We consider the non-horizontal distributions of orbital angular momentum biphotons through free-space atmospheric channels, in which case the non-Kolmogorov turbulent effects shall be considered. By considering the case of initial non-perfect resource, i.e., the orbital angular momentum biphotons are initially prepared in an extended Werner-like state, we investigate the non-Kolmogorov effects on the propagations of nonclassical correlations, including quantum entanglement and quantum discord. It is found that universal decay laws of entanglement and discord also exist for non-Kolmogorov turbulence but with their decay curves different from that of entanglement for Kolmogorov turbulence reported by Leonhard et al. [Phys. Rev. A 91, 012345 (2015) [CrossRef] ]. We show that the universal decay laws are dependent on the power-law exponent of the non-Kolmogorov spectrum and compare the differences of decay properties between entanglement and discord caused by non-Kolmogorov turbulence.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Orbital angular momenta (OAM) of photons have been playing an important role in quantum information science (QIS) [1–7], which provide a new degree of freedom for the encoding of nonclassical information, such as entanglement . However, the beam’s wavefront will be distorted during free-space propagation due to the refractive index fluctuation of turbulent atmosphere, leading to random phase aberrations  as well as the unavoidable decay of quantum information encoded. Therefore, the understanding of turbulent effects on OAM states is crucial. In recent years, there has been much theoretical [10–15] and experimental [16–19] research concerning the influences of atmospheric turbulence and the protection of OAM photons from decoherence caused by a turbulent atmosphere.
On the other hand, the quantitative exploration of quantum information encoded in OAM photons needs first to specify a proper measure of nonclassical information. Quantum entanglement is a kind of nonclassical correlation considered as a fundamental resource in QIS , and it is usually measured by Wootters’ concurrence . However, it has been found that entanglement is not the unique resource utilized in QIS and there exist other resources such as quantum discord [21–23]. A quantifier of discord, termed as local quantum uncertainty (LQU) , has been recently put forward by Girolami et al., which meets all the known bona fide criteria. The LQU is exactly computable as Wootters’ concurrence and, more importantly, it has a geometrical and observable interpretation.
Recently, the entanglement decay of OAM states in Kolmogorov atmospheric turbulence has been reported via concurrence . However, the atmospheric turbulence will show obvious non-Kolmogorov characteristics when light beams propagate along the vertical direction [25,26]. The entanglement decay has been investigated with a multiple phase screen approach for non-Kolmogorov turbulence with inner and outer scale . In Ref. , the maximally entangled state of two OAM photons has been considered, and concurrence is described by the ratio of the beam waist radius and the spatial coherence radius. Besides, the beams are horizontally propagating, and the power-law exponent of non-Kolmogorov spectrum is only chosen to be 11/3 (Kolmogorov) and 3.6 with weak turbulence. In this paper, we consider the non-horizontal distributions of orbital angular momentum biphotons through free space atmospheric channels, in which case the non-Kolmogorov turbulent effects have to be considered. By considering the case of initial non-perfect resource, i.e., the OAM biphotons are initially prepared in an extended Werner-like state, we investigate the non-Kolmogorov effects on the propagations of nonclassical correlations, including quantum entanglement and quantum discord. We instead explore the nonclassical correlations versus the ratio of phase correlation length and the Fried-like parameter, and find that universal decay laws of entanglement and discord also exist for non-Kolmogorov turbulence, but with their decay curves different from that of entanglement for Kolmogorov turbulence reported by Ref. . The dependence of universal decay laws on the power-law exponent of the non-Kolmogorov spectrum and the differences of decay properties between entanglement and discord caused by non-Kolmogorov turbulence are also discussed.
This paper is organized as follows. In Section 2, we describe the model and derive the photonic OAM state influenced by the non-Kolmogorov turbulent atmosphere. In Section 3, the evolutions of the concurrence and the LQU of the initial extended Werner-like state as well as their decay laws are discussed. Conclusions are presented in Section 4.
2. ENTANGLED OAM STATE THROUGH ATMOSPHERIC TURBULENCE
The model considered is sketched in Fig. 1. Two Laguerre–Gaussian (LG) beams , carrying pairs of entangled photons with opposite azimuthal quantum number and , are emitted non-horizontally from a source at the ground and then propagate through the atmospheric turbulence in two symmetrical channels. We assume the propagating direction is along the axis and the photons are detected at the altitude , which form a zenith angle of the communication channel. This model can be established for quantum communication including quantum key distribution and quantum teleportation. For instance, Charlie (the sender) in the valley uses the resource and sends the entangled states through turbulent atmosphere to Alice and Bob (the two receivers) on two mountains with the altitude . The propagation distance is of the order of kilometers (km), and we will consider a distance of approximately 1 km to illustrate the transfer of an entangled resource in weak-to-moderate turbulence. Also, we consider that the mountains are about 0.5 km high and the zenith angle is about . The wavefunction of the LG beam is given by 
The radial part of the LG beam can be expressed as12] 1, root-mean-square (rms) wind speed and denoting a nominal value of at the ground.
