seen that for pump cone states with radii of |p_p| = 2π 0.05rad/mm and 2π/0.04rad/mm, which are smaller than the radius of the down-conversion two-photon cone Δ_q, the width of two-photon angular position correlation

Δ (θ_{s}^{2} + θ_{i}^{2})

is much larger than 0, and P (l, −l) is mainly concentrated near l = 0. As |p_p| becomes comparable to or larger than Δ_q, the width of P (l, −l) gradually increases and, thus a higher dimensional OAM-entangled two-photon state arises.

Fig. 3 Dependence of the two-photon OAM entanglement on the radius of the pump cone sate, |p_p|. (a) Probability distribution P (l, −l) for two-photon OAM entanglement for different pump cone states. (b) The dependence of the entanglement entropy on |p_p|. In this simulation, the OAM of the pump state l_p is set to 0, and the length of the nonlinear crystal is fixed at 1mm.

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The entanglement of a two-part system can be characterized by the so-called von Neumann entropy (or entanglement entropy) for the reduced state, which quantifies the number of entangled bits within the state. The entanglement entropy depends both on the entanglement dimension and the degree of entanglement. Fig. 3(b) shows the relationship between the von Neumann entropy for two-photon OAM entanglement and |p_p| where the entanglement entropy is expressed as [32,33]

E = - \sum_{l} P (l, - l) \log_{2} P (l, - l) .

In accordance with the |p_p| dependence of OAM entanglement in Fig. 3(b), the entanglement entropy increases with |p_p|. That is, as |p_p| increases, the entanglement dimension or degree of entanglement (or the both) will increase. In the present simulation, because we limit the OAM to a range from −15 to 15, the entanglement entropy will finally saturate at 4.95. From this figure we can see that as |p_p| is larger than 2000 rad/mm, a near maximal two-photon entanglement with dimension of 31 can be obtained. It should be noted that before the steep increase of the entanglement entropy, there is a small non-rising region. This is because for small values of |p_p| that are less than |q_s(i)|, the approximation in equations (12) and (13) will not be fulfilled. In this region, the dependence of the entanglement entropy on p_p is negligible and the down-converted two photons mainly occupy the l = 0 mode.

Figure 4(a) shows the dependence of the mode probability distribution P (l, −l) of the two-photon OAM entanglement on the length of the nonlinear crystal L, where |p_p| is 2π/0.04(rad/mm) and l_p is 0. Fig. 4(b) shows the dependence of the entanglement entropy on the crystal length for different pump cone states. From this figure we can see that the entanglement dimension and degree of entanglement increase with L, the entanglement entropy gradually saturates at a rate depending on |p_p|. In contrast to the |p_p| dependence of the OAM entanglement, which is ascribed to the increase of the pump cone radius, the L dependence of OAM entanglement is caused by a decrease in the radius of the two-photon down-conversion cone. From equation (5) it can be predicted that the thicker the nonlinear crystal is, the less phase-mismatch the nonlinear parametric interaction can tolerate and, thus, the smaller the radius of the down-conversion cone is. In this case, the increase in L will lead to a stronger angular position correlation $Δ (θ_{s}^{2} + θ_{i}^{2})$ (as shown in equation (13)). According to Heisenberg’s uncertainty principle, a wider OAM correlation bandwidth will be obtained [26,27].

Fig. 4 Dependence of the two-photon OAM entanglement on the length of the nonlinear crystal. (a) Probability distribution P (l, −l) of the OAM entanglement for different crystal lengths. Here |p_p| = 2π 0.004rad/mm is shown as an example. (b) Evolution of the entanglement entropy with respect to the length of the nonlinear crystal for different pump cone states.

