## Abstract

We propose a feasible scheme to remotely prepare three-dimensional (3D) optical vortex lines. Our scheme relies on the complete description of high-dimensional orbital angular momentum (OAM) entanglement in terms of the Laguerre-Gaussian modes. It is theoretically demonstrated that by simply changing the pump beam waist, we can remotely prepare the target photons in the vortex states of 3D interesting morphology, appearing as twisted vortex strands, separated vortex loops, and vortex link or knot. Furthermore, we employ the biphoton Klyshko picture to illustrate the conservation law of the OAM index *ℓ* and the spreading effect of the radial mode index *p*, where the Schmidt numbers are calculated to show the high-dimensional capacity of the quantum channels involved in the present remote state preparation.

© 2014 Optical Society of America

## 1. Introduction

The aim of quantum communication is to manipulate and transmit quantum information with higher efficiency and better security. The quantum teleportation, a portocol originally proposed by Bennett *et al.* [1], describes how Alice helps Bob to remotely prepare a qubit state unknown to either by consuming 1 entanglement bit (ebit) and 2 classical bits (cbits). In contrast, in the protocol of remote state preparation (RSP) [2], similar task can be done by consuming only 1 ebit and 1 cbit, provided that Alice knows the state in advance but Bob does not. Both protocols rely on the entanglement to work, however, unlike the teleportation, the Bell-state measurement is not required in RSP. Besides, the communication cost is one question that concerned by many researchers [3–5]. Considering the trade-off between classical communication and entanglement, RSP has attracted a growing research interest in recent years, and several RSP demonstrations, such as the low-entanglement RSP [6], the optimal RSP of mixed states [7], RSP in high-dimension space [8], and continuous variable RSP [9], have been reported. Besides, RSP of arbitrary photon polarization states [10] and hybrid vector-polarization states [11] have been demonstrated, respectively.

Here we propose theoretically another feasible scheme to remotely prepare three-dimensional optical vortices. Optical vortices are generally discussed in connection with the angular momentum of light [12]. In addition to spin angular momentum (SAM) associated with polarization, a light beam with helical phase front of exp(*iℓϕ*) possesses a well-defined orbital angular momentum (OAM) of *ℓh̄* per photon, where *ϕ* is the azimuthal angle and *ℓ* is the OAM quantum number [13]. The OAM beam is the simplest class of light field carrying optical vortex along the beam axis. Generally, an optical vortex is a space around which the optical phase advances or retards by a multiple of 2*π*. At the vortex center, the phase is singular and the intensity is zero. They occur at points in two-dimensional (2D) fields and along lines in three-dimensional (3D) space. It was theoretically discovered that the vortex lines can be manipulated to form links or knots [14, 15], which was later verified in experiment using specific superpositions of the Laguerre-gaussian (LG) beams [16–18]. The OAM entanglement has been well established between photon pairs generated by spontaneous parametric down-conversion (SPDC), and recently, it was demonstrated that the entanglement can be also observed between the separate vortex links towards the macroscopic and finite volumes [19]. In this work, we combine the concept of optical vortices with the protocol of RSP, and demonstrate a theoretical scheme to remotely prepare the 3D structures of optical vortex lines, based on the OAM-entangled photon pairs. We demonstrate theoretically that a variety of interesting topological singularities including the twisted vortex strands, separated vortex loops, vortex link and knot can be remotely prepared by simply changing the pump beam waist. We also employ the biphoton Klyshko picture and calculate the Schmidt numbers to illustrate the effective capacity of the quantum channels involved in our RSP scheme.