As the increase of the propagation altitude , the change of the index-of-refraction structure constant is shown in Fig. 2. The index-of-refraction structure parameter is , which is important in atmospheric optics for the propagation of laser beam in the atmospheric turbulence . It can be found in Fig. 2 that the index-of-refraction structure constant increases with the increase of at a short distance and then decreases fast, which means that the strength of the atmospheric turbulence decreases sharply when the propagation altitude is high enough . It should be pointed out that, for fixed distance , the index-of-refraction structure increases when increases, which corresponds to stronger turbulence. In this work, we consider weak-to-moderate turbulence with and accordingly choose the proper values of . The turbulent strength changes as the vertical distance increases during non-horizontal propagation, which is different from the case of horizontal propagation in Refs. [15,27].
Here, we assume that the waist of the input LG modes is , the radial quantum number is , the azimuthal quantum numbers are and , and the wavelength of the LG beams is chosen as 1550 nm, which is in the transmission window of the atmosphere. We study the effects of atmospheric turbulence at a distance near 1 km, and the diffraction of beams can be ignored for . The initial state of the photon pair is prepared in an extended Werner-like state defined as33–37]. In realistic experiments, quantum states prepared are usually not determined but mixed with an ensemble of pure states. The purity parameter is crucial for whether a Werner-like state is entangled or not, since it is unentangled only for . The parameter determines the degree of the entanglement for Bell-like states in Eq. (8), which is maximally entangled for as considered in Ref.  at and . The action of the turbulent atmosphere on the entangled photon pair can be treated as a linear operator , in terms of which the received state at the detectors can be expressed as 15] 31], with the spatial coherence radius of a spherical wave,
The phase structure function of non-Kolmogorov turbulence will reduce to that of Kolmogorov turbulence as in Ref. . In the following, we will have .
In the basis , the density matrix of the input state from Eq. (7) can be written in an X form,
According to Ref. , the post-selected state at the detectors is given by
It is worth noting that since and vanish. Other non-vanishing matrix elements can be numerically evaluated by the integrals in Eq. (10).
3. NONCLASSICAL CORRELATION OF OAM STATE IN ATMOSPHERIC TURBULENCE
We proceed to study the nonclassical information contained in the output state. The first kind of nonclassical information we investigate is entanglement. To quantify the degree of entanglement, a measure of entanglement defined as concurrence  will be used, which varies from for a separable state to for a maximally entangled state. For a two-qubit state , the concurrence of this state is given by 
Then, we plot the concurrence as a function of the propagation distance and the initial state parameter in Fig. 3(a). It is seen that the concurrence decays with the increase of the propagation distance to vanishing in a non-asymptotical manner, a phenomenon that is termed as entangled sudden death (ESD) [38,39]. As clearly shown for near , the decay rate at a short distance is much faster than that at long distance. The reason why concurrence decays fast at the beginning and then slows down is that the strength of the atmospheric turbulence is strong when the OAM beams propagate near the ground, and then the strength decreases fast with the altitude increasing, which is shown in Fig. 2. Besides, we plot the concurrence as a function of the propagation distance and the initial state purity in Fig. 3(b). The concurrence goes to vanishing more quickly as the purity decreases. For with , the concurrence is always vanishing, since the initial state is unentangled, and no entanglement can be created by independent turbulent channels.
To explore the ESD phenomenon more precisely, we display the vanishing distance as a function of and of in Figs. 4(a) and 4(b), respectively. In Fig. 4(a), the vanishing distance increases rapidly as increases to , and then it decreases fast as increases from to , exhibiting a symmetric distribution as well as a peak at . It is demonstrated in Fig. 4(b) that the vanishing distance also decreases non-asymptotically to vanishing as the purity decreases from 1 to 1/3 for .
Another important type of nonclassical information in QIS is quantum discord [21–23]. In order to quantify the degree of discord, we use a measure termed as local quantum uncertainty (LQU) , which is defined by the minimal Wigner–Yanase skew information ,
In Fig. 5(a), we plot the LQU as a function of the propagation distance and the initial state parameter . Similar to the concurrence, the LQU also decays fast at the beginning and the decay gradually becomes slow. Different from the concurrence, the LQU does not vanish even when the OAM photons in the atmospheric turbulence have propagated at a long distance. When the propagation distance is long enough, the LQU becomes insensitive to the increase of . The LQU as a function of and is also plotted in Fig. 5(b). The LQU decreases only in an asymptotical manner when the purity decreases from 1 to 0, which is also different from that of the concurrence shown in Fig. 3(b).