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Similar to Fig. 3(b), there is also a non-rising region for the entanglement entropy in Fig. 4. The width of this non-rising region increases as |p_p| decreases. In this region, if the radius of the two-photon down-conversion cone is larger than or comparable to the radius of the pump cone, then equation (13) will not be satisfied, the entanglement entropy will be zero, and the two-photon state will mainly occupy the l = 0 mode. Furthermore, both Figs. 3 and 4 as well as equation (13) show that, the rate of increase for the two-photon OAM entanglement with L is slower than with |p_p|.

Finally, we study _.the dependence of the OAM entanglement on l_p, the OAM carried by the pump state. As shown in equation (10), the introduction of l_p can be viewed as a displacement operation that shifts the two-photon OAM entanglement from A (l, −l) to A (l, l_p −l), which are the diagonals of A (l_s, l_i). The dependence of the mode probability distribution P (l, l_p −l) of two-photon OAM entanglement is shown Fig. 5(a), where L and |p_p| are 1 mm 0.04 rad/mm, respectively. The mode probability distribution P (l, l_p −l) is centered at l_p/2, and the two-photon OAM entanglement varies with l_p. The corresponding dependence of the entanglement entropy on l_p for different pump cone states is shown in Fig. 5(b). The entanglement entropy shows no explicit dependence on l_p. Compared with the L dependence and |p_p| dependence of the OAM entanglement, changes in l_p do not affect the angular position correlation for the down-converted two photons; thus, changes in l_p are a less efficient way to enhance OAM entanglement.

Fig. 5 Dependence of the two-photon OAM entanglement on the OAM of the pump cone state. (a) Probability distribution P (l, l_p − l) of the OAM entanglement for different l_p, the OAM carried by the pump state. The parameters for this simulation are |p_p| = 0.05 rad/mm, and L_z = 1 mm. (b) Dependence of the entanglement entropy on l_p for four different pump cone states.

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4. Conclusion

In conclusion, we have provided a variant theoretical interpretation of the down-converted two-photon OAM entanglement within the azimuthal region and pointed out several approaches to enhance the entanglement dimension and degree of entanglement. The pump state is decomposed into a set of pump cone states characterized by their radii. We discuss the two-photon correlation in the azimuthal region by fixing the transverse momentum projections of the signal and idler states. The angular position correlation and OAM correlation between the down-converted two photons are closely linked, they form an EPR pair [16] just as position and momentum do [34]. The OAM entanglement can be viewed as a unitary transformation operated on the two-photon angular position correlation, which is always used to quantify entanglement [35]. The entanglement dimension and degree, characterized by the entanglement entropy, shows a strong dependence on the radius of the pump cone state and the length of the nonlinear crystal. Such phenomena can be explained by applying the Heisenberg uncertainty principle to the between angular position correlation and OAM correlation. In contrast, varying the OAM of the pump state just results in shifting the entanglement spectrum, which turns out to be a less efficient way to increase the OAM entanglement. This implies that for actual experiments in which an incident plane-wave like pump light with rotational symmetry is used, the angular position correlation of the down-converted two photons can be strengthened by choosing a lens of shorter focal length and a thicker nonlinear crystal. In addition, to reduce the influence of the pump cone state near the central propagation axis, a ring-like pump beam may favor the generation of high-dimensional OAM entanglement.

Funding

Young Fund of Jiangsu Natural Science Foundation of China (SJ216025); National Fundation Incubation Project (NY217024); Scientific Research Foundation of Nanjing University of Posts and Telecommunications (NY215034); National Natural Science Foundation of China (NSFC) (No. 61475075); Open Subject of National Laboratory of Solid State Microstructures of Nanjing University (M31021).

Acknowledgments

We thank prof. Jozef Gruska from Faculty of Informatics, Masaryk University and Jim Bailey, PhD, from Liwen Bianji, Edanz Editing China (www.liwenbianji.cn/ac), for editing the English text of a draft of this manuscript.

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Manipulating orbital angular momentum entanglement by using the Heisenberg uncertainty principle

Abstract

1. Introduction

2. Theory

3. Results and Discssion

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (15)

Optics Express