## 2. RSP of 3D optical vortices

The proposed experimental scheme is depicted in Fig. 1. The OAM-entangled photon pairs are produced *v*ia the type-I collinear SPDC by a UV pump laser incident on a BBO crystal. The degenerate down-converted signal and idler beams are then separated by a non-polarizing beam splitter (BS). We start with the complete description of OAM entanglement in terms of the LG modes. The LG mode is a natural choice to describe twisted photons carrying OAM. In the cylindrical coordinates (*R*, *ϕ*, *z*), the normalized form of a LG mode is mathematically expressed by,

*w*is the beam waist width, ${\text{L}}_{p}^{\left|\ell \right|}(\cdot )$ is the generalized Laguerre polynomials,

*p*and

*ℓ*are the radial and azimuthal mode indices, respectively. With a fundamental Gaussian pump, the two-photon state can be described completely in terms of the LG modes as follows [20–22],

_{2}

*F*

_{1}(·) is the Gauss hypergeometric function, and

*γ*(

_{s}*γ*) denotes the ratio of the pump beam waist

_{i}*w*to the signal (idler) beam waist

_{p}*w*(

_{s}*w*). Generally, the SPDC photon pairs are not produced in the maximally entangled state. For the OAM index, there is a limited spiral bandwidth, namely, the LG modes of lower OAM

_{i}*ℓ*are generated more frequently than the higher-order ones. While for the azimuthal index,

*p*and

_{s}*p*are not perfectly correlated, namely, a single

_{i}*p*is generally spread to a spectrum of

_{i}*p*, as indicated by Eq. (3).

_{s}In our scheme, the signal photons at Alice’s side serve as the trigger photons while the idler photons at Bob’s side serve as the target photons. In our simulations, the trigger photons are measured and projected into the state of vortex link or knot, which was readily weaved by using specific superpositions of the LG modes, namely [16],

*a*

_{1}= 1 and

*w*= 1 mm. The link can be formed with

*ℓ*= 2, and the relative amplitudes:

*a*

_{0}= 0.35,

*a*

_{2}= −0.35,

*a*

_{3}= 0.36. In contrast, the knot can be formed with

*ℓ*= 3, and

*a*

_{0}= 0.25,

*a*

_{2}= −0.26,

*a*

_{3}= 0.29. According to Eqs. (2) and (4), we know the remotely prepared state of the target photons can be derived as follows,

Owing to the OAM conservation, we always have couples of azimuthal indices for signal and idler modes, namely, *ℓ _{i}* = −

*ℓ*. In contrast, the correlation between

_{s}*p*and

_{s}*p*breaks down in a practical experiment, as the crystal size places an effective upper limit to pump beam waist [21]. A most usually encountered case is that the pump beam size approaches that of signal and idler ones, namely,

_{i}*w*=

_{p}*w*= 1 mm in our case.

It is experimentally feasible for Alice to project her own trigger photons onto the vortex link state |*ψ*_{link}〉 or knot state |*ψ*_{knot}〉, if she has a spatial light modulator (SLM), a single-mode fiber (SMF) and a single-photon detector (APD) at hand. Based on the LabVIEW simulation according to Eq. (4), we show in the left panels of Fig. 2 the desired holograms addressed by SLM that are used to make vortex link and knot, and plot in the middle panels their 3D structures, respectively. Having been informed about which hologram, Fig. 2(a) or 2(d), has been used by Alice *v*ia a classical link, Bob achieves the expected vortex lines encoded in his target photons. Let’s assume that Bob has an intensified CCD (ICCD) camera, which is triggered by the output from Alice’s APD, and can be translated along the beam axis to record the intensity of the beam at different planes. Besides, the ICCD camera has been over-saturated in order to measure the vortex positions within the beam cross section. Then the resulting image contains points of darkness corresponding to the vortices as they intersected the plane of the ICCD [16]. In this scenario, Bob is able to reconstruct the 3D vortices of the target photons. According to Eq. (5) with *w _{p}* = 1 mm, we present in the right panels of Fig. 2 the numerical simulations of 3D vortex lines. We have plotted the phase files at three transverse planes that the vortex lines intersect at

*z*= −10 mm,

*z*= 0 mm and

*z*= 10 mm, which are utilized to show conceptually the locations of the vortex points. Around each vortex point, there is a spiral phase change of 2

*π*, and, therefore, giving an individual topological charge

*Q*= 1. One can observe the differences between the topological singularities of the target and trigger photons, as are visualized by Figs. 2(b) and 2(c), Figs. 2(e) and 2(f). This is because the vortex link and knot states in Eq. (4) are the superpositions of only four specific LG modes. But the remotely prepared states of Eq. (5) are represented by the superpositions of many LG modes as a summation over all

*p*, resulted from the spreading effect of

_{i}*p*index. As a result, the vortex link and knot are erased by the superpositions of the spread LG modes for the idler photons. For an easy visualization, we also plotted the corresponding intensity patterns (

*w*= 1 mm) at these observing planes, as shown in Fig. 3. One can see that these vortex points are rotating clockwise along the direction of beam propagation, and the vortex lines are slightly twisted in the 3D space.