Moreover, we also explore the effect of azimuthal quantum number on the decay of concurrence and LQU. The decays of the concurrence and LQU with the increase of with different azimuthal quantum numbers are shown in Figs. 6(a) and 6(b). Both concurrence and LQU decay more slowly with the increase of the azimuthal quantum number . It has been reported that the decay of concurrence may finally collapse onto a universal curve with the increase of for the case of Kolmogorov atmospheric turbulence . Here, we have shown that the larger azimuthal quantum number OAM photons carry, the more quantum-ness can be preserved through non-Kolmogorov atmospheric turbulence, which is rather different from that of Kolmogorov atmospheric turbulence.
The effects of the power-law exponent of the non-Kolmogorov spectrum on the decay of concurrence and LQU are also shown in Figs. 7(a) and 7(b), respectively. In Fig. 7(a), we see that the decay rates of concurrence and LQU become faster with the value of increasing, which indicates the OAM photons are more deeply influenced by the atmospheric turbulence with the higher power-law exponent . The ESD also becomes clearer as the power-law exponent increases (see purple dashed curve for ). In Fig. 7(b), the LQU decays faster at first, with increasing while the decay rates become almost the same, and the curves for different become almost parallel at long propagation distance , which is also different from that of the concurrence in Fig. 7(a).12],
We then plot the concurrence as a function of the ratio for weak () and moderate () turbulence in Figs. 8(a) and 8(b). We gain two universal decay laws for and in the case of a large . The function of the universal curve in Fig. 8(a) is , and in Fig. 8(b) is , which are different from that for Kolmogorov turbulence reported in Ref. . Besides, we also would like to explore the existence of a decay law for discord in Fig. 9. Two universal decay curves and are also found for weak [ in Fig. 9(a)] and moderate [ in Fig. 9(b)] turbulence, respectively. To discuss the power-law exponent of the non-Kolmogorov spectrum on the decay law of entanglement and discord, in Fig. 10, we plot concurrence and LQU as functions of and . It can be found that the universal decay to vanishing becomes more fast when increases with stronger turbulence.
In conclusion, we have investigated the decay properties of the nonclassical correlations (entanglement and quantum discord) of two photonic qubits propagating through the non-Kolmogorov atmospheric turbulence via concurrence and LQU. By considering that the photonic OAM beams are initially prepared in an extended Werner-like state and non-horizontally propagate through turbulence, we show that entanglement and quantum discord have different decay properties in non-Kolmogorov atmospheric turbulence for different cases, including the change of the propagation distance, the azimuthal quantum number, and the power-law exponent of the non-Kolmogorov spectrum. Comparing the decay properties of the concurrence and LQU with the increase of the propagation distance, we find that there exists ESD in the propagation of the entanglement while the LQU decays merely in an asymptotical manner. The effects of value of the azimuthal quantum number are also explored. The nonclassical correlations with a larger azimuthal quantum number can be more robust against the influence of atmospheric turbulence.
In addition, we also investigate the effects of the power-law exponent of the non-Kolmogorov spectrum. With the increase of the power-law exponent, the decay rates of the concurrence and LQU both become faster at the beginning, due to a larger indicating stronger turbulence. The difference is that the entanglement suffers from ESD in shorter distances while quantum discord keeps almost constant at long distances. Moreover, we explore the decay properties of concurrence and LQU versus the ratio of the phase correlation length to the Fried-like parameter and find that universal decay laws of entanglement and discord also exist for non-Kolmogorov turbulence but with their decay curves different from that of entanglement for Kolmogorov turbulence reported by . The universal decay laws are dependent on the power-law exponent of the non-Kolmogorov spectrum, and the universal decay to vanishing becomes faster with stronger turbulence.
Before ending, we shall remark that the results here, such as an increased stability for higher-order OAM modes, may be valid only for weak-to-moderate turbulence. For strong turbulence, different results have been shown by Roux et al. . It would also be interesting to explore the turbulence in a strong regime, which is beyond the scope of this work. We believe our results are the first theoretical attempt to explore quantum discord in non-Kolmogorov atmospheric turbulence and may contribute to the understanding of OAM quantum communication in free-space communication with new quantum resources.
National Natural Science Foundation of China (NSFC) (11504140; 11504139); Natural Science Foundation of Jiangsu Province (BK20140128, BK20140167); Fundamental Research Funds for the Central Universities (JUSRP51517).
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