_{p}It is worth noting that the spreading effect of *p* index has an evident dependence on the ratio of the pump beam waist to the signal (idler), *γ _{s}* (

*γ*). In our scheme, the signal and idler beam waists are readily defined according to Eq. (4), where

_{i}*w*= 1 mm. So it is interesting for us to examine the 3D vortex morphology attained by the target photons, by changing the waists of the pump beam not limited to

*w*=

_{p}*w*. Our simulation results are presented in Fig. 4, which indicates that different kinds of 3D vortex structures can be remotely prepared. Figures 4(a) and 4(d) are merely the two- and three-stand vortices (

*w*= 1 mm) that are reproduced from Figs. 2(c) and 2(f) for comparison. By changing the pump beam waist width from

_{p}*w*= 1 mm to 1.5 mm, our simulations in Figs. 4(b) and 4(e) reveal the formation of 3D separated vortex loops. Specifically, there are two separated loops appearing for

_{p}*ℓ*= 2, and they orient almost perpendicularly, namely, one is lying horizontally and the other is standing vertically. In contrast, there are three vortex loops emerging for

*ℓ*= 3, and they distribute around the beam axis at the vertices of a regular triangle. As the pump beam waist is further increased to

*w*= 3 mm, these vortex loops expand in the 3D space, and more interestingly, the two loops (

_{p}*ℓ*= 2) get interlinked to form a vortex link, while the three loops (

*ℓ*= 3) get tangled to form a vortex knot, see Figs. 4(c) and 4(f), respectively. Along this line, one can theoretically conclude that in the limit of infinite pump width (or sufficiently large),

*p*and

_{s}*p*will be delta correlated, and the photon pairs can be maximally entangled, namely, ${C}_{{p}_{s},{p}_{i}}^{{\ell}_{s},{\ell}_{i}}={\delta}_{{p}_{s},{p}_{i}}{\delta}_{{\ell}_{s},-{\ell}_{i}}$. Therefore it is possible for Alice to remotely prepare Bob’s photon to attain the same states of vortex link and knot as Figs. 2(b) and 2(e), respectively.

_{i}## 3. Biphoton Klyshko picture and the Schmidt number

As is well known, high-dimensional entangled systems are desirable for enhancing the robustness and security of quantum communications [23], and for increasing the information capacity of a photon [24]. So it is also crucial for us to investigate the high-dimensional quantum nature of our RSP protocol. A most convenient and direct approach to quantify the amount of entanglement in a pure two-photon entangled state is the Schmidt number, which is given as
$\kappa =1/{\text{Tr}}_{A}({\rho}_{A}^{2})=1/{\text{Tr}}_{B}({\rho}_{B}^{2})$, with *ρ _{A}* =

*Tr*(|Ψ〉 〈Ψ|) and

_{B}*ρ*=

_{B}*Tr*(|Ψ〉 〈Ψ|) being the reduced density matrices for the two subsystems, respectively. The Schmidt number also characterizes the number of effective entangled modes involved in the entangled state [25] or the dimensionality of that Hilbert space [26]. According to Eqs. (4) and (5), we know that the two-photon state post-selected by our RSP scheme can be written as,

_{A}After a straightforward calculation based on Eq. (6), we obtain the Schmidt numbers *κ* different pump beam waists *w _{p}*, as are listed in Table 1. One can see that in all cases the Schmidt numbers are obviously beyond the two-dimensional thinking with polarization entanglement, which therefore suggests that there may be many entangled spatial modes participating in process of our RSP protocol.

The associated high-dimensional quantum channels can also be understood in light of the biphoton Klyshko picture. In a conventional Klyshko picture, the signal and idler apparatus are unfolded with respect to the crystal and the straight lines represent the advanced light rays [27]. In contrast, the straight lines (red) in Fig. 5 represent the biphoton OAM channels sustained by the LG modes [28, 29]. Because of the mode orthogonality of 〈*ℓ*, *p*|*ℓ′*, *p′*〉 = *δ _{ℓ,ℓ′}δ_{p,p′}*, each pair of the LG modes, |

*ℓ*,

_{s}*p*〉|

_{s}*ℓ*,

_{i}*p*〉, can thus be treated as a biphoton quantum channel, some examples of the LG mode channels are shown by the red lines. In the Klyshko’s advanced wave model, the single-photon detector (APD) is substituted by a standard light source, and the connected SMF transmits a Gaussian light with zero OAM, namely, |

_{i}*ψ*

_{0}〉 = |0〉. This light goes back to SLM and the reflected light acquires the vortex link or knot state described by Eq. (4), namely, |

*ψ*

_{1}〉 = |

*ψ*

_{link,knot}〉. The BBO crystal is replaced by a mirror, which is assumed to have a mode-dependent reflectivity of ${C}_{\ell ,-\ell}^{{\ell}_{s},{\ell}_{i}}$ when taking the limited spiral bandwidth into consideration. After reflecting off the BBO crystal, the azimuthal index

*ℓ*is flip, namely,

*ℓ*= −

_{i}*ℓ*, to conserve the total OAM. In contrast, the radial index

_{s}*p*is spread, see the forked lines in Fig. 5. As a result, the target photons take the desired state of |

*ψ*

_{2}〉 = |

*ψ*

_{rp}〉, which is identical to Eq. (5). Thus, if we have an ICCD in the optical path of the trigger photons, then we would be able to obtain the 3D vortex lines of Fig. 4, for example, the two vortex strands are shown in Fig. 5. One can image that there are actually infinite LG mode channels involved in the Klyshko picture, as

*p*can be spread to take any possible integers, however, the effective channel capacity has a finite value, for example,

_{i}*κ*= 9.90 and 8.35 when

*w*= 1 mm. Besides, one can see from Table 1 that

_{p}*κ*decreases as

*w*increases. This can also be well interpreted by the spreading effect of

_{p}*p*index. For a smaller

*w*, the spreading effect is evident such that there are many new LG modes contributing together to the RSP. While for a larger

_{p}*w*, the spreading effect becomes less evident and only a few of the LG modes dominates in the RSP. In the limit of infinite pump width, as

_{p}*p*and

_{s}*p*are delta correlated, as can be inferred from Eqs. (4) and (5), there are only four pairs of specific LG modes participating in the RSP. For example, one can see that the Schmidt numbers are

_{i}*κ*= 4.07 and 4.09 with a sufficiently large pump beam waist, e.g.,

*w*= 20, which have approached the theoretical limit

_{p}*κ*= 4 for the 4D maximally entangled states.

## 4. Conclusion

In summary, we have combined the RSP protocol with optical vortices to remotely prepare a variety of 3D interesting vortex structures, including the vortex strands, vortex loops, vortex link and knot. Our analysis is based on the complete description of two-photon OAM entanglement in terms of the LG modes, and our results can be well interpreted in the frame of the conservation law of the OAM index *ℓ* and the spreading effect of the radial index *p*. We have illustrated the quantum channels involved in our RSP protocol with the biphoton Klyshko picture, and calculated the Schmidt numbers to show their high dimensions. Our work may raise the possibility of studying the initial-value problems through the RSP protocol: the non-trivial 3D topological information of optical vortices at Bob’s side are encrypted, non-locally, in the 2D section taken as the holograms at Alice’s side, which may also has the potential in understanding the theory of ghost imaging [27].

## Acknowledgments

L.C. thanks Dr. Jonathan Leach at Heriot-Watt University for previous discussions about the OAM spreading effect. Figures 2(a) and 2(d) were plotted using LabVIEW with the help of the Optics group at University of Glasgow. This work is supported by the National Natural Science Foundation of China (NSFC) ( 11104233), the Fundamental Research Funds for the Central Universities ( 2011121043, 2012121015), the Program for New Century Excellent Talents in Fujian Province University, and the program for New Century Excellent Talents in University of China ( NCET-13-0495).